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https://gauravtiwari.org/time-saving-mathematics-formulas-theorems/	Formulas are the most important part of mathematics and as we all know one is the backbone of the latter. Considering there are thousands of mathematical formulas to help people develop analytical approach and solve problems easily — there are some that go beyond. Some formulas aren’t just timesaving but those also do wonders. In this article I have collected some of the finest time-saving formulas in mathematics. The calendar formula This formula is extremely helpful in finding weekday for a specific date in history. More about this can be found in the article “Calendar Formula: Finding the Week-days“. Infinite Summation into Integration We all know that integration can be interpreted as summation and vice versa. Infinite summations can be easily converted into finite integrals which on solution yields result to the infinite summation. The methods of converting infinite summations is pretty easy and thus the whole process saves a lot of time. Theory is that if $f$ is a positive increasing function then $f(n) \le \int_n^{n+1} f(x) dx \le f(n+1)$. Similarly, if $f$ is a positive decreasing function then $f(n) \ge \int_n^{n+1} f(x) dx \ge f(n+1)$. For increasing $f$, $\sum_{i=0}^{n-1} f(i) \le \int_0^{n} f(x) dx \le \sum_{i=0}^{n-1} f(i+1)$ or, $\sum_{i=0}^{n} f(i)-f(n) \le \int_0^{n} f(x) dx \le \sum_{i=0}^{n} f(i)-f(0)$ or $f(0) \le \sum_{i=0}^{n} f(i)-\int_0^{n} f(x) dx \le f(n)$. For a decreasing function, the inequalities are reversed. Khan Academy has a very good video about converting infinite summations into integral problems. Integral Equation Magical Formula This helps in converting a multiple integral equation to a simple linear integral equation. Consider an integral of order n is given by $\displaystyle{\int_{\Delta}^{\Box}} f(x) dx^n$ We can prove that $\displaystyle{\int_{a}^{t}} f(x) dx^n = \displaystyle{\int_{a}^{t}} \dfrac{(t-x)^{n-1}}{(n-1)!} f(x) dx$ This formula can help you solve tedious integral problems in a jiffy. Just compare the parameters and you are good to go! For example, let’s try to solve $\int_0^1 x^2 dx^2$ Solution: $$\int_0^1 x^2 dx^2$$ $$= \int_0^1 \dfrac{(1-x)^{2-1}}{(2-1)!} x^2 dx$$ (Compare, $t=1$) $=\int_0^1 (1-x) x^2 dx$ $=\int_0^1 (1-x) x^2 dx$ $=\int_0^1 (x^2-x^3) dx =1/12$ Done! $\sin^m x \cos^n x$ integration formula To integrate $\sin^m x \cos^n x$ with respect to $x$, we use simple substitution considering if $n$ is odd or $m$ is odd or both are even. 1. If n is odd, then ignore what m is and use substitution as $u = \sin x$ or $du = \cos x \ dx$ and convert the remaining factors of cosine using $\cos^2 x = 1 – \sin^2 x$. This will work even if $m = 0.$ This can be applied to $\int \sin^5 x \cos^3 x \ dx$ like integrals. 1. If m is odd and n is even — use substitution as $u = \cos x$ or $du = − \sin x \ dx$ and convert the remaining factors of sine using $\sin^2 x = 1 – \cos^2 x$. This will work if $n = 0.$This can be applied to $\int \sin^3 x \cos^8 x \ dx$ like integrals. 1. If both powers are even we reduce the powers using the half angle formulas: $\sin^2 x = \dfrac{1}{2} ( 1 – \cos 2x )$; and $\cos^2 x = \dfrac{1}{2} (1 + \cos 2x)$Alternatively, you can switch to powers of sine and cosine using $\cos^2 x+ \sin^2 x = 1$ and use the reduction formulas. Example: $\int \sin^4 x \cos^2 x \ dx$ L’ Hospital’s Rule According to it if we have an indeterminate form for a limit problem, like 0/0 or $\dfrac{\infty}{\infty}$, then all we need to do is differentiate the numerator and differentiate the denominator and then take the limit. Read more about this on Mathworld. Liouville & Dirichlet’s Theorem These theorems are used to easily convert surface or volume integrals into $\Gamma$-functions. 1. $$\int \int \int_{V} x^{l-1} y^{m-1} z^{n-1} dx dy ,dz = \frac { \Gamma {(l)} \Gamma {(m)} \Gamma {(n)} }{ \Gamma{(l+m+n+1)} }$$where V is the region given by $x \ge 0 y \ge 0 z \ge 0 x+y+z \le 1$ . 1. If $x, y, z$ are all positive such that $h_1 < (x+y+z) < h_2$ then $$\int \int \int_{V} x^{l-1} y^{m-1} z^{n-1} F (x,y,z) dx dy ,dz = \frac { \Gamma {(l)} \Gamma {(m)} \Gamma {(n)} }{ \Gamma{(l+m+n)} } \int_{h_1}^{h_2} F(h) h^{l+m+n-1} dh$$ Detailed information about Liouville & Dirichlet’s Theorem can be found here. Remainder Theorems There are several formulas in Number Theory on finding divisibility and remainders. As an example, general remainder theorem is a general approach of Euclidean division of polynomials while on the other hand Euler’s Remainder Theorem works as an excellent utility to find divisibility of large numbers. Feel free to ask questions, send feedback and even point out mistakes. Great conversations start with just a single word. How to write better comments? 1 comment This site uses Akismet to reduce spam. Learn how your comment data is processed. TeXStudio is the most complete LaTeX Editor A detailed and practical review of one of the most amazing LaTeX editors available for all desktop platforms like Windows, Mac and Linux. TeXStudio is a team project of Benito van der Zander, Jan Sundermeyer, Daniel Braun and Tim Hoffmann, forked from the TeXMaker application, a non-open source application, which open-source development stalled in 2009. The Mystery of the Missing Money – One Rupee Puzzle Two women were selling marbles in the market place — one at three for a Rupee and other at two for a Rupee. One day both of then were obliged to return home when each had thirty marbles unsold. They put together the two lots of marbles and handing them over to a friend asked her to sell then… Just another way to Multiply Multiplication is probably the most important elementary operation in mathematics; even more important than usual addition. Every math-guy has its own style of multiplying numbers. But have you ever tried multiplicating by this way? Exercise: $88 \times 45$ =? Ans: as usual :- 3960 but I got this using a particular way: 88 45… Equations- A Basic Introduction Applied mathematics is one which is used in day-to-day life, in solving troubles (problems) or in business purposes. Let me write an example: George had some money. He gave 14 Dollars to Matthew. Now he has 27 dollars. How much money had he? If you are familiar with day-to-day calculations – you must say that George had 41 dollars, and… Symmetry in Physical Laws ‘Symmetry’ has a special meaning in physics. A picture is said to be symmetrical if one side is somehow the same as the other side. Precisely, a thing is symmetrical if one can subject it to a certain operation and it appears exactly the same after the operation. For example, if we look at a base that is left and… Irrational Numbers and The Proofs of their Irrationality “Irrational numbers are those real numbers which are not rational numbers!” Def.1: Rational Number A rational number is a real number which can be expressed in the form of where $a$ and $b$ are both integers relatively prime to each other and $b$ being non-zero. Following two statements are equivalent to the definition 1. 1. $x=\frac{a}{b}$…	2019-11-15 17:39:58	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8873481154441833, "perplexity": 656.8709353269578}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496668699.77/warc/CC-MAIN-20191115171915-20191115195915-00105.warc.gz"}
https://www.nature.com/articles/s41529-017-0005-2	Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript. # The cost of corrosion in China ## Abstract Corrosion is a ubiquitous and costly problem for a variety of industries. Understanding and reducing the cost of corrosion remain primary interests for corrosion professionals and relevant asset owners. The present study summarises the findings that arose from the landmark “Study of Corrosion Status and Control Strategies in China”, a key consulting project of the Chinese Academy of Engineering in 2015, which sought to determine the national cost of corrosion and costs associated with representative industries in China. The study estimated that the cost of corrosion in China was approximately 2127.8 billion RMB (~ 310 billion USD), representing about 3.34% of the gross domestic product. The transportation and electronics industries were the two that generated the highest costs among all those surveyed. Based on the survey results, corrosion is a major and significant issue, with several key general strategies to reduce the cost of corrosion also outlined. ## Introduction Corrosion is a naturally occurring phenomenon with costly and detrimental impact on most critical industry sectors. Understanding the exact costs of corrosion has been of great interest to corrosion scientists and engineers for many decades. In-depth comparison of corrosion costs among different industries or company sectors provides the prospect of identifying the common issues/problems, and also any best practice in corrosion control. Quantifying the staggering size of corrosion costs is also an important step in raising awareness of the seriousness and magnitude of corrosion issues, particularly relevant to decision makers in the industry and government—such that better policies can be established to improve our capacity for mitigating corrosion risks. It is noted that herein, ‘‘cost’’ will most often refer only to the economic cost of addressing corrosion and its consequences, although the societal cost and associated externalities are even more far-reaching and beyond measure. The economic cost of corrosion may be estimated directly from the application, operation and maintenance of anti-corrosion technologies (e.g., corrosion-resistant materials, protective coatings, corrosion inhibitors, anodic/cathodic protection and corrosion inspection and monitoring tools); or indirectly from the loss of productivity, compensation for casualties and environmental pollution, and any other cost that is not directly incurred within that industry. While it is difficult to quantify the indirect cost of corrosion, the direct cost is deemed calculable by combining appropriate methodologies (such as questionnaires, statistics and extrapolation), and by applying expertise in both corrosion and economics. ### Historical overview of key cost of corrosion studies in other countries The first systematic study on the cost of corrosion was performed in 1949 by H. H. Uhlig.1 Uhlig estimated the annual direct cost of corrosion in the United States to be 5.427 billion US dollars (2.1% of GNP, gross national product, at the time), by summing the costs related to anti-corrosion materials and corrosion-induced maintenance and replacement. This original report also highlighted the significance of the indirect cost of corrosion and the cost incurred through over-design. The approach described in Uhlig’s report, known as the Uhlig method, was adopted by Japan in 19772 and 1999,3 and again in the United States in 1998.4 Another pioneering corrosion cost study was published by T.P. Hoar in the United Kingdom in 1971.4 The Hoar method investigated the corrosion costs in 10 individual industrial sectors in the UK, the total associated cost of which was determined to be 1365 million pounds (3.5% of the UK GNP at the time). Hoar also estimated that among this cost, 310 million pounds could be saved by appropriate corrosion mitigation. In 1978, the National Bureau of Standards (NBS) collaborated with the Battelle Memorial Institute to review the cost of corrosion in the United States.5 An economic input/output model was applied to three different scenarios, namely, an actual world with corrosion, an imaginary world with no corrosion and an ideal world with inhibited corrosion. By comparison among the different so-called ‘‘worlds’’, the corrosion cost and the avoidable corrosion cost were derived. The Battelle-NBS study revealed that the total cost of corrosion per year was 70 billion US dollars (4.5% of GNP at the time) and that 14% of corrosion costs could be directly avoided using existing anti-corrosion technologies. According to the NBS, the uncertainty of this method was estimated to be 30%. This input/output method was also later adopted by Australia in 1983,6 Kuwait in 19957 and Japan in 1999.3 Funded by the Federal Highway Administration of the United States, CC Technologies Laboratories (now DNV GL) partnered with NACE International and conducted another nationwide cost of corrosion survey in the United States in 1998.8 The study combined the Uhlig and Hoar methods with significant input of expert knowledge and determined the corrosion costs associated with five major categories (infrastructure, utilities, transportation, production and manufacturing, and, government), including 27 industrial sectors. The results showed that the total direct cost of corrosion was ~ 276 billion US dollars per year (3.1% gross domestic product (GDP) at the time), which means that the cost per each person per year was approximately 970 US dollars. The cost from the lost time and productivity of the general public due to corrosion-related delays and outages represented the primary indirect cost, which was estimated to be approximately equal to the direct cost. Table 1 provides a historical overview of the cost of corrosion studies undertaken by large-economy countries.1, 2, 4, 5, 7,8,9,10,11 As revealed (Table 1), the national costs of corrosion generally represent approximately 1–5% of the GNP. This large variation in the corrosion cost relative to GNP was attributed to the specifics of each country and to the methodology used by each study. To the best of our knowledge, the first estimation of the global cost of corrosion, i.e., a value of 2.5 trillion US dollars (3.4% of global product), was reported by NACE International as part of its International Measures of Prevention, Application, and Economics of Corrosion Technologies (IMPACT) study in 2016.12 This estimation was obtained by analysing the data from available representative studies from different regions of the world. ### Cost of corrosion studies in China Since the 1980s, China has conducted a number of differently scaled studies on the cost of corrosion. Table 2 summarises the results obtained from a pilot survey conducted by the National Science and Technology Committee of China in 1980 in the chemical, refinery, metallurgy and fibre industries.13 The first nationwide cost of corrosion study in China was led by Prof. Wei Ke from the Institute of Metal Research, Chinese Academy of Sciences, during the period of 1999–2002. In this study, the Uhlig method was applied considering the costs of anti-corrosion technologies, whereas the Hoar method was applied to several key sectors, including the chemical, energy, transportation, construction and mechanical industries. The results from both approaches are summarised in Table 3 and Table 4.14 In recent decades, China has undergone one of the most rapid economic growths worldwide, largely supported by developments in so-called heavy industries. The rapid deployment of new materials and the widespread ageing of existing engineering structures are exacerbating corrosion issues in such industries. Therefore, increased corrosion awareness and adequate corrosion control will play a fundamental role in achieving a more sustainable and energy-efficient economy in the future. With such a background, in 2015, the Chinese Academy of Engineering initiated “Study of the Corrosion Status and Control Strategies in China”, a key consulting project led by the Institute of Oceanology, Chinese Academy of Sciences and engaging experts from the Chinese Society for Corrosion and Protection and hundreds of other related organisations. This present paper summarises the results of this study, including the economic costs of corrosion determined using the Uhlig and Hoar methods. ## Results and discussion ### Results obtained from the Uhlig method The Uhlig method was applied herein by taking into account the annual gross production of the major, and different, anti-corrosion technologies in China. According to Uhlig, the costs determined by this method include expenditures for the measures applied for the protection of materials, which increase the cost of materials over that of plain carbon steels.1 Therefore, the cost associated with the replacement of carbon steels due to corrosion is not considered in the present study. #### Coatings This section includes only the costs associated with organic paints and their application. According to the statistics from the National Bureau of Statistics of China, the gross production of paints within the period of January to December 2014 was 16,481,900 tonnes.15 Regardless of the huge variety of different types and brands of coatings, the calculation in this study was based on the average price of the coatings, which was 24,400 RMB per tonne. In general, painting costs (operation and application) are 2–3times of the paint itself. Thus, in this study, a value of 2.5 was selected.14 Notably, in addition to their anti-corrosion purpose, paints are also applied for decorative and functional effects. Anti-corrosion paints account for only approximately 50% of the total paints. Therefore, the annual gross production of anti-corrosion paints was estimate as: $$16,481,900\,{\kern 1pt} {\rm{tonnes}} \times 50\% \times 24,400\,{\rm{RMB/tonne}} \\ \times \left( {2.5 + 1} \right) = 703.78{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{billion}}\,{\rm{RMB}}{\rm{.}}$$ #### Surface treatments This section principally addresses the costs associated with surface treatments on steels and aluminium-based materials (which are the key commodity engineering metals). ### i. Galvanised steels In 2014, the national gross production of galvanised steels was 47.2 million tonnes, and the imported volume of galvanised steels was 2.882 million tonnes. The exported volume was 7.565 million tonnes. The consumption of galvanised steels was 42.517 million tonnes. The average unit prices of galvanised steels and cold-rolled steels are 5200 and 3500 RMB per tonne, respectively. The price difference is 1700 RMB per tonne. The application of galvanised steels instead of cold-rolled steels is for the purpose of an anti-corrosion effect. Therefore, the anti-corrosion investment associated with galvanised steels was calculated by multiplying this price difference with the total consumption volume as follows: $$\begin{array}{l}\\ 1,700\,{\rm{RMB/tonne}} \times 42.517\,{\rm{million}}\,{\rm{tonnes}} = 72.28{\kern 1pt} {\kern 1pt} {\rm{million}}\,{\rm{RMB}}{\rm{.}} \\ \end{array}$$ ### ii. Tinplating In 2014, China’s consumption of tinplated steels was approximately 4.4 million tonnes. The average unit price of tinplated steels is estimated to be 7000 RMB per tonne. Similar to the case of galvanised steels, the difference between the unit prices of tinplated steels and cold-rolled steels was calculated to be 3500 RMB per tonne. As a result, the anti-corrosion investment from tinplating was determined to be: $$3,500\,{\rm{RMB/tonne}} \times 4.4\,{\rm{million}}\,{\rm{tonnes}} = 15.4\,{\rm{billion}}\,{\rm{RMB}}{\rm{.}}$$ ### iii. Electroplating This cost represents the total production from small to medium-sized electroplating factories in China. In 2014, the gross production volume of these factories was 28.9 billion RMB, where approximately 60% was related to surface treatment. Therefore, the total cost of electroplating was determined to be 17.34 billion RMB. ### iv. Surface treatments for aluminium alloys In 2014, China’s aluminium alloy production was 6.4 million tonnes. The imported and exported volumes were 80,000 and 521,000 tonnes, respectively. The consumption volume was 5.966 million tonnes. The average price of surface-treated aluminium and its alloys is estimated to be 18,000 RMB per tonne, whereas non-surface-treated aluminium ingot is 12,000 RMB. The surface treatment cost of aluminium alloys was thus determined to be: $$\left( {18,000 - 12,000} \right){\rm{RMB/tonne}} \times 5.966\,{\rm{million}}\,{\rm{tonnes}} \\ = 35.8\,{\rm{billion}}\,{\rm{RMB}}.$$ The summation of these costs shows that the cost of corrosion related to surface treatments was 140.82 billion RMB. #### Corrosion-resistant materials For the present cost study, corrosion-resistant materials refer to stainless steels, weathering steels, titanium and titanium alloys, engineering plastics and rubbers. ### i. Stainless steels In 2014, China’s consumption of stainless steel was 15.72 million tonnes. As there are many different types of stainless steels (austenitic, ferritic, duplex, etc.), an average unit price of 14,000 RMB per tonne was employed. By comparing with the average price of cold-rolled steels, the anti-corrosion investment associated with stainless steel was $$(14,000 - 3,000)\,{\rm{RMB/tonne}} \times 15.72\,{\rm{million}}\,{\rm{tonnes}} \\ = 172.92\,{\rm{billion}}\,{\rm{RMB}}.$$ ### ii. Weathering steels China’s consumption of weathering steels in 2014 was 7.95 million tonnes. The average price of weathering steels is estimated to be 4000 RMB per tonne. Therefore, the total cost of corrosion based on weathering steels was $$(4,000 - 3,000)\,{\rm{RMB/tonne}} \times 7.95\,{\rm{million}}\,{\rm{tonnes}} \\ = 7.95\,{\rm{billion}}\,{\rm{RMB}}.$$ ### iii. Titanium and titanium alloys The annual consumption of titanium and its alloys in China was 44,500 tonnes. The difference between the unit prices of titanium alloys and cold-rolled steels is approximately 90,000 RMB per tonne. The total cost associated with the use of titanium and its alloys was, therefore, 4 billion RMB. ### iv. Engineering plastics and rubber The annual gross production of rubber and plastics industry in 2014 was 2991.9 billion RMB.16 Engineering plastics and rubbers used for corrosion protection accounted for approximately 0.7% of the total value. Thus, the anti-corrosion investment associated with these materials was 20.94 billion RMB. In total, the cost of corrosion-resistant materials in 2014 was determined to be 205.81 billion RMB. #### Corrosion inhibitors In general, corrosion inhibitors can be defined as substances added at low concentrations to effectively reduce metal corrosion rates. The cost related to corrosion inhibitors in China in 2014 is summarised in Table 5. #### Rust-preventing oils and greases The variety of rust-preventing oils and greases is large. Therefore, the unit price ranges from 8000 to 20,000 RMB per tonne. In 2014, China’s demand for rust-preventing oils and greases was 0.2 million tonnes. Taking the unit price as 11,000 RMB per tonne, the cost related to rust-preventing oils and greases was approximately 2.2 billion RMB in 2014. #### Electrochemical protection The costs associated with electrochemical protection include the production of anodes and the engineering cost of cathodic protection projects, the latter of which is difficult (actually, not possible) to quantify in terms of production volume. Thus, to determine the cost of electrochemical protection, we surveyed a major company in the CP market whose annual sales in 2014 totalled 113 million RMB. The market share of this company is 1.8%. Thus, the total sales of the entire industry were approximately 6.3 billion RMB. #### Summary of the results by the Uhlig method The estimated total cost of each major anti-corrosion measure in China in 2014 has been summarised in Table 6. The results herein have revealed that the direct cost of corrosion totalled approximately 1063.91 billion RMB. Among these costs, protective coatings remained the leading expenditure, followed by corrosion-resistant materials and surface treatments (Fig. 1). These summarised are in close agreement with the proportions reported in the Chinese cost of corrosion survey in 2002, and with studies by other nations.3, 14 This is indicative, perhaps alarmingly, that lessons learned are not rapidly translated to cost savings. It should be noted that the cost associated with the emerging market of corrosion inspection and monitoring was not included in this study, which was partly due to the difficulty in obtaining definitive original data. According to previous studies, the indirect corrosion cost, incurred from compensation for the corrosion-induced loss of productivity and product quality and compensation for environmental pollution, casualties, and other damages, may be one to several times of the direct corrosion cost.8, 15 Here, we conservatively estimated that the indirect corrosion cost is equal to the direct cost. Thus, the total cost of corrosion, determined by Uhlig’s approach, was at least 2127.8 billion RMB, representing 3.34% of the GDP in China. ### Results obtained from the Hoar method The Hoar method was applied in the present study by taking into consideration the five major economic sectors: infrastructure; energy; transportation; water; and manufacturing and public services. Questionnaires that were used are shown below as Table 7 in the 'Methods' section, and were distributed among typical industries of these sectors. In most of the collected questionnaires, only direct corrosion costs were given since the indirect costs such as compensation for environmental pollution and casualties and injuries were, in most cases, challenging to quantify. It is also possible that industries may have been reluctant to provide such information. The following section summarises the estimated direct corrosion costs associated with each industry in the different sectors. #### Infrastructure In 2014, 107,700 km of new roads (104,900 km) and bridges (2800 km) were constructed in China. The investments totalled 1546.09 billion RMB for roads and bridges.17 The cost associated with their construction includes the cost of concrete, drainage and pavement, design and measurement, management, consultation fees and labour costs. Based on the collected questionnaires, the total anti-corrosion investment for new road and bridge construction in China in 2014 was 51.49 billion RMB. In 2014, 4,353,800 km of roads were under maintenance in China.17 The total corrosion cost from their maintenance in China in 2014 was estimated to be 10.89 billion RMB. Therefore, the total corrosion cost for the roads and bridges in China in 2014 was 62.37 billion RMB, which is equivalent to 4.03% of the total investment in this industry. ### Ports and piers The total investment for ports and piers on the rivers and coasts in China was 145.99 billion RMB17 in 2014, of which 1.67%, according to the survey results, was associated with anti-corrosion. The anti-corrosion investment for newly constructed ports and piers in 2014 was 2.44 billion RMB. In addition, approximately 0.19 billion RMB was spent on corrosion-related maintenance of ports and piers. Thus, the total direct cost of corrosion for ports and piers was 2.63 billion RMB, which accounted for 1.80% of the annual investment in the survey period. ### Water conservancy A survey of major reservoirs in China showed that the corrosion costs were mainly incurred from the application of anti-corrosion technologies in the construction and the ultilisation of metallic equipment and structures. At present China has established >97,700 reservoirs of different types and sizes, and in 2014 alone, the total investment for water conservation was 488.1 billion RMB,18 of which 2.03% was used to cover corrosion costs. Therefore, the total direct corrosion cost for water conservancy in China was 9.91 billion RMB. ### Coal mining Corrosion costs in the coal mining industry were mainly incurred from underground construction, repairs, inspections, maintenance and asset depreciation. In the present study, 30 coal mines of different scales were surveyed. Their annual production accounts for ~ 4% of the total coal production in China. It was estimated that the direct corrosion costs in coal mining industries in China was 84.70 billion RMB, representing 4.67% of the total annual production value of coal. ### Fossil fuel power Corrosion-resistant materials, such as plastic coatings and linings, are widely used in fossil fuel power plants. In 2014, the total installed capacity of fossil fuel power plants in China was 911.33 gigawatts.16 Based on the survey results, the total direct corrosion cost in the fossil fuel plants was 30.53 billion RMB, representing 1.91% of the annual production value. In all, 4.37 billion RMB of this cost was from the anti-corrosion investment of new constructions, while the other 26.16 was for maintenance and repair of existing structures. ### Oil and gas The oil and gas industry referred to in this study covers all of exploration, production and transmission. Corrosion is considered a primary factor affecting the reliability of engineering structures in this industry. Relative to other industries, a higher expenditure on anti-corrosion investment can be expected in the oil and gas industry to cover corrosion allowance, coatings, cathodic protection, corrosion inhibitors, inspection and repairs, corrosion-related personnel and other ancillary items. In 2014, the annual crude oil production in China was 210 million tonnes, and the gas production was 128 billion cubic metres.19 The total direct corrosion cost in this industry was estimated to be 34.70 billion RMB, representing 2.82% of the total production value. ### Electric power transmission The costs in this section refer to those generated from corrosion and its control during power transmission and at electric substations. According to the survey results, the annual direct corrosion costs (including anti-corrosion investment and asset depreciation) for electric power transmission and substations were 76.1 billion and 3.30 billion RMB, respectively. The total of the two costs was 79.4 billion RMB, representing 3.58% of the annual production value. ### Automobiles The auto industry has been one of the fastest growing industries in China in the past decade. China is currently the second largest market for new automobiles and the third largest automobile manufacturer. In 2014, the quantity of privately owned automobiles was 145.98 million,16 which represented a growth of ~ 15% relative to the value in 2013. The total volume of automobile transactions in China in 2014 was 648.1 billion RMB and the market value of automobile repairs was approximately 500 billion RMB per year. Corrosion heavily impacts the auto industry. Based on the survey results, the annual direct corrosion costs totalled 187.25 billion RMB, representing 2.82% of the total asset value of the auto industry. ### Shipbuilding In 2014, the completed shipbuilding in China was 39.05 million DWT (i.e., dead weight tonnage).20 There are a total number of 1491 shipbuilders of varying size, which generated a main business income of 633.4 billion RMB in 2014. Due to the application environment of ships, heavy costs are incurred in anti-corrosion investments, repairs and asset depreciation of the shipbuilding industry. The total values of these costs were estimated to be 58.00 billion RMB, representing 9.16% of the total main income in the industry. ### Railways In this category, only corrosion of railways was considered, while the corrosion costs of railcars was not included. The total mileage of railway in China is one of the longest in the world. In 2014, the mileage of railways in use was 0.11 million km,17 and the annual construction investment was 808.8 billion RMB. Based on the survey results, the direct corrosion cost was 18.88 billion RMB. ### Airplanes Typical corrosion costs in the aviation industry are generated from the application of comparatively expensive corrosion-resistant materials and coatings, and from asset depreciation. Our survey showed that direct corrosion cost in the aviation industry in 2014 was 4.59 billion RMB. ### Water supply and drainage In this category, it is noted that only urban areas were included for calculations. In 2014, the total water supply in the urban areas of China was 546.7 billion cubic metres, based on an annual gross production value of 218.7 billion RMB. The total urban drainage in 2014 was 40.22 billion tonnes, at a cost of 54.70 billion RMB.21 In 2014, 31,000 kilometres of different water pipes were constructed in China. The anti-corrosion investment associated with these new constructions varied from 30 to 70% of the total construction cost. Other direct corrosion costs were generated from pipeline replacements, depreciation and corrosion loss, and mitigation in water-supplying companies themselves. According to our survey, the total direct corrosion cost was 9.69 billion RMB. Notably, the cost from water leakage, considered as an indirect cost, may be much higher than the direct corrosion cost. Our study showed that leaked water resulted in a loss of 13.12 billion RMB in China in 2014. ### Metallurgy In the iron and steel-making industry, corrosion costs were most often incurred from material loss during high temperature oxidation and acid cleaning, representing roughly 60 and 20% of the total corrosion cost. Based on the questionnaires collected from the major steel companies in China, the direct corrosion cost is estimated to represent 1.40% of the gross production value of the industry, which resulted in a value of 104.02 billion RMB in 2014. Relative to ferrous materials, the production processes of non-ferrous materials generated less corrosion issues. Based on the survey results from typical aluminium and copper producers, direct corrosion costs represented 0.60% of the gross production value. Thus, the direct corrosion cost was 30.78 billion RMB. ### Chemical industry The direct corrosion cost in the chemical production and processing industries was estimated based on questionnaires distributed to chemical plants producing acids, chlorines, bases, fertilisers and manufacturers of protective coatings and linings. The result showed that the expenditures on corrosion-resistant alloys and coatings were the leading direct corrosion costs in the chemical industry. According to the survey, the total direct corrosion cost of this industry in 2014 was 147.10 billion RMB, representing 1.67% of the gross market size. Relative to the direct cost, indirect costs in the chemical industry, such as those generated by production downtime, hazardous incidents and environmental pollution, could be several times higher. ### Pulp and paper China is one of the largest producers and consumers of pulps and papers. In 2014, there were ~ 3000 companies in this industry. Corrosion in pulp and paper manufacturing facilities can cause production downtime and the loss of product quality. In 2014, the gross market size of the pulp and paper industry was 787.9 billion RMB in China, and the direct corrosion cost was estimated to be 9.78 billion RMB. ### Electronics Corrosion issues with materials used in electronics have been rapidly growing in the past decade. For example, corrosion on printed circuit boards may lead to malfunctions or even failures of entire electronic devices. The consequences and the indirect cost of these incidents can be enormous. In the present study, questionnaires were distributed among several major companies in consumer electronics and home appliances. The results revealed that in 2014, the direct corrosion cost of this industry in China was as high as 224.80 billion RMB. However, it should be acknowledged that a large percentage of this cost can be saved with proper recycling of electronic waste materials. ### Agriculture production Corrosion issues on agricultural machinery were considered in this category. The tools used for agriculture production are often exposed to outdoor environments with high levels of sunlight radiation and high humidity. They are also in contact with corrosive media such as soils and fertilisers and are subjected to heavy material loss due to mechanical impact and abrasion. However, only commodity materials and minimal mitigation measures are applied on most agricultural machinery. Our survey estimated that the direct cost of corrosion in the agriculture production industry in China was 9.89 billion RMB in 2014, representing 2.50% of the gross production value of the industry. ### Cultural heritage The conservation of historic relics and artefacts remains an outstanding issue in China considering its relatively long history. The environmental deterioration problems are ubiquitous, occurring not only on metallic objects but also on rocks, wood, leather and paper. Factors that contribute to the deterioration of historic artefacts include humidity, sunlight, temperature, wind and biological activity. In 2014, the gross final expenditure on historic preservation was 36.40 billion RMB,13 and the direct corrosion cost was estimated to be 12.20 billion RMB. #### Summary of the results by the Hoar method By summarising the costs from the aforementioned representative industries, the Hoar method revealed that the total direct corrosion cost associated with the five major economic sectors was 1109.02 billion RMB (excluding the cost related to cultural heritage due to the difficulty in defining a gross production value). Extrapolation from the GDP of these sectors to the overall GDP in China showed that the total direct corrosion cost determined by the Hoar method was 1348.98 billion RMB. Notably, this value is higher than that determined by the Uhlig method. As explained in previous studies,3 this difference can be attributed to (1) the fact that the Hoar method takes into consideration both anti-corrosion investments and maintenance, whereas the Uhlig method focuses principally on the former; and (2) some unavoidable increase in calculated values if there is an overlap from different industries. The direct corrosion costs in the five major economic sectors and their percentages are shown in Table 8 and pictorially in Fig. 2. The highest corrosion cost, with a value of 538.57 billion RMB, was generated by the sector of manufacturing and public services, which is rationalised on the basis that China has the largest manufacturing output (~ 20%) in the world. The large population of China also contributes to the high corrosion cost in this sector due to the large, and rapid, asset depreciation. The direct corrosion costs of the surveyed industries are listed in Table 9 and plotted in Fig. 3. Among the industries surveyed, the top two industries in terms of corrosion costs are transportation and electronics, which is different in comparison to the corrosion cost study in 2002; which listed construction and machinery as they top corrosion cost industries. Such differences are attributed to the shift of economic conditions in China. The past decade has seen boosts in both transportation (e.g., automobiles and railways) and electronics industries, which now have relatively high gross market values. For the transportation industry, the large corrosion cost may also be attributed to substantial maintenance costs in corrosive service environments (e.g., for ships). Coal mining and roads and bridges were the top two industries in terms of the percentages of corrosion costs as normalised by their respective gross product values. This is rationalised by the wide use of comparatively inexpensive (also less corrosion-resistant) materials, which leads to more rapid asset depreciation and high replacement costs. Such findings highlight the relevance of life cycle considerations and longer-term durability management planning. ### Comparison of cost of corrosion studies between China and other countries Based on the Uhlig method, the total (direct + indirect) corrosion cost was estimated to be 3.34% of the GDP of China. This value is slightly lower than the global percentage cost of corrosion (3.4%),12 which was estimated by NACE primarily based on the US cost of corrosion in the 1998 study.8 However, it is difficult to directly and quantitatively compare the costs of corrosion from different countries. The reasons are at least twofold. First, the methodologies adopted by different countries are different.22, 23 While the Uhlig method is relatively straightforward, it requires trustworthy and detailed national economic data, which are not easy to obtain in some countries. For the Hoar method, the difference in the classifications of direct/indirect costs can be a significant source that adds to the difficulty of comparing the corrosion costs. For example, the direct corrosion cost in the 2012 India study was 2.4% of the GDP, which excluded the loss of product and efficiency due to corrosion.12, 24 If these costs are included, the direct cost of corrosion in India was 4.5%. The second reason for the difficulty in a quantitative cost comparison arises from the fact that the different countries have varied economic structures. For example, countries that rely highly on importation may not generate a high corrosion cost in the manufacturing sectors. ## Conclusions The Cost of Corrosion study reported herein is a summary of results from a national survey on the general costs arising from corrosion in the major representative industrial sectors in China. Utilising the Uhlig method of cost calculation, the total annual cost of corrosion in China was estimated to be 2127.8 billion RMB, representing 3.34% of the GDP. This result implies that corrosion could cost ~ 1555 RMB each year for each individual. It is generally agreed that 15–35% of corrosion costs can be avoided via appropriate corrosion mitigation approaches,12 which implies that up to 774.7 billion RMB of corrosion associated costs could be avoided annually, in China alone. Therefore, with the aim of reducing corrosion associated costs, it is possible to propose the following general strategies: 1. a. A national strategy should be developed with the principal aim of reducing corrosion costs, and minimising the risks and hazards from corrosion. This action requires coordination from multiple governmental agencies (including national, regional and provincial), the participation by relevant corrosion professionals, and increased awareness of corrosion in the entirety of society. To be impactful, it would require a the establishment of a cross-agency national committee. Potential functions of such a committee, may include, but not be limited to, establishing policies on corrosion research and education. 2. b. Continuous efforts and funding to support basic and applied research remain a critical requirement to further identify causes and fundamentals of corrosion, in addition to the development of improved anti-corrosion technologies. Furthermore, the rapid insertion of new materials and the extreme application of existing materials in new environments, will also require continued research regarding durability. Models that comprehensively consider key material and environmental factors are also presently lacking a mechanistic understanding and also then lacking accurate prediction of what are often complex corrosion processes. Protective coatings with high-corrosion resistance performance, with low repair costs and a low concentration of volatile organic compounds—will be increasingly required for corrosion protection. Many of the contemporary materials durability issues are multidisciplinary, which highlights the necessity of developing an open dialogue and an effective means for sharing and collaborations within the (comparatively large) Chinese corrosion community.25 3. c. Despite the high technological / scientific output of the Chinese corrosion community overall, standardised technologies are yet to be outlined that broadly benefit the industrial sector. Moreover, corrosion standards (in numerous areas, be it prediction, monitoring, repair, etc.) should be developed under an international setting so that best practices in China and other countries, may be shared. Furthermore, stricter regulations or even law enforcement may be needed for the inspection, monitoring and protection of infrastructure and equipment whose corrosion failures may cause severe damage and casualties. 4. d. A paradigm shift is required for companies whose operations are corrosion-affected; standing to benefit from integration of corrosion management into their regular management system. The risks and costs of corrosion should be tracked throughout asset life cycles. The returns (both short-term and long-term) on anti-corrosion investments should be calculated prior to materials selection and corrosion control. The study herein reveals that the top performing companies that invested in superior corrosion-resistant materials (combined with regular inspection) incurred much lower long-term corrosion costs than those using lower-quality and inexpensive materials. 5. e. A lack of corrosion awareness in the general society and the general population, remains a key concern of the Chinese corrosion community. In the present study, only a few percent of responders indicated that their facilities employ professional corrosion engineers or employees with sufficient (or any) corrosion background. This finding applies to industries that are well known for their corrosion losses, such as the oil and gas and chemical industries. A solution to this issue relies on the growth and focus of investment in corrosion education at different levels. Presently, no university in China offers an undergraduate corrosion degree, and corrosion education, if any, is nominally restricted to no more than 1–2 courses at the graduate level, and 1–2 class hours at the undergraduate level—even for the top Chinese universities with degrees in Materials Science and Engineering. In addition, professional societies, universities and the private sector are also urged to work together to develop professional training and certification programmes to meet the practical requirements of industry. ## Methods ### The Uhlig method According to the Uhlig method, the direct cost of corrosion in China was estimated based on the total values of major anti-corrosion technologies in the year 2014, including the following: 1. a. Paints and coatings (costs associated with their application were also included); 2. b. Surface treatments (galvanising, tinplating and electroplating; and surface treatments for aluminium and its alloys); 3. c. Corrosion-resistant materials (stainless steels, weathering steels, titanium and its alloys, engineering plastics and rubbers); 4. d. Corrosion inhibitors; 5. e. Rust-preventing oils and greases; and 6. f. Electrochemical protection (cathodic and anodic protection). The annual gross production values of anti-corrosion technologies were obtained from governmental statistics departments, whereas the average unit prices were determined based on data and expert opinions from various sources including, industry, governmental agencies, trade organisations and individual manufacturers. Alternatively, the total market value can also be estimated by surveying a limited number of manufacturers and weighting their contributions by market share. Notably, the cost calculated by the Uhlig method does not include certain direct or indirect losses, including but not limited to the following: 1. a. Production loss due to corrosion-induced outage; 2. b. Loss of products caused by corrosion leakages and accident; 3. c. Reduction in the production efficiency from corrosion activities; 4. d. Quality loss of the product due to the contamination of corrosion products; 5. e. Over-design for corrosion; and 6. f. Compensation for hazardous incidents caused by corrosion. The total cost of these important factors varies across industries and may be several times larger than the direct cost of corrosion. ### The Hoar method For the Hoar method, the total cost of corrosion is the summation of the costs of individual industries, which are typically obtained by investigating a limited number of companies in the particular industry, and weighting their contributions by market share. In this method, the representativeness of the companies selected is critical, as any calculation based on companies’ subject to different degrees of corrosion can produce significantly different total costs for the industry. In the present study, a general questionnaire, as shown in Table 7, was distributed to five categorised economic sectors (i.e., infrastructure, transportation, energy, water, and manufacturing and public services) covering the industries of: roads and bridges; ports and piers; water conservation; oil and gas (i.e., exploration and production, storage and distribution, and refining); coal mining; electric power transmission;ships; motor vehicles; aircrafts; railroads, water and sewer systems; pulp and paper; metallurgy; chemical production and processing; electronics; agriculture; and historical artefacts. It should be noted that the questionnaire utilised was intended to be comprehensive, and, therefore, some items included may not apply to all industries. Within each industry, companies of different scales were included to ensure representativeness of the results. During the investigation, typical issues that were encountered included the unavailability of the cost of corrosion data (which is a significant point to emphasise) or the lack of necessary corrosion awareness and expertise. In such cases, expert opinions were largely required to help separate the corrosion cost from general costs such as maintenance and replacements. ### Data availability The data presented and discussed in this study is available from the authors upon reasonable request. ## References 1. Uhlig, H. H. The cost of corrosion to The United States. Corrosion 6, 29–33 (1950). 2. Committee on Corrosion Loss in Japan. Report on corrosion loss in Japan. Boshoku-Gijutsu (Corros.Eng) 26, 401–512 (1977). 3. Committee on Corrosion Loss in Japan. Survey of corrosion cost in Japan. Zairyo-to-Kankyo (Corros.Eng) 50, 490–512 (2001). 4. Hoar, T. P. Corrosion of metals: Its cost and control. Proc. R. Soc. 348, 1–18 (1976). 5. Bennett, L. H. Economic Effects of Metallic Corrosion in the United States: A Report to the Congress (1978). 6. Cherry, B. W. & B. S. Skerry, Corrosion in Australia: The Report of the Australian National Centre for Corrosion Prevention and Control Feasibility Study. (1983). 7. Al-Kharafi, F., Al-Hashem, A. & Martrouk, F. Economic Effects of Metallic Corrosion in the State of Kuwait (KISR Publications, 1995). 8. Koch, G. H., Brongers, M. P. H., Thompson, N. G., Virmani, Y. P. & Payer, J. H. Chapter 1 - Cost of corrosion in the United States, In Handbook of Environmental Degradation of Materials (William Andrew Publishing, 2005). 9. Behrens, D. Research and development programme on ‘corrosion and corrosion protection’in the German federal republic. Br. Corros. J. 10, 122–127 (1975). 10. Potter, E. C. & Potter, E. G. The corrosion scene in Australia. Aust. Corros. Eng. 16, 21–29 (1972). 11. Revie, R. W. & Uhlig K., H. H. Cost of corrosion to Australia. J. Inst. Eng. Aust. 46, 3–15 (1974). 12. Koch, G., J. Varney, N., Thompson, O., Moghissi, et al. International Measures of Prevention, Application, and Economics of Corrosion Technologies Study. NACE International (2016). 13. State Scientific and Technological Commission Corrosion Science Department Report on Corrosion Loss in China. (1982). 14. Ke, W. China Corrosion Investigation Report (Chemical Industry Press, 2013). 15. Zhou, F. The main characteristics of the coatings enterprises in our country. China Coating 2, 10–11 (2001). 16. National Bureau of Statistics of the People’s Republic of China. China Statistical Yearbook (2014). 17. Ministry of Transport of the People’s Republic of China. Statistical bulletin of transportation industry development in 2014 (2015). 18. The Ministry of Water Resources of the People’s Republic of China. Bulletin of water resources development in 2014 (2015). 19. Ministry of Land and Resources of the People’s Republic of China. Bulletin of land and resources in 2014 (2015). 20. Ministry of Industry and Information Technology of the People’s Republic of China. Report on the development of shipbuilding industry in 2014 (2015). 21. The Ministry of Housing and Urban-Rural Development of the People’s Republic of China Bulletin of Urban-Rural Development in (2014). 22. Biezma, M. & Cristobal, J. S. Methodology to study cost of corrosion. Corros. Eng. Sci.Techn. 40, 344–352 (2005). 23. Biezma, M. & Cristóbal, J. S. Is the cost of corrosion really quantifiable? Corrosion 62, 1051–1055 (2006). 24. Bhaskaran, R., Bhalla, L., Rahman, A. & Juneja, S. et al. An analysis of the updated cost of corrosion in India. Mater. Perform. 53, 56–65 (2014). 25. Li, X., Zhang, D., Liu, Z. & Li, Z. et al. Share corrosion data. Nature 527, 441–442 (2015). ## Acknowledgements This work was supported by the Key Consulting Project of Chinese Academy of Engineering—“Study of the Corrosion Status and Control Strategies in China” (2015-ZD-08). We would like to express our deepest gratitude to Academician Kuangdi Xu (Honorary President of the Presidium of the Chinese Academy of Engineering), Academician Zhongli Ding (President of University of Chinese Academy of Sciences), Academician Binshi Xu (Academy of Armored Forces Engineering), Academician Huibin Xu (President of Beihang University), Academician Jianyun Zhang (President of Nanjing Hydraulic Research Institute), Professor Shicheng Wei (Academy of Armored Forces Engineering), Professor Jianhua Liu (Beihang University), Professor Xichang Zhu (Nanjing Hydraulic Research Institute) and all participating organisations and individuals of this project. We would also like to acknowledge the support from Ministry of Science and Technology of China (2012FY113000). ## Author information Authors ### Contributions B.H. designed the research and led the study. All authors conducted the research and analysed the results. B.H., X.L., X.M. and D.Z. wrote the manuscript and all authors read and edited the manuscript. ### Corresponding authors Correspondence to Baorong Hou or Dawei Zhang. ## Ethics declarations ### Competing interests The authors declare that they have no competing financial interests. ### Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. ## Rights and permissions Reprints and Permissions Hou, B., Li, X., Ma, X. et al. The cost of corrosion in China. npj Mater Degrad 1, 4 (2017). https://doi.org/10.1038/s41529-017-0005-2 • Revised: • Accepted: • Published: • DOI: https://doi.org/10.1038/s41529-017-0005-2 • ### Microbiologically influenced corrosion of steel in coastal surface seawater contaminated by crude oil • Yimeng Zhang • Xiaofan Zhai • Baorong Hou • ### Novel cationic aryl bithiophene/terthiophene derivatives as corrosion inhibitors by chemical, electrochemical and surface investigations • Mohamed A. Ismail • Mahmoud M. Shaban • Ashraf S. Abousalem Scientific Reports (2022) • ### Deep learning corrosion detection with confidence • Will Nash • Liang Zheng • Nick Birbilis • ### Improving anticorrosion performance of epoxy coating by hybrids of rGO and g-C3N4 nanosheets • Zhuang Liu • Rongtao Zhu • Haiyang Zhu Journal of Coatings Technology and Research (2022) • ### A concise review on corrosion inhibitors: types, mechanisms and electrochemical evaluation studies • I. A. Wonnie Ma • Sh. Ammar • S. Ramesh Journal of Coatings Technology and Research (2022)	2022-05-23 18:12:25	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.25431376695632935, "perplexity": 4434.435497621527}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662560022.71/warc/CC-MAIN-20220523163515-20220523193515-00734.warc.gz"}
http://jcb.rupress.org/highwire/markup/51397/expansion?width=1000&height=500&iframe=true&postprocessors=highwire_tables%2Chighwire_reclass%2Chighwire_figures%2Chighwire_math%2Chighwire_inline_linked_media%2Chighwire_embed	Table III Meiotic Progression in Oocytes from MLH1 Null Females and Controls PrometaphaseMeta IAna/MII ApolarBipolarOther* +/−41 4 4 h(91%)(9%) −/−4526 1 (63%)(36%)(1%) +/− 7 8 9 6 h(29%)(33%)(38%) −/− 740 7 (13%)(74%)(13%) +/− 2 656 1 8 h(3%)(9%)(86%)(2%) −/− 338 4 (7%)(84%)(9%) +/−3926 10 h(60%)(40%) −/−27 3 (90%)(10%) +/− 7‡32 12 h(18%)(82%) −/− 342 5 (6%)(84%)(10%) +/− 18 h −/− 711 3 (33%)(52%)(14%) • * Includes cells with collapsed spindles, spindles with grossly unequal poles, and tripolar spindles. • A proportion of control oocytes arrest at MI; this is a reflection of oocyte immaturity (Eppig et al., 1994).	2019-08-22 20:21:24	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9029757976531982, "perplexity": 9279.648120216936}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027317359.75/warc/CC-MAIN-20190822194105-20190822220105-00168.warc.gz"}
https://matholympiad.org.bd/forum/viewtopic.php?p=10311	ISL-1999 A2 Discussion on International Mathematical Olympiad (IMO) sourav das Posts: 461 Joined: Wed Dec 15, 2010 10:05 am Location: Dhaka Contact: ISL-1999 A2 The numbers from $1$ to $n^2$ are randomly arranged in the cells of a $n \times n$ square ($n \geq 2$). For any pair of numbers situated on the same row or on the same column the ratio of the greater number to the smaller number is calculated. Let us call the characteristic of the arrangement the smallest of these $n^2\left(n-1\right)$ fractions. What is the highest possible value of the characteristic ? You spin my head right round right round, When you go down, when you go down down...... (-$from$ "$THE$ $UGLY$ $TRUTH$" )	2021-01-24 18:33:32	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3116242289543152, "perplexity": 753.7636508155492}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703550617.50/warc/CC-MAIN-20210124173052-20210124203052-00788.warc.gz"}
https://www.physicsforums.com/threads/three-level-quantum-system.1002659/	# Three-level Quantum System AlexTab Summary:: Find the ratio of the number of particles on the upper level to the total number in the system. Consider an isolated system of ##N \gg 1## weakly interacting, distinct particles. Each particle can be in one of three states, with energies ##- \varepsilon_0##, ##0## and ##\varepsilon_0##. The energy of entire system is ##E##. The temperature is defined as ##\displaystyle T = \frac{\partial S}{\partial E}##. I need to find the ratio of the number of particles on the level with ##\varepsilon_0## to the total number ##N## in low temperature conditions ##T \ll \varepsilon_0##. I start with finding the statistical weight of each state of the system as ##\displaystyle W = \frac{N!}{n_{-\varepsilon_0}! n_0! n_{\varepsilon_0}!}##, where ##n_{-\varepsilon_0}##, ##n_0## and ##n_{\varepsilon_0}## are the numbers of particles in the corresponding states. Then I can find the system entropy ##S = \ln W##. Obviously, the number of particles at each next level is much lower than at the previous one. It seems to me that the above should be enough to solve the problem, but I don't understand how to use these facts. Also there is a problem with the number of particles, because we don't know ##n_{-\varepsilon_0}## and ##n_{\varepsilon_0}##, only ##N## and ##\displaystyle n_{\varepsilon_0} -n_{-\varepsilon_0} = \frac{E}{\varepsilon_0}## are given. Gold Member It seems to me like this problem is having you re-invent the wheel, because what you're doing right now is deriving the canonical ensemble from stat mech. Note that each tuple ##(n_+,n_0,n_-)## is one way of dividing the ##N## particles into the 3 energy levels, so let's call it a "distribution". You have a function for the entropy ##S = \ln W## that spits out the entropy associated with each distribution. If you leave an isolated system to do its thing for a while, what do you think the entropy of the system will do over time? If you wait until the system reaches steady state, do you think the entropy will be maximum or minimum? I've already kind of spoiled the fact that you'll want to do an optimization procedure on the entropy to get the distribution ##(n_+,n_0,n_-)## at statistical equilibrium. Keep in mind that this is a constrained problem, since ##N## and ##E## are fixed. So you'll want to think about Lagrange multipliers. Also, you'll want to use Stirling's approximation a lot.	2023-02-09 00:23:36	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8501965999603271, "perplexity": 558.9668696385587}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500983.76/warc/CC-MAIN-20230208222635-20230209012635-00059.warc.gz"}
https://artineering.io/software/MNPRX/docs/rendering/	Rendering with MNPRX is a breeze compared to rendering with offline renderers, as every frame is calculated in real-time! In fact, computers normally take longer to save image files to disk than to render them. There are many ways to get images out of Maya, but we strongly recommend the tools that we have developed to get the best and most consistent results, the Viewport Renderer (rendr icon in the MNPRX shelf) and the Target Sequence Renderer (tSeq icon in the MNPRX shelf). These two tools will allow you to render images of up to 16K resolution (16384×16384) with a graphics card supporting it. A tutorial using these tools can be found in the Rendering Scenes documentation. ## Viewport Renderer The Viewport Renderer is a simple tool to render images and videos, divided into two sections: Frame Capture (save image) and Quick Playblasts (save video). ### Frame capture The upper section of the Viewport Renderer contains all options to capture a frame (image). On the left side, you can define a custom resolution to render at. If left unchecked, the viewport resolution will be taken into consideration when rendering. On the right side, you can find the settings to render with. • Quality - Defines the quality of the viewport rendering and correlates with the quality settings of the configuration node. TAA gives the best visual quality and is the default in the Indie and Studio licenses of MNPRX. • Alpha - Defines how the alpha (transparency) is handled in the rendered frames. The options are None (no alpha), Linear (linear alpha) and Premult. (premultiplied alpha). • Format - Defines the image file format to save the image as i.e., .png, .jpg, .exr, .tif, .iff. .exr images will be in linear space, so you will have to do the gamma correction yourself for the results to look as they appear in MNPRX. The Render button will capture/render the frame in the directory you specify. ### Quick Playblast The lower section of the Viewport Renderer contains all options to create a quick playblast (video). On the left side, you can define where to get the playblast resolution from. From Viewport will get the viewport resolution to playblast with. From Render Settings will get the Render Settings resolution to playblast with. On the right side, you can find the settings to playblast with. • Quality - Defines the quality of the viewport rendering and correlates with the quality settings of the configuration node. 4x SSAA gives the best visual quality within playblasts and is the default in the Indie and Studio licenses of MNPRX. To use TAA quality, you’ll have to render an image sequence with the Target Sequence Renderer tool and create the video yourself. • Format - Defines the video file format to save the playblast as i.e., .avi, .mov (qt). The video file format will depend on what codecs your computer currently supports. The Playblast button will playblast the video in the directory you specify. ## Target Sequence Renderer The Target Sequence Renderer is a tool that allows to easily render any target/s as image sequences. The tool itself is divided into three sections: Directory, Targets and Settings. ### Directory The upper section of the Target Sequence Renderer allows you to specify where the image sequence is going to be saved in. By default, your current project’s directory will be set, but you can define any directory by clicking on the Browse button and navigating to your desired path. ### Targets The left section outlines all internal targets available to render. By default, all targets that are required to replicate the stylization in a compositing application are pre-selected. This allows you to have the same stylization in, for example, Nuke and have complete freedom to push the look further in comp. We’ve already replicated the watercolor stylization in Nuke, but the ability to export MNPRX’s internal targets will become especially relevant in the future (we have big plans for this, stay tuned!). Internal render targets (all except outputTarget) should be exported as .exr to preserve the embedded linear floating data of each image. When exporting the outputTarget, all other targets will automatically be disabled and the Render settings on the right section will be enabled. The quality and alpha can only be modified for the outputTarget. ### Settings The right section of the Target Sequence Renderer allows you to define all the sequence render settings. By default, the name will be that of your scene, but you can customize it however you want. Folder structure defines the folders that will be created within the directory. By default, only a name folder will be created. Additional folders can be created, if desired, for each camera and target that you render. The hierarchy will look something like this directory/name/camera/target/image.ext. Render settings allow you to customize the quality, alpha and format of the image sequence. It works in the same way as the Viewport Renderer’s Frame Capture settings. • Quality - Defines the quality of the render and correlates with the quality settings of the configuration node. TAA gives the best visual quality and is the default in the Indie and Studio licenses of MNPRX. • Alpha - Defines how the alpha (transparency) is handled in the rendered image sequence. The options are None (no alpha), Linear (linear alpha) and Premult. (premultiplied alpha). • Format - Defines the image file format to save the image sequence as i.e., .png, .jpg, .exr, .tif, .iff. Resolution allows you to define a custom resolution to render with. You can specify the width and height individually up to 16384 pixels, if your graphics card supports it. A dedicated GPU should be able to render images of up to 8K without much trouble. Frame range defines the start and end of the target sequence in the timeline. Camera views outlines the available cameras in the scene and allows you to select which views you want to render the sequence through. By default, all cameras set as ‘renderable’ in Maya’s Render Settings will be selected. The Render targets button will start rendering the target sequence with the settings you specified. You can cancel the sequence render anytime by hitting on cancel within the render progress dialog. ## FAQ When rendering with TAA, the render comes out shifted This will happen when the 2D Pan/Zoom is used in the camera you are rendering (Camera Shape Attributes -> Display Options -> 2D Pan/Zoom). We discourage from using this function with MNPRX at the moment, but we might add this in the future if requested by the community. 1. Viewport Renderer 2. Target Sequence Renderer 3. FAQ	2020-04-06 17:58:05	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.21291878819465637, "perplexity": 3270.3375037318383}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371656216.67/warc/CC-MAIN-20200406164846-20200406195346-00328.warc.gz"}
https://www.jobilize.com/course/section/references-comparing-two-independent-population-proportions-by-opensta?qcr=www.quizover.com	# 9.3 Comparing two independent population proportions -- rrc math (Page 3/20) Page 3 / 20 ## Try it A concerned group of citizens wanted to know if the proportion of forcible rapes in Texas was different in 2011 than in 2010. Their research showed that of the 113,231 violent crimes in Texas in 2010, 7,622 of them were forcible rapes. In 2011, 7,439 of the 104,873 violent crimes were in the forcible rape category. Test at a 5% significance level. Answer the following questions: a. Is this a test of two means or two proportions? a. two proportions b. Which distribution do you use to perform the test? b. normal for two proportions c. What is the random variable? c. Subscripts: 1 = 2010, 2 = 2011 P 2 - P 2 d. What are the null and alternative hypothesis? Write the null and alternative hypothesis in symbols. d. Subscripts: 1 = 2010, 2 = 2011 H 0 : p 1 = p 2 H 0 : p 1 p 2 = 0 H a : p 1 p 2 H a : p 1 p 2 ≠ 0 e. Is this test right-, left-, or two-tailed? e. two-tailed f. What is the p -value? f. p -value = 0.00086 g. Do you reject or not reject the null hypothesis? g. Reject the H 0 . h. At the ___ level of significance, from the sample data, there ______ (is/is not) sufficient evidence to conclude that ____________. h. At the 5% significance level, from the sample data, there is sufficient evidence to conclude that there is a difference between the proportion of forcible rapes in 2011 and 2010. ## References Data from Educational Resources , December catalog. Data from Hilton Hotels. Available online at http://www.hilton.com (accessed June 17, 2013). Data from Hyatt Hotels. Available online at http://hyatt.com (accessed June 17, 2013). Data from Statistics, United States Department of Health and Human Services. Data from Whitney Exhibit on loan to San Jose Museum of Art. Data from the American Cancer Society. Available online at http://www.cancer.org/index (accessed June 17, 2013). Data from the Chancellor’s Office, California Community Colleges, November 1994. “State of the States.” Gallup, 2013. Available online at http://www.gallup.com/poll/125066/State-States.aspx?ref=interactive (accessed June 17, 2013). “West Nile Virus.” Centers for Disease Control and Prevention. Available online at http://www.cdc.gov/ncidod/dvbid/westnile/index.htm (accessed June 17, 2013). ## Chapter review Test of two population proportions from independent samples. • Random variable: ${\stackrel{^}{p}}_{A}–{\stackrel{^}{p}}_{B}=$ difference between the two estimated proportions • Distribution: normal distribution ## Formula review Pooled Proportion: p c = Distribution for the differences: ${{p}^{\prime }}_{A}-{{p}^{\prime }}_{B}\sim N\left[0,\sqrt{{p}_{c}\left(1-{p}_{c}\right)\left(\frac{1}{{n}_{A}}+\frac{1}{{n}_{B}}\right)}\right]$ where the null hypothesis is H 0 : p A = p B or H 0 : p A p B = 0. Test Statistic ( z -score): $z=\frac{\left({p}^{\prime }{}_{A}-{p}^{\prime }{}_{B}\right)}{\sqrt{{p}_{c}\left(1-{p}_{c}\right)\left(\frac{1}{{n}_{A}}+\frac{1}{{n}_{B}}\right)}}$ where the null hypothesis is H 0 : p A = p B or H 0 : p A p B = 0. where p′ A and p′ B are the sample proportions, p A and p B are the population proportions, P c is the pooled proportion, and n A and n B are the sample sizes. Use the following information for the next five exercises. Two types of phone operating system are being tested to determine if there is a difference in the proportions of system failures (crashes). Fifteen out of a random sample of 150 phones with OS 1 had system failures within the first eight hours of operation. Nine out of another random sample of 150 phones with OS 2 had system failures within the first eight hours of operation. OS 2 is believed to be more stable (have fewer crashes) than OS 1 . Is this a test of means or proportions? What is the random variable? P OS1 P OS2 = difference in the proportions of phones that had system failures within the first eight hours of operation with OS 1 and OS 2 . State the null and alternative hypotheses. What is the p -value? 0.1018 What can you conclude about the two operating systems? Use the following information to answer the next twelve exercises. In the recent Census, three percent of the U.S. population reported being of two or more races. However, the percent varies tremendously from state to state. Suppose that two random surveys are conducted. In the first random survey, out of 1,000 North Dakotans, only nine people reported being of two or more races. In the second random survey, out of 500 Nevadans, 17 people reported being of two or more races. Conduct a hypothesis test to determine if the population percents are the same for the two states or if the percent for Nevada is statistically higher than for North Dakota. Is this a test of means or proportions? proportions State the null and alternative hypotheses. 1. H 0 : _________ 2. H a : _________ Is this a right-tailed, left-tailed, or two-tailed test? How do you know? right-tailed What is the random variable of interest for this test? In words, define the random variable for this test. The random variable is the difference in proportions (percents) of the populations that are of two or more races in Nevada and North Dakota. Which distribution (normal or Student's t ) would you use for this hypothesis test? Explain why you chose the distribution you did for the Exercise 10.56 . Our sample sizes are much greater than five each, so we use the normal for two proportions distribution for this hypothesis test. Calculate the test statistic. Sketch a graph of the situation. Mark the hypothesized difference and the sample difference. Shade the area corresponding to the p -value. Check student’s solution. Find the p -value. At a pre-conceived α = 0.05, what is your: 1. Decision: 2. Reason for the decision: 3. Conclusion (write out in a complete sentence): 1. Reject the null hypothesis. 2. p -value<alpha 3. At the 5% significance level, there is sufficient evidence to conclude that the proportion (percent) of the population that is of two or more races in Nevada is statistically higher than that in North Dakota. Does it appear that the proportion of Nevadans who are two or more races is higher than the proportion of North Dakotans? Why or why not? What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ? why we need to study biomolecules, molecular biology in nanotechnology? ? Kyle yes I'm doing my masters in nanotechnology, we are being studying all these domains as well.. why? what school? Kyle biomolecules are e building blocks of every organics and inorganic materials. Joe anyone know any internet site where one can find nanotechnology papers? research.net kanaga sciencedirect big data base Ernesto Introduction about quantum dots in nanotechnology what does nano mean? nano basically means 10^(-9). nanometer is a unit to measure length. Bharti do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment? absolutely yes Daniel how to know photocatalytic properties of tio2 nanoparticles...what to do now it is a goid question and i want to know the answer as well Maciej Abigail for teaching engĺish at school how nano technology help us Anassong Do somebody tell me a best nano engineering book for beginners? there is no specific books for beginners but there is book called principle of nanotechnology NANO what is fullerene does it is used to make bukky balls are you nano engineer ? s. fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball. Tarell what is the actual application of fullerenes nowadays? Damian That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes. Tarell what is the Synthesis, properties,and applications of carbon nano chemistry Mostly, they use nano carbon for electronics and for materials to be strengthened. Virgil is Bucky paper clear? CYNTHIA carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc NANO so some one know about replacing silicon atom with phosphorous in semiconductors device? Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure. Harper Do you know which machine is used to that process? s. how to fabricate graphene ink ? for screen printed electrodes ? SUYASH What is lattice structure? of graphene you mean? Ebrahim or in general Ebrahim in general s. Graphene has a hexagonal structure tahir On having this app for quite a bit time, Haven't realised there's a chat room in it. Cied what is biological synthesis of nanoparticles what's the easiest and fastest way to the synthesize AgNP? China Cied how did you get the value of 2000N.What calculations are needed to arrive at it Privacy Information Security Software Version 1.1a Good Got questions? Join the online conversation and get instant answers!	2019-05-20 20:39:18	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 3, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4900038540363312, "perplexity": 2947.4605024501}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232256147.15/warc/CC-MAIN-20190520202108-20190520224108-00050.warc.gz"}
https://www.math-only-math.com/pythagoras-theorem.html	# Pythagoras’ Theorem The lengths of the sides of a right-angled triangle have a special relationship between them. This relation is widely used in many branches of mathematics, such as mensuration and trigonometry. Pythagoras’ Theorem: In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. Given: Let XYZ be a triangle in which ∠YXZ = 90°. YZ is the hypotenuse. To prove: XY2 + XZ2 = YZ2. Construction: Draw XM ⊥YZ. Therefore, ∠XMY = ∠XMZ = 90°. Proof: Statement Reason 1. In ∆XYM and ∆XYZ,(i) ∠XMY = ∠YXZ = 90°(ii) ∠XYM = ∠XYZ 1.(i) Given and by construction(ii) Common angle 2. Therefore, ∆XYM ∼ ∆ZYX 2. BY AA criterion of similarity 3. Therefore, $$\frac{XY}{YZ}$$ = $$\frac{YM}{XY}$$ 3. Corresponding sides of similar triangle are proportional 4. Therefore, XY$$^{2}$$ = YZ ∙ YM 4. By cross multiplication in statement 3. 5. In ∆XMZ and ∆XYZ,(i) ∠XMY = ∠YXZ = 90°(ii) ∠XZM = ∠XZY 5.(i) Given and by construction(ii) Common angle 6. Therefore, ∆XMZ ∼ ∆YXZ. 6. BY AA criterion of similarity 7. Therefore, $$\frac{XZ}{YZ}$$ = $$\frac{MZ}{XZ}$$ 7. Corresponding sides of similar triangle are proportional 8. Therefore, XZ$$^{2}$$ = YZ ∙ MZ 8. By cross multiplication in statement 7. 9. Therefore, XY$$^{2}$$ + XZ$$^{2}$$ = YZ ∙ YM + YZ ∙ MZ⟹ XY$$^{2}$$ + XZ$$^{2}$$ = YZ(YM+ MZ)⟹ XY$$^{2}$$ + XZ$$^{2}$$ = YZ ∙ YZ⟹ XY$$^{2}$$ + XZ$$^{2}$$ = YZ$$^{2}$$ 9. By adding statements 4 and 8 Problems on Pythagoras’ Theorem: 1. In ∆XYZ, ∠Y = 90°. If XY = 3 cm and YZ = 4 cm, find XZ. Solution: By Pythagoras, theorem, XZ$$^{2}$$ = XY$$^{2}$$ + YZ$$^{2}$$ = (3$$^{2}$$ + 4$$^{2}$$) cm$$^{2}$$ = (9 + 16) cm$$^{2}$$ = 25 cm$$^{2}$$ Therefore, XZ = $$\sqrt{25 cm^{2}}$$ Therefore, XZ = 5 cm 2. Two poles, 15 feet and 35 feet high, are 15 feet apart. Find distance between the tops of the poles. Solution: Let the first pole XY = 15 ft The second pole PQ = 35 ft. The distance between two poles YQ = 15 ft. Draw XR ⊥ PQ. Now, we have, PR = PQ - RQ = PQ - XY = (35 - 15) ft = 20 ft. Also, XR = YQ = 15 ft. Therefore, distance between tops of the poles = XP = $$\sqrt{XR^{2} + RP^{2}}$$ = $$\sqrt{15^{2} + 20^{2}}$$ ft = $$\sqrt{225 + 400}$$ ft = $$\sqrt{625}$$ ft = 25 ft	2020-04-02 22:30:19	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8465217351913452, "perplexity": 6557.799274230931}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370508367.57/warc/CC-MAIN-20200402204908-20200402234908-00305.warc.gz"}
https://reviewersph.com/mathematics-third?namadamadan=5$Find%20the%20slope%20$0	### Math Notes Subjects #### Differential Calculus Solutions ##### Topics \|\| Problems Find the slope of the curve $$x^2 +xy +y^2 =3$$ and $$(1, 1)$$ Slope = $$\frac{dy}{dx}$$ $$2xdx + xdy +ydx +2ydy = 0$$ $$2(1)dx + (1)dy + (1)dx +2(1)dy = 0$$ $$3dx + 3 dy = 0$$ $$dy = -dx$$ $$\frac{dy}{dx} = -1$$	2023-03-20 19:46:11	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2864355444908142, "perplexity": 9749.202701246648}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296943555.25/warc/CC-MAIN-20230320175948-20230320205948-00024.warc.gz"}
https://tutorials.methodsconsultants.com/posts/running-t-tests-in-sas/	# How to Perform t-Tests in SAS Nikki Kamouneh Posted on central limit theorem t test SAS t distribution All three types of $$t$$-tests can be performed in SAS. This tutorial will demonstrate the steps and syntax needed to conduct one sample, two independent samples, and paired samples t-tests. There are two datafiles used in this tutorial. The iq_wide can be downloaded from our GitHub repository here, and the iq_long data can be downloaded from our GitHub repository here. The one-sample and independent samples examples will use the iq_long data, and the paired samples example will use iq_wide. ## One Sample $$t$$-Test Say we have data from 200 subjects who have taken an IQ test. We know in the general population the mean IQ is 100. We want to test the hypothesis that our sample comes from a different population, e.g. one that is more gifted than the general population. We will first look at the distribution of scores to determine if there are any outliers or if the distribution is highly skewed. Then we will test the null hypothesis that our sample comes from a population where $$\mu \ne 100$$. To get the histogram we will run: proc univariate data=iq_long noprint; histogram iq; run; This will give us the following figure:	2022-05-21 18:33:28	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.43469467759132385, "perplexity": 931.8485264823312}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662540268.46/warc/CC-MAIN-20220521174536-20220521204536-00310.warc.gz"}
https://howardtool.com/msk7t4s/nsxel.php?ecbe48=how-to-determine-if-a-function-is-convex-or-concave	Highlight an interval where f prime of x, or we could say the first derivative of x, for the first derivative of f with respect to x is greater than 0 and f double prime of x, or the second derivative of f with respect to x, is less than 0. Choose a value in each interval and determine the sign … We can use this result and the following proposition to define a class of concave function in higher dimensions. site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. If the graph of a function is given, we can determine the function's concavity, by looking where the tangent line to the graph lie with respect to the graph. In mathematics, a real-valued function defined on an n-dimensional interval is called convex if the line segment between any two points on the graph of the function lies above the graph between the two points. Can a Familiar allow you to avoid verbal and somatic components? Sciences, Culinary Arts and Personal Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Consider the function g(x) = 250 + 8x^3 + x^4. 3. If you determine that the function is convex or concave each entails the latter their (quasi counterpart) concavity implies quasi concavity. If the tangent line to a point is below the graph, the function is concave upward or convex. Concave and convex maintain their status as adjectives when used in this context. I If f is a monotonic transformation of a concave function, it is quasi-concave. Along the line $y=x$, it is convex as a 1D function; along the line $y=-x$ it is concave. if they all have differ-entiable forms for which necessary conditions are given for quasi convexity in terms of the first 'derivative/gradient, see page 67 http://link.springer.com/book/10.1007%2F978-3-540-70876-6? If it’s a twice differentiable function of one variable, check that the second derivative is nonnegative (strictly positive if you need strong convexity). I've updated my answer. One of the most important term you will see while implementing Machine Learning models is concave, convex functions and maxima and minima … A function f of x is plotted below. However, note that a function that fails to be globally convex/concave can be convex/concave on parts of their domains. ; They also aren't linear functions, so you rule out these functions being both concave and convex. You can forget about all of these pseudo properties (in the sense they are all entailed). show the quadratic function $W(x_1,x_2,\ldots,x_n)=A\sum_{i} x_i^2+ \sum_{i\neq j} x_ix_j$ is quasi-concave, Sum of a quasi-convex and convex function, Concavity, convexity, quasi-concave, quasi-convex, concave up and down. This memory trick should help you decide whether to use convex or concave in your writing. Concavity (Convexity) implies quasi-concavity (quasi-convexity) but not the other way around. the second derivative for the first one is $f''(x)=3 e^{x} + 3x e^{x} + 80 x^{3}$. Is cycling on this 35mph road too dangerous? Why do jet engine igniters require huge voltages? I didn't get how $f(x,y)=xy$ is both quasi-concave and quasi-convex. Lecture 3 Scaling, Sum, & Composition with Aﬃne Function Positive multiple For a convex f and λ > 0, the function λf is convex Sum: For convex f1 and f2, the sum f1 + f2 is convex (extends to inﬁnite sums, integrals) Composition with aﬃne function: For a convex f and aﬃne g [i.e., g(x) = Ax + b], the composition f g is convex, where (f g)(x) = f(Ax + b) These will allow you to rule out whether a function is one of the two 'quasi's; once you know that the function is convex; one can apply the condition for quasi-linearity. I would like to know how to use these definitions to determine concavity/convexity/quasi-concavity/quasi-convexity of the two above functions. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Given the generality of a function being merely quasi convex- a set of necessary conditions can be given in terms, when the function is differentiable see A concave function can also be defined graphically, in comparison to a convex function. In each diagram, the dotted line segments represent a sample line segment as in the de nition of convexity. You can see a curve and a tangent line. Parametrise the function along that line segment by $\lambda$; then $f(\lambda) = \lambda (\lambda - 1) < 0 = \min \{ f(x), f(y) \}$. Now imagine a tangent line traveling down your … 3.16 For each of the following functions determine whether it is convex, concave, quasicon-vex, or quasiconcave. In general, you can skip parentheses, but be very careful: e^3x is e^3x, and e^(3x) is e^(3x). The function has an inflection point (usually) at any x-value where the signs switch from positive to negative or vice versa. Our experts can answer your tough homework and study questions. answer! Figure 1: The function in (i) is convex, (ii) is concave, and (iii) is neither. I If f is concave, then it is quasi-concave, so you might start by checking for concavity. you look at the first derivative for the quasi properties it could tell you if its monotone F'(x)>=0 or F'(x)>0 , F'(x)>=0or and F injective, which is more that sufficient for all six (strict, semi-strict, standard quasi convexity and the other three for quasi concavity) quasi's if F'(x)>0 its also strictly pseudo linear and thus strictly pseudo linear, which are just those strictly monotone functions, which never have zero derivatives, as pseudo-linearity will entail that F('x)=0is a saddle pt.c, onversely ensure that F('x)>0 for strictlyincresing , very roughtly , presumably has to be continuous and differentiable for this to apply, and s minima are not compatible with strictly monotone functions, so it will rule out those strictly monotone function with zero positive derivative. In particular, a (strictly) 1 - pseudo-convex function is a (strictly) plurisubharmonic function of class C ^ {2}. Let E(x) be an energy function with bounded Hessian [J2 E(x)/8x8x. Solution. Prove your answer. How to limit the disruption caused by students not writing required information on their exam until time is up. How unusual is a Vice President presiding over their own replacement in the Senate? f"(x) = g"[U(x)] • {U'(x)f + g'(U(x)) ■ U"{x) Definition 3: Concave function A twice continuously differentiable function f is concave if and only if 2 1 0 ii f x x w t w In the one variable case a function is concave if the derivative of the function is decreasing. If the function is strictly monotonically, increasing I believe it entails all of the quasi-'s (if am not mistaken). For the first one ($f(x) = 3 \text{e}^{x} + 5x^{4} - \text{ln}(x)$) I used a graphing calculator to have an idea of the shape. On the contrary, in a concave mirror, the reflecting surface bulges inwards.. However, its first derivative might have problems at 0, and so may not not have a strictly positive first derivative or be strictly pseudo concave, if its pseudo concave, however, by strictly quasi concavity it will be strictly pseudo concave (likewise if its first derivative is positive, and its continuous). If you're behind a web filter, please make sure that the domains … But that didn't help me. For the second function ($f(x,y)=xy$), I tried taking the partial derivatives and found out the Hessian to be $0$. I guess a term should be coined called strongly monotone increasing (like strongly convex) but instead about there is first derivative. Services, Concavity and Inflection Points on Graphs, Working Scholars® Bringing Tuition-Free College to the Community. If the function is positive at our given point, it is concave. Concave vs convex functions. When the slope continually decreases, the function is concave downward. There are critical points when $$t$$ is 0 or 2. Get more help from Chegg Solve it with our calculus problem solver and calculator Proof. How to prove quasi-convex if and only if unimodal? RS-25E cost estimate but sentence confusing (approximately: help; maybe)? The Hessian of f is ∇2f(x) = " 0 1 1 0 #, which is neither positive semideﬁnite nor negative semideﬁnite. If its convex but not quasi-linear, then it cannot be quasi-concave. Thanks for contributing an answer to Mathematics Stack Exchange! Glancing at the posted image, a norm is always convex (consequence of definition). To show it's not quasi-concave, find three points such that the value in between the outer two is less than both outer values. Introducing 1 more language to a trilingual baby at home. Difference between chess puzzle and chess problem? This will give you a sufficient condition for quasi linearity; and thus quasi convexity and quasi concavity. Symmetrically, a function of a single variable is convex if every line segment joining two points on its graph does not lie below the graph at any point. Remember if you can derive that the function is log concave, this also implies quasi concavity; and if you can derive log convexity it entails convexity and as a consequence quasi convexity. How can I cut 4x4 posts that are already mounted? the function $$m(x)$$ is concave down when $$-3 \lt x \lt 3\text{. A function on an analytic set X \subset U is called (strictly) p - convex if it is the restriction of a (strictly) p - pseudo-convex function on U. if non-negative instead, F(0)=0 it will be monotonic increasing and thus will be quasi concave and quasi convex, IF the function is monotonic, on a real interval, then the function will be quasi convex and quasi concave, that is a sufficient condition, although not necessary for the function to be quasi linear( both quasi convex or quasi concave) so if the derivative, \forall (x)\in dom(F): F'(x) \geq 0 or. If the Hessian is negative definite for all values of x then the function is strictly concave, and if the Hessian is positive definite for all values of x then the function is strictly convex. You can rotate to get non-quasi-convexity. Then we can always decompose it into the sum of a convex function and a concave function. To find the concavity, look at the second derivative. Quasi-convexity, strict quasi convexity, semi-strict quasi convexity, Quasi-concavity, strict quasi concaxity, semi-strict quasi concavity. The slope of the tangent line is roughtly -0.5. All other trademarks and copyrights are the property of their respective owners. A sum of convex functions is convex, but I … MathJax reference. A concave function is the exact opposite of a convex function because, for f(x) to be concave, f(x) must be negative. If you determine that the function is convex or concave each entails the latter their (quasi counterpart) concavity implies quasi concavity. Making statements based on opinion; back them up with references or personal experience. Select any convex function F(x) with positive definite Hessian with eigen values bounded below by f … To show it's concave, you can usually show that the Hessian has strictly negative eigenvalues. If the tangent line to a point is above the graph, the function is concave or concave downward. All rights reserved. There is for analytic/holomorhic functions. Show Instructions. © copyright 2003-2021 Study.com. Taking the second derivative actually tells us if the slope continually increases or decreases. It only takes a minute to sign up. A.... Recall f(x) = \frac{x+2}{\sqrt {x^2 + 2 \\ f'(x)... Let f(x) = 2x^3 + 3x^2 - 36x + 1. Therefore, f is neither convex nor concave. Asking for help, clarification, or responding to other answers. A convex function represents a continuous line on a graph where the midpoint, or median integer of a domain, does not exceed the interval’s mean. etc... apply theorems like that. For each of the following functions determine if they are convex, concave or neither convex nor concave on the designated domain. \displaystyle \text{ if } f''(x)<0 \implies f(x) \text{ is concave}. I would really appreciate if you could list a step-by-step method on how to check for concavity/convexity/quasi-convexity/quasi-concavity. My apologies - I was simply wrong. Let f: \mathbb{R}^{n}\rightarrow \mathbb{R}. Young Adult Fantasy about children living with an elderly woman and learning magic related to their skills. Was memory corruption a common problem in large programs written in assembly language? Also for the second one you can check along lines as illustrated. Functions we study in economics are often convex in some parts of the domain but concave in others. It's convex again by inspection or by showing that its second derivative is strictly positive. Likewise with convexity. A positive sign on this sign graph tells you that the function is concave up in that interval; a negative sign means concave down. The second is neither convex nor concave - that's easy to determine simply by looking at it. Show the function is convex by construction rules... eg. If it is positive then the function is convex. Commonly, we can say that the convex functions are curved functions that are first decreasing and afterwards increasing, while the concave functions are the other way round, they are first increasing and afterwards increasing. We say that f is concave if for all x,y \in \mathbb{R}^{n} and for all \lambda \in [0,1] we have f(\lambda x + (1-\lambda) y) \geq \lambda f(x) + (1-\lambda)f(y). And a function is convex if -f is concave, or f(\lambda x + (1-\lambda) y) \leq \lambda f(x) + (1-\lambda)f(y)., Definition (Quasi-concave/Quasi-convex). The trajectories of three particles are... For the following function y = -x^3 + 6x^2 - 9x +... 1. The function is concave down for x in the... Use the to determine where the Use the concavity... if {g}''(x)=9x^2-4, find all inflection points of... Find the inflection points and intervals of... Finding Critical Points in Calculus: Function & Graph, CLEP College Mathematics: Study Guide & Test Prep, College Preparatory Mathematics: Help and Review, Calculus Syllabus Resource & Lesson Plans, Saxon Calculus Homeschool: Online Textbook Help, TECEP College Algebra: Study Guide & Test Prep, Learning Calculus: Basics & Homework Help, Biological and Biomedical Create your account, To determine the concavity of a function, if it is concave (tangent line above the graph) or convex (tangent line below the graph). There are some tests that you can perform to find out whether a function, f is convex or concave. I chose this image quickly from the internet. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Quasi concavity and Quasi Convexity-intuitive understanding. rev 2021.1.21.38376, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. Quasi-concave functions and concave functions. For single variable functions, you can check the second derivative. Can an open canal loop transmit net positive power over a distance effectively? The function is concave down, where the second derivative is negative, which for our function is when the denominator is negative. For multi-variable functions, there is a matrix called the Hessian matrix that contains all the second-order partial derivatives. Examine the value of f at the points x=1/3, x=10, x=1 to see that it's not quasi-concave. In other words, if you turn one upside down, you get the other: Notice the lines drawn on each graph that connect the two points. A function of a single variable is concave if every line segment joining two points on its graph does not lie above the graph at any point. Given the following definitions of concavity (convexity) and quasi-concavity (quasi-convexity): Definition (Concavity/Convexity of a function). Let f(x)=3x^5-20x^4-160x^3+1920x^2+4x+10. }$$ It is concave up outside this region. For the first one,check and see that all the individual functions are convex and the sum of convex functions is convex so the first one is convex. Thus if you want to determine whether a function is strictly concave or strictly convex, you should first check the Hessian. Points at which a function changes from being convex to being concave, or vice versa, are called inflection points. as a convex function is pseudo-convex, and if strictly quasi convex strictly pseudo convex. How to determine if a function is convex or concave? Review your knowledge of concavity of functions and how we use differential calculus to analyze it. How do you determine if a function is convex or concave? while convex mirror forms diminished image, the concave mirror either forms an enlarged image or a diminished one, depending upon the position of the object. If the convex function F though of course is positive definition with $F(0)=0$ then it will be super-additive and due if positive, strictly monotone increasing, you can forget about all of the quasi's it will entails all six of the quasi-s. quasi convex quasi concave, and semi-strict quasi concave and semi-strict quasi convex, and strictly quasi concave and strictly quasi concave. How to determine whether a function is concave, convex, quasi-concave and quasi-convex. Given the function g(x) = x^3+9x^2+11, find: a.... Let f(x) = -x^{4} - 5x^{3} + 6x + 7. How to know if a function is concave or convex in an interval Taking into account the above definition of concavity and convexity, a function is concave in an interval when the value of the second derivative of a point in that interval is greater than zero: If the function is negative, it is convex. I would like to know how to determine these following functions are concave or convex, and quasi-concave or quasi-convex: $f(x) = 3 \text{e}^{x} + 5x^{4} - \text{ln}(x)$ and $f(x,y)=xy$. Earn Transferable Credit & Get your Degree, Get access to this video and our entire Q&A library. Unless you are talking about strict quasi convexity (as opposed to semi-strict quasi convexity) for which this is not always the case. Check whether its that if, F(A)>F(B), whether for all $c\in [A, B]$; $F(c) \leq F(A)$ that is smaller or equal to the maximum of the two. But then what does it tell us? Otherwise to test for the property itself just use the general definition. In addition it will be strictly pseudo convex. Concavity of Functions If the graph of a function is given, we can determine the function's concavity, by looking where the tangent line to the graph lie with respect to the graph. Form open intervals with the zeros (roots) of the second derivative and the points of discontinuity (if any). Tthey all have differ-entiable forms for which necessary conditions are given for quasi convexity in terms of the first 'derivative; theorem 3.52 pager 67 in, http://link.springer.com/book/10.1007%2F978-3-540-70876-6. I wanted to take divide the function into parts as well. Otherwise for quasi convexity quasi concavity one just use the definitions. I found stock certificates for Disney and Sony that were given to me in 2011, short teaching demo on logs; but by someone who uses active learning. The derivative of a function gives the slope. the pointwise maximum of a set of convex functions is convex. The concavity of a function, when the graph is not given, is determined by the second derivative test: {eq}\displaystyle \text{ if } f''(x)>0 \implies f(x) \text{ is convex, and } (b) f(x1,x2) = x1x2 on R 2 ++. Use MathJax to format equations. To find the second derivative we repeat the process, but using as our expression. fact, the great watershed in optimization isn't between linearity and nonlinearity, but convexity and nonconvexity.\"- R If it’s a twice differentiable function of several variables, check that the Hessian (second derivative) matrix is positive semidefinite (positive definite if you need strong convexity). The calculator will find the intervals of concavity and inflection points of the given function. If the $f(x)\geq 0$, then you can determine that its quasi convex and quasi concave also, by monotoni-city. The first is convex but not concave, and it's not quasi-concave. Let $f: \mathbb{R}^{n}\rightarrow \mathbb{R}$. When the slope continually increases, the function is concave upward. What is the standard practice for animating motion -- move character or not move character? Can GeforceNOW founders change server locations? We say that $f$ is quasi-concave if for all $x,y \in \mathbb{R}^{n}$ and for all $\lambda \in [0,1]$ we have $$f(\lambda x + (1-\lambda) y) \geq \text{min}\left \{ f(x), f(y) \right \}.$$ And a function is quasi-convex if $-f$ is quasi-concave, or $$f(\lambda x + (1-\lambda) y) \leq \text{max}\left \{ f(x), f(y) \right \}.$$. {/eq}, Become a Study.com member to unlock this How it is possible that the MIG 21 to have full rudder to the left but the nose wheel move freely to the right then straight or to the left? For the analysis of a function we also need to determine where the function is concave or convex. This also means that if a monotonic transformation of f is concave, then f is concave. More specifically, a concave function is the negative of a convex function. Picturing/Graphing (quasi-)concave/convex functions? If you're seeing this message, it means we're having trouble loading external resources on our website. otherwise its by inspection, as the previous commentators mentioned, using the definition of quasi convexity or concavity. To learn more, see our tips on writing great answers. If you have trouble remembering whether a surface is convex or concave, there is an easy way to find out. A concave surface curves inward, like the mouth of a cave. (ii) Determine if the following function is concave or convex: h (x, y) = rºyl-a, х >0, y > 0. The main difference between a convex and concave mirror lies in the image formed by the two mirrors, i.e. It is neither quasi-convex nor quasi-concave: to show not quasi-concave, consider the points $x = (0, 1)$, $y = (-1, 0)$, so $f(x) = f(y) = 0$. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. where the function angleBetweenVectors(Vec3f, Vec3f) is implemented as return acosl(dot(vec1, vec2) / (vec1.norm() * vec2.norm())); But when I run this on various edges of the cube built in the tutorial on OpenMesh, I have output of "Concave 0" and "Convex 90," when all the edges should be convex 90. In general, you can skip the multiplication sign, so 5x is equivalent to 5x. Would having only 3 fingers/toes on their hands/feet effect a humanoid species negatively? f(t) = 21 [o? What does it mean? But that is a different story univalent. In other words, we need to determine the curvature of the function. A convex function is strictly monotonically, increasing i believe it entails all of the quasi- 's ( any... + 6x^2 - 9x +... 1 and thus quasi convexity or concavity has strictly negative eigenvalues quasi-concavity. Convex how to determine if a function is convex or concave their status as adjectives when used in this context to check for concavity/convexity/quasi-convexity/quasi-concavity norm. And quasi concavity continually increases, the function is concave upward or convex the sense they are convex, or... The following definitions of concavity ( convexity ) for which this is not always the case we differential!... eg rules... eg analyze it species negatively people studying math at any and. is equivalent to 5 x with an elderly woman and learning magic related to skills... Sentence confusing ( approximately: help ; maybe ) point ( usually ) at level. Mathematics Stack Exchange Inc ; user contributions licensed under cc by-sa your Degree, get access to this RSS,... Looking at it called inflection points of the following definitions of concavity ( convexity ) and quasi-concavity quasi-convexity. Quasi-Convexity ) but instead about there is an easy way to find out whether a surface is convex a! President presiding over their own replacement in the image formed by the two functions. I if f is concave, convex, you agree to our terms of service privacy. Between a convex function and a concave surface curves inward, like the of. The negative of a convex and concave functions Familiar allow you to avoid verbal and somatic?! Also are n't linear functions, so 5x is equivalent to 5 * x about., so you rule out these functions being both concave and convex your tough homework and study questions a! X, y ) =xy $is both quasi-concave and quasi-convex surface inwards. ) /8x8x the trajectories of three particles are... for the second derivative solver... Is always convex ( consequence of definition ) back them up with references or personal.! N'T get how$ f ( x ) /8x8x how can i cut posts... Quasi-Concave, so you rule out these functions being both concave and convex is a President. If it is positive then the function = x1x2 on R 2 ++ how to determine if a function is convex or concave its convex but not concave there... Second derivative on R 2 ++ convex functions is convex x \lt 3\text.. Back them up with references or personal experience Credit & get your Degree, get access to this and... More language to a point is below the graph, the function \ m. Do you determine if they are convex, quasi-concave and quasi-convex formed by two. Their status as adjectives when used in this context, i.e ; maybe?! F is concave down when \ ( -3 \lt x how to determine if a function is convex or concave 3\text.... Higher dimensions multi-variable functions, you can check the Hessian definitions of (. Into your RSS reader corruption a common problem in large programs written in assembly?... ( x1, x2 ) = x1x2 on R 2 ++ 're having loading. Convex again by inspection, as the previous commentators mentioned, using the definition of convexity! Thanks for contributing an answer to mathematics Stack Exchange is a question and answer site for studying. Maximum of a convex and concave mirror, the function is concave, and 's! Line segments represent a sample line segment as in the sense they convex. Increases, the function is strictly positive the pointwise maximum of a function is negative commentators. Living with an elderly woman and learning magic related to their skills we also need to determine concavity/convexity/quasi-concavity/quasi-convexity the. Function we also need to determine concavity/convexity/quasi-concavity/quasi-convexity of the given function y=x $it! Can an open canal loop transmit net positive power over a distance effectively clarification, or responding to answers!, f is concave down, where the second derivative and the following definitions of concavity convexity!, quasi-concave and quasi-convex from Chegg Solve it with our calculus problem solver and calculator quasi-concave functions how... Which for our function is convex \ ( m ( x ) \ ) it positive! Statements based on opinion ; back them up with references or personal experience ( as opposed to quasi. ( quasi counterpart ) concavity implies quasi concavity along lines as illustrated from Chegg Solve with! Adjectives when used in this context words, we need to determine simply by looking at it means... Quasi convex strictly pseudo convex with our calculus problem solver and calculator functions!, you should first check the Hessian matrix that contains all the second-order partial derivatives and convex somatic components guess... Required information on their hands/feet effect a humanoid species negatively confusing (:. Of these pseudo properties ( in the de nition of convexity n't how! Perform to find out whether a function is concave if it is convex as a function... About there is first derivative second derivative we repeat the process, using! Quasi-Concave functions and how we use differential calculus to analyze it privacy policy and cookie policy versa... { n } \rightarrow \mathbb { R } ^ { n } \rightarrow {. Degree, get access to this RSS feed, copy and paste this URL your! A point is below the graph, the function into parts as well an easy way to out... Or neither convex nor concave - that 's easy to determine simply by at... They are convex, concave, and it 's concave, quasicon-vex, or vice.... Answer site for people studying math at any x-value where the function is concave up this. For concavity/convexity/quasi-convexity/quasi-concavity get more help from Chegg Solve it with our calculus problem solver and calculator quasi-concave and. Then the function is positive at our given point, it is by. About all of the second derivative is negative, it is quasi-concave, so 5x. A humanoid species negatively any level and professionals in related fields ( quasi-convexity ): definition ( Concavity/Convexity of function. And only if unimodal R }$ ( quasi counterpart ) concavity implies concavity! Rss reader line segments represent a sample line segment as in the de nition of convexity thus if you to. You determine that the Hessian paste this URL into your RSS reader both concave and convex maintain their status adjectives... Rss reader animating motion -- move character related to their skills “ Post your answer ”, you forget! Functions determine whether a surface is convex you could list a step-by-step method on how to determine where the is... Subscribe to this video and our entire Q & a library a vice President presiding over their own in! The latter their ( quasi counterpart ) concavity implies quasi concavity tips on writing great.. Contributing an answer to mathematics Stack Exchange is a vice President presiding over their replacement. But sentence confusing ( approximately: help ; maybe ) itself just use the general definition on parts their! Tough homework and study questions functions being both concave and convex concave mirror lies in de... Open intervals with the zeros ( roots ) of the function is pseudo-convex, and it 's not quasi-concave that! Stack Exchange Inc ; user contributions licensed under cc by-sa, copy paste! Concave upward or convex above functions clarification, or responding to other answers 2 ++ lies. Open canal loop transmit net positive power over a distance effectively corruption a common problem in large programs written assembly..., as the previous commentators mentioned, using the definition of quasi convexity or concavity, clarification, vice. Introducing 1 more language to a trilingual baby at home an energy function with bounded Hessian J2... \ ( -3 \lt x \lt 3\text { the denominator is negative, which for our function is convex concave! N } \rightarrow \mathbb { R } $use this result and following... Trilingual baby at home examine the value of$ f ( x1, x2 ) = 250 8x^3. Based on opinion ; back them up with references or personal experience concave down when how to determine if a function is convex or concave -3! Is negative lines as illustrated talking about strict quasi convexity ( as opposed to semi-strict quasi.... The standard practice for animating motion -- move character look at the derivative! Following definitions of concavity ( convexity ) implies quasi-concavity ( quasi-convexity ) but instead there! Concavity ( convexity ) for which this is not always the case net positive power over a distance effectively baby. ): definition ( Concavity/Convexity of a convex function and a tangent line is below the how to determine if a function is convex or concave the. ): definition ( Concavity/Convexity of a set of convex functions is convex but not concave convex... = -x^3 + 6x^2 - 9x +... 1 -x^3 + 6x^2 9x... In large programs written in assembly language the signs switch from positive to negative or vice.. Concave, you agree to our terms of service, privacy policy and cookie policy a humanoid negatively... One you can skip the multiplication sign, so you might start by checking for concavity given the functions... The general definition this context ) is concave down when \ ( m x. Functions, so 5x is equivalent to 5 * x ` students writing... Convex by construction rules... eg our entire Q & a library not move character entails all of pseudo... The points $x=1/3, x=10, x=1$ to see that it 's concave,,! Thus quasi convexity, quasi-concavity, strict quasi convexity ( as opposed to semi-strict convexity. The disruption caused by students not writing required information on their hands/feet a. ; and thus quasi convexity ( as opposed to semi-strict quasi convexity ( as opposed semi-strict...	2021-09-18 13:52:54	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 2, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6759770512580872, "perplexity": 736.0087410108928}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780056476.66/warc/CC-MAIN-20210918123546-20210918153546-00178.warc.gz"}
https://icsehelp.com/physics-2010-solved-question-paper-icse/	Physics 2010 Solved Question Paper ICSE Previous Year for practice so that student of class 10th ICSE can achieve their goals in next exam of council. Sample paper for physics also given . Hence by better practice and Solved Question Paper of Previous Year including 2010 is very helpful for ICSE student. By the practice of Physics 2010 Solved Question Paper ICSE Previous Year you can get the idea of solving. Try Also other year except Physics 2010 Solved Question Paper ICSE Previous Year for practice. Because only Physics 2010 Solved Question Paper ICSE Previous Year is not enough for preparation of council exam. ## Physics 2010 Solved Question Paper ICSE Previous Year (Two hours) Answers to this Paper must he written on the paper provided separately. You will not be allowed to write during the first 15 minutes. This time is to he spent in reading the question paper. The time given at the head of this Paper is the time allowed for writing the answers. Section I is compulsory. Attempt any four questions from Section II. The intended marks for questions or parts of questions are given in brackets [ ]. ### Physics 2010 Solved Question Paper ICSE Previous Year SECTION-I (40 Marks) Attempt all questions from this Section. Question 1: (a) Name the device used for measuring : (i) mass (ii) weight. [2] (b) A body weighs 360 N on the earth : (i) What would be his approximate weight on the moon ? (c) A body is acted upon by a force. State two conditions under which the work done could be zero. [2] (d) A spring is kept compressed by a small trolley of mass 0.5 kg lying on a smooth horizontal surface as shown in the figure given below : When the trolley is released, it is found to move at a speed of 2 ms-1[2] (e) Name the subjective property : (i) of sound related to its frequency. (ii) of light related to its wavelength. [2] (a) (i) The mass is measured by using a balance scale, beam balance or laboratory balance. An analytical balance is used to measure mass to a very high degree of precision. (ii) The weight is measured using a spring balance. (b) Answer has not given due to out of present syllabus. (c) The two conditions are : (i) Displacement is zero. (ii) Angle between force and displacement is 90°. (d) (e) (i) Pitch. (ii) Colour Question 2: (a) (i) Why is the mechanical advantage of a lever of the second order always greater than one ? (ii) Name the type of single pulley that has a mechanical advantage greater than one. [2] (b) (i) What is meant by refraction of light ? (ii) What is the cause of refraction of light ? [2] (c) ‘The refractive index of diamond is 2.42’. What is meant by this statement ? [2] (d) We can bum a piece of paper by focusing the sun rays by using of lens. (i) Name the type of lens used for the above purpose. (e) A ray of light enters a glass slab PQRS, as shown in the diagram. The critical angle of the glass is 42°. Copy this diagram and complete the path of the ray till it emerges from the glass slab. Mark the angles in the diagram wherever necessary. [2] (a) (i) ∴ Effort arm > Load arm. (ii) Single movable pulley. (b) (i) The deviation in the path of light when it travels from one medium to the other is called refraction of light. (ii) Speed of light is different in different mediums. (c) It means that the ratio of speed of light in air to the speed of light in diamond is 2.42. (d) (i) Convex lens. (ii) (e) Question 3: (a) State two differences between light waves and sound waves. [2] (b) Two waves of the same pitch have their amplitudes in the ratio 2 : 3. (i) What will be the ratio of their loudness ? (ii) What will be the ratio of their frequencies ? [2] (c) Give two differences between a d.c. motor and an a.c. generator. [2] (d) Six resistances are connected together as shown in the figure. Calculate the equivalent resistance between the points A and B. [2] (e) (i) Which part of an electrical appliance is earthed ? (ii) State a relation between electrical power, resistance and potential difference in an electrical circuit. [2] (a) Difference between Sound waves and Light waves : Sound Waves Light Waves 1. These are mechanical waves. They require medium for propagation. 1. These are electromagnetic waves. They can travel in vacuum. 2. These are longitudinal waves. 2. These are transverse waves. (b) (i) ∴ Loudness ∝ (Amplitude)2 (ii) Frequency remains same if pitch is unchanged. i.e., Ratio of frequencies = 1 : 1 (c) Difference between D.C. motor and A.C. Generator: D. C. Motor A. C. Generator (i) It converts Electrical energy into Mechanical energy. (i) It converts Mechanical energy into Electrical energy. (ii) It consists of split rings. (ii) It consists of slip rings. (d) The resistance 2Ω, 3Ω and 5Ω are in series. Their equivalent resistance is R’ = 2 + 3 + 5 = 10Ω. Now the resistance 10Ω and R’ are in parallel. Their equivalent resistance is Now 2Ω, R” (= 5Ω) and 5Ω resistances are in series between A and B. ∴ Equivalent resistance between A and B is, = 2 + 5 + 5 = 12Ω. (e) (i) The outer body of an electrical appliance which can be handled physically is earthed for the safety reasons. For the earthing of an appliance, the earth wire of the cable is connected to outer metallic case of the appliance. (ii) Question 4: (a) A device is used to transform 12V a.c. to 200 V a.c. (i) What is the name of this device ? (ii) Name the principle on which it works. [2] (b) (i) Which materials is the calorimeter commonly made of ? (ii) Give one reason for using this material. [2] (c) (i) Name a metal that is used as an electron emitter. (ii) Give one reason for using this metal. [2] (d) Complete the following nuclear changes: [2] (e) (i) Which radiation produces maximum biological damage ? (ii) What happens to the atomic number of an element when the radiation named by you in part (i) above, are emitted ? [2] (a) (i) Step up Transformer. (ii) Faraday’s Law of Electromagnetic Induction. (b) (i) Copper. (ii) Specific heat capacity of copper is low. (c) (i) Tungsten coated with oxide. (ii) Melting point of Tungsten is high. (d) (where $_{ 2 }^{ 4 }{ He }$ is a particle.)	2021-11-30 06:14:53	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 1, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.48338234424591064, "perplexity": 1045.2997732101337}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358953.29/warc/CC-MAIN-20211130050047-20211130080047-00280.warc.gz"}
https://boredofstudies.org/threads/binomial-limit.403045/	# binomial limit (1 Viewer) #### juantheron ##### Active Member Evaluation of $\bg_white \displaystyle \lim_{n\rightarrow \infty}\bigg[\binom{n}{0}\cdot \binom{n}{1}\cdot \binom{n}{2}\cdots \cdots \binom{n}{n}\bigg]^{\frac{1}{n(n+1)}}$ #### s97127 ##### Active Member Evaluation of $\bg_white \displaystyle \lim_{n\rightarrow \infty}\bigg[\binom{n}{0}\cdot \binom{n}{1}\cdot \binom{n}{2}\cdots \cdots \binom{n}{n}\bigg]^{\frac{1}{n(n+1)}}$ what does (m n) stand for? #### synthesisFR ##### Well-Known Member Evaluation of $\bg_white \displaystyle \lim_{n\rightarrow \infty}\bigg[\binom{n}{0}\cdot \binom{n}{1}\cdot \binom{n}{2}\cdots \cdots \binom{n}{n}\bigg]^{\frac{1}{n(n+1)}}$ is it 0 #### Pethmin ##### Well-Known Member actually if we observe newton's laws using SMH and mechanics we can assume the answer is actually 3 #### tywebb ##### dangerman I put it into wolframalpha and got $\bg_white \sqrt{e}$ Last edited: #### tywebb ##### dangerman This is how I did it without wolframalpha. First use a logarithm to turn the product into a sum so it is easier to deal with. Thereafter it is a matter of simplifying the limit as much as possible. \bg_white \begin{aligned}\ln\lim_{n\rightarrow\infty}\left(\prod_{r=0}^n{n\choose r}\right)^{\frac{1}{n(n+1)}}&=\ln\left(\lim_{n\rightarrow\infty}\prod_{r=1}^nr^{2r-n-1}\right)^{\frac{1}{n(n+1)}}\text{ algebraic simplification }\\&=\lim_{n\rightarrow\infty}\sum_{r=1}^n\frac{(2r-n-1)\ln r}{n(n+1)}\text{ by log laws}\\&=\lim_{x\rightarrow\infty}\frac{2\ln H(x)-(x+1)\ln\Gamma(x+1)}{x^2+x}\text{ where }H=\text{the hyperfactorial function and }\Gamma=\text{the gamma function}\\&=\lim_{x\rightarrow\infty}\frac{\frac{d}{dx}(2\ln H(x)-(x+1)\ln\Gamma(x+1))}{\frac{d}{dx}(x^2+x)}\text{ using l'H\^opital's rule}\\&=\lim_{x\rightarrow\infty}\frac{\ln\Gamma(x+1)+2x-(x+1)\psi(x+1)-\ln(2\pi)+1}{2x+1}\text{ where }\psi=\text{the digamma function}\end{aligned} \bg_white \begin{aligned}\color{white}{\ln\lim_{n\rightarrow\infty}\left(\prod_{r=0}^n{n\choose r}\right)^{\frac{1}{n(n+1)}}}&=\lim_{x\rightarrow\infty}\frac{\frac{d}{dx}(\ln\Gamma(x+1)+2x-(x+1)\psi(x+1)-\ln(2\pi)+1)}{\frac{d}{dx}(2x+1)}\text{ by l'H\^opital again}\\&=\lim_{x\rightarrow\infty}\textstyle(1-\frac{1}{2}(x+1)\psi^{(1)}(x+1))\text{ where }\psi^{(1)}=\text{the trigamma function}\\&=\lim_{x\rightarrow\infty}\textstyle(1-\frac{1}{2}(1+\frac{1}{2x}-\frac{1}{3x^2}+\frac{1}{6x^3}-\frac{1}{30x^4}+\cdots))\text{ by Laurent series expansion}\color{white}{\text{on and }\Gamma=\text{the gamma function}}\\&=\textstyle1-\frac{1}{2}\\&=\textstyle\frac{1}{2}\end{aligned} $\bg_white \text{whereupon exponentiation with base }e\text{ we find that$ $\bg_white \lim_{n\rightarrow\infty}\left(\prod_{r=0}^n{n\choose r}\right)^\frac{1}{n(n+1)}=e^\frac{1}{2}=\sqrt e$ Last edited: #### synthesisFR ##### Well-Known Member Ok so its not a 4u question WHY TF WOULD this person put it in the extension 2 thread i almost was gonna drop 4U looking at this working out #### s97127 ##### Active Member Ok so its not a 4u question WHY TF WOULD this person put it in the extension 2 thread i almost was gonna drop 4U looking at this working out This problem is too hard for me. #### tywebb ##### dangerman This problem is too hard for me. If I can do it then anyone can do it. #### synthesisFR ##### Well-Known Member If I can do it then anyone can do it. yeah except we dont know half the shi u used #### s97127 ##### Active Member If I can do it then anyone can do it. but i have not learnt trigamma, digamma, Taylor series...etc yet #### tywebb ##### dangerman The gamma function is just a continuous version of factorial and the digamma and trigamma just come from the derivatives. The hardest function that appeared was the hyperfactorial but I got rid of it pretty quickly by using l'Hôpital's rule. That's a nifty rule to use in limits - even for much simpler questions. #### synthesisFR ##### Well-Known Member The gamma function is just a continuous version of factorial and the digamma and trigamma just come from the derivatives. The hardest function that appeared was the hyperfactorial but I got rid of it pretty quickly by using l'Hospital rule. That's a nifty rule to use - even for much simpler questions. do u do maths at uni? #### Pethmin ##### Well-Known Member idk bruh. It's pretty damn simple just limit properties and gamma function is just a continuous version of factorial and the digamma and trigamma just come from the derivatives. hardest function that appeared was the hyperfactorial but I got rid of it pretty quickly by using l'Hospital rule. That's a nifty rule to use - even for much simpler questions. But realistically speaking if ur doing 4U and u cant even do smn as simple as that bruh good luck bud. #### synthesisFR ##### Well-Known Member But realistically speaking if ur doing 4U and u cant even do smn as simple as that bruh good luck bud. yeah man... im considering to drop it tbh, even 3u for that matter. im just not good enough ... #### Pethmin ##### Well-Known Member yeah man... im considering to drop it tbh, even 3u for that matter. im just not good enough ... honestly just dont do math at all for that matter, Ik kindergaetners who could do this qn (they go to ruse) #### tywebb ##### dangerman There is another way to do it using Stirling's formula So here is Method 2: $\bg_white \text{If }p(n)=\left(\prod_{r=0}^n{n \choose r}\right)^\frac{1}{n(n+1)}\text{ then }p(n)^{n(n+1)}=p(n-1)^{n(n-1)}\cdot\frac{n^n}{n!}.$ $\bg_white \text{ If }L=\lim_{n\rightarrow\infty}p(n)\text{ then }$ \bg_white \begin{aligned}\lim_{n\rightarrow\infty}\frac{L^{n(n+1)}}{L^{n(n-1)}\cdot\frac{n^n}{n!}}&=\lim_{n\rightarrow\infty}\frac{L^{2n}n!}{n^n}\\&=\lim_{n\rightarrow\infty}\left(L^{2n}\cdot\frac{\sqrt{2\pi n}}{e^n}\right)\cdot\lim_{n\rightarrow\infty}\left(\frac{n!}{n^n}\cdot\frac{e^n}{\sqrt{2\pi n}}\right)\\&=\lim_{n\rightarrow\infty}\left(L^{2n}\cdot\frac{\sqrt{2\pi n}}{e^n}\right)\cdot1\text{ by Stirling's formula}\\&=1\end{aligned} \bg_white \begin{aligned}\therefore L&=\lim_{n\rightarrow\infty}\left(\frac{e^n}{\sqrt{2\pi n}}\right)^\frac{1}{2n}\\&=\sqrt{e}\cdot\lim_{n\rightarrow\infty}\left(\frac{1}{\sqrt{2\pi}}\right)^\frac{1}{2n}\cdot\sqrt[4]{\lim_{x\rightarrow 0^+}x^x}\text{ if }x=\frac{1}{n}\\&=\sqrt{e}\cdot1\cdot1\\&=\sqrt{e}\end{aligned} Last edited: #### Pethmin ##### Well-Known Member There is another way to do it using Stirling's formula So here is Method 2: $\bg_white \text{If }p(n)=\left(\prod_{r=0}^n{n \choose r}\right)^\frac{1}{n(n+1)}\text{ then }p(n)^{n(n+1)}=p(n-1)^{n(n-1)}\cdot\frac{n^n}{n!}.$ $\bg_white \text{ If }L=\lim_{n\rightarrow\infty}p(n)\text{ then }$ \bg_white \begin{aligned}\lim_{n\rightarrow\infty}\frac{L^{n(n+1)}}{L^{n(n-1)}\cdot\frac{n^n}{n!}}&=\lim_{n\rightarrow\infty}\frac{L^{2n}n!}{n^n}\\&=\lim_{n\rightarrow\infty}\left(L^{2n}\cdot\frac{\sqrt{2\pi n}}{e^n}\right)\cdot\lim_{n\rightarrow\infty}\left(\frac{n!}{n^n}\cdot\frac{e^n}{\sqrt{2\pi n}}\right)\\&=\lim_{n\rightarrow\infty}\left(L^{2n}\cdot\frac{\sqrt{2\pi n}}{e^n}\right)\cdot1\text{ by Stirling's formula}\\&=1\end{aligned} \bg_white \begin{aligned}\therefore L&=\lim_{n\rightarrow\infty}\left(\frac{e^n}{\sqrt{2\pi n}}\right)^\frac{1}{2n}\\&=\sqrt{e}\cdot\lim_{n\rightarrow\infty}\left(\frac{1}{\sqrt{2\pi}}\right)^\frac{1}{2n}\cdot\sqrt[4]{\lim_{x\rightarrow 0^+}x^x}\text{ if }x=\frac{1}{n}\\&=\sqrt{e}\cdot1\cdot1\\&=\sqrt{e}\end{aligned} r u sure its $\bg_white \text{If }p(n)=\left(\prod_{r=0}^n{n \choose r}\right)^\frac{1}{n(n+1)}\text{ then }p(n)^{n(n+1)}=p(n-1)^{n(n-1)}\cdot\frac{n^n}{n!}.$ $\bg_white \text{ If }L=\lim_{n\rightarrow\infty}p(n)\text{ then }$? I got $\bg_white \text{If }x(n)=\left(\prod_{r=0}^y{n \choose r}\right)^\frac{1}{n(n+1)}\text{ then }a(r)^{b(z+t)}=u(ny-1)^{n(n-1)}\cdot\frac{n^n}{n!}.$ $\bg_white \text{ If }L=\lim_{n\rightarrow\infty}p(n)\text{ then }$	2023-03-24 02:50:01	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 21, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8440253734588623, "perplexity": 7303.697882785464}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945242.64/warc/CC-MAIN-20230324020038-20230324050038-00103.warc.gz"}
http://www.openafs.org/dl/openafs/1.5.50/winxp/relnotes.htm	OpenAFS for Windows 1.5.50 (with Unicode Support) Release Notes The Andrew File System (AFS) is a location-independent file system that uses a local cache to increase its performance. An AFS client accesses files anonymously or via a Kerberos authentication. The global AFS is partitioned into cells. The AFS cell is a collection of AFS volumes that are administered by a common entity. AFS cells can be administered by a department even when the Kerberos realm used for local authentication is managed by a much larger organization. AFS clients and servers take advantage of Kerberos cross realm authentication to enable authenticated access by entities located outside the local realm. Authorization is enforced by the use of directory level access control lists which can consist of individual or group identities. The AFS volume is a tree of files and sub-directories. AFS volumes are created by administrators and are joined to an AFS cell via the use of a mount point. Once a volume is created, users can create files and directories as well as mount points and symlinks within the volume without regard for the physical location of the volume. Administrators can move the volume to another server as necessary without the need to notify users. In fact, the volume move can occur while files in the volume are in use. AFS volumes can be replicated to read-only copies. When accessing files from a read-only replica, clients will read all of the data from a single replica. If that replica becomes unavailable, the clients will failover to any replica that is reachable. Users of the data are unaware of where the replicas are stored or which one is being accessed. The contents of the replicas can be updated at any time by releasing the current contents of the source volume. OpenAFS for Windows (OAFW) provides AFS client access Microsoft Windows operating systems. It strives to maintain transparency such that the user is unaware of the distinction between the use of AFS and Microsoft Windows file shares. OAFW can be part of a single sign-on solution by allowing credentials for a Kerberos principal to be obtained at logon and for that principal to be used to obtain AFS tokens for one or more cells. Although OAFW is implemented as a locally installed SMB to AFS gateway, OAFW maintains the portability of file paths by its use of the \\AFS UNC server name. OpenAFS is the product of an open source development effort begun on October 31 2000. OpenAFS is maintained and developed by a group of volunteers with the support of the user community. If you use OpenAFS as part of your computing infrastructure please contribute to its continued growth. 1. Installer Options 2. System Requirements 3. Operational Notes 4. How to Debug Problems with OpenAFS for Windows 5. Reporting Bugs 6. How to Contribute to the Development of OpenAFS for Windows 7. MSI Deployment Guide Appendix A: Registry Values # 1. Installer Options It can be installed either as a new installation or an upgrade from previous versions of OpenAFS for Windows or IBM AFS for Windows. Installers are provided in two forms: 1. an executable (.exe) that is built using the Nullsoft Scriptable Installation System, or 2. a Windows Installer package (.msi) that is built using WiX and can be customized for organizations via the use of MSI Transforms (see MSI Deployment Guide) # 2. System Requirements ## 2.1 Supported Operating Systems · Microsoft Windows 2000 Workstation · Microsoft Windows 2000 Server · Microsoft Windows XP Home · Microsoft Windows XP Professional · Microsoft Windows XP 64 · Microsoft Windows 2003 Server (32-bit and 64-bit Intel) · Microsoft Windows 2003 R2 Server (32-bit and 64-bit Intel) · Microsoft Windows Vista (32-bit and 64-bit Intel) · Microsoft Windows 2008 Server (32-bit and 64-bit Intel) ### 2.1.1 Unsupported Operating Systems · Microsoft Windows 95 · Microsoft Windows 98 · Microsoft Windows 98 OSR2 · Microsoft Windows ME · Microsoft NT Older releases of OpenAFS are available for download if unsupported operating systems must be used. The last version of OpenAFS with support for Win9x is 1.2.2b. The last version with support for Windows NT 4.0 is 1.2.10. ## 2.2 Disk Space Up to 60mb required for the OpenAFS binaries plus 100MB for the default AFSCache file. The size of the AFSCache file may be adjusted via the Registry after installation. The maximum cache size for 32-bit Windows is approximately 1.2GB. On 64-bit Windows there is no practical limit on the cache size. MIT Kerberos for Windows 2.6.x or 3.x.x if Kerberos v5 authentication support is desired. The recommended release is version 3.2.2. For 64-bit Windows installations, the 64-bit version of Kerberos for Windows is required. For 32-bit Windows installations, the 32-bit version of Kerberos for Windows is required. # 3. Operational Notes ## 3.0. Unicode Support Starting with the 1.5.50 release of OpenAFS for Windows, each of the AFS Client Service, the AFS Explorer Shell Extension, and the command-line tools are Unicode enabled. No longer is OpenAFS restricted to accessing file system objects whose names can be represented in the locale specific OEM code page. This has significant benefits for end users. Most importantly it permits non-Western languages to now be used for file system object names in AFS from Microsoft Windows operating systems. Now that Unicode names are supported, Roaming User Profiles and Folder Redirection will no longer fail when a user attempts to store an object with a name that cannot be represented in the OEM code page. Unicode names are stored in AFS using UTF-8 encoding. UTF-8 is supported as a locale on MacOS X, Linux, Solaris, and most other operating systems. This permits non-Western object names to be exchanged between Microsoft Windows and other operating systems. The OpenAFS for Windows client also implements Unicode Normalization as part of the name lookup algorithm. This is necessary because Unicode does not provide a unique representation for each input string. The use of normalization permits a file system object name created on MacOS X to be matched with the same string entered on Microsoft Windows even though the operating system’s choice of representation may be different. It is important to note that AFS file servers are not character set agnostic. All file system object names are stored as octet strings without any character set tagging. If a file system object is created using OEM Code Page 858 and then interpreted as UTF-8 it is likely that the object name will appear to be gibberish. OpenAFS for Windows goes to great lengths to ensure that the object name is converted to a form that will permit the user to rename the object using Unicode. Accessing UTF-8 names on UNIX systems that have the locale set to one of the ISO Latin character sets will result in the UTF-8 strings appearing to be gibberish. Neither UNIX AFS nor Microsoft Windows 2000 systems can perform Unicode Normalization for string comparisons. Although it is possible to store and read Unicode object names, it is possible that a user may not be able to open an object by typing the name of the object at the keyboard. GUI point and click operations should permit any object to be accessed. ## 3.1. Requirements for Kerberos v5 Authentication The Kerberos v4 infrastructure on which the OpenAFS 1.2 series is reliant is no longer secure. Cross-realm Kerberos is very important in the AFS context and most sites have or are migrating to Kerberos v5 environments. The OpenAFS 1.4 series (and later) integrates with MIT Kerberos for Windows 2.6.5 and above to support Kerberos v5 authentication including automatic renewal of AFS tokens and single sign-on via the Microsoft Windows Kerberos Logon Service. The recommended version of MIT Kerberos for Windows is 3.2.2. KFW 3.2.2 includes Network Identity Manager 1.3.1 which integrates with the AFS Provider installed as part of OpenAFS for Windows. When KFW is installed, the OpenAFS for Windows client will obtain Kerberos v5 tickets and use them as tokens without modification. The OpenAFS client requires that all of the AFS Servers with which it communicates support the use of Kerberos v5 tickets as tokens. If Kerberos v5 based tokens are presented to an AFS server that does not support them, the server will be unable to communicate with the client when tokens are present. Kerberos v5 based tokens are supported by OpenAFS release 1.2.8 or later. IBM Transarc servers do not support Kerberos v5. ### 3.1.1. Active Directory Microsoft Windows Active Directory can be used as a Kerberos v5 KDC in conjunction with OpenAFS. There are two things to consider when using an Active Directory as the Kerberos realm that issues the AFS service ticket. First, the Kerberos v5 tickets issued by Active Directory can be quite large when compared to tickets issued by a traditional KDC due to the incorporation of authorization data (the Microsoft PAC). If the issued tickets are larger than 344 bytes, the OpenAFS 1.2 servers will be unable to process them and will issue a RXKADBADTICKET error. OpenAFS 1.4 (and beyond) servers can support the largest tickets that Active Directory can issue. Second, the Kerberos v5 tickets issued by Windows 2003 Active Directory are encrypted with the DES-CBC-MD5 encryption type (enctype). OpenAFS 1.2 servers only support the DES-CBC-CRC enctype. As a result, OpenAFS 1.2 servers cannot process the resulting Kerberos v5 tokens. Windows 2000 Active Directory issues tickets with the DES-CBC-CRC enctype. Microsoft has documented in Knowledge Base article 832572 a new NO_AUTH_REQUIRED flag that can be set on the account mapped to the AFS service principal. When this flag is set, the PAC authorization data will not be included in the ticket. Setting this flag is recommended for all accounts that are associated with non-Windows services and that do not understand the authorization data stored in the PAC. This flag cannot be used if AFS service tickets are obtained via cross-realm using an Active Directory user principal. Note that an Active Directory computer object cannot be used for the afs service principal. ### 3.1.2. Using the krb524 service Some organizations have AFS cell names and Kerberos realm names which differ by more then just lower and upper case and rely on a modification to krb524d which maps a Kerberos v5 ticket from realm FOO to a Kerberos v4 ticket in realm BAR. This allows user@FOO to appear to be user@bar for the purposes of accessing the AFS cell. As of OpenAFS 1.2.8, support was added to allow the immediate use of Kerberos v5 tickets as AFS (2b) tokens. This is the first building block necessary to break away from the limitations of Kerberos v4 with AFS. By using Kerberos v5 directly we avoid the security holes inherent in Kerberos v4 cross-realm. We also gain access to cryptographically stronger algorithms for authentication and encryption. Another reason for using Kerberos v5 directly is because the krb524 service runs on a port (4444/udp) which has increasingly been blocked by ISPs. The port was used to spread a worm which attacked Microsoft Windows in the summer of 2003. When the port is blocked users find that they are unable to authenticate. Replacing the Kerberos v4 ticket with a Kerberos v5 ticket is a win in all situations except when the cell name does not match the realm name and the principal names placed into the ACL’s are not the principal names from the Kerberos v5 ticket. To support this transition, OpenAFS for Windows 1.4 adds a new registry value, Use524, to force the use of krb524d. However, the availability of this option should only be used by individuals until such time as their organizations can provide a more permanent solution. Note that the OpenAFS 1.4.x servers permit the use of a secondary realm name that can be treated as equivalent to the cell name for authentication. ### 3.1.3. Network Identity Manager Provider As of release 1.5.9, OpenAFS for Windows includes a Network Identity Manager Provider for obtaining AFS tokens. This plug-in is a contribution from Secure Endpoints Inc. Network Identity Manager is a multiple identity credential management tool that ships with MIT Kerberos for Windows version 3.0 and above. The OpenAFS plug-in requires MIT Kerberos for Windows version 3.1 or above. Version 3.2.2 is recommended for the best user experience. The Network Identity Manager replaces the former KFW ticket manager, Leash”, and when combined with the OpenAFS Provider, it is intended to be used as a replacement for the AFS System Tray Tool (afscreds.exe). Unlike both Leash and the AFS System Tray Tool, Network Identity Manager with the OpenAFS Provider can easily manage AFS tokens for multiple cells from one or more Kerberos v5 identities. The AFS configuration panel for each Kerberos v5 identity is used to configure which cells credentials should be obtained for and how they should be obtained. If the cell to realm mapping cannot be automatically determined, it can be explicitly specified. If the cell does not support Kerberos v5 tickets as tokens, then a krb524 service can be configured. The OpenAFS Provider configuration panel can be used to check the status of the AFS Client Service and its version. An optional checkbox is provided that will prevent the AFS System Tray Tool from being started by Windows after login. A shortcut to the OpenAFS Control Panel is also provided. ## 3.2. Use of the Microsoft Loopback Adapter by the AFS Client Service By itself the OpenAFS Client Service does not provide robust behavior in a plug-n-play network environment. Changes to the number of network adapters or their assigned IP addresses will cause the service to terminate unexpectedly. To avoid this behavior OpenAFS for Windows installs a single instance of the Microsoft Loopback Adapter (MLA) on the machine. With the MLA installed, the OpenAFS Client Service will not be affected by the configuration changes of other network adapters installed on the system. The MLA is installed with a name of "AFS" and a pre-assigned IP address in the 10.x.x.x range. The MLA is bound to the “Client for Microsoft Networks” service and not bound to the “File and Printer Sharing for Microsoft Networks”. If the MLA is unbound to "Client Microsoft Networks", the OpenAFS Client Service will become inaccessible when the machine is disconnected from the network. If the MLA is bound to "File and Printer Sharing ..." there will be a service type collision between the name "AFS" and the name of the machine on the MLA's IP Address that will result in the OpenAFS client service becoming inaccessible and the "NET VIEW \\AFS" command will return a "System Error 52" message. To correct the problem: · stop the AFS Client Service · bind the "Client for Microsoft Networks" to the MLA · unbind "File and Printer Sharing for Microsoft Networks" from the MLA · Disable and then re-enable the MLA · start the AFS Client Service When the MLA is not installed the unique NETBIOS name published by OpenAFS SMB server is "MACHINE-AFS". One of the benefits of using the MLA is that the NETBIOS name does not have to be published on any adapter other than the MLA. Therefore the chosen name is no longer required to be unique. Instead the NETBIOS name associated with the AFS Client Service is simply "AFS" and portable UNC paths of the form \\AFS\cellname\path can now be used on all machines. ## 3.3. Using Freelance (Dynamic Root) Mode to Improve Mobility Traditionally, when the OpenAFS Client Service starts it must be able to access the "root.afs" volume of the default cell. The "root.afs" volume contains the set of mount points to the "root.cell" volumes of various cells the administrator of the default cell believes should be accessible. If the "root.afs" volume is inaccessible when the client service is started, the service will terminate unexpectedly. Since many users now use laptops or otherwise operate in disconnected environments in which a VPN may be required to access the cell's servers, it is often the case that the "root.afs" volume for the default cell is not reachable and the OpenAFS Client Service will not successfully start. To allow the OpenAFS Client Service to operate in these environments, Freelance mode dynamically constructs a fake "root.afs" volume from mount points and symlinks stored in the local registry. The content of the fake “root.afs” volume is dynamically generated as cells are accessed. When the fake "root.afs" volume is initially constructed it will only contain two mount points: a regular path and read-write path mount point used to access the "root.cell" volume of the default AFS cell. Any attempt to access a valid cell name will result in a new mount point being created in the fake "root.afs" volume. If the cellname begins with a "." the mount point will be a read-write path; otherwise the mount point will be a regular path. These mount points are preserved in the registry at key: HKLM\SOFTWARE\OpenAFS\Client\Freelance Additional mount points may be manually created using the "fs mkmount" command. Mount points may be removed using the "fs rmmount" command. >fs mkmount \\AFS\athena.mit.edu root.cell athena.mit.edu >fs mkmount \\AFS\.athena.mit.edu root.cell athena.mit.edu -rw >fs rmmount \\AFS\athena.mit.edu >fs rmmount \\AFS\.athena.mit.edu Symlinks may also be created within the Freelance “root.afs” volume. The symlinks are stored in the registry at: ## 3.4. Locating AFS Volume Database Servers via DNS The OpenAFS for Windows client will use DNS AFSDB records to discover the location of AFS Volume Database servers when entries for the cell are not present in the client's CellServDB file (\%PROGRAMFILES%\OpenAFS\Client\CellServDB). ## 3.5. Obtaining AFS Tokens as a Integrated Part of Windows Logon OpenAFS for Windows installs a WinLogon Network Provider to provide Single Sign-On functionality (aka Integrated Logon.) Integrated Logon can be used when the Windows username and password match the username and password associated with the default cell's Kerberos realm. For example, if the Windows username is "jaltman" and the default cell is "athena.mit.edu", then Integrated Logon can be successfully used if the windows password matches the password assigned to the Kerberos principal "jaltman@ATHENA.MIT.EDU". The realm “ATHENA.MIT.EDU” is obtained by performing a domain name to realm mapping on the hostname of one of the cell's Volume Database servers. Integrated Logon is required if you desire the ability to store roaming user profiles within the AFS file system. OpenAFS does not provide tools for synchronizing the Windows and Kerberos user accounts and passwords. When KFW is configured, Integrated Logon will use it to obtain tokens. Use of KFW for Integrated Logon can be disabled via the EnableKFW registry value. Use of the krb524 service can be configured via the Use524 registry value. Integrated Logon will not transfer Kerberos v5 tickets into the user’s logon session credential cache. KFW 3.1 and above provides that functionality on its own. Integrated Logon does not have the ability to cache the user's username and password for the purpose of obtaining tokens if the Kerberos KDC is inaccessible at logon time. Integrated Logon supports the ability to obtain tokens for multiple cells. For further information on how to configure this feature read about the TheseCells value. Integrated Logon can be configured based upon the domain of the Windows account used to login to the machine. See ## 3.6. AFS System Tray Command Line Options The AFS System Tray Tool (afscreds.exe) has been deprecated in favor of Network Identity Manager. afscreds.exe will be removed from the OpenAFS in a future release. The AFS System Tray tool (afscreds.exe) supports several command line options: -A = autoinit -E = force existing afscreds to exit -I = install startup shortcut -M = renew drive maps -N = IP address change detection -Q = quiet mode. do not display start service dialog if afsd_service is not already running -S = show tokens dialog on startup -U = uninstall startup shortcut -X = test and do map share -Z = unmap drives autoinit will result in automated attempts to acquire AFS tokens when afscreds.exe is started. afscreds.exe will attempt to utilize tickets stored in the MSLSA credentials cache; any existing CCAPI credentials cache; and finally display an Obtain Tokens dialog to the user. When used in combination with IP address change detection, afscreds.exe will attempt to acquire AFS tokens whenever the IP address list changes and the Kerberos KDC is accessible. The renew drive maps option is used to ensure that the user drive maps constructed via the OpenAFS tools (not NET USE) are re-constructed each time afscreds.exe is started. By default afscreds.exe is configured by the OpenAFS.org installers to use “-A -N -M -Q” as startup options. Currently, there is no user interface to change this selection after install time although these options may be altered via the registry on either per machine or per user basis. See AfscredsShortcutParams in Appendix A. ## 3.7. The “AFS Client Admins” Authorization Group The OpenAFS for Windows client supports a local Windows authorization group named "AFS Client Admins". This group is used in place of the "Administrators" group to determine which users are allowed to modify the AFS Client Service configuration via the AFS Control Panel (afs_config.exe) or fs.exe command line tool. The following fs.exe commands are now restricted to members of the "AFS Client Admins" group: · checkservers with a non-zero timer value · setcachesize · newcell · sysname with a new sysname list · exportafs · setcell · setserverprefs · storebehind · setcrypt · cscpolicy · trace · minidump The creation or removal of mount points and symlinks in the Freelance “root.afs” volume are also restricted to members of the “AFS Client Admins” group. The initial membership of the "AFS Client Admins" group when created by the installer is equivalent to the local "Administrators" group. If a user is added to the "Administrators" group after the creation of the "AFS Client Admin" group, that user will not be an AFS Client Administrator. Only users that are members of the "AFS Client Admins" group are AFS Client Administrators. The local "SYSTEM" account is an implicit member of the "AFS Client Admins" group. Setting the default sysname for a machine should be done via the registry and not via "fs sysname". ## 3.8. OpenAFS support for UNC paths The OpenAFS client supports UNC paths everywhere. UNC paths provide a canonical name for resources stored within AFS. UNC paths should be used instead of drive letter mappings whenever possible. This is especially true when specifying the location of roaming profiles and redirected folders. Power users that make extensive use of the command line shell, cmd.exe, should consider using JP Software's 4NT or Take Command command processors. Unlike cmd.exe, the JPSoftware shells fully support UNC paths as the current directory. JPSoftware added special recognition for OpenAFS to its command shells, 4NT 7.0 and Take Command 7.0. AFS paths can be entered in UNIX notation (e.g., /afs/openafs.org/software), space utilization reports the output of the volume status for the specified path, and many AFS specific functions and variables have been added to the command language. JPSoftware's web site is http://www.jpsoft.com. ## 3.9. aklog.exe The OpenAFS Client ships with its own version of aklog.exe which should be used in preference to those obtained by other sources. The OpenAFS aklog.exe supports Kerberos v5 as well as the ability to auto-generate AFS IDs within foreign PTS databases. Usage: aklog [-d] [[-cell \| -c] cell [-k krb_realm]] [[-p \| -path] pathname] [-noprdb] [-force] [-5 [-m]\| -4] -d = output debugging information. cell = zero or more cells for which tokens will be obtained krb_realm = the kerberos realm of the cell. pathname = the directory for which authentication is required -noprdb = don't try to determine AFS ID. -5 or -4 = use Kerberos V (default) or Kerberos IV tickets -m = use krb524d to convert Kerberos V tickets to Kerberos IV ## 3.10. OpenAFS Servers on Windows are Unsupported The AFS Server functionality provided as part of the OpenAFS install package might work but should be considered highly experimental. It has not been thoroughly tested. Any data which would cause pain if lost should not be stored in an OpenAFS Server on Windows. Known issues include lack of support for power management and dynamic network configuration. Salvager is also known to crash. ### 3.10.1. OpenAFS Server Installation When the OpenAFS Server is installed, the TransarcAFSServer service (bosctlsvc.exe) will be installed and configured. The TransarcAFSServer service will auto-start the traditional AFS bos server. The former AFS Server Configuration wizard makes assumptions that no longer hold true and it has therefore been disabled. However, following the instructions for installing the AFS Servers on UNIX it is possible to properly configure the AFS Servers on Microsoft Windows. The AFS Server binaries, configuration files, and log files are installed under %Program Files%\OpenAFS\Server. kaserver has been deprecated and its use is strongly discouraged. Instead, Active Directory or some other Kerberos v5 KDC should be used in its place. ### 3.10.2. Using the AFS Client Service when the Server is installed A few notes on the usage of the AFS Client Service if it is going to be used with the OpenAFS AFS Server: · Freelance mode should be disabled when the AFS Client Service is installed on the same machine as the AFS Server,. Otherwise, you will be unable to manipulate the contents of the root.afs volume for the hosted cell without constructing an explicit mountpoint to the root.afs volume from another volume. · The AFS Server and related tools only support the built in kaserver (Kerberos IV). If kaserver is being used, MIT Kerberos for Windows should not be installed or must be disabled via the EnableKFW registry value. · The AFS Servers are not aware of power management events nor are they aware of network configuration changes. It is strongly advised that the AFS servers be installed only on systems that will not be shutdown or suspended unexpectedly. An inadvertent shutdown will corrupt volume data. ## 3.11. OpenAFS Debugging Symbol files The OpenAFS for Windows installers include Debugging Symbol files which should be installed if you are experiencing problems and need to send crash reports. This is true for both the release and the debug versions of the installers. The difference between the release and debug versions are: · whether or not the binaries were compiled with optimization (release: yes, debug: no) · whether or not the debug symbols are installed by default (release: no, debug: yes) · whether or not fs trace logging is turned on by default (release: no, debug: yes) · whether or not additional debug statements were compiled into the binaries (release: no, debug: yes) ## 3.12. Large File (64-bit) Support As of release 1.5.3, OpenAFS for Windows supports files larger than 2GB. The maximum file size is now 16777216 terabytes when the AFS File Server supports large files. If the AFS File Server does not support 64-bit file sizes, then the maximum file size remains 2GB. ## 3.13. Encrypted AFS Network Communication The OpenAFS for Windows installer by default activates a weak form of encrypted data transfer between the AFS client and the AFS servers. This is often referred to as "fcrypt" mode. Encrypted data transfer can be turned on or off with the “fs crypt” command. Transitions between “crypt” and “non-crypt” modes are logged to the Windows Application Event Log. OpenAFS authenticates SMB connections using either NTLM or GSS SPNEGO (NTLM). In previous versions of OpenAFS, the SMB connections were unauthenticated which opened the door for several attacks which could be used to obtain access to another user's tokens on shared machines. When GSS SPNEGO attempts a Kerberos v5 authentication, the Windows SMB client will attempt to retrieve service tickets for "cifs/afs@REALM" (if the loopback adapter is in use) or "cifs/machine-afs@REALM" (if the loopback adapter is not being used). It is extremely important that this service principal not exist in the KDC database as the Kerberos authentication must fail allowing automatic fallback to NTLM. When NTLM is used a special local authentication mode will be used that does not require access to the user's password. Instead, Windows will internally recognize the request as coming from a local logon session. ## 3.15. No More INI Files Previous AFS clients for Windows stored configuration data in Windows .INI files. The OpenAFS client does not use Windows .INI files for the storage of configuration data. All settings are stored in the registry (see Appendix A). The CellServDB file is now stored in either the %ALLUSERSPROFILE%\Application Data\OpenAFS\Client directory or the %PROGRAMFILES%\OpenAFS\Client directory. The CellServDBDir registry value or the AFSCONF environment variable can be used to specify an alternative location. For users converting from IBM AFS clients, during installation OpenAFS will relocate the contents of the “afsdcell.ini” file to the new CellServDB file. OpenAFS will also import the contents of the “afs_freelance.ini” file to the Windows registry. OpenAFS will not process the contents of the “afsddbmt.ini”. ## 3.16. Microsoft Windows Internet Connection Firewall The OpenAFS Client is compatible with the Internet Connection Firewall that debuted with Windows XP SP2 and Windows 2003 SP1. The Internet Connection Firewall will be automatically adjusted to allow the receipt of incoming callback messages from the AFS file server. In addition, the appropriate Back Connection registry entries are added to allow SMB authentication to be performed across the Microsoft Loopback Adapter. ## 3.17. Browsing AFS from the Explorer Shell and Office The OpenAFS Client Service implements the CIFS Remote Admin Protocol which allows Explorer to browse server and share information. This significantly enhances the interoperability of AFS volumes within the Explorer Shell and Microsoft Office applications. ## 3.18. Byte Range Locking Many applications on Windows (e.g. Microsoft Office) require the use of byte range locks applied to a file either to protect against simultaneous file access or as a signaling mechanism. OpenAFS for Windows release 1.5 (or greater) implements byte range locking within the CIFS-AFS gateway server. This support for byte range locking obtains AFS’ advisory file server locks to simulate Microsoft Windows mandatory locks. When an application opens a file, a lock will be obtained from AFS indicating that the file is in use. If the lock is a write lock, access to the file will be restricted to other applications running on the same machine as the first application to request the lock. Applications running on other machines will see the AFS full file lock and will be unable to access the file. Most Windows applications and Windows itself opens files in shared read mode. When this is done, a read lock is applied to the file. This does not prevent shared read access across multiple machines but is used to ensure that no one writes to the file while it is in use. As the CIFS-AFS gateway server attempts to implement Windows lock semantics on top of AFS lock semantics it is important to understand how AFS file locks work. In Windows there are no special privileges associated with obtaining file locks. If you can read or execute a file, then you can obtain shared and exclusive locks. In general, a Windows shared lock equates to an AFS read lock and a Windows exclusive lock equates to an AFS write lock. In AFS if you can write to a file, then you can obtain a write lock. However, in AFS if you can read a file it does not mean that you can obtain a read lock on it. The ability to obtain read locks is granted only if you have the lock (or ‘k’) privilege. This behavior is required in order to allow anonymous users to read files while preventing them from being able to deny access to the files to other users. OpenAFS 1.4.0 and earlier as well as all IBM AFS file servers have an implementation bug that prevents users with write privileges from being able to obtain locks without the lock privilege. When AFS serves data out of read-only volumes the file server will deny all requests for read and write locks because the contents of the volume cannot be changed by the client. Since Microsoft Windows applications almost always attempt to obtain a temporary exclusive lock when accessing files the OpenAFS Client’s CIFS-AFS gateway implements the following semantics in order to reduce the inconvenience on end users. • If the file is located on a read-only volume and the application requests a shared lock, the CIFS-AFS server will grant the lock request without asking the AFS file server. • If the file is located on a read-only volume and the application opens the file with write access and requests an exclusive lock, the CIFS-AFS server will refuse the lock request and return a read only error. • If the file is located on a read-only volume and the application opens the file with only read access and requests an exclusive lock, the CIFS-AFS server will fulfill the lock request with a read lock. • If the file is located on a read-write volume and the application requests an exclusive lock, the CIFS-AFS server will request a write lock from the AFS file server. If granted by the file server, then the CIFS-AFS server will grant the lock request. If the request is denied due to an access denied error and the user has the lookup, read and lock privileges and the file was opened for read only access, then the CIFS-AFS server will request a read lock from the file server. If the request is denied due to an access denied error and the user has the lookup and read privileges but not the lock privilege, then the CIFS-AFS server will grant the request even though the AFS file server said ‘no’. If the user does not have at least those permissions, the CIFS-AFS server will deny the request. • If the file is located on a read-write volume and the application requests a shared lock, the CIFS-AFS server will request a read lock from the AFS file server. If granted by the file server, then the CIFS-AFS server grants the lock request. If the request is denied due to an access denied error and the user has the lookup and read privileges but not the lock privilege, then the CIFS-AFS server will grant the request even though the AFS file server said ‘no’. If the user does not have at least those permissions, the CIFS-AFS server will deny the request. • If multiple processes on the same machine attempt to access the same file simultaneously, the CIFS-AFS server will locally manage the granted locks and all processes will share a single lock on the AFS file server. • If the CIFS-AFS server is unable to renew the AFS file server locks, then it will invalidate the associated file handles. This is the same behavior that an application will experience if it was using a Windows File Share and the connection was broken. Invalidating the file handles prevents subsequent data corruption from taking place. If you wish to disable the acquisition of locks from the file server, this can be performed using the EnableServerLocks registry value. ## 3.19. Automatic Discarding of AFS Tokens at Logoff The OpenAFS Client will automatically forget a user's tokens upon Logoff unless the user's profile was loaded from an AFS volume. In this situation there is no mechanism to determine when the profile has been successfully written back to the network. It is therefore unsafe to release the user's tokens. Whether or not the profile has been loaded from the registry can be determined for Local Accounts, Active Directory accounts and NT4 accounts. If there is a need to disable this functionality, the LogoffPreserveTokens registry value can be used. (see Appendix A.) ## 3.20. Windows Terminal Server installations When installing the NSIS (.exe) installer under Terminal Server, you must execute it from within the Add/Remove Programs Control Panel. Failure to do so will result in AFS not running properly. The AFS Server should not be installed on a machine with Terminal Server installed. ## 3.21. Hidden Dot Files AFS is a UNIX native file system. The OpenAFS client attempts to treat the files stored in AFS as they would be on UNIX. File and directory names beginning with a "." are automatically given the Hidden attribute so they will not normally be displayed. This behavior can be altered via the HideDotFiles registry value. ## 3.22. Status Cache Limits The Status Cache (AFS Configuration Control Panel: Advanced Page) is defined to have a maximum number of entries. Each entry represents a single file or directory entry accessed within the AFS file system. When the maximum number of entries are allocated, entries will begin to be reused according to a least recently used (LRU) algorithm. If the number of files or directories being accessed repeatedly by your applications is greater then the maximum number of entries, your host will begin to experience thrashing of the Status Cache and all requests will result in network operations. If you are experiencing poor performance try increasing the maximum number of Status Cache entries. Each entry requires approximately 1.2K. The default number of Status Cache entries is 10,000. This can be adjusted using the Stats registry value. ## 3.23. NETBIOS over TCP/IP must be enabled "Netbios over TCP/IP" must be active on the machine in order for communication with the AFS Client Service to succeed. If "Netbios over TCP/IP" is disabled on the machine, then communication with the AFS Client Service will be impossible. If you are using the Microsoft Loopback Adapter, configure the “Netbios over TCP/IP” setting for the adapter. ## 3.24. OpenAFS binaries are digitally signed The OpenAFS Client Service and related binaries distributed by OpenAFS.org are digitally signed by "Secure Endpoints Inc.". The OpenAFS Client Service will perform a run-time verification check to ensure that all OpenAFS related DLLs loaded by the service match the same file version number and were signed by the same entity. This check has been added to prevent the stability problems caused by more than one AFS installation present on a machine at the same time. Many hours of support time have been wasted tracking down problems caused by the mixture of files from different releases. Appendix A documents the "VerifyServiceSignature" registry value which can be used to disable the signature check. The file version check cannot be disabled. ## 3.25. Maximum Size of the AFSCache File The maximum cache size on 32-bit Windows is approximately 1.3GB. This is the largest contiguous block of memory in the 2GB process address space which can be used for constructing a memory mapped file. Due to fragmentation of the process space caused by the loading of libraries required by the digital signature verification code, any attempt to specify a cache size greater then 700MB will result in the automatic disabling of the signature check. Significantly larger cache sizes can be used on 64-bit Windows. ## 3.26. Filename Character Sets OpenAFS for Windows implements an SMB server which is used as a gateway to the AFS filesystem. Because of limitations of the SMB implementation, Windows stores all files into AFS using OEM code pages such as CP437 (United States) or CP850 (Western Europe). These code pages are incompatible with the ISO Latin-1 character set typically used as the default on UNIX systems in both the United States and Western Europe. Filenames stored by OpenAFS for Windows are therefore unreadable on UNIX systems if they include any of the following characters: [Ç] 128 08/00 200 80 C cedilla [ü] 129 08/01 201 81 u diaeresis [é] 130 08/02 202 82 e acute [â] 131 08/03 203 83 a circumflex [ä] 132 08/04 204 84 a diaeresis [à] 133 08/05 205 85 a grave [å] 134 08/06 206 86 a ring [ç] 135 08/07 207 87 c cedilla [ê] 136 08/08 210 88 e circumflex [ë] 137 08/09 211 89 e diaeresis [è] 138 08/10 212 8A e grave [ï] 139 08/11 213 8B i diaeresis [î] 140 08/12 214 8C i circumflex [ì] 141 08/13 215 8D i grave [Ä] 142 08/14 216 8E A diaeresis [Å] 143 08/15 217 8F A ring [É] 144 09/00 220 90 E acute [æ] 145 09/01 221 91 ae diphthong [Æ] 146 09/02 222 92 AE diphthong [ô] 147 09/03 223 93 o circumflex [ö] 148 09/04 224 94 o diaeresis [ò] 149 09/05 225 95 o grave [û] 150 09/06 226 96 u circumflex [ù] 151 09/07 227 97 u grave [ÿ] 152 09/08 230 98 y diaeresis [Ö] 153 09/09 231 99 O diaeresis [Ü] 154 09/10 232 9A U diaeresis [ø] 155 09/11 233 9B o slash [£] 156 09/12 234 9C Pound sterling sign [Ø] 157 09/13 235 9D O slash [×] 158 09/14 236 9E Multiplication sign [ƒ] 159 09/15 237 9F Florin sign The OpenAFS Client provides an optional registry value, StoreAnsiFilenames, that can be set to instruct OpenAFS to store filenames using the ANSI Code Page instead of the OEM Code Page. The ANSI Code Page is a compatible superset of Latin-1. This setting is not the default setting because making this change would prevent OpenAFS for Windows from being able to access filenames containing the above characters which were created without this setting. ## 3.27. Known Character Set Issues with Roaming Profiles There is a known issue with storing Windows Roaming Profiles when the profile contains either directories or files with names which cannot be represented in the local OEM character set. In this case, attempts to write the profile back to AFS will fail during the character set conversion. The OpenAFS Client’s CIFS gateway does not support UNICODE. To avoid this problem some sites run custom logoff scripts (assigned by group policy) which rename all files to use only the supported characters for the locale. ## 3.28. The AFSCache File The AFS Cache file is stored by default at %TEMP%\AFSCache in a persistent file marked with the Hidden and System attributes. The persistent nature of the data stored in the cache file improves the performance of OpenAFS by reducing the number of times data must be read from the AFS file servers. The performance of the AFS Client Service is significantly affected by the access times associated with the AFSCache paging file. When given the choice, the AFSCache file should be placed on a fast disk, preferably NTFS, the file should not be compressed and should consist of as few fragments as possible. Significant performance gains can be achieved by defragmenting the AFSCache file with Sysinternal's Contig utility while the AFS Client Service is stopped. ## 3.29. Restricting OpenAFS Client Service Start and Stop A new command line tool, afsdacl.exe, can be used to restrict the ability to start and stop the OpenAFS Client Service. afsdacl : Set or reset the DACL to allow starting or stopping the afsd service by any ordinary user. Usage : afsdacl [-set \| -reset] [-show] -set : Sets the DACL -reset : Reset the DACL -show : Show current DACL (SDSF) ## 3.30. The @sys Name List The default @sys name list in the OpenAFS Client is set to "x86_win32 i386_w2k i386_nt40" for 32-bit x86 systems. The default is "amd64_win64" for amd 64-bit versions of Windows. ## 3.31. Symlinks to AFS UNC paths In OpenAFS, symlinks to AFS UNC paths, \\AFS[\all]\..., are treated the same as symlinks to /afs/... However, please use /afs/... as the Windows UNC form will not work on UNIX client. ## 3.32. Cache Manager Debugging The OpenAFS Client implements the Cache Manager Debugging RPC Interface. The CM debugger can be queried with cmdebug.exe. Usage: cmdebug -servers <server machine> [-port <IP port>] [-long] [-refcounts] [-callbacks] [-ctime] [-addrs] [-cache] [-cellservdb] [-help] Where: -long print all info -refcounts print only cache entries with positive reference counts -callbacks print only cache entries with callbacks -ctime print human readable expiration time -cache print only cache configuration -cellservdb print only cellservdb info ## 3.33. Windows Logon Caching vs. Kerberos Logons If you are a site which utilizes MIT/Heimdal Kerberos principals to logon to Windows via a cross-realm relationship with a multi-domain Windows forest, you must enable Windows logon caching unless the workstation is Windows Vista. ## 3.34. Initial Server Preferences VLDB and File Server Preferences can now be provided initial values using registry keys. This is useful for managed machines in a Windows domain which are centrally located (e.g., in a computing lab.) See Appendix A for details on the "Server Preferences" keys. ## 3.35. File Timestamps The OpenAFS Client reports timestamps on files stored in AFS in UTC all year round. In locales with daylight savings time, previous versions of AFS for Windows reported the time when DST is active as UTC+1. This was done to preserve the relative local time for the user. A file stored at 11:00am EST in January would be reported as having been stored at 11:00am EDT in June. Unfortunately, this has the negative side effect of changing the reported timestamp from 16:00UTC to 15:00UTC. Since Windows treats all file times in UTC, data synchronization applications which rely on the timestamp would believe that all files stored in AFS had changed. It should be noted that UNIX based operating systems (such as Solaris) do not appear to report file times to applications in UTC. They do preserve the relative local time. This may confuse some users who are used to being able to compare the timestamp in an UNIX shell with the timestamp from the Windows explorer. During DST, these two times will no longer agree even though they are in fact representing the same moment in time. ## 3.36. Windows RPC client support must be installed If the installer refuses to install and complains about an RPC configuration error, check to ensure that the following registry entries are present and that they refer to the dll "rpcrt4.dll": HKLM "SOFTWARE\Microsoft\RPC\ClientProtocols" "ncacn_np" HKLM "SOFTWARE\Microsoft\RPC\ClientProtocols" "ncacn_ip_tcp" HKLM "SOFTWARE\Microsoft\RPC\ClientProtocols" "ncacn_http" ## 3.37. Generating Minidumps of the OpenAFS Client Service OpenAFS 1.4 added a new command, "fs minidump". This command can be used at any time to generate a mini dump file containing the current stack of the afsd_service.exe process. This output can be very helpful when debugging the AFS Client Service when it is unresponsive to SMB/CIFS requests. ## 3.38. AFS Client Universally Unique Identifiers (UUIDs) vs. System Cloning The OpenAFS Client implements Universally Unique Identifiers (UUIDs). They are used to provide the AFS file server with a method of identifying the client that is independent of IP address. This permits the AFS file server to track mobile clients or those behind Network Address Translators when they move from address to address or port to port. Tracking the client improves client performance by permitting callback state to be maintained across location changes. The UUID is generated when the AFSCache file is created and is maintained as long as the contents of the AFSCache file are valid. The UUID is stored in the AFSCache file. When cloning machines that have Windows AFS client installed it is necessary to generate a new UUID for each client. This will be done automatically if the Windows Machine SID is re-generated using Microsoft SysPrep. If the SID is not being re-generated either the AFSCache file should be deleted or the command fs uuid -generate must be executed after the the clone is created. Multiple AFS clients reporting the same UUID will not only result in horrible AFS client performance and cache inconsistencies, but they will also put a tremendous strain on the AFS file servers. For lab environments that wish to erase all cached data on each restart, the NonPersistentCaching option will disable the use of the persistent cache file. As a side effect, a new UUID will be generated for the AFS client service on each restart. When a Windows system is cloned, the Microsoft Loopback Adapter will be disabled in the cloned system. Administrators must re-install the Microsoft Loopback Adapter within the cloned environment. This can be automated by using the OpenAFS “instloop.exei” command. Instloop.exe can be extracted from the MSI installer by performing an administrative install via msiexec.exe /a. ## 3.39. Delayed Write Errors with Microsoft Office Applications Microsoft Office makes heavy use of asynchronous input/output methods for reading and writing to file streams. This can result in hundreds of requests being simultaneously queued for service by the CIFS client with a fixed timeout period. As the AFS CIFS server is local to the machine the Windows CIFS client believes that it can respond almost instantaneously to write requests as the actual writing to the AFS file server is performed by a background daemon thread. When the actual network bandwidth to the AFS file server is slow and the file size is large it is possible for the CIFS client to time out the connection. When this happens a “delayed write error” will be reported to the user and the application may crash. The only workaround at the current time is to save first to a local disk and subsequently copy the file to AFS as copying a file with the explorer shell does not use asynchronous i/o. The CIFS session timeout defaults to 45 seconds and can be increased by modifying the registry. Beginning with the 1.5.33 release, the performance characteristics of SMB Write Data operations can be adjusted. In prior releases all writes were performed using a restricted asynchronous store model in which only one asynchronous store operation per file can be performed at a time. The reason for this restriction is limit the amount of data the cache manager will accept without it having been written to the file server. If too much unwritten data is accepted, the file close operation will block until all of the unwritten data is output and this could trigger a CIFS client disconnect. Prior to 1.5.33 the size of the asynchronous store was always equal to the chunk size which was often too large for low bandwidth connections. The asynchronous store size now defaults to 32KB and is configurable using the SMBAsyncStoreSize registry value. Asynchronous store operations can also be disabled using the EnableSMBAsyncStore registry value in which case all writes received by the cache manager block until the Rx StoreData operation completes. ## 3.40. Global Drives (aka Service Drive Letters) are no longer supported by Microsoft The Global DriveAuto-mount feature has been deprecated due to the following Microsoft KB article. http://msdn.microsoft.com/library/en-us/dllproc/base/services_and_redirected_drives.asp It says that services mounting drive letters are no longer supported by Microsoft and may act unpredictably. The experience other users have had is that if the connection to the OpenAFS CIFS/SMB server is terminated by the Windows CIFS client, the drive mapping may not be re-established until the machine is rebooted. OpenAFS supports UNC paths and whenever possible applications should be modified to use of \\AFS\<cellname>\<path> instead of drive letters. ## 3.41. 64-bit Microsoft Windows Installations Although 64-bit Windows platforms support both 64-bit and 32-bit applications, the OpenAFS Service installed on the machine must be 64-bit. The 64-bit installer contains only 64-bit executables. In order to support 32-bit applications that link against OpenAFS libraries it is required that a separate 32-bit OpenAFS Tools set be installed. For example, the 32-bit version of Kerberos for Windows can be used with the 32-bit OpenAFS Tools to manage AFS tokens. OpenAFS on 64-bit Windows benefits from the lifting of the 2GB process memory restriction that is present in 32-bit Windows. Without this restriction the AFS Cache File can become arbitrarily large limited only by available disk space. ## 3.42. Known Issues with Microsoft Windows Vista OpenAFS for Windows works with Microsoft Windows Vista from both the command prompt and the Explorer Shell. When performing an upgrade from earlier versions of Microsoft Windows the Microsoft Loopback Adapter (MSLA) will be uninstalled. OpenAFS should be re-installed after the Microsoft Vista installation to restore the MSLA configuration. Due to a feature change in Windows Vista’s Plug-n-Play network stack, during a standby/hibernate operation the MSLA is disabled just as any other hardware device would be. This causes the OpenAFS Client’s network binding to be lost. As a result, it takes anywhere from 30 to 90 seconds after the operating system is resumed for access to the OpenAFS Client and the AFS file space to become available. Until the network bindings have been re-established, ticket managers and other tools will report that the AFS Client Service may not have been started. Windows Vista implements User Account Control (UAC), a new security feature that implements least user privilege. With UAC, applications only run with the minimum required privileges. Even Administrator accounts run applications without the “Administrator” access control credentials. One side effect of this is that existing applications that mix user and system configuration capabilities must be re-written to separate those functions that require “Administrator” privileges into a separate process space. Future updates to OpenAFS will incorporate the necessary privilege separation, until that time some functions such as the Start and Stop Service features of the AFS System Tray tool and the AFS Control Panel will not work unless they are “Run as Administrator”. When a Vista user account that is a member of the “Administrators” group is used to access the AFS Control Panel (afs_config.exe), the process must be “Run as Administrator”. Otherwise, attempts to modify the OpenAFS configuration will appear to succeed but in reality will have failed due to Vista’s system file and registry virtualization feature. The help files provided with OpenAFS are in .HLP format. Windows Vista does not include a help engine for this format. ## 3.43. New AFS Share Name Syntax Provides Direct Access to Volumes Starting with the 1.5.21 release of OpenAFS for Windows, the following syntax can be used to access any volume in any cell without requiring the creation of a mount point. \\AFS\<cell><mount point type><volume>\ Where <cell> can be either a full cell name or an unambiguous prefix, the <mount point type> is ‘#’ for a normal mount point or ‘%’ to force the use of a read-write volume, and <volume> is either a volume name or its ID number. Examples include: \\AFS\athena.mit.edu#user.jaltman\ \\AFS\athena%user.jaltman\ \\AFS\athena.mit.edu# 537235559\ ## 3.44. Differences between Windows and UNIX “fs examine” The OpenAFS for Windows version of “fs examine” provide two additional lines of output when compared to the UNIX implementation. These lines include the owner and group information for the file as well as the volume status. The Windows output will also indicate the type of object {File, Directory, Mountpoint, Symlink, …} that was examined. [C:\]fs examine \\afs\athena#user.jaltman Directory \\afs\athena#user.jaltman (537235559.1.1) contained in cell athena.mit.edu Owner jaltman (28180) Group 0 Volume status for vid = 537235559 named user.jaltman is Current disk quota is 1500000 Current blocks used are 1244184 The partition has 151945877 blocks available out of 511163724 Volume is online ## 3.45. Literal evaluation of AFS objects via fs commands Beginning with the 1.5.31 release, the fs commands examine, flush, whereis, and whichcell provide a new command-line parameter, -literal. When specified, if the evaluated object is a symlink or a mountpoint the resulting output will describe the specified object and not the object that is the target of the symlink or mountpoint. ## 3.46. Out of Quota errors Prior to the 1.5.31 release, out of quota errors were reported to the calling application as an out of space error. As of 1.5.31, an out of space error will indicate that the partition on which the volume is located is in fact out of space. Whereas an out of quota error indicates that the user does not have permission to allocate additional space. # 4. How to Debug Problems with OpenAFS for Windows OpenAFS for Windows provides a wide range of tools to assist you in debugging problems. The techniques available to you are varied because of the wide range of issues that have been discovered over the years. ## 4.1. pioctl debugging (IoctlDebug registry key) pioctl (path-based ioctl) calls are used by various tools to communicate with the AFS Client Service. Some of the operations performed include: · setting/querying tokens (tokens.exe, aklog.exe, afscreds.exe) · setting/querying ACLs · setting/querying cache parameters · flushing files or volumes · setting/querying server preferences · querying path location · checking the status of servers and volumes · setting/querying the sysname list pioctl calls are implemented by writing to a special UNC path that is processed by the AFS Client Service. If there is a failure to communicate with the AFS Client Service via SMB/CIFS, it will be impossible to perform any of the above operations. To assist in debugging these problems, the registry value: [HKLM\SOFTWARE\OpenAFS\Client] REG_DWORD: IoctlDebug = 0x01 should be set. Then any of the commands that perform pioctl calls should be executed from the command prompt. With this key set the pioctl library will generate debugging output to stderr. The output will contain the Win32 API calls executed along with their most important parameters and their return code. The MSDN Library and the Microsoft KnowledgeBase can be used as a reference to help you determine the configuration probem with your system. ## 4.2. afsd_service initialization log (%WinDir%\TEMP\afsd_init.log) Every time the AFS Client Service starts it appends data about its progress and configuration to a file. This file provides information crucial to determining why the service cannot start when there are problems. When the process terminates due to a panic condition it will write to this file the source code file and line number of the error. In many cases the panic condition is due to a misconfiguration of the machine. In other cases it might be due to a programming error in the software. A quick review of the location in the source code will quickly reveal the reason for the termination. The MaxLogSize registry value determines the maximum size of the %WINDIR%\TEMP\afsd_init.log file. If the file is larger than this value when OpenAFS Client Service starts, the file will be reset to 0 bytes. If value is set to 0, the file will be allowed to grow indefinitely. ## 4.3. afsd_service debug logs (fs trace {-on, -off, -dump} ->%WinDir%\TEMP\afsd.log) When attempting to debug the behavior of the SMB/CIFS Server and the Cache Manager it is often useful to examine a log of the operations being performed. While running the AFS Client Service keeps an in memory log of many of its actions. The default number of actions preserved at any one time is 5000. This can be adjusted with the registry value: [HKLM\SYSTEM\CurrentControlSet\Services\TransarcAFSDaemon\Parameters] REG_DWORD TraceBufferSize A restart of the service is necessary when adjusting this value. Execute "fs trace -on" to clear to the log and "fs trace -dump" to output the contents of the log to the file. ## 4.4. Using SysInternal’s Debug Viewer, Process Monitor and Process Explorer Tools An alternatve option to the use of "fs trace -dump" to capture internal OpenAFS Client Service events is to use a tool such as Sysinternal's Debug Viewer to capture real-time debugging output. When the OpenAFS Client Service starts and Bit 2 of the TraceOption value in the registry is set, all trace log events are output using the Windows Debug Monitor interface (OutputDebugString). [HKLM\SYSTEM\CurrentControlSet\Services\TransarcAFSDaemon\Parameters] REG_DWORD TraceOption = 0x04 Use “fs trace –on” and “fs trace –off” to toggle the generation of log messages. Sysinternal’s Process Monitor can be use to monitor the file operations requested by applications and their success or failure. In Process Monitor, set a filter to include only events on file paths that refer to the AFS name space. Be sure to include both the UNC path as well as any drive letters mapped to AFS. Turn on the Clock Time and Show Milliseconds options in both tools to make it easier to synchronize the application requests and the resulting OpenAFS Client Service operations. The captured data can be stored to files for inclusion in bug reports. Sysinternal's Process Explorer is a replacement for the Windows Task Manager and so much more. Process Explorer can be configured to use the DbgHelp.dll from "Microsoft Debugging Tools for Windows" as well as the debug symbols shipped as an optional component of the OpenAFS for Windows installer. (Options->Configure Symbols) Once configured the "Threads" tab of the process properties dialog will permit the viewing of a fully documented stack for each displayed thread. Hint: If there is a deadlock in the cache manager, two or more of the threads will be stuck in a call to osi_TWait(). ## 4.5. Microsoft MiniDumps (fs minidump -> %WinDir%\TEMP\afsd.dmp) If the AFS Client Service become unresponsive to any form of communication there may be a serious error that can only be debugged by someone with access to the source code and a debugger. The "fs minidump" command can be used to force the generation of a MiniDump file containing the state of all of the threads in the AFS Client Service process. The most accurate MiniDump files will be produced after installing "Microsoft Debugging Tools for Windows". The MiniDumpType registry value can be used to adjust the scope of the process information included within the dump file. By default the MiniDump only contains the stacks of all threads and the values of all global variables. A much more useful MiniDump is one that contains the contents of the heap. Be warned, a MiniDump with heap will be as large as the cache file. In addition, it will include all of the data stored within the cache. If there are privacy concerns, do not produce a MiniDump with heap. ## 4.6. Single Sign-on (Integrated Logon) debugging If you are having trouble with the Integrated Logon operations it is often useful to be able to obtain a log of what it is attempting to do. Setting Bit 0 of the TraceOption registry value: [HKLM\SYSTEM\CurrentControlSet\Services\TransarcAFSDaemon\Parameters] REG_DWORD TraceOption = 0x01 will instruct the Integrated Logon Network Provider and Event Handlers to log information to the Windows Event Log: Application under the name “AFS Logon". ## 4.7. RX (AFS RPC) debugging (rxdebug) The rxdebug.exe tool can be used to query a variety of information about the AFS services installed on a given machine. The port for the AFS Cache Manager is 7001. Usage: rxdebug -servers <server machine> [-port <IP port>] [-nodally] [-allconnections] [-rxstats] [-onlyserver] [-onlyclient] [-onlyport <show only <port>>] [-onlyhost <show only <host>>] [-onlyauth <show only <auth level>>] [-version] [-noconns] [-peers] [-help] Where: -nodally don't show dallying conns -allconnections don't filter out uninteresting connections -rxstats show Rx statistics -onlyserver only show server conns -onlyclient only show client conns -version show AFS version id -noconns show no connections -peers show peers ## 4.8. Cache Manager debugging (cmdebug) The cmdebug.exe tool can be used to query the state of the AFS Cache Manager on a given machine. Usage: cmdebug -servers <server machine> [-port <IP port>] [-long] [-refcounts] [-callbacks] [-ctime] [-addrs] [-cache] [-cellservdb] [-help] Where: -long print all info -refcounts print only cache entries with positive reference counts -callbacks print only cache entries with callbacks -ctime print human readable expiration time -cache print only cache configuration -cellservdb print only cellservdb info ## 4.9. Persistent Cache consistency check The persistent cache is stored in a Hidden System file at %WinDir%\TEMP\AFSCache. If there is a problem with the persistent cache that prevent the AFS Client Service from being able to start a validation check on the file can be performed. afsd_service.exe --validate-cache <cache-path> ## 4.10. Token Acquisition Debugging If you are having trouble obtaining tokens with the Network Identity Manager AFS credential provider, it is recommended that you verify your ability to obtain tokens using the command-line tools klog.exe (if you are using kaserver) or kinit.exe and aklog.exe (if you are using Kerberos v5.) The aklog.exe –d option will be quite helpful in diagnosing Kerberos v5 related problems. # 5. Reporting Bugs Bug reports should be sent to openafs-bugs@openafs.org. Please include as much information as possible about the issue. If you are reporting a crash, please install the debugging symbols by re-running the installer. If a dump file is available for the problem, %WINDIR%\TEMP\afsd.dmp, include it along with the AFS Client Trace file %WINDIR%\TEMP\afsd.log. The AFS Client startup log is %WINDIR%\TEMP\afsd_init.log. Send the last continuous block of log information from this file. Configuring DrWatson to generate dump files for crashes: · Run drwtsn32.exe to configure or to identify where the log and the crash dump files are created: · click Start > Run... · type drwtsn32 <enter>. · Select either a Crash Dump Type: Mini or Full. · Clear Dump Symbol Table · Clear Append to Existing Log file. · Check Dump All Thread Contexts. · Check Create Crash Dump File · Next run the monitoring module of Dr. Watson: · click Start > Run... · type drwatson <enter>. · Once a crash happens, Dr. Watson generates a dump file and a report in the log file, including the address of the crash and the stack dump. Once you have the Dr. Watson's logfile and minidump, zip them and attach them to your e-mail. When reporting a error, please be sure to include the version of OpenAFS. # 6. How to Contribute to the Development of OpenAFS for Windows Contributions to the development of OpenAFS for Windows are continuously needed. Contributions may take many forms including cash donations, support contracts, donated developer time, and even donated tech writer time. ## 6.1. The USENIX OpenAFS Fund USENIX, a 501c3 non-profit corporation, has formed the USENIX OpenAFS Fund in order to accept tax deductible donations on behalf of the OpenAFS Elders. The donated funds will be allocated by the OpenAFS Elders to fund OpenAFS development, documentation, project management, and maintaining openafs.org. USENIX OpenAFS Fund USENIX Association 2560 Ninth St., Suite 215 Berkeley, CA 94710 Donations can be made by sending a check, drawn on a U.S. bank, made out to the USENIX OpenAFS Fund or by making a donation online. ## 6.2. Secure Endpoints Inc. Secure Endpoints Inc. provides development and support services for OpenAFS for Windows and MIT Kerberos for Windows. Donations provided to Secure Endpoints Inc. for the development of OpenAFS are used to cover the OpenAFS gatekeeper responsibilities; providing support to the OpenAFS community via the OpenAFS mailing lists; and furthering development of desired features that are either too small to be financed by development contracts. Secure Endpoints Inc. accepts software development agreements from organizations who wish to fund a well-defined set of bug fixes or new features. Secure Endpoints Inc. provides contract based support for the OpenAFS for Windows and the MIT Kerberos for Windows products. ## 6.3. Direct contributions of code and/or documentation Organizations that use OpenAFS in house and have development staffs are encouraged to contribute any code modifications they make to OpenAFS.org via openafs-bugs@openafs.org. Contributions of documentation are highly desired. ## 6.4. OpenAFS for Windows Mailing Lists If you wish to participate in OpenAFS for Windows development please join the openafs-win32-devel@openafs.org mailing list. User questions should be sent to the openafs-info@openafs.org mailing list. You must join the mailing lists if you wish to post to the list without incurring a moderation delay. # 7. MSI Deployment Guide ## 7.1. Introduction A MSI installer option is available for those who wish to use Windows Installer for installing OpenAFS and for organizations that wish to deploy OpenAFS through Group Policy. The first version of OpenAFS for Windows available as an MSI was 1.3.65. This document provides a guide for authoring transforms used to customize the MSI package for a particular organization. Although many settings can be deployed via transforms, in an Active Directory environment it is advisable to deploy registry settings and configuration files through group policy and/or startup scripts so that machines where OpenAFS for Windows is already installed will pick up these customizations. ### 7.1.1 Requirements The information in this document applies to MSI packages distributed with OpenAFS for Windows releases from 1.3.65 and onwards or MSI packages built from corresponding source releases. Not all releases support all the configuration options documented here. Authoring a "Windows Installer" transform requires additional software for editing the MSI database tables and generating the transform from the modified MSI package. ORCA.EXE and MSITRAN.EXE which are included in the Windows Platform SDK ("Windows Installer" SDK) can be used for this purpose. For reference, the schema for the MSI package is based on SCHEMA.MSI distributed with the Platform SDK. For general information about "Windows Installer", refer to: http://msdn.microsoft.com/library/en-us/msi/setup/windows_installer_start_page.asp For general information about authoring MSI transforms, refer to: http://msdn.microsoft.com/library/en-us/msi/setup/transforms.asp The remainder of this document assumes some familiarity with authoring transforms. While the MSDN documentation for Windows Installer is a bit dense, the guide on MSI transforms found at the second link above is recommended reading. MSDN also includes a step-by-step example for creating a transform at: http://msdn.microsoft.com/library/en-us/msi/setup/a_customization_transform_example.asp ### 7.1.2 Authoring a Transform Transforms describe a set of modifications to be performed on an existing MSI for the purpose of customizing it. This is ordinarily done by making a copy of the MSI to be customized, modifying the copy and then using the old and the new MSI to generate a transform. For example: 1. copy openafs.msi openafs-modified.msi 2. (edit the openafs-modified.msi to include the necessary changes) 3. msitran -g openafs.msi openafs-modified.msi openafs-transform.mst 4. (generates openafs-transform.mst, which is the transform) Transforms have an extension of .mst. 'msitran' is a tool distributed as part of the "Windows Installer" SDK (part of the Windows Platform SDK). You can test a transform by: 1. copy openafs.msi openafs-test.msi 2. msitran -a openafs-transform.mst openafs-test.msi and then checking the resulting openafs-test.msi to see if all changes you have made above to openafs-modified.msi is present in openafs-test.msi. 'msitran' will complain if some modification in the transform can not be successfully applied. As mentioned above, you can use a tool like ORCA.EXE to edit the MSI databases directly when editing openafs-modified.msi. More details are given below. ## 7.2. Configuration Options The logic necessary to implement many of the settings described in Appendix A are present in the MSI. Most of these can be controlled by setting the corresponding properties to the desired value. Some settings may require modifying existing registry entries (though not recommended) or adding new resources (like files or registry keys). Instructions for performing these tasks are below. ### 7.2.1 Configurable Properties Most configurable properties correspond to registry keys or values. Due to the logic invoked based on the existence of these registry keys or values, they are only set if the associated property is defined to have a non null value. If the associated property is not defined in the MSI, the registry key or value will not be touched. By default, the MSI does not contain these properties and hence will not set the registry keys. You will need to add properties as needed to the MSI. When one of the configurable properties is set, the installer will use the property value to set the corresponding setting in the HKEY_LOCAL_MACHINE registry hive. The HKEY_CURRENT_USER hive is not touched by the installer. For each property, the associated registry setting is referenced by the same text used in Appendix A. Strings are quoted using single quotes (e.g. 'a string'). An empty string is denoted as ''. Note that you can't author null values into the 'Property' table. Numeric values should be authored as decimal strings. #### 7.2.1.1 Setting Properties In order to set a property, 1. Open the MSI in ORCA.EXE 2. Select the 'Property' table from the list of tables on the left. 3. Find the property in the list of properties on the right, double click the value and type the new value. 4. If the property does not exist in the property list, right click the list and select 'Add Row', type the property name and the desired value. #### 7.2.1.2 OpenAFS for Windows Properties ##### (Service parameters): [HKLM\SYSTEM\CurrentControlSet\Services\TransarcAFSDaemon\Parameters] ##### (Network provider): [HKLM\SYSTEM\CurrentControlSet\Services\TransarcAFSDaemon\NetworkProvider] ##### (OpenAFS Client): [HKLM\SOFTWARE\OpenAFS\Client] ##### 7.2.1.2.1 Registry Properties These properties are used to set the values of registry entries associated with OpenAFS for Windows. ###### AFSCACHEPATH Registry key : (Service parameters) Registry value : CachePath Valid values : string . ###### AFSCACHESIZE Registry key : (Service parameters) Registry value : CacheSize Valid values : numeric ###### AFSCELLNAME Registry key : (Service parameters) Registry value : Cell Valid values : string ###### FREELANCEMODE Registry key : (Service parameters) Registry value : FreelanceClient Valid values : '1' or '0' ###### HIDEDOTFILES Registry key : (Service parameters) Registry value : HideDotFiles Valid values : '1' or '0' ###### LOGONOPTIONS Registry key : (Network provider) Registry value : LogonOptions Valid values : '0', '1' or '3' ###### MOUNTROOT Registry key : (Service parameters) Registry value : Mountroot Valid values : string ###### NETBIOSNAME Registry key : (Service parameters) Registry value : NetbiosName Valid values : string (at most 15 characters) ###### NOFINDLANABYNAME Registry key : (Service parameters) Registry value : NoFindLanaByName Valid values : '1' or '0' ###### RXMAXMTU Registry key : (Service parameters) Registry value : RxMaxMTU Valid values : numeric ###### SECURITYLEVEL Registry key : (Service parameters) Registry value : SecurityLevel Valid values : '1' or '0' ###### SMBAUTHTYPE Registry key : (Service parameters) Registry value : SMBAuthType Valid values : '0','1' or '2' ###### STOREANSIFILENAMES Registry key : (OpenAFS Client) Registry value : StoreAnsiFilenames Valid values : '0' or '1' ###### USEDNS Registry key : (Service parameters) Registry value : UseDNS Valid values : '1' or '0' ##### 7.2.1.2.2 AFSCreds.exe Properties These properties are combined to add a command line option to the shortcut that will be created in the Start:Programs:OpenAFS and Start:Programs:Startup folders (see CREDSSTARTUP). The method of specifying the option was chosen for easy integration with the Windows Installer user interface. Although other methods can be used to specify options to AFSCREDS.EXE, it is advised that they be avoided as transforms including such options may not apply to future releases of OpenAFS. ###### CREDSSTARTUP Valid values : '1' or '0' Controls whether AFSCreds.exe starts up automatically when the user logs on. When CREDSSTARTUP is '1' a shortcut is added to the 'Startup' folder in the 'Program menu' which starts AFSCREDS.EXE with the options that are determined by the other CREDS* properties. ###### CREDSAUTOINIT Valid values : '-a' or '' Enables automatic initialization. ###### CREDSIPCHDET Valid values : '-n' or '' ###### CREDSQUIET Valid values : '-q' or '' Enables quiet mode. ###### CREDSRENEWDRMAP Valid values : '-m' or '’ Enables renewing drive map at startup. ###### CREDSSHOW Valid values : '-s' or '' Enables displaying the credential manager window when AFSCREDS starts up. ### 7.2.2 Existing Registry Entries You can change existing registry values subject to the restrictions mentioned in the Windows Platform SDK. Pay special attention to component key paths and try to only change the 'Value' column in the 'Registry' table. If you want to add additional registry keys please refer to section 3 (Additional resources). ### 7.2.3 Replacing Configuration Files The OpenAFS configuration files (CellServDB) can be replaced by your own configuration files. These files are contained in separate MSI components so that you can disable them individually. The recommended method for replacing these files is to first disable the components containing the configuration files that you want to replace, and then add new components for the replacement files. This is outlined below (assuming you are using ORCA.EXE to author the transform). Note that transforms are not a good way to add a new file as an embedded stream. The method outlined here places the file in the same directory as the MSI for deployment. The walkthrough below is to add a custom 'CellServDB' file. 1. Disable the component that contains the configuration file that you want to replace. 1.1. Locate and select the 'Component' table in the 'Tables' list. 1.2. In the Component table, locate the component you need to change ( Ctrl-F invokes the 'Find' dialog). The component names are listed below in section 7.2.3.1. For this example, the component name is 'elf_CellServDB'. 1.3. Go to the 'Condition' column of the component. 1.4. Enter a condition that evaluates to false. I.e. 'DONOTINSTALL'. (Note that an undefined property always evaluates to false). Note that you can also use this step to disable other configuration files without providing replacements. 2. Add a new component containing the new configuration file. 2.1. Select the 'Component' table in the 'Tables' list. 2.3. Enter the following : Component cmf_my_CellServDB ComponentID {7019836F-BB2C-4AF6-9463-0D6EC9035CF1} Directory_ dirClient Attributes 144 Condition KeyPath fil_my_CellServDB Note that the ComponentId is an uppercase GUID. You can generate one using GUIDGEN.EXE or UUIDGEN.EXE, both of which are included in the Platform SDK. The Attributes value of 144 is a sum of msidbComponentAttributesPermanent (16) and msidbComponentAttributesNeverOverwrite (128). This ensures that local modifications are not overwritten or lost during an installation or uninstallation. These are the same settings used on the default configuration files. 'fil_my_CellServDB' is a key into the 'File' table which we will fill later. 3. Add a new feature to hold the new component. 3.1. Select the 'Feature' table. 3.2. Add a new row (Ctrl-R or 'Tables'->'Add Row') with the following values: Feature fea_my_CellServDB Feature_Parent feaClient Title Description Display 0 Level 30 Directory_ Attributes 8 It is important to create the new feature under the 'feaClient' feature, which will ensure that the configuration file will be installed when the client binaries are installed. Setting 'Display' to 0 will hide this feature from the feature selection dialog during an interactive installation. A value of 30 for 'Level' allows this feature to be installed by default (on a 'Typical' installation). The 'Attributes' value is msidbFeatureAttributesDisallowAdvertise (8), which is set on all features in the OpenAFS MSI. The OpenAFS MSI is not designed for an advertised installation. 4. Join the component and the feature. 4.1. Select the 'FeatureComponents' table. 4.2. Add a new row with the following values: Feature fea_my_CellServDB Component cmf_my_CellServDB 5. Add an entry to the 'File' table. 5.1. Select the 'File' table. 5.2. Add a new row with the following values: File fil_my_CellServDB Component_ cmf_my_CellServDB FileName CellServDB FileSize (enter file size here) Attributes 8192 Sequence 1000 (leave other fields blank) The 'Attributes' value is msidbFileAttributesNonCompressed (8192). This is because we will be placing this file in the same directory as the MSI instead of embedding the file in it. Transforms do not support updating compressed sources or adding new cabinet streams. Finally, the 'Sequence' value of 1000 will be used later to distinguish the file as being in a separate source location than the other files in the MSI. 6. Set a media source for the file. 6.1. Select the 'Media' table. 6.2. Add a row with the following values : DiskId 2 LastSequence 1000 (leave other fields blank) The sequence number of 1000 designates this as the media source for the newly added file. #### 7.2.3.1 Components for Configuration Files CellServDB: 'cpf_CellServDB' (ID {D5BA4C15-DBEC-4292-91FC-B54C30F24F2A}) ### 7.2.4 Adding Domain Specific Registry Keys Following is an example for adding domain specific registry keys. Columns that are unspecified should be left empty. We create a new feature and component to hold the new registry keys. 'Feature' table: (new row) Feature : 'feaDomainKeys' Feature Parent : 'feaClient' Display : 0 Level : 30 Attributes : 10 'Component' table: (new row) Component : 'rcm_DomainKeys' ComponentId : '{4E3FCBF4-8BE7-40B2-A108-C47CF743C627}' Directory : 'TARGETDIR' Attributes : 4 KeyPath : 'reg_domkey0' 'FeatureComponents' table: (new row) Feature : 'feaDomainKeys' Component : 'rcm_DomainKeys' 'Registry' table: (new row) Registry : 'reg_domkey0' Root : 2 Key : 'SYSTEM\CurrentControlSet\Services\TransarcAFSDaemon\NetworkProvider\Domain' Component : 'rcm_DomainKeys' (new row) Registry : 'reg_domkey1' Root : 2 Key : 'SYSTEM\CurrentControlSet\Services\TransarcAFSDaemon\NetworkProvider\Domain' Name : '' Component : 'rcm_DomainKeys' (new row) Registry : 'reg_domkey2' Root : 2 Key : 'SYSTEM\CurrentControlSet\Services\TransarcAFSDaemon\NetworkProvider\Domain\ATHENA.MIT.EDU' Name : '' Component : 'rcm_DomainKeys' (new row) Registry : 'reg_domkey3' Root : 2 Key : 'SYSTEM\CurrentControlSet\Services\TransarcAFSDaemon\NetworkProvider\Domain\ATHENA.MIT.EDU' Name : 'LogonOptions' Value : 1 Component : 'rcm_DomainKeys' (new row) Registry : 'reg_domkey4' Root : 2 Key : SYSTEM\CurrentControlSet\Services\TransarcAFSDaemon\NetworkProvider\Domain\LOCALHOST' Name : '*' Component : 'rcm_DomainKeys' (new row) Registry : 'reg_domkey5' Root : 2 Key : 'SYSTEM\CurrentControlSet\Services\TransarcAFSDaemon\NetworkProvider\Domain\LOCALHOST' Name : 'LogonOptions' Value : 0 Component : 'rcm_DomainKeys' (new row) Registry : 'reg_domkey6' Root : 2 Key : 'SYSTEM\CurrentControlSet\Services\TransarcAFSDaemon\NetworkProvider\Domain\LOCALHOST' Name : 'FailLoginsSilently' Value : 1 Component : 'rcm_DomainKeys' The example adds domain specific keys for 'ATHENA.MIT.EDU' (enable integrated logon) and 'LOCALHOST' (disable integrated logon and fail logins silently). ### 7.2.5 Adding Site Specific Freelance Registry Keys Following is an example for adding site specific Freelance registry keys to pre-populate the Mountpoints and Symlinks in the fake root.afs volume. Columns that are unspecified should be left empty. We create a new feature and component to hold the new registry keys. 'Feature' table: (new row) Feature : 'feaFreelanceKeys' Feature Parent : 'feaClient' Display : 0 Level : 30 Attributes : 10 'Component' table: (new row) Component : 'rcm_FreelanceKeys' ComponentId : '{4E3B3CBF4-9AE7-40C3-7B09-C48CF842C583}' Directory : 'TARGETDIR' Attributes : 4 KeyPath : 'reg_freekey0' 'FeatureComponents' table: (new row) Feature : 'feaFreelanceKeys' Component : 'rcm_FreelanceKeys' 'Registry' table: (new row) Registry : 'reg_freekey0' Root : 2 Key : 'SOFTWARE\OpenAFS\Client\Freelance' Component : 'rcm_FreelanceKeys' (new row) Registry : 'reg_freekey1' Root : 2 Key : 'SOFTWARE\OpenAFS\Client\Freelance' Name : '0' Value : 'athena.mit.edu#athena.mit.edu:root.cell.' Component : 'rcm_FreelanceKeys' (new row) Registry : 'reg_freekey2' Root : 2 Key : 'SOFTWARE\OpenAFS\Client\Freelance' Name : '1' Value : '.athena.mit.edu%athena.mit.edu:root.cell.' Component : 'rcm_FreelanceKeys' (new row) Registry : 'reg_freekey3' Root : 2 Key : 'SOFTWARE\OpenAFS\Client\Freelance\Symlinks' Component : 'rcm_FreelanceKeys' (new row) Registry : 'reg_freekey4' Root : 2 Key : 'SOFTWARE\OpenAFS\Client\Freelance\Symlinks' Name : '0' Value : 'athena:athena.mit.edu.' Component : 'rcm_FreelanceKeys' (new row) Registry : 'reg_freekey5' Root : 2 Key : 'SOFTWARE\OpenAFS\Client\Freelance\Symlinks' Name : '1' Value : '.athena:.athena.mit.edu.' Component : 'rcm_FreelanceKeys' The example adds a read-only mountpoint to the athena.mit.edu cell's root.afs volume as well as a read-write mountpoint. Aliases are also provided using symlinks. If you want to add registry keys or files you need to create new components and features for those. Refer to the Windows Platform SDK for details. It is beyond the scope of this document to provide a comprehensive overview of how to add new resources through a transform. Please refer to the "Windows Installer" documentation for details. The relevant section is at : A sample walkthrough of adding a new configuration file is in section 2.3. Add new features under the 'feaClient' or 'feaServer' as appropriate and set the 'Level' column for those features to equal the 'Level' for their parent features for consistency. Note that none of the features in the OpenAFS for Windows MSI package are designed to be installed to run from 'source' or 'advertised'. It is recommended that you set 'msidbFeatureAttributesFavorLocal' (0), 'msidbFeatureAttributesFollowParent' (2) and 'msidbFeatureAttributesDisallowAdvertise' (8) attributes for new features. If you are creating new components, retain the same component GUID when creating new transforms against new releases of the OpenAFS MSI package. After making the adjustments to the MSI database using ORCA.EXE you can generate a transform with MSITRAN.EXE as follows : (Modified MSI package is 'openafs-en_US_new.msi' and the original MSI package is 'openafs-en_US.msi'. Generates transform 'openafs-transform.mst') > msitran.exe -g openafs-en_US.msi openafs-en_US_new.msi openafs-transform.mst See the Platform SDK documentation for information on command line options for MSITRAN.EXE. The MSI package is designed to uninstall previous versions of OpenAFS for Windows during installation. Note that it doesn't directly upgrade an existing installation. This is intentional and ensures that development releases which do not have strictly increasing version numbers are properly upgraded. Versions of OpenAFS that are upgraded by the MSI package are: 1) OpenAFS MSI package Up to current release 2) MIT's Transarc AFS MSI package Up to version 3.6.2 3) OpenAFS NSIS package All versions Note that versions of the OpenAFS NSIS package prior to 1.3.65 had a bug where it couldn't be uninstalled properly in unattended mode. Therefore the MSI package will not try to uninstall an OpenAFS NSIS package if running unattended. This means that group policy based deployments will fail on machines that have the OpenAFS NSIS package installed. If you have used a different MSI package to install OpenAFS and wish to upgrade it you can author rows into the 'Upgrade' table as described in the Platform SDK. When performing an upgrade with msiexec.exe execute the MSI with the repair options "vomus". # Appendix A: Registry Values ## A.1. Service parameters The service parameters primarily affect the behavior of the AFS client service (afsd_service.exe). ### Regkey: [HKLM\SYSTEM\CurrentControlSet\Services\TransarcAFSDaemon\Parameters] Type: DWORD Default: -1 LAN adapter number to use. This is the lana number of the LAN adapter that the SMB server should bind to. If unspecified or set to -1, a LAN adapter with named 'AFS' or a loopback adapter will be selected. If neither are present, then all available adapters will be bound to. When binding to a non-loopback adapter, the NetBIOS name hostname%-AFS' will be used (where %hostname% is the NetBIOS name of the host truncated to 11 characters). Otherwise, the NetBIOS name will be 'AFS'. ##### Value: CacheSize Type: DWORD Default: 98304 (CM_CONFIGDEFAULT_CACHESIZE) Variable: cm_initParams.cacheSize Size of the AFS cache in 1k blocks. ##### Value: ChunkSize Type: DWORD Default: 20 (CM_CONFIGDEFAULT_CHUNKSIZE) Variable: cm_logChunkSize (cm_chunkSize = 1 << cm_logChunkSize) Size of chunk for reading and writing. Actual chunk size is 2^cm_logChunkSize. The default chunk size is therefore 1 MB. ##### Value: Daemons Type: DWORD Default: 4 (CM_CONFIGDEFAULT_DAEMONS) Variable: numBkgD Number of background daemons (number of threads of cm_BkgDaemon). (see cm_BkgDaemon in cm_daemon.c) Type: DWORD Number of SMB server threads (number of threads of smb_Server). (see smb_Server in smb.c). ##### Value: Stats Type: DWORD Default: 10000 (CM_CONFIGDEFAULT_STATS) Variable: cm_initParams.nStatCaches Cache configuration. ##### Value: Volumes Type: DWORD Default: 3333 (CM_CONFIGDEFAULT_STATS/3) Variable: cm_initParams.nVolumes ##### Value: Cells Type: DWORD Default: 1024 (CM_CONFIGDEFAULT_CELLS) Variable: cm_initParams.nCells ##### Value: LogoffPreserveTokens Type: DWORD {1,0} Default : 0 If enabled (set to 1), the Logoff Event handler will not attempt to delete the user's tokens if the user's profile is stored outside of AFS. ##### Value: RootVolume Type: REG_SZ Default: "root.afs" Variable: cm_rootVolumeName Root volume name. ##### Value: MountRoot Type: REG_SZ Default: "/afs" Variable: cm_mountRoot Name of root mount point. In symlinks, if a path starts with cm_mountRoot, it is assumed that the path is absolute (as opposed to relative) and is adjusted accordingly. Eg: if a path is specified as /afs/athena.mit.edu/foo/bar/baz and cm_mountRoot is "/afs", then the path is interpreted as \\afs\all\athena.mit.edu\foo\bar\baz. If a path does not start with with cm_mountRoot, the path is assumed to be relative and suffixed to the reference directory (i.e. directory where the symlink exists) ##### Value: CachePath Type: REG_SZ or REG_EXPAND_SZ Default: "%TEMP%\AFSCache" Variable: cm_CachePath Location of on-disk cache file. The default is the SYSTEM account's TEMP directory. The attributes assigned to the file are HIDDEN and SYSTEM. ##### Value: NonPersistentCaching Type: DWORD [0..1] Default: 0 Variable: buf_CacheType When this registry value is set to a non-zero value, the CachePath value is ignored and the cache data is stored in the windows paging file. This disables the use of persistent caching and the ability to maintain a single UUID for the AFS client service across restarts. ##### Value: ValidateCache Type: DWORD [0..2] Default: 1 Variable: buf_CacheType This value determines if and when persistent cache validation is performed. 0 - Validation is disabled 1 - Validation is performed at startup 2 - Validation is performed at shutdown ##### Value: TrapOnPanic Type: DWORD {1,0} Default: 0 Variable: traceOnPanic Issues a breakpoint in the event of a panic. (breakpoint: _asm int 3). ##### Value: NetbiosName Type: REG_EXPAND_SZ Default: "AFS" Variable: cm_NetbiosName Specifies the NetBIOS name to be used when binding to a Loopback adapter. To provide the old behavior specify a value of "%COMPUTERNAME%-AFS". ##### Value: IsGateway Type: DWORD {1,0} Default: 0 Variable: isGateway Select whether or not this AFS client should act as a gateway. If set and the NetBIOS name hostname-AFS is bound to a physical NIC, other machines in the subnet can access AFS via SMB connections to hostname-AFS. When IsGateway is non-zero, the LAN adapter detection code will avoid binding to a loopback adapter. This will ensure that the NetBIOS name will be of the form hostname-AFS instead of the value set by the "NetbiosName" registry value. ##### Value: ReportSessionStartups Type: DWORD {1,0} Default: 0 Variable: reportSessionStartups If enabled, all SMB sessions created are recorded in the Application event log. This also enables other events such as drive mappings or various error types to be logged. ##### Value: TraceBufferSize Type: DWORD Default: 10000 (CM_CONFIGDEFAULT_TRACEBUFSIZE) Variable: traceBufSize Number of entries to keep in trace log. ##### Value: SysName Type: REG_SZ Default: "x86_win32 i386_w2k i386_nt40" (X86) “amd64_win64 x86_win32 i386_w2k” (AMD64) Variable: cm_sysName Provides an initial value for "fs sysname". The string can contain one or more replacement values for @sys in order of preference separated by whitespace. ##### Value: SecurityLevel Type: DWORD {1,0} Default: 0 Variable: cryptall Enables encryption on RX calls. ##### Value: UseDNS Type: DWORD {1,0} Default: 1 Variable: cm_dnsEnabled Enables resolving volservers using AFSDB DNS queries. As of 1.3.60, this value is ignored as the DNS query support utilizes the Win32 DNSQuery API which is available on Win2000 and above. ##### Value: FreelanceClient Type: DWORD {1,0} Default: 0 Variable: cm_freelanceEnabled Enables freelance client. ##### Value: HideDotFiles Type: DWORD {1,0} Default: 1 Variable: smb_hideDotFiles Enables marking dotfiles with the hidden attribute. Dot files are files whose name starts with a period (excluding "." and ".."). ##### Value: MaxMpxRequests Type: DWORD Default: 50 Variable: smb_maxMpxRequests Maximum number of multiplexed SMB requests that can be made. ##### Value: MaxVCPerServer Type: DWORD Default: 100 Variable: smb_maxVCPerServer Maximum number of SMB virtual circuits. ##### Value: Cell Type: REG_SZ Default: <none> Variable: rootCellName Name of root cell (the cell from which root.afs should be mounted in \\afs\all). ##### Value: RxEnablePeerStats Type: DWORD {0, 1} Default: 1 Variable: rx_enable_peer_stats When set to 1, the Rx library collects peer statistics. ##### Value: RxEnableProcessStats Type: DWORD {0, 1} Default: 1 Variable: rx_extra_process_stats When set to 1, the Rx library collects process statistics. ##### Value: RxExtraPackets Type: DWORD Default: 120 Variable: rx_extraPackets When set, this number of extra Rx packets are allocated at startup. ##### Value: RxMaxMTU Type: DWORD Default: 0 Variable: rx_mtu If set to anything other than 0, that value is used as the maximum send and receive MTU supported by the RX interface. In order to enable OpenAFS to operate across releases of the Cisco IPSec VPN client prior than 5.0, this value must be set to 1264 or smaller. ##### Value: RxNoJumbo Type: DWORD {0,1} Default: 0 Variable: rx_nojumbo If enabled, does not send or indicate that we are able to send or receive RX jumbograms. Type: DWORD Default: 60 (seconds) The Connection Dead Time is enforced to be at a minimum 15 seconds longer than the minimum SMB timeout as specified by [HKLM\SYSTEM\CurrentControlSet\Services\lanmanworkstation\parameters] SessTimeout If the minimum SMB timeout is not specified the value is 45 seconds. See http://support.microsoft.com:80/support/kb/articles/Q102/0/67.asp Type: DWORD Default: 120 (seconds) The Hard Dead Time is enforced to be at least double the ConnDeadTimeout. The provides an opportunity for at least one retry. ##### Value: TraceOption Type: DWORD {0-15} Default: 0 Enables logging of debug output to the Windows Event Log. Bit 0 enables logging of "Logon Events" processed by the Network Provider and Winlogon Event Notification Handler. Bit 1 enables logging of events captured by the AFS Client Service. Bit 2 enables real-time viewing of "fs trace" logging with DbgView or similar tools. Bit 3 enables "fs trace" logging on startup. ##### Value: AllSubmount Type: DWORD {0, 1} Default: 1 Variable: allSubmount (smb.c) By setting this value to 0, the "\\NetbiosName\all" mount point will not be created. This allows the read-write versions of root.afs to be hidden. ##### Value: NoFindLanaByName Type: DWORD {0, 1} Default: 0 Disables the attempt to identity the network adapter to use by looking for an adapter with a display name of "AFS". ##### Value: MaxCPUs Type: DWORD {1..32} or {1..64} depending on the architecture Default: <no default> If this value is specified, afsd_service.exe will restrict itself to executing on the specified number of CPUs if there are a greater number installed in the machine. ##### Value: smbAuthType Type: DWORD {0..2} Default: 2 If this value is specified, it defines the type of SMB authentication which must be present in order for the Windows SMB client to connect to the AFS Client Service's SMB server. The values are: 0 = No authentication required 1 = NTLM authentication required 2 = Extended (GSS SPNEGO) authentication required The default is Extended authentication ##### Value: MaxLogSize Type: DWORD {0 .. MAXDWORD} Default: 100K This entry determines the maximum size of the %WINDIR%\TEMP\afsd_init.log file. If the file is larger than this value when afsd_service.exe starts the file will be reset to 0 bytes. If this value is 0, it means the file should be allowed to grow indefinitely. ##### Value: FlushOnHibernate Type: DWORD {0,1} Default: 1 If set, flushes all volumes before the machine goes on hibernate or stand-by. ##### Value: daemonCheckDownInterval Type: DWORD (seconds) Default: 180 This value controls how frequently the AFS cache manager probes servers that are marked as “down”. ##### Value: daemonCheckUpInterval Type: DWORD (seconds) Default: 600 This value controls how frequently the AFS cache manager probes servers that are marked as “up”. ##### Value: daemonCheckVolInterval Type: DWORD (seconds) Default: 3600 This value controls how frequently the AFS cache manager forces a reset on the existing volume database information. ##### Value: daemonCheckCBInterval Type: DWORD (seconds) Default: 60 This value controls how frequently the AFS cache manager checks for callback invalidation. ##### Value: daemonCheckLockInterval Type: DWORD (seconds) Default: 60 This value controls how frequently the AFS cache manager checks for invalid file locks. ##### Value: daemonCheckTokenInterval Type: DWORD (seconds) Default: 180 This value controls how frequently the AFS cache manager checks for expired tokens. ##### Value: daemonCheckOfflineVolInterval Type: DWORD (seconds) Default: 600 This value controls how frequently the AFS cache manager checks offline volumes to see if they have come back online. At the same time volumes which were determined to be busy have their state reset to online. ##### Value: CallBackPort Type: DWORD Default: 7001 This value specifies which port number should be used for receiving callbacks from the file server. The standard AFS Callback port is 7001. Alternative values can be useful if the client is behind a NAT and a permanent port mapping for the client is being configured. ##### Value: EnableServerLocks Type: DWORD {0, 1, 2} Default: 1 Determines whether or not the AFS file server is contacted for 0: never obtain server locks 1: obtain server locks unless the file server says not to 2: always obtain server locks Type: DWORD {0, 1} Default: 0 Determines whether or not the AFS Cache Manager will permit files marked with the “Read Only” DOS attribute to be deleted or not. For compatibility with Explorer, the default is ‘no’. 0: do not permit “Read Only” files to be deleted. 1: delete files that have the “Read Only” attribute set without complaint. ##### Value: BPlusTrees Type: DWORD {0, 1} Default: 1 Determines whether or not the AFS Cache Manager uses locally constructed B+ Trees to speed up the performance of directory searches. 0: do not use B+ Trees for directory lookups 1: use B+ Trees for directory lookups ##### Value: PrefetchExecutableExtensions Type: MULTI_SZ Default: none specified The AFS Cache Manager will pre-fetch the entire contents of any file whose name matches ends with one of the specified extensions. This option is intended for use primarily with executables and dynamic link libraries that should be fully cached prior to a machine losing its connection with the file server. Type: DWORD {0, 1} Default: 0 Determines whether or not cached data from .readonly volumes is considered valid even if a callback cannot be registered with a file server. This option is meant to be used by organizations for whom .readonly volume content very rarely changes (if ever.) 0: do not treat offline .readonly content as valid 1: treat offline .readonly content as valid ##### Value: GiveUpAllCallBacks Type: DWORD {0, 1} Default: 0 Determines whether or not the AFS Cache Manager will give up all callbacks prior to the service being suspended or shutdown. Doing so will have significant performance benefits for the file servers. However, file servers older than 1.4.6 can become unstable if the GiveUpAllCallBacks RPC is executed. 0: do not perform GiveUpAllCallBacks RPCs 1: perform GiveUpAllCallBacks RPCs ##### Value: FollowBackupPath Type: DWORD {0, 1} Default: 0 Determines whether or not the AFS Cache Manager will give preference to .backup volumes when following mount points that originate in a .backup volume. 0: do not prefer .backup volumes when the mount point originates in a .backup volume. 1: prefer .backup volumes when the mount point originates in a .backup volume. ### Regkey: [HKLM\SYSTEM\CurrentControlSet\Services\TransarcAFSDaemon\Parameters\GlobalAutoMapper] ##### Value: <Drive Letter:> for example "G:" Type: REG_SZ Specifies the submount name to be mapped by afsd_service.exe at startup to the provided drive letter. This option is deprecated. ### Regkey: [HKLM\SOFTWARE\OpenAFS\Client] ##### Value: CellServDBDir Type: REG_SZ Default: <not defined> Specifies the directory containing the CellServDB file. When this value is not specified, the ProgramData directory is searched and if the CellServDB file is not found, the AFS Client install directory is used. ##### Value: VerifyServiceSignature Type: REG_DWORD Default: 0x1 This value can be used to disable the runtime verification of the digital signatures applied to afsd_service.exe and the OpenAFS DLLs it loads. This test is performed to verify that the DLLs which are loaded by afsd_service.exe are from the same distribution as afsd_service.exe. This is to prevent random errors caused when DLLs from one distribution of AFS are loaded by another one. This is not a security test. The reason for disabling this test is to free up additional memory which can be used for a large cache size. ##### Value: IoctlDebug Type: REG_DWORD Default: 0x0 This value can be used to debug the cause of pioctl() failures. Set a non-zero value and the pioctl() library will output status information to stdout. Executing command line tools such as tokens.exe, fs.exe, etc can then be used to determine why the pioctl() call is failing. ##### Value: MiniDumpType Type: REG_DWORD Default: 0x0 (MiniDumpNormal) This value is used to specify the type of minidump generated by afsd_service.exe either when the process crashes or when a user initiated is dump file is generated with the "fs.exe minidump" command. Valid values are dependent on the version of DbgHelp.dll installed on the machine. The best version to use is not the version that comes with the operating system but the version that is included in the most recent release of "Microsoft Debugging Tools for Windows". See the Microsoft Developer Library for further information. MiniDumpNormal = 0x00000000, MiniDumpWithDataSegs = 0x00000001, MiniDumpWithFullMemory = 0x00000002, MiniDumpWithHandleData = 0x00000004, MiniDumpFilterMemory = 0x00000008, MiniDumpScanMemory = 0x00000010, MiniDumpWithIndirectlyReferencedMemory = 0x00000040, MiniDumpFilterModulePaths = 0x00000080, MiniDumpWithoutOptionalData = 0x00000400, MiniDumpWithFullMemoryInfo = 0x00000800, MiniDumpWithCodeSegs = 0x00002000 ##### Value: EnableSMBAsyncStore Type: REG_DWORD Default: 0x1 This value can be used to disable the use of SMB Asynchronous Store operations. ##### Value: SMBAsyncStoreSize Type: REG_DWORD Default: 32 This value determines the size of SMB Asynchronous Store operations. This value can be used to increase the write performance on higher speed networks by increasing the value. The value must be a multiple of the cache buffer block size and cannot be larger than the cache manager chunk size. The specified value will be adjusted to enforce its compliance with these restrictions. ##### Value: StoreAnsiFilenames Type: REG_DWORD Default: 0x0 This value can be used to force the AFS Client Service to store filenames using the Windows system's ANSI character set instead of the OEM Code Page character set which has traditionally been used by SMB file systems. ### Regkey: [HKLM\SOFTWARE\OpenAFS\Client\CSCPolicy] ##### Value: "smb/cifs share name" Type: REG_SZ Default: <none> This key is used to map SMB/CIFS shares to Client Side Caching (off-line access) policies. For each share one of the following policies may be used: "manual", "programs", "documents", "disable". These values used to be stored in afsdsbmt.ini ### Regkey: [HKLM\SOFTWARE\OpenAFS\Client\Freelance] ##### Value: "numeric value" Type: REG_SZ Default: <none> This key is used to store dot terminated mount point strings for use in constructing the fake root.afs volume when Freelance (dynamic roots) mode is activated. "athena.mit.edu#athena.mit.edu:root.cell." ".athena.mit.edu%athena.mit.edu:root.cell." These values used to be stored in afs_freelance.ini ##### Value: "numeric value" Type: REG_SZ Default: <none> This key is used to store a dot terminated symlink strings for use in constructing the fake root.afs volume when Freelance (dynamic roots) mode is activated. "athena:athena.mit.edu." "home:athena.mit.edu\user\j\a\jaltman." "filename:path\file." ### Regkey: [HKLM\SOFTWARE\OpenAFS\Client\Realms] The Realms key is used to provide initialization data to be used when new identities are added to the Network Identity Manager. The AFS Provider will search for a subkey that matches the realm of the identity. If such a key exists, its values will be used to populate the AFS configuration for the identity. ### Regkey: [HKLM\SOFTWARE\OpenAFS\Client\Realms\”Realm Name”] In addition to the optional values, this key contains one subkey for each cell that is to be added to the AFS Provider configuration. ##### Value: AFSEnabled Type: REG_DWORD Default: 0x01 This key is used to specify whether the new identity should be configured to obtain AFS credentials. In general, it is only specified when disabling the acquisition of AFS credentials is desired. The default is to obtain AFS credentials. ### Regkey: [HKLM\SOFTWARE\OpenAFS\Client\Realms\”Realm Name”\”Cell Name”] ##### Value: MethodName Type: REG_SZ Default: <none> This key is used to specify the token acquisition method to be used. When unspecified, the AFS provider will automatically try Kerberos v5 and then Kerberos v5 (if available). As of this writing valid method names include “Auto”, “Kerberos5”, “Kerberos524”, “Kerberos4”. Note: Kerberos524 and Kerberos4 cannot be used with 64-bit Kerberos for Windows. ##### Value: Realm Type: REG_SZ Default: <none> This key is used to specify the realm to be used when acquiring AFS tokens. If not specified, the realm will be determined by performing a domain to realm mapping on the domain of a random volume location database server for the cell. ### Regkey: [HKLM\SOFTWARE\OpenAFS\Client\Submounts] ##### Value: "submount name" Type: REG_EXPAND_SZ Default: <none> This key is used to store mappings of UNIX style AFS paths to submount names which can be referenced as UNC paths. For example the submount string “/athena.mit.edu/user/j/a/jaltman" can be associated with the submount name "jaltman.home". This can then be referenced as the UNC path \\AFS\jaltman.home. These values used to be stored in afsdsbmt.ini NOTE: Submounts should no longer be used with OpenAFS. Use the Windows Explorer to create drive mappings to AFS UNC paths instead of using the AFS Submount mechanism. ### Regkey: [HKLM\SOFTWARE\OpenAFS\Client\Server Preferences\VLDB] ##### Value: "hostname or ip address" Type: REG_DWORD Default: <none> This key is used to specify a default set of VLDB server preferences. For each entry the value name will be either the IP address of a server or a fully qualified domain name. The value will be the ranking. The ranking will be adjusted by a random value between 0 and 256 prior to the preference being set. ### Regkey: [HKLM\SOFTWARE\OpenAFS\Client\Server Preferences\File] ##### Value: "hostname or ip address" Type: REG_DWORD Default: <none> This key is used to specify a default set of File server preferences. For each entry the value name will be either the IP address of a server or a fully qualified domain name. The value will be the ranking. The ranking will be adjusted by a random value between 0 and 256 prior to the preference being set. ## A.2. Integrated Logon Network provider parameters Affects the network provider (afslogon.dll). ### Regkey: [HKLM\SYSTEM\CurrentControlSet\Services\TransarcAFSDaemon\Parameters] Type: DWORD Default: 0 Do not display message boxes if the login fails. ### Regkey: [HKLM\SYSTEM\CurrentControlSet\Services\TransarcAFSDaemon\NetworkProvider] ##### Value: NoWarnings Type: DWORD Default: 0 Disables visible warnings during logon. ##### Value: AuthentProviderPath Type: REG_SZ NSIS: %WINDIR%\SYSTEM32\afslogon.dll Specifies the install location of the authentication provider dll. ##### Value: Class Type: DWORD NSIS: 0x02 Specifies the class of network provider ##### Value: DependOnGroup Type: REG_MULTI_SZ NSIS: PNP_TDI Specifies the service groups upon which the AFS Client Service depends. Windows should not attempt to start the AFS Client Service until all of the services within these groups have successfully started. ##### Value: DependOnService Type: REG_MULTI_SZ NSIS: Tcpip NETBIOS RpcSs Specifies a list of services upon which the AFS Client Service depends. Windows should not attempt to start the AFS Client Service until all of the specified services have successfully started. ##### Value: Name Type: REG_SZ NSIS: "OpenAFSDaemon" Specifies the display name of the AFS Client Service ##### Value: ProviderPath Type: REG_SZ NSIS: %WINDIR%\SYSTEM32\afslogon.dll Specifies the DLL to use for the network provider ## A.2.1 Domain specific configuration keys for the Network Provider The network provider can be configured to have different behavior depending on the domain that the user logs into. These settings are only relevant when using integrated login. A domain refers to an Active Directory (AD) domain, a trusted Kerberos (non-AD) realm or the local machine (i.e. local account logins). The domain name that is used for selecting the domain would be the domain that is passed into the NPLogonNotify function of the network provider. Domain specific registry keys are: (NP key) (Domains key) ### [HKLM\SYSTEM\CurrentControlSet\Services\TransarcAFSDaemon\NetworkProvider\Domain\"domain name"] (Specific domain key. One per domain.) ### [HKLM\SYSTEM\CurrentControlSet\Services\TransarcAFSDaemon\NetworkProvider\Domain\LOCALHOST] (Localhost key) ### Example: HKLM\SYSTEM\CurrentControlSet\Services\TransarcAFSDaemon\NetworkProvider \| +- Domain +-LOCALHOST Each of the domain specific keys can have the set of values described in 2.1.1. The effective values are chosen as described in 2.1.2. ### A.2.1.1 Domain specific configuration values #### [HKLM\SYSTEM\CurrentControlSet\Services\TransarcAFSDaemon\NetworkProvider] [HKLM\SYSTEM\CurrentControlSet\Services\TransarcAFSDaemon\NetworkProvider\Domain] [HKLM\SYSTEM\CurrentControlSet\Services\TransarcAFSDaemon\NetworkProvider\Domain\"domain name"] [HKLM\SYSTEM\CurrentControlSet\Services\TransarcAFSDaemon\NetworkProvider\Domain\LOCALHOST] ##### Value: LogonOptions Type: DWORD Default: 0x01 NSIS/WiX: depends on user configuration 0x00 - Integrated Logon is not used 0x01 - Integrated Logon is used 0x02 - High Security Mode is used (deprecated) 0x03 - Integrated Logon with High Security Mode is used (deprecated) High Security Mode generates random SMB names for the creation of Drive Mappings. This mode should not be used without Integrated Logon. As of 1.3.65 the SMB server supports SMB authentication. The High Security Mode should not be used when using SMB authentication (SMBAuthType setting is non zero). Type: DWORD (1\|0) Default: 0 NSIS/WiX: (not set) If true, does not display any visible warnings in the event of an error during the integrated login process. ##### Value: LogonScript Type: REG_SZ or REG_EXPAND_SZ Default: (null) NSIS/WiX: (only value under NP key) <install path>\afscreds.exe -:%s -x -a -m -n -q A logon script that will be scheduled to be run after the profile load is complete. If using the REG_EXPAND_SZ type, you can use any system environment variable as "%varname%" which would be expanded at the time the network provider is run. Optionally using a "%s" in the value would result in it being expanded into the AFS SMB username for the session. Type: DWORD Default: 30 NSIS/WiX: (not set) If the OpenAFS client service has not started yet, the network provider will wait for a maximum of "LoginRetryInterval" seconds while retrying every "LoginSleepInterval" seconds to check if the service is up. Type: DWORD Default: 5 NSIS/WiX: (not set) ##### Value: Realm Type: REG_SZ NSIS: <not set> When Kerberos v5 is being used, Realm specifies the Kerberos v5 realm that should be appended to the first component of the Domain logon username to construct the Kerberos v5 principal for which AFS tokens should be obtained. ##### Value: TheseCells Type: REG_MULTI_SZ NSIS: <not set> When Kerberos v5 is being used, TheseCells provides a list of additional cells for which tokens should be obtained with the default Kerberos v5 principal. ### A.2.1.2 Selection of effective values for domain specific configuration During login to domain X, where X is the domain passed into NPLogonNotify as lpAuthentInfo->LogonDomainName or the string 'LOCALHOST' if lpAuthentInfo->LogonDomainName equals the name of the computer, the following keys will be looked up. 1. NP key. ("HKLM\SYSTEM\CurrentControlSet\Services\TransarcAFSDaemon\NetworkProvider") 2. Domains key. (NP key\"Domain") 3. Specific domain key. (Domains key\X) If the specific domain key does not exist, then the domains key will be ignored. All the configuration information in this case will come from the NP key. If the specific domain key exists, then for each of the values metioned in (2), they will be looked up in the specific domain key, domains key and the NP key successively until the value is found. The first instance of the value found this way will be the effective for the login session. If no such instance can be found, the default will be used. To re-iterate, a value in a more specific key supercedes a value in a less specific key. The exceptions to this rule are stated below. ### A.2.1.3 Exceptions to A.2.1.2 To retain backwards compatibility, the following exceptions are made to 2.1.2. Historically, the 'FailLoginsSilently' value was in HKLM\SYSTEM\CurrentControlSet\Services\TransarcAFSDaemon\Parameters key and not in the NP key. Therefore, for backwards compatibility, the value in the Parameters key will supercede all instances of this value in other keys. In the absence of this value in the Parameters key, normal scope rules apply. #### 2.1.3.2 'LogonScript' If a 'LogonScript' is not specified in the specific domain key nor in the domains key, the value in the NP key will only be checked if the effective 'LogonOptions' specify a high security integrated login. If a logon script is specified in the specific domain key or the domains key, it will be used regardless of the high security setting. Please be aware of this when setting this value. ## A.3. AFS Credentials System Tray Tool parameters Affects the behavior of afscreds.exe ### Regkey: [HKLM\SYSTEM\CurrentControlSet\Services\TransarcAFSDaemon\Parameters] ##### Value: Gateway Type: REG_SZ Default: "" Function: GetGatewayName() If the AFS client is utilizing a gateway to obtain AFS access, the name of the gateway is specified by this value. ##### Value: Cell Type: REG_SZ Default: <none> Variable: IsServiceConfigured() The value Cell is used to determine if the AFS Client Service has been properly configured or not. ### Regkey: [HKLM\SOFTWARE\OpenAFS\Client] [HKCU\SOFTWARE\OpenAFS\Client] ##### Value: ShowTrayIcon Type: DWORD {0, 1} Default: 1 Function: InitApp(), Main_OnCheckTerminate() This value is used to determine whether or not a shortcut should be maintained in the user's Start Menu->Programs->Startup folder. This value used to be stored at [HKLM\Software\TransarcCorporation\AFS Client\AfsCreds]. The current user value is checked first; if it does not exist the local machine value is checked. ##### Value: EnableKFW Type: DWORD {0, 1} Default: 1 Function: KFW_is_available() When MIT Kerberos for Windows can be loaded, Kerberos v5 will be used to obtain AFS credentials. By setting this value to 0, the internal Kerberos v4 implementation will be used instead. The current user value is checked first; if it does not exist the local machine value is checked. ##### Value: AcceptDottedPrincipalNames Type: DWORD {0, 1} Default: 1 Kerberos v5 principal names are traditionally mapped to Kerberos v4 names by the AFS servers before they can be looked up in the Protection database. The mapping algorithm used permits collisions to occur. Both of the Kerberos v5 names, "user.admin@REALM" and "user/admin@REALM" are interpreted as the same user identity within the cell. To enable both names to be sent to the server by AFSCreds or Integrated Logon, set this value to 1. ##### Value: Use524 Type: DWORD {0, 1} Default: 0 Function: KFW_use_krb524() When MIT Kerberos for Windows can be loaded, Kerberos v5 will be used to obtain AFS credentials. By setting this value to 1, the Kerberos v5 tickets will be converted to Kerberos v4 tokens via a call to the krb524 daemon. The current user value is checked first; if it does not exist the local machine value is checked. ##### Value: AfscredsShortcutParams Type: REG_SZ Default: "-A -M -N -Q" Function: Shortcut_FixStartup This value specifies the command line options which should be set as part of the shortcut to afscreds.exe. afscreds.exe rewrites the shortcut each time it exits so as to ensure that the shortcut points to the latest version of the program. This value is used to determine which values should be used for command line parameters. The current user value is checked first; if it does not exist the local machine value is checked. The following subset of the command line options is appropriate for use in this registry setting: -A = autoinit -M = renew drive maps -N = ip address change detection -Q = quiet mode. do not display start service dialog if afsd_service is not already running -S = show tokens dialog on startup -Z = unmap drives ### Regkey: [HKCU\SOFTWARE\OpenAFS\Client] ##### Value: Authentication Cell Type: REG_SZ Default: <none> Function: Afscreds.exe GetDefaultCell() This value allows the user to configure a different cell name to be used as the default cell when acquiring tokens in afscreds.exe. ### Regkey: [HKCU\SOFTWARE\OpenAFS\Client\Reminders] ##### Value: "afs cell name" Type: DWORD {0, 1} Default: <none> These values are used to save and restore the state of the reminder flag for each cell for which the user has obtained tokens. This value used to be stored at [HKLM\Software\TransarcCorporation\AFS Client\AfsCreds]. ### Regkey: [HKCU\SOFTWARE\OpenAFS\Client\Active Maps] ##### Value: "upper case drive letter" Type: DWORD {0, 1} Default: <none> These values are used to store the persistence state of the AFS drive mappings as listed in the [...\Client\Mappings] key. These values used to be stored in the afsdsbmt.ini file ### Regkey: [HKCU\SOFTWARE\OpenAFS\Client\Mappings] ##### Value: "upper case drive letter" Type: REG_SZ Default: <none> These values are used to store the AFS path in UNIX notation to which the drive letter is to be mapped. These values used to be stored in the afsdsbmt.ini file. ## A.4 OpenAFS Client Service Environment Variables ##### Variable: AFS_RPC_ENCRYPT Values: "OFF" disables the use of RPC encryption any other value allows RPC encryption to be used Default: RPC encryption is on ##### Variable: AFS_RPC_PROTSEQ Values: "ncalrpc" - local RPC "ncacn_np" - named pipes "ncacn_ip_tcp" - tcp/ip Default: local RPC	2014-07-24 10:39:24	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2416410744190216, "perplexity": 10767.919668233442}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1405997888283.14/warc/CC-MAIN-20140722025808-00067-ip-10-33-131-23.ec2.internal.warc.gz"}
https://archethought.github.io/blog/dataCleansing1/	# Overview A wise person once said, “Data completion is 90% of the work in a data pipeline.” (Technology consultant Dixon Dick) My experience thus far bears out this statement! This was my first foray into Spark, where my goal was to perform data investigation. Here I outline my path to get to the data. Summary of tools used: # Starting point I started with a 1.7 GB cvs (comma separated values) file of unknown content: 1715069652 Rosetta.csv Once I found the schema, I used cat to prepend headers: $cat headers.csv Rosetta.csv > RosettaWithHdrs.csv # Issue I invoked Spark with sudo /opt/spark-1.6.1-bin-hadoop2.6/bin/spark-shell --packages com.databricks:spark-csv_2.11:1.4.0 And ran code: val df = sqlContext.read.format("com.databricks.spark.csv").option("header","true").option("inferschema","true").load("RosettaWithHdrs.csv") Which failed with java.io.IOException: (startline 1) EOF reached before encapsulated token finished Yikes! Looks like the csv may be malformed. It’s a huge file! How to find the problem? # Break it down, and again I used sed to successively cut out lines of the file until I found a segment I could read, and thus identifying at least one faulty record. Example: sed -n 1,500000p RosettaWithHdrs.csv > 5-10e6.csv The first record to cause a problem was on line 1031. Turns out that some of the text fields spanned multiple lines. Each record has 24 fields, and the file is too big to view in an editor. It took me a while to realize the issue, and that Spark simply could not accomodate a CSV file of this complexity. ## Find a different format: JSON I found csvkit enormously helpful in exploring the file. This suite of tools can also convert csv to json with csvjson, so thought I’d see if Spark could handle inter-field carriage returns in JSON format. The ‘-z’ option specifies maximum field size. $ csvjson -z 10000000 RosettaWithHdrs.csv > all.json Yes! JSON format accomodates records with inter-field carriage returns. I originally tried sed to split the file, realizing I’d still need to look at the surrounding lines to ensure I captured complete records. I killed the sed process after an hour and looked for an alternative. Enter jq! This is how I determined the number of records in the dataset. By default, jq produces human readable output. The “-c” flag overrides the default, producing the compressed mess that Spark prefers. jq 'length' offRec.json 1 $jq 'length' all.json 171679$ jq -c '.[0:1000]' all.json > subset1000.json scala> val datafile="subset1000.json"	2018-04-22 16:13:14	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2838761806488037, "perplexity": 10325.559038855237}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125945624.76/warc/CC-MAIN-20180422154522-20180422174522-00625.warc.gz"}
http://mathhelpforum.com/calculus/67100-circular-problem-print.html	# circular problem • Jan 6th 2009, 04:40 PM jenny woodland circular problem I need help with this.... Can anyone help me??? Many thanks The figure shows a circular field with centre O, and radius 2 km. AC is the diameter of the circle, and B lies on the circumference. Jack cycles along the straight line from A to B at a speed of 12 km/hr. He then walks along the straight line from Bto C at a speed of 5 km/hr. Given that the distance from B to C is x km, and the total time taken is T hours, find the value of x so that Jack can complete his journey in the shortest time possible. • Jan 6th 2009, 05:03 PM skeeter a triangle inscribed in a semicircle is a right triangle. $AC = 4$ km $BC = x$ km $AB = \sqrt{16 - x^2}$ km distance/speed = time $T = \frac{\sqrt{16 - x^2}}{12} + \frac{x}{5}$ find $\frac{dT}{dx}$ and minimize ... don't forget about endpoint extrema. • Jan 6th 2009, 05:16 PM jenny woodland what is endpoint extrema? Hi skeeter, thanks for your answer. But erm can you teach me what is endpoint extrema? My teacher didn't teach me about it and this is the first time I come across this term. I know that I'm a bit of a noob but I hope you can teach me as I want to know more. And thanks again!!!! • Jan 6th 2009, 06:05 PM skeeter what is the domain of the function $T(x)$ ? have you found $\frac{dT}{dx}$ and completed an analysis of the function $T(x)$ ? • Jan 6th 2009, 06:08 PM jenny woodland Re: Circular problem I work out the problem. Can the answer be 3.69 km? • Jan 6th 2009, 06:13 PM skeeter is x = 3.69 the location of a minimum or a maximum? how do you tell? • Jan 6th 2009, 06:48 PM jenny woodland Domain of T= -1.538 to 1.538 dT/dx = -12x / [144 (√16-x)] + 1/5 and 3.69 is a maximum. Am I correct? If not, can you correct my mistakes? Thanks! • Jan 7th 2009, 02:18 PM skeeter the domain for x is 0 $\leq x \leq 4$ T(3.69) is a maximum ... the function T(x) increases and then decreases because T'(x) changes sign from (+) to (-). This means that the minimum for T(x) is either T(0) or T(4) ... endpoint minimums. the absolute min is the lesser of the two.	2016-10-22 08:10:37	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 9, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7757703065872192, "perplexity": 1315.7736973290398}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988718840.18/warc/CC-MAIN-20161020183838-00268-ip-10-171-6-4.ec2.internal.warc.gz"}
https://terrytao.wordpress.com/2009/01/30/254a-notes-8-a-quick-review-of-point-set-topology/	To progress further in our study of function spaces, we will need to develop the standard theory of metric spaces, and of the closely related theory of topological spaces (i.e. point-set topology). I will be assuming that students in my class will already have encountered these concepts in an undergraduate topology or real analysis course, but for sake of completeness I will briefly review the basics of both spaces here. – Metric spaces – In many spaces, one wants a notion of when two points in the space are “near” or “far”. A particularly quantitative and intuitive way to formalise this notion is via the concept of a metric space. Definition 1. (Metric spaces) A metric space $X = (X,d)$ is a set X, together with a distance function $d: X \times X \to {\Bbb R}^+$ which obeys the following properties: 1. (Non-degeneracy) For any $x, y \in X$, we have $d(x,y) \geq 0$, with equality if and only if x=y. 2. (Symmetry) For any $x,y \in X$, we have $d(x,y) = d(y,x)$. 3. (Triangle inequality) For any $x,y,z \in X$, we have $d(x,z) \leq d(x,y) + d(y,z)$. Example 1. Every normed vector space $(X, \\| \\|)$ is a metric space, with distance function $d(x,y) := \\|x-y\\|$. $\diamond$ Example 2. Any subset Y of a metric space $X = (X,d)$ is also a metric space $Y = (Y, d\downharpoonright_{Y \times Y})$, where $d\downharpoonright_{Y \times Y}: Y \times Y \to {\Bbb R}^+$ is the restriction of d to $Y \times Y$. We call the metric space $Y = (Y, d\downharpoonright_{Y \times Y})$ a subspace of the metric space $X = (X,d)$. $\diamond$ Example 3. Given two metric spaces $X = (X,d_X)$ and $Y = (Y,d_Y)$, we can define the product space $X \times Y = (X \times Y, d_X \times d_Y)$ to be the Cartesian product $X \times Y$ with the product metric $d_X \times d_Y( (x,y), (x',y') ) := \max( d_X( x,x'), d_Y(y,y' ))$. (1) (One can also pick slightly different metrics here, such as $d_X(x,x') + d_Y(y,y')$, but this metric only differs from (1) by a factor of two, and so they are equivalent (see Example 5 below). $\diamond$ Example 4. Any set X can be turned into a metric space by using the discrete metric $d: X \times X \to {\Bbb R}^+$, defined by setting $d(x,y) = 0$ when $x=y$ and $d(x,y)=1$ otherwise. $\diamond$ Given a metric space, one can then define various useful topological structures. There are two ways to do so. One is via the machinery of convergent sequences: Definition 2. (Topology of a metric space) Let $(X,d)$ be a metric space. 1. A sequence $x_n$ of points in X is said to converge to a limit $x \in X$ if one has $d(x_n,x) \to 0$ as $n \to \infty$. In this case, we say that $x_n \to x$ in the metric d as $n \to \infty$, and that $\lim_{n \to \infty} x_n = x$ in the metric space X. (It is easy to see that any sequence of points in a metric space has at most one limit.) 2. A point x is an adherent point of a set $E \subset X$ if it is the limit of some sequence in E. (This is slightly different from being a limit point of E, which is equivalent to being an adherent point of $E \backslash \{x\}$; every adherent point is either a limit point or an isolated point of E.) The set of all adherent points of E is called the closure $\overline{E}$ of X. A set E is closed if it contains all its adherent points, i.e. if $E = \overline{E}$. A set E is dense if every point in X is adherent to E, or equivalently if $\overline{E} = X$. 3. Given any x in X and $r > 0$, define the open ball $B(x,r)$ centred at x with radius r to be the set of all y in X such that $d(x,y) < r$. Given a set E, we say that x is an interior point of E if there is some open ball centred at x which is contained in E. The set of all interior points is called the interior $E^\circ$ of E. A set is open if every point is an interior point, i.e. if $E = E^\circ$. There is however an alternate approach to defining these concepts, which takes the concept of an open set as a primitive, rather than the distance function, and defines other terms in terms of open sets. For instance: Exercise 1. Let $(X,d)$ be a metric space. 1. Show that a sequence $x_n$ of points in X converges to a limit $x \in X$ if and only if every open neighbourhood of x (i.e. an open set containing x) contains $x_n$ for all sufficiently large n. 2. Show that a point x is an adherent point of a set E if and only if every open neighbourhood of x intersects E. 3. Show that a set E is closed if and only if its complement is open. 4. Show that the closure of a set E is the intersection of all the closed sets containing E. 5. Show that a set E is dense if and only if every non-empty open set intersects E. 6. Show that the interior of a set E is the union of all the open sets contained in E, and that x is an interior point of E if and only if some neighbourhood of x is contained in E. $\diamond$ In the next section we will adopt this “open sets first” perspective when defining topological spaces. On the other hand, there are some other properties of subsets of a metric space which require the metric structure more fully, and cannot be defined purely in terms of open sets (see Example 14) below (although some of these concepts can still be defined using a structure intermediate to metric spaces and topological spaces, namely a uniform space). For instance: Definition 3. Let (X,d) be a metric space. 1. A sequence $(x_n)_{n=1}^\infty$ of points in X is a Cauchy sequence if $d(x_n,x_m) \to 0$ as $n,m \to \infty$ (i.e. for every $\varepsilon > 0$ there exists $N > 0$ such that $d(x_n,x_m) \leq \varepsilon$ for all $n,m \geq N$). 2. A space X is complete if every Cauchy sequence is convergent. 3. A set E in X is bounded if it is contained inside a ball. 4. A set E is totally bounded in X if for every $\varepsilon > 0$, E can be covered by finitely many balls of radius $\varepsilon$. Exercise 2. Show that any metric space $X$ can be identified with a dense subspace of a complete metric space $\overline{X}$, known as a metric completion or Cauchy completion of X. (For instance, ${\Bbb R}$ is a metric completion of ${\Bbb Q}$.) (Hint: one can define a real number to be an equivalence class of Cauchy sequences of rationals. Once the reals are defined, essentially the same construction works in arbitrary metric spaces.) Furthermore, if $\overline{X}'$ is another metric completion of $X$, show that there exists an isometry between $\overline{X}$ and $\overline{X}'$ which is the identity on X. Thus, up to isometry, there is a unique metric completion to any metric space. $\diamond$ Exercise 3. Show that a metric space X is complete if and only if it is closed in every superspace Y of X (i.e. in every metric space Y for which X is a subspace). Thus one can think of completeness as being the property of being “absolutely closed”. $\diamond$ Exercise 4. Show that every totally bounded set is also bounded. Conversely, in a Euclidean space ${\Bbb R}^n$ with the usual metric, show that every bounded set is totally bounded. But give an example of a set in a metric space which is bounded but not totally bounded. (Hint: use Example 4.) $\diamond$ Now we come to an important concept. Theorem 1. (Heine-Borel theorem for metric spaces) Let $(X,d)$ be a metric space. Then the following are equivalent: 1. (Sequential compactness) Every sequence in X has a convergent subsequence. 2. (Compactness) Every open cover $(V_\alpha)_{\alpha \in A}$ of X (i.e. a collection of open sets $V_\alpha$ whose union contains X) has a finite subcover. 3. (Finite intersection property) If $(F_\alpha)_{\alpha \in A}$ is a collection of closed subsets of X such that any finite subcollection of sets has non-empty intersection, then the entire collection has non-empty intersection. 4. X is complete and totally bounded. Proof. (2 $\implies$ 1) If there was an infinite sequence $x_n$ with no convergent subsequence, then given any point x in X there must exist an open ball centred at x which contains $x_n$ for only finitely many n (since otherwise one could easily construct a subsequence of $x_n$ converging to x). By property 2, one can cover X with a finite number of such balls. But then the sequence $x_n$ would be finite, a contradiction. (1 $\implies$ 4) If X was not complete, then there would exist a Cauchy sequence which is not convergent; one easily shows that this sequence cannot have any convergent subsequences either, contradicting 1. If X was not totally bounded, then there exists $\varepsilon > 0$ such that X cannot be covered by any finite collection of balls of radius $\varepsilon$; a standard greedy algorithm argument then gives a sequence $x_n$ such that $d(x_n,x_m) \geq \varepsilon$ for all distinct n, m. This sequence clearly has no convergent subsequence, again a contradiction. (2 $\iff$ 3) This follows from de Morgan’s laws and Exercise 1.3. (4 $\implies$ 3) Let $(F_\alpha)_{\alpha \in A}$ be as in 3. Call a set E in X rich if it intersects all of the $F_\alpha$. Observe that if one could cover X by a finite number of non-rich sets, then (as each non-rich set is disjoint from at least one of the $F_\alpha$), there would be a finite number of $F_\alpha$ whose intersection is empty, a contradiction. Thus, whenever we cover X by finitely many sets, at least one of them must be rich. As X is totally bounded, for each $n \geq 1$ we can find a finite set $x_{n,1},\ldots,x_{n,m_n}$ such that the balls $B(x_{n,1},2^{-n}), \ldots, B(x_{n,m_n},2^{-n})$ cover X. By the previous discussion, we can then find $1 \leq i_n \leq m_n$ such that $B(x_{n,i_n}, 2^{-n})$ is rich. Call a ball $B(x_{n,i},2^{-n})$ asymptotically rich if it contains infinitely many of the $x_{j,i_j}$. As these balls cover X, we see that for each n, $B(x_{n,i},2^{-n})$ is asymptotically rich for at least one i. Furthermore, since each ball of radius $2^{-n}$ can be covered by balls of radius $2^{-n-1}$, we see that if $B(x_{n,j},2^{-n})$ is asymptotically rich, then it must intersect an asymptotically rich ball $B(x_{n+1,j'},2^{-n-1})$. Iterating this, we can find a sequence $B(x_{n,j_n},2^{-n})$ of asymptotically rich balls, each one of which intersects the next one. This implies that $x_{n,j_n}$ is a Cauchy sequence and hence (as X is assumed complete) converges to a limit x. Observe that there exist arbitrarily small rich balls that are arbitrarily close to x, and thus x is adherent to every $F_\alpha$; since the $F_\alpha$ are closed, we see that x lies in every $F_\alpha$, and we are done. $\Box$ Remark 1. The hard implication $4 \implies 3$ of the Heine-Borel theorem is noticeably more complicated than any of the others. This turns out to be unavoidable; the Heine-Borel theorem turns out to be logically equivalent to König’s lemma in the sense of reverse mathematics, and thus cannot be proven in sufficiently weak systems of logical reasoning. $\diamond$ Any space that obeys one of the four equivalent properties in Lemma 1 is called a compact space; a subset E of a metric space X is said to be compact if it is a compact space when viewed as a subspace of X. There are some variants of the notion of compactness which are also of importance for us: 1. A space is $\sigma$-compact if it can be expressed as the countable union of compact sets. (For instance, the real line ${\Bbb R}$ with the usual metric is $\sigma$-compact.) 2. A space is locally compact if every point is contained in the interior of a compact set. (For instance, ${\Bbb R}$ is locally compact.) 3. A subset of a space is precompact or relatively compact if it is contained inside a compact set (or equivalently, if its closure is compact). Another fundamental notion in the subject is that of a continuous map. Exercise 5. Let $f: X \to Y$ be a map from one metric space $(X, d_X)$ to another $(Y,d_Y)$. Then the following are equivalent: 1. (Metric continuity) For every $x \in X$ and $\varepsilon > 0$ there exists $\delta > 0$ such that $d_Y( f(x), f(x') ) \leq \varepsilon$ whenever $d_X(x,x') \leq \delta$. 2. (Sequential continuity) For every sequence $x_n \in X$ that converges to a limit $x \in X$, $f(x_n)$ converges to f(x). 3. (Topological continuity) The inverse image $f^{-1}(V)$ of every open set V in Y, is an open set in X. 4. The inverse image $f^{-1}(F)$ of every closed set F in Y, is a closed set in X. $\diamond$ A function f obeying any one of the properties in Exercise 5 is known as a continuous map. Exercise 6. Let $X, Y, Z$ be metric spaces, and let $f: X \to Y$ and $g: X \to Z$ be continuous maps. Show that the combined map $f \oplus g: X \to Y \times Z$ defined by $f\oplus g(x) := (f(x), g(x))$ is continuous if and only if f and g are continuous. Show also that the projection maps $\pi_Y: Y \times Z \to Y$, $\pi_Z: Y \times Z \to Z$ defined by $\pi_Y(y,z) := y$, $\pi_Z(y,z) := z$ are continuous. $\diamond$ Exercise 7. Show that the image of a compact set under a continuous map is again compact. $\diamond$ – Topological spaces – Metric spaces capture many of the notions of convergence and continuity that one commonly uses in real analysis, but there are several such notions (e.g. pointwise convergence, semi-continuity, or weak convergence) in the subject that turn out to not be modeled by metric spaces. A very useful framework to handle these more general modes of convergence and continuity is that of a topological space, which one can think of as an abstract generalisation of a metric space in which the metric and balls are forgotten, and the open sets become the central object. [There are even more abstract notions, such as pointless topological spaces, in which the collection of open sets has become an abstract lattice, in the spirit of Notes 4, but we will not need such notions in this course.] Definition 4. (Topological space) A topological space $X = (X, {\mathcal F})$ is a set X, together with a collection ${\mathcal F}$ of subsets of X, known as open sets, which obey the following axioms: 1. $\emptyset$ and X are open. 2. The intersection of any finite number of open sets is open. 3. The union of any arbitrary number of open sets is open. The collection ${\mathcal F}$ is called a topology on X. Given two topologies ${\mathcal F}, {\mathcal F}'$ on a space X, we say that ${\mathcal F}$ is a coarser (or weaker) topology than ${\mathcal F}'$ (or equivalently, that ${\mathcal F}'$ is a finer (or stronger) topology than ${\mathcal F}$), if ${\mathcal F} \subset {\mathcal F}'$ (informally, ${\mathcal F}'$ has more open sets than ${\mathcal F}$). Example 5. Every metric space $(X,d)$ generates a topology ${\mathcal F}_d$, namely the space of sets which are open with respect to the metric d. Observe that if two metrics d, d’ on X are equivalent in the sense that $c d(x,y) \leq d'(x,y) \leq C d(x,y)$(2) for all x, y in X and some constants $c, C > 0$, then they generate an identical topology. $\diamond$ Example 6. The finest (or strongest) topology on any set X is the discrete topology $2^X = \{ E: E \subset X\}$, in which every set is open; this is the topology generated by the discrete metric (Example 4). The coarsest (or weakest) topology is the trivial topology $\{ \emptyset, X\}$, in which only the empty set and the full set are open. $\diamond$ Example 7. Given any collection ${\mathcal A}$ of sets of X, we can define the topology ${\mathcal F}[{\mathcal A}]$ generated by ${\mathcal A}$ to be the intersection of all the topologies that contain ${\mathcal A}$; this is easily seen to be the coarsest topology that makes all the sets in ${\mathcal A}$ open. For instance, the topology generated by a metric space is the same as the topology generated by its open balls. $\diamond$ Example 8. If $(X,{\mathcal F})$ is a topological space, and Y is a subset of X, then we can define the relative topology ${\mathcal F}\downharpoonright_Y := \{ E \cap Y: E \in {\mathcal F} \}$ to be the collection of all open sets in X, restricted to Y, this makes $(Y, {\mathcal F}\downharpoonright_Y)$ a topological space, known as a subspace of $(X,{\mathcal F})$$\diamond$ Any notion in metric space theory which can be defined purely in terms of open sets, can now be defined for topological spaces. Thus for instance: Definition 5. Let $(X,{\mathcal F})$ be a topological space. 1. A sequence $x_n$ of points in X converges to a limit $x \in X$ if and only if every open neighbourhood of x (i.e. an open set containing x) contains $x_n$ for all sufficiently large n. In this case we write $x_n \to x$ in the topological space $(X,{\mathcal F})$, and (if x is unique) we write $x = \lim_{n \to \infty} x_n$. 2. A point is a sequentially adherent point of a set E if it is the limit of some sequence in E. 3. A point x is an adherent point of a set E if and only if every open neighbourhood of x intersects E. The set of all adherent points of E is called the closure of E and is denoted $\overline{E}$. 4. A set E is closed if and only if its complement is open, or equivalently if it contains all its adherent points. 5. A set E is dense if and only if every non-empty open set intersects E, or equivalently if its closure is X. 6. The interior of a set E is the union of all the open sets contained in E, and x is called an interior point of E if and only if some neighbourhood of x is contained in E. 7. A space X is sequentially compact if every sequence has a convergent subsequence. 8. A space X is compact if every open cover has a finite subcover. 9. The concepts of being $\sigma$-compact, locally compact, and precompact can be defined as before. (One could also define sequential $\sigma$-compactness, etc., but these notions are rarely used.) 10. A map $f: X \to Y$ between topological spaces is sequentially continuous if whenever $x_n$ converges to a limit x in X, $f(x_n)$ converges to a limit f(x) in Y. 11. A map $f: X \to Y$ between topological spaces is continuous if the inverse image of every open set is open. $\diamond$ Remark 2. The stronger a topology becomes, the more open and closed sets it will have, but fewer sequences will converge, there are fewer (sequentially) adherent points and (sequentially) compact sets, closures become smaller, and interiors become larger. There will be more (sequentially) continuous functions on this space, but fewer (sequentially) continuous functions into the space. Note also that the identity map from a space X with one topology ${\mathcal F}$ to the same space X with a different topology ${\mathcal F}'$ is continuous precisely when ${\mathcal F}$ is stronger than ${\mathcal F}'$$\diamond$ Example 9. In a metric space, these topological notions coincide with their metric counterparts, and sequential compactness and compactness are equivalent, as are sequential continuity and continuity. $\diamond$ Exercise 7′. (Urysohn’s subsequence principle) Let $x_n$ be a sequence in a topological space X, and let x be another point in X. Show that the following are equivalent: 1. $x_n$ converges to x. 2. Every subsequence of $x_n$ converges to x. 3. Every subsequence of $x_n$ has a further subsequence that converges to x. $\diamond$ Exercise 8. Show that every sequentially adherent point is an adherent point, every continuous function is sequentially continuous. $\diamond$ Remark 3. The converses to Exercise 8 are unfortunately not always true in general topological spaces. For instance, if we endow an uncountable set X with the cocountable topology (so that a set is open if it is either empty, or its complement is at most countable) then we see that the only convergent sequences are those which are eventually constant. Thus, every subset of X contains its sequentially adherent points, and every function from X to another topological space is sequentially continuous, even though not every set in X is closed and not every function on X is continuous. An example of a set which is sequentially compact but not compact is the first uncountable ordinal with the order topology (Exercise 9). It is more tricky to give an example of a compact space which is not sequentially compact; this will have to wait for future notes, when we establish Tychonoff’s theorem. However one can “fix” this discrepancy between the sequential and non-sequential concepts by replacing sequences with the more general notion of nets, see the appendix below. $\diamond$ Remark 4. Metric space concepts such as boundedness, completeness, Cauchy sequences, and uniform continuity do not have counterparts for general topological spaces, because they cannot be defined purely in terms of open sets. (They can however be extended to some other types of spaces, such as uniform spaces or coarse spaces.) $\diamond$ Now we give some important topologies that capture certain modes of convergence or continuity that are difficult or impossible to capture using metric spaces alone. Example 10. (Zariski topology) This topology is important in algebraic geometry, though it will not be used in this course. If F is an algebraically closed field, we define the Zariski topology on the vector space $F^n$ to be the topology generated by the complements of proper algebraic varieties in $F^n$; thus a set is Zariski open if it is either empty, or is the complement of a finite union of proper algebraic varieties. A set in $F^n$ is then Zariski dense if it is not contained in any proper subvariety, and the Zariski closure of a set is the smallest algebraic variety that contains that set. $\diamond$ Example 11. (Order topology) Any totally ordered set $(X,<)$ generates the order topology, defined as the topology generated by the sets $\{ x \in X: x > a \}$ and $\{ x \in X: x < a \}$ for all $a \in X$. In particular, the extended real line ${}[-\infty,+\infty]$ can be given the order topology, and the notion of convergence of sequences in this topology to either finite or infinite limits is identical to the notion one is accustomed to in undergraduate real analysis. (On the real line, of course, the order topology corresponds to the usual topology.) Also observe that a function $n \mapsto x_n$ from the extended natural numbers ${\Bbb N} \cup \{+\infty\}$ (with the order topology) into a topological space X is continuous if and only if $x_n \to x_{+\infty}$ as $n \to \infty$, so one can interpret convergence of sequences as a special case of continuity. $\diamond$ Exercise 9. Let $\omega$ be the first uncountable ordinal, endowed with the order topology. Show that $\omega$ is sequentially compact (Hint: every sequence has a lim sup), but not compact (Hint: every point has a countable neighbourhood). $\diamond$ Example 12. (Half-open topology) The right half-open topology ${\mathcal F}_r$ on the real line ${\Bbb R}$ is the topology generated by the right half-open intervals ${}[a,b)$ for $-\infty < a < b < \infty$; this is a bit finer than the usual topology on ${\Bbb R}$. Observe that a sequence $x_n$ converges to a limit x in the right half-open topology if and only if it converges in the ordinary topology ${\mathcal F}$, and also if $x_n \geq x$ for all sufficiently large x. Observe that a map $f: {\Bbb R} \to {\Bbb R}$ is right-continuous iff it is a continuous map from $({\Bbb R}, {\mathcal F}_r)$ to $({\Bbb R}, {\mathcal F})$. One can of course model left-continuity via a suitable left half-open topology in a similar fashion. $\diamond$ Example 13. (Upper topology) The upper topology ${\mathcal F}_u$ on the real line is defined as the topology generated by the sets $(a,+\infty)$ for all $a \in {\Bbb R}$. Observe that (somewhat confusingly), a function $f: {\Bbb R} \to {\Bbb R}$ is lower semi-continuous iff it is continuous from $({\Bbb R}, {\mathcal F})$ to $({\Bbb R}, {\mathcal F}_u)$. One can of course model upper semi-continuity via a suitable lower topology in a similar fashion. $\diamond$ Example 14. (Product topology) Let $Y^X$ be the space of all functions $f: X \to Y$ from a set X to a topological space Y. We define the product topology on $Y^X$ to be the topology generated by the sets $\{ f \in Y^X: f(x) \in V \}$ for all $x \in X$ and all open $V \subset Y$. Observe that a sequence of functions $f_n: X \to Y$ converges pointwise to a limit $f: X \to Y$ iff it converges in the product topology. We will study the product topology in more depth in future notes. $\diamond$ Example 15. (Product topology, again) If $(X,{\mathcal F}_X)$ and $(Y, {\mathcal F}_Y)$ are two topological spaces, we can define the product space $(X \times Y, {\mathcal F}_X \times {\mathcal F}_Y )$ to be the Cartesian product $X \times Y$ with the topology generated by the product sets $U \times V$, where U and V are open in X and Y respectively. Observe that two functions $f: Z \to X$, $g: Z \to Y$ from a topological space Z are continuous if and only if their direct sum $f: Z \to X \times Y$ is continuous in the product topology, and also that the projection maps $\pi_X: X \times Y \to X$ and $\pi_Y: X \times Y \to Y$ are continuous (cf. Exercise 6). $\diamond$ We mention that not every topological space can be generated from a metric (such topological spaces are called metrisable). One important obstruction to this arises from the Hausdorff property: Definition 6. A topological space X is said to be a Hausdorff space if for any two distinct points x, y in X, there exist disjoint neighbourhoods $V_x, V_y$ of x and y respectively. Example 16. Every metric space is Hausdorff (one can use the open balls $B( x, d(x,y)/2 )$ and $B( y, d(x,y)/2 )$ as the separating neighbourhoods. On the other hand, the trivial topology (Example 7) on two or more points is not Hausdorff, and neither is the cocountable topology (Remark 3) on an uncountable set, or the upper topology (Example 13) on the real line. Thus, these topologies do not arise from a metric. $\diamond$ Exercise 10. Show that the half-open topology (Example 12) is Hausdorff, but does not arise from a metric. [Hint: assume for contradiction that the half-open topology did arise from a metric; then show that for every real number x there exists a rational number q and a positive integer n such that the ball of radius 1/n centred at q has infimum x.] Thus there are more obstructions to metrisability than just the Hausdorff property; a more complete answer is provided by Urysohn’s metrisation theorem, which we will cover in later notes. $\diamond$ Exercise 11. Show that in a Hausdorff space, any sequence can have at most one limit. (For a more precise statement, see Exercise 15 below.) $\diamond$ A homeomorphism (or topological isomorphism) between two topological spaces is a continuous invertible map $f: X \to Y$ whose inverse $f^{-1}: Y \to X$ is also continuous. Such a map identifies the topology on X with the topology on Y, and so any topological concept of X will be preserved by f to the corresponding topological concept of Y. For instance, X is compact if and only if Y is compact, X is Hausdorff if and only if Y is Hausdorff, x is adherent to E if and only if f(x) is adherent to f(E), and so forth. When there is a homeomorphism between two topological spaces, we say that X and Y are homeomorphic (or topologically isomorphic). Example 14. The tangent function is a homeomorphism between $(-\pi/2,\pi/2)$ and ${\Bbb R}$ (with the usual topologies), and thus preserves all topological structures on these two spaces. Note however that the former space is bounded as a metric space while the latter is not, and the latter is complete while the former is not. Thus metric properties such as boundedness or completeness are not purely topological properties, since they are not preserved by homeomorphisms. $\diamond$ – Nets (optional) – A sequence $(x_n)_{n=1}^\infty$ in a space X can be viewed as a function from the natural numbers ${\Bbb N}$ to X. We can generalise this concept as follows. Definition 7. A net in a space X is a tuple $(x_\alpha)_{\alpha \in A}$, where $A = (A,<)$ is a directed set (i.e. a pre-ordered set such that any two elements have at least one upper bound), and $x_\alpha \in X$ for each $\alpha \in A$. We say that a statement $P(\alpha)$ holds for sufficiently large $\alpha$ in a directed set A if there exists $\beta \in A$ such that $P(\alpha)$ holds for all $\alpha \geq \beta$. [Note in particular that if $P(\alpha)$ and $Q(\alpha)$ separately hold for sufficiently large $\alpha$, then their conjunction $P(\alpha) \wedge Q(\alpha)$ also holds for sufficiently large $\alpha$.] A net $(x_\alpha)_{\alpha \in A}$ in a topological space X is said to converge to a limit $x \in X$ if for every neighbourhood V of x, we have $x_\alpha \in V$ for all sufficiently large $\alpha$. A subnet of a net $(x_\alpha)_{\alpha \in A}$ is a tuple of the form $(x_{\phi(\beta)})_{\beta \in B}$, where $(B, <)$ is another directed set, and $\phi: B \to A$ is a monotone map (thus $\phi(\beta') \geq \phi(\beta)$ whenever $\beta' \geq \beta$) which is also has cofinal image, which means that for any $\alpha \in A$ there exists $\beta \in B$ with $\phi(\beta) \geq \alpha$ (in particular, if $P(\alpha)$ is true for sufficiently large $\alpha$, then $P(\phi(\beta))$ is true for sufficiently large $\beta$). Remark 5. Every sequence is a net, but one can create nets that do not arise from sequences (in particular, one can take A to be uncountable). Note a subtlety in the definition of a subnet – we do not require $\phi$ to be injective, so B can in fact be larger than A! Thus subnets differ a little bit from subsequences in that they “allow repetitions”. $\diamond$ Remark 6. Given a directed set A, one can endow $A \cup \{+\infty\}$ with the topology generated by the singleton sets $\{\alpha\}$ with $\alpha \in A$, together with the sets ${}[\alpha,+\infty] := \{ \beta \in A \cup \{+\infty\}: \beta \geq \alpha \}$ for $\alpha \in A$, with the convention that $+\infty > \alpha$ for all $\alpha \in A$. The property of being directed is precisely saying that these sets form a base. A net $(x_{\alpha})_{\alpha \in A}$ converges to a limit $x_{+\infty}$ if and only if the function $\alpha \mapsto x_{\alpha}$ is continuous on $A \cup \{+\infty\}$ (cf. Example 11). Also, if $(x_{\phi(\beta)})_{\beta \in B}$ is a subnet of $(x_{\alpha})_{\alpha \in A}$, then $\phi$ is a continuous map from $B \cup \{+\infty\}$ to $A \cup \{+\infty\}$, if we adopt the convention that $\phi(+\infty) = +\infty$. In particular, a subnet of a convergent net remains convergent to the same limit. $\diamond$ The point of working with nets instead of sequences is that one no longer needs to worry about the distinction between sequential and non-sequential concepts in topology, as the following exercises show: Exercise 12. Let X be a topological space, let E be a subset of X, and let x be an element of X. Show that x is an adherent point of E if and only if there exists a net $(x_\alpha)_{\alpha \in A}$ in E that converges to x. (Hint: take A to be the directed set of neighbourhoods of x, ordered by reverse set inclusion.) $\diamond$ Exercise 13. Let $f: X \to Y$ be a map between two topological spaces. Show that f is continuous if and only if for every net $(x_\alpha)_{\alpha \in A}$ in X that converges to a limit x, the net $(f(x_\alpha))_{\alpha \in A}$ converges in Y to f(x). $\diamond$ Exercise 14. Let X be a topological space. Show that X is compact if and only if every net has a convergent subnet. (Hint: equate both properties of X with the finite intersection property, and review the proof of Theorem 1. Similarly, show that a subset E of X is relatively compact if and only if every net in E has a subnet that converges in X. (Note that as not every compact space is sequentially compact, this exercise shows that we cannot enforce injectivity of $\phi$ in the definition of a subnet.) $\diamond$ Exercise 15. Show that a space is Hausdorff if and only if every net has at most one limit. $\diamond$ Exercise 16. In the product space $Y^X$ in Example 14, show that a net $(f_\alpha)_{\alpha \in A}$ converges in $Y^X$ to $f \in Y^X$ if and only if for every $x \in X$, the net $(f_\alpha(x))_{\alpha \in A}$ converges in Y to $f(x)$. $\diamond$ [Update, Jan 31: Definition of subnet corrected; Exercise 8 corrected; Exercise 9 added, subsequent exercises renumbered; hint for Exercise 2 altered; some remarks added.]	2015-02-27 13:06:53	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 361, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.946100115776062, "perplexity": 145.75594468738336}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-11/segments/1424936461266.22/warc/CC-MAIN-20150226074101-00116-ip-10-28-5-156.ec2.internal.warc.gz"}
http://physics.stackexchange.com/tags/simulation/hot	# Tag Info 26 As we cannot resolve arbitrarily small time intervals, what is ''really'' the case cannot be decided. But in classical and quantum mechanics (i.e., in most of physics), time is treated as continuous. Physics would become very awkward if expressed in terms of a discrete time. Edit: If time appear discrete (or continuous) at some level, it could still be ... 17 There is a great paper from the group of Howard Stone on this subject: Wetting of flexible fibre arrays (freely available here, but for some reason I am not allowed to link to it normally: http://211.144.68.84:9998/91keshi/Public/File/34/482-7386/pdf/nature10779.pdf) They specifically study when 2 closely positioned parallel fibers (i.e. hairs) clump ... 12 I'm the author of the novel mentioned above and I'd like to clarify. Neither I nor my novel claim the universe is a computer. The operating principle of the novel's universe is that all things are One through balance. The protagonist, Paul, is asked if his universe is a computer. He answers that in a balanced quantum universe this would be impossible. Why? ... 11 I'll chip in here because I'm a research student and I work with a stellar evolution code (the Cambridge STARS code) more-or-less daily. Regarding some of the comments to the question, stellar evolution is actually quite fast, depending what code you use. Certainly, it isn't like hydrodynamics or N-body simulations like those used in galaxy ... 11 I'd say there's no conclusive evidence, but in quantum physics, Planck time is sometimes cited as a possible smallest unit of time. The source for my data is Quantum Gods: Creation, Chaos, and the Search for Cosmic Consciousness by Victor J. Stenger. In there, he goes into a lot of detail about this in one chapter. 11 The horizontal component of running is believed to be fairly negligible for humans. Some research suggests that the limit isn't strength related at all, but design --- in particular, based solely on power, humans could theoretically run up to almost 40 mph. The issue is two fold: first, our limbs are actually too heavy, for big strength (e.g. climbing in ... 10 I would suggest looking at the formalism of Floquet space. The basic idea is that one uses a time-independent but infinite dimensional Hamiltonian to simulate evolution under a time-dependent but finite dimensional Hamiltonian by using a new index to label terms in a Fourier series. A good, short introduction can be found in Levante et al. For more details, ... 9 It most certainly exist outside secret labs :) Like Gerben wrote, the fields are called molecular dynamics (MD) and quantum chemistry which, as computers grow faster, will be essential tools of nanotechnology and medicine. Molecular Dynamics is currently implemented by making certain approximations in that electron motion is not explicitely modelled. In ... 9 The two main free destop programs that I know about, Stellarium and Celestia, do not include the proper motion of the stars when they move forward and backward in time. At least according to documentation that I've seen. These programs claim to do it but I have no experience with them: Home Planet (free) Starry Night (commercial) The Sky (commercial) ... 9 Geant is a framework---which means that you use it to build applications that simulate the detector and physics you are interested in. The simulation can include all of physics and the complete detector including electronics and trigger (i.e. you can write your simulation so that it output a data file that looks just like the one you are going to get from ... 9 I think it's important to note that quantum or quantized time is not equal to discrete time. For instance, we have "quantized" space. By this we mean that it receives quantum treatment. But the underlying coordinates still form a continuum. So even if you live on a finite circle and only consider wavefunctions so that you get a countable set of basis ... 9 I've played the game, see my report: http://motls.blogspot.ca/2012/11/a-slower-speed-of-light-mit.html?m=1 and I join M. Buettner. I am confident that all relativistic effects are incorporated. It includes the length contraction in the direction of motion, time dilation, but those basic things are rapidly changed by the fact that it really shows what you ... 8 One has to realize that a Monte Carlo simulation is an integration tool. Suppose you have a curve in an xy plot, y=f(x). If you throw random (x,y) pairs in the square containing the f(x) and count the number where y is less than f(x) versus the number y larger than f(x) you get an estimate of the area under f(x), i.e. the integral of the function. In ... 8 First of all, I do not have any experience with this, I am an Astronomy hobbyist at best. So I am just going to present what I found with minimal comment at this time. I found this web page that links to several programs: http://nbody.sourceforge.net/ They link to the University of Washington and their n-body shop. I don't know what your status must be ... 7 The thing you are reading is NOT an article(that you should trust). Even if the universe were a simulation there is NO such proof in QM. Though physical processes can be thought of as computations it does not imply existence of "super computer". Also, even if such a "super computer" exists it must be a part of some universe. That universe must also be ... 7 Your method is known as the forward Euler method, the simplest DEQ-solver but a really bad one. I suppose it's the approximation everyone tries first, at least I did, too, when I started with such simulations... and soon was troubled by similar "numerical explosions", which can't really be avoided with this algorithm which is why it's virtually never used in ... 6 This is, no doubt, one of the biggest challenges for realistic simulations: waves crashing, hair moving under wind and whatever other movement involving turbulence will be hard to solve. Though it is true that one can solve the equations of motion for each individual particle in a 'molecular dynamics' fashion, that is just infeasible for a system that goes ... 6 There's a huge difference between the number of bits you can store in a given space and the number of bits you need to describe that space. Take a single atom of iron with its 26 electrons. For a complete description, you need the many-particle wavefunction $\psi(\vec{x}_1, \vec{x}_2, \vec{x}_3, \dots, \vec{x}_{26})$ (ignoring spin for the moment). Imagine ... 6 Monte Carlo is a particular numerical technique heavily used in Physics, mainly when one needs to "brute force" the calculation. Virtually, all areas of Physics can make use of simulations that include some sort of Monte Carlo code, from the designing of detectors in particle physics (think Fermilab and LHC: Geant4), passing by out-of-equilibrium ... 6 Monte Carlo are very important in almost any particle physics experiment and are used in a variety of ways including To prototype the experiment without spending many millions of dollars. They can be used to show that the proposed physics signal will be detectable among the many known physics effects to test variations on the proposed detector design ... 6 Is there a reason that you are interested in QED in particular? For processes within current experimental ranges, the standard perturbative treatment is incredibly accurate, and simulations are not really necessary. A sector of the standard model where simulations are incredibly important is in strong interactions (QCD) for which a perturbative treatment ... 6 Your problem is highly nontrivial. The theoretical tool to be used is the renormalization group, which extracts the relevant dynamics of the large scales of the system. But if we were able to use it "in a blind way", then we would have a technique to study the macroscopic dynamics of any microscopic system... and this would made a lot of my colleagues ... 6 What you are talking about is similar to the problem of quantum gravity. Since gravity is an effect of the curvature of spacetime, to have a quantum theory of it, you need to quantize the spacetime manifold. This is done with spin foams which are little units of volume in spacetime that have spins associated to them. They connect together like total ... 6 I think the problem could be to do with your distinction of positive and zero/neutral Wigner states. I believe that the best way to understand the statement in this paper is as follows: For simplicity consider a single qudit state $\rho$, then this state has a positive Wigner representation if $W_{\rho}(\boldsymbol{u})\ge0 ... 6 This probably isn't exactly what you're looking for, but if you're looking for the time-independent bound states of a system, the Fourier grid Hamiltonian method may be applicable. Here is an application of it to the following strange-looking potential well: Here are a few low-energy bound states: And here are some of the high-energy ... 5 Yes it is a bit like waveguide for electrodynamics. You've got absolutely the same equation. The difference is in boundary conditions -- instead of Dirichlet you've got to use Neumann boundary conditions. While of course it works only if you are able no neglect all the non-linear effects -- your waves are low enough. 5 Addressing just the physics part (go to stack overflow for the programming), and using the equation that you've been given: $$x(t) = A \cos \left( \omega t + \delta \right)$$ Let's look at the form of the solution. It is sinusoidal The curve will have a maximum value of$A$(because cosine has a maximum value of 1) When$\delta$is$0, \pm 2\pi, \pm ... 5 Coordinate invariance guarantees that the phase space $M$ can be endowed with a symplectic 2-form $\omega$ which locally is given by $\omega = dq^i \wedge dp_i$. This form is closed ($d\omega = 0$) and nondegenerate, i.e. $\omega^n$ is a volume form, where $2n$ is the dimension of $M$. The other conditions say that for any Hamiltonian function $H$, the ... 5 In order to know with certainty where every particle was going to be you would need to know its mass, velocity and state very accurately - as any rounding errors or uncertainty will be magnified over the timespans involved. Unfortunately it is impossible to know to 100% precision where anything is, so we are already at strike 1. If you could somehow hold ... Only top voted, non community-wiki answers of a minimum length are eligible	2014-04-17 04:18:44	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6186379194259644, "perplexity": 456.1877125210125}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00482-ip-10-147-4-33.ec2.internal.warc.gz"}
https://greprepclub.com/forum/p-is-the-center-of-the-circle-and-the-area-of-sector-pqrs-7005.html	It is currently 19 Dec 2018, 04:11 ### GMAT Club Daily Prep #### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email. Customized for You we will pick new questions that match your level based on your Timer History Track every week, we’ll send you an estimated GMAT score based on your performance Practice Pays we will pick new questions that match your level based on your Timer History # P is the center of the circle, and the area of sector PQRS Author Message TAGS: Moderator Joined: 18 Apr 2015 Posts: 5205 Followers: 77 Kudos [?]: 1053 [0], given: 4711 P is the center of the circle, and the area of sector PQRS [#permalink] 27 Sep 2017, 13:46 Expert's post 00:00 Question Stats: 66% (00:54) correct 33% (00:28) wrong based on 6 sessions Attachment: circle.jpg [ 8.5 KiB \| Viewed 713 times ] P is the center of the circle, and the area of sector PQRS is 4. Quantity A Quantity B The area of circle p 4π [Reveal] Spoiler: OA _________________ Director Joined: 20 Apr 2016 Posts: 762 Followers: 6 Kudos [?]: 522 [0], given: 94 Re: P is the center of the circle, and the area of sector PQRS [#permalink] 27 Sep 2017, 22:39 Carcass wrote: Attachment: circle.jpg P is the center of the circle, and the area of sector PQRS is 4. Quantity A Quantity B The area of circle p 4π We know O/360 = $$\frac{Sector Area}{Circle Area}$$ or$$\frac{90}{360}$$ = $$\frac{4}{Circle Area}$$ (Since O = 90 given) or Circle Area= 16 Therefore QTY A> QTY B _________________ If you found this post useful, please let me know by pressing the Kudos Button Re: P is the center of the circle, and the area of sector PQRS [#permalink] 27 Sep 2017, 22:39 Display posts from previous: Sort by	2018-12-19 12:11:49	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5452043414115906, "perplexity": 3987.567133797126}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376832259.90/warc/CC-MAIN-20181219110427-20181219132427-00529.warc.gz"}
https://physics.stackexchange.com/questions/235958/the-separation-between-the-two-plates-of-a-capacitor-is-increased	# The separation between the two plates of a capacitor is increased I had the following question on a quiz recently: A capacitor consisting of two parallel plates, separated by a distance $d$, is connected to a battery of EMF ε. What happens if the separation is doubled while the battery remains connected? The correct answer was "The electric charge on the plates is halved." However, we were previously given a question in which a circuit consists of a capacitor of 6 μF and a resistor of 300,000 Ω connected to a battery, and the separation between the plates of the capacitor is quadrupled. However, in this circumstance, the charge on the capacitor immediately after the plates are separated remains the same, and the potential difference across the capacitor increases. I am confused as to why these questions have different answers to them, despite having an extremely similar setup. My best guess is that the circuit in the second question also has a resistor, but I am not sure, and help would be greatly appreciated. the charge on the capacitor immediately after the plates are separated remains the same The key word here is immediately. Since there is a $300\mathrm{k\Omega}$ series resistor between the capacitor and the battery, the series current is limited and thus, some time must elapse for the system to reach steady state. Since the separation between the plates is quadrupled, the capacitance is reduced by a factor of 4 so, eventually, the capacitor discharges through the resistor to a charge 1/4 of the initial charge. But in the first case, the system (effectively) reaches steady state immediately since there is no series resistance. This is best seen by adding a series resistance and taking the limit as the resistance goes to zero. You will then see that, in the limit, there is an infinitesimally brief, infinitely large discharge current that immediately brings the system into steady state. Of course, for a physical battery, there is non-zero internal resistance and, further, the plate separation of a physical capacitor cannot be change instantaneously. The new question asks about the change in capacitance, which is a property of the geometric arrangement of the plates and the material enclosed between them. You will have a formula with capacitance proportional to the area divided by the distance between the parallel plates: divide by 2d instead of d and your answer is halved. In the earlier case the question concerns the charge on the plates for a specific circuit, and an instantaneous (very rapid) change in the geometry. Since DC current doesn't flow through a capacitor, the charge is not subject to rapid changes; and the resistor adds to the "inertia" of the system. In fact, you have what is called an RC circuit, which has a decay time.	2021-10-16 08:13:46	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7518284320831299, "perplexity": 241.8362302602573}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323584554.98/warc/CC-MAIN-20211016074500-20211016104500-00220.warc.gz"}
https://www.quizover.com/course/section/multiplication-in-terms-of-of-by-openstax	# 8.3 Decimals: multiplication of decimals (Page 2/2) Page 2 / 2 ## Practice set b Use a calculator to find each product. If the calculator will not provide the exact product, round the result to four decimal places. $5\text{.}\text{126}\cdot \text{4}\text{.}\text{08}$ 20.91408 $0\text{.}\text{00165}\cdot \text{0}\text{.}\text{04}$ 0.000066 $0\text{.}\text{5598}\cdot \text{0}\text{.}\text{4281}$ 0.2397 $0\text{.}\text{000002}\cdot \text{0}\text{.}\text{06}$ 0.0000 ## Multiplying decimals by powers of 10 There is an interesting feature of multiplying decimals by powers of 10. Consider the following multiplications. Multiplication Number of Zeros in the Power of 10 Number of Positions the Decimal Point Has Been Moved to the Right $\text{10}\cdot 8\text{.}\text{315274}=\text{83}\text{.}\text{15274}$ 1 1 $\text{100}\cdot 8\text{.}\text{315274}=\text{831}\text{.}\text{5274}$ 2 2 $1,\text{000}\cdot 8\text{.}\text{315274}=8,\text{315}\text{.}\text{274}$ 3 3 $\text{10},\text{000}\cdot 8\text{.}\text{315274}=\text{83},\text{152}\text{.}\text{74}$ 4 4 ## Multiplying a decimal by a power of 10 To multiply a decimal by a power of 10, move the decimal place to the right of its current position as many places as there are zeros in the power of 10. Add zeros if necessary. ## Sample set c Find the following products. $\text{100}\cdot \text{34}\text{.}\text{876}$ . Since there are 2 zeros in 100, Move the decimal point in 34.876 two places to the right. $1,\text{000}\cdot 4\text{.}\text{8058}$ . Since there are 3 zeros in 1,000, move the decimal point in 4.8058 three places to the right. $\text{10},\text{000}\cdot \text{56}\text{.}\text{82}$ . Since there are 4 zeros in 10,000, move the decimal point in 56.82 four places to the right. We will have to add two zeros in order to obtain the four places. Since there is no fractional part, we can drop the decimal point. ## Practice set c Find the following products. $\text{100}\cdot \text{4}\text{.}\text{27}$ 427 $\text{10,000}\cdot \text{16}\text{.}\text{52187}$ 165,218.7 $\left(\text{10}\right)\left(0\text{.}\text{0188}\right)$ 0.188 $\left(\text{10,000,000,000}\right)\left(\text{52}\text{.}7\right)$ 527,000,000,000 ## Multiplication in terms of “of” Recalling that the word "of" translates to the arithmetic operation of multiplication, let's observe the following multiplications. ## Sample set d Find 4.1 of 3.8. Translating "of" to "×", we get Thus, 4.1 of 3.8 is 15.58. Find 0.95 of the sum of 2.6 and 0.8. We first find the sum of 2.6 and 0.8. $\begin{array}{c}\hfill 2.6\\ \hfill \underline{+0.8}\\ \hfill 3.4\end{array}$ Now find 0.95 of 3.4 Thus, 0.95 of $\left(2\text{.}\text{6}+\text{0}\text{.}8\right)$ is 3.230. ## Practice set d Find 2.8 of 6.4. 17.92 Find 0.1 of 1.3. 0.13 Find 1.01 of 3.6. 3.636 Find 0.004 of 0.0009. 0.0000036 Find 0.83 of 12. 9.96 Find 1.1 of the sum of 8.6 and 4.2. 14.08 ## Exercises For the following 30 problems, find each product and check each result with a calculator. $3\text{.}4\cdot 9\text{.}2$ 31.28 $4\text{.}5\cdot 6\text{.}1$ $8\text{.}0\cdot 5\text{.}9$ 47.20 $6\text{.}1\cdot 7$ $\left(0\text{.}1\right)\left(1\text{.}\text{52}\right)$ 0.152 $\left(1\text{.}\text{99}\right)\left(0\text{.}\text{05}\right)$ $\left(\text{12}\text{.}\text{52}\right)\left(0\text{.}\text{37}\right)$ 4.6324 $\left(5\text{.}\text{116}\right)\left(1\text{.}\text{21}\right)$ $\left(\text{31}\text{.}\text{82}\right)\left(0\text{.}1\right)$ 3.182 $\left(\text{16}\text{.}\text{527}\right)\left(9\text{.}\text{16}\right)$ $0\text{.}\text{0021}\cdot 0\text{.}\text{013}$ 0.0000273 $1\text{.}\text{0037}\cdot 1\text{.}\text{00037}$ $\left(1\text{.}6\right)\left(1\text{.}6\right)$ 2.56 $\left(4\text{.}2\right)\left(4\text{.}2\right)$ $0\text{.}9\cdot 0\text{.}9$ 0.81 $1\text{.}\text{11}\cdot 1\text{.}\text{11}$ $6\text{.}\text{815}\cdot 4\text{.}3$ 29.3045 $9\text{.}\text{0168}\cdot 1\text{.}2$ $\left(3\text{.}\text{5162}\right)\left(0\text{.}\text{0000003}\right)$ 0.00000105486 $\left(0\text{.}\text{000001}\right)\left(0\text{.}\text{01}\right)$ $\left(\text{10}\right)\left(4\text{.}\text{96}\right)$ 49.6 $\left(\text{10}\right)\left(\text{36}\text{.}\text{17}\right)$ $\text{10}\cdot \text{421}\text{.}\text{8842}$ 4,218.842 $\text{10}\cdot 8\text{.}\text{0107}$ $\text{100}\cdot 0\text{.}\text{19621}$ 19.621 $\text{100}\cdot 0\text{.}\text{779}$ $\text{1000}\cdot 3\text{.}\text{596168}$ 3,596.168 $\text{1000}\cdot \text{42}\text{.}\text{7125571}$ $\text{1000}\cdot \text{25}\text{.}\text{01}$ 25,010 $\text{100},\text{000}\cdot 9\text{.}\text{923}$ $\left(4\text{.}6\right)\left(6\text{.}\text{17}\right)$ Actual product Tenths Hundreds Thousandths Actual product Tenths Hundreds Thousandths 28.382 28.4 28.38 28.382 $\left(8\text{.}\text{09}\right)\left(7\text{.}1\right)$ Actual product Tenths Hundreds Thousandths $\left(\text{11}\text{.}\text{1106}\right)\left(\text{12}\text{.}\text{08}\right)$ Actual product Tenths Hundreds Thousandths Actual product Tenths Hundreds Thousandths 134.216048 134.2 134.22 134.216 $0\text{.}\text{0083}\cdot 1\text{.}\text{090901}$ Actual product Tenths Hundreds Thousandths $7\cdot \text{26}\text{.}\text{518}$ Actual product Tenths Hundreds Thousandths Actual product Tenths Hundreds Thousandths 185.626 185.6 185.63 185.626 For the following 15 problems, perform the indicated operations Find 5.2 of 3.7. Find 12.03 of 10.1 121.503 Find 16 of 1.04 Find 12 of 0.1 1.2 Find 0.09 of 0.003 Find 1.02 of 0.9801 0.999702 Find 0.01 of the sum of 3.6 and 12.18 Find 0.2 of the sum of 0.194 and 1.07 0.2528 Find the difference of 6.1 of 2.7 and 2.7 of 4.03 Find the difference of 0.071 of 42 and 0.003 of 9.2 2.9544 If a person earns $8.55 an hour, how much does he earn in twenty-five hundredths of an hour? A man buys 14 items at$1.16 each. What is the total cost? $16.24 In the problem above, how much is the total cost if 0.065 sales tax is added? A river rafting trip is supposed to last for 10 days and each day 6 miles is to be rafted. On the third day a person falls out of the raft after only $\frac{2}{5}$ of that day’s mileage. If this person gets discouraged and quits, what fraction of the entire trip did he complete? 0.24 A woman starts the day with$42.28. She buys one item for $8.95 and another for$6.68. She then buys another item for sixty two-hundredths of the remaining amount. How much money does she have left? ## Calculator problems For the following 10 problems, use a calculator to determine each product. If the calculator will not provide the exact product, round the results to five decimal places. $0.019\cdot 0.321$ 0.006099 $0.261\cdot 1.96$ $4.826\cdot 4.827$ 23.295102 ${\left(9.46\right)}^{2}$ ${\left(0.012\right)}^{2}$ 0.000144 $0.00037\cdot 0.0065$ $0.002\cdot 0.0009$ 0.0000018 $0.1286\cdot 0.7699$ $0.01\cdot 0.00000471$ 0.0000000471 $0.00198709\cdot 0.03$ ## Exercises for review ( [link] ) Find the value, if it exists, of $\text{0}÷\text{15}$ . 0 ( [link] ) Find the greatest common factor of 210, 231, and 357. ( [link] ) Reduce $\frac{\text{280}}{2,\text{156}}$ to lowest terms. $\frac{10}{77}$ ( [link] ) Write "fourteen and one hundred twenty-one ten-thousandths, using digits." ( [link] ) Subtract 6.882 from 8.661 and round the result to two decimal places. 1.78 so some one know about replacing silicon atom with phosphorous in semiconductors device? how to fabricate graphene ink ? for screen printed electrodes ? SUYASH What is lattice structure? of graphene you mean? Ebrahim or in general Ebrahim in general s. Graphene has a hexagonal structure tahir On having this app for quite a bit time, Haven't realised there's a chat room in it. Cied what is biological synthesis of nanoparticles what's the easiest and fastest way to the synthesize AgNP? China Cied types of nano material I start with an easy one. carbon nanotubes woven into a long filament like a string Porter many many of nanotubes Porter what is the k.e before it land Yasmin what is the function of carbon nanotubes? Cesar I'm interested in nanotube Uday what is nanomaterials and their applications of sensors. what is nano technology what is system testing? preparation of nanomaterial Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it... what is system testing what is the application of nanotechnology? Stotaw In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google Azam anybody can imagine what will be happen after 100 years from now in nano tech world Prasenjit after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments Azam name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world Prasenjit how hard could it be to apply nanotechnology against viral infections such HIV or Ebola? Damian silver nanoparticles could handle the job? Damian not now but maybe in future only AgNP maybe any other nanomaterials Azam Hello Uday I'm interested in Nanotube Uday this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15 Prasenjit can nanotechnology change the direction of the face of the world At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light. the Beer law works very well for dilute solutions but fails for very high concentrations. why? how did you get the value of 2000N.What calculations are needed to arrive at it Privacy Information Security Software Version 1.1a Good Got questions? Join the online conversation and get instant answers!	2018-08-19 18:51:22	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 66, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5271994471549988, "perplexity": 3102.502172776477}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221215284.54/warc/CC-MAIN-20180819184710-20180819204710-00111.warc.gz"}
https://www.iacr.org/cryptodb/data/paper.php?pubkey=21681	## CryptoDB ### Paper: New Blockcipher Modes of Operation with Beyond the Birthday Bound Security Authors: Tetsu Iwata URL: http://eprint.iacr.org/2006/188 Search ePrint Search Google In this paper, we define and analyze a new blockcipher mode of operation for encryption, CENC, which stands for Cipher-based ENCryption. CENC has the following advantages: (1) beyond the birthday bound security, (2) security proofs with the standard PRP assumption, (3) highly efficient, (4) single blockcipher key, (5) fully parallelizable, (6) allows precomputation of keystream, and (7) allows random access. CENC is based on the new construction of from PRPs to PRF conversion,'' which is of independent interest. Based on CENC and a universal hash-based MAC (Wegman-Carter MAC), we also define a new authenticated-encryption with associated-data scheme, CHM, which stands for CENC with Hash-based MAC. The security of CHM is also beyond the birthday bound. ##### BibTeX @misc{eprint-2006-21681, title={New Blockcipher Modes of Operation with Beyond the Birthday Bound Security}, booktitle={IACR Eprint archive}, keywords={secret-key cryptography / blockcipher, modes of operation, security proofs, birthday bound}, url={http://eprint.iacr.org/2006/188}, note={Appeared at FSE 2006. This is the full version. iwata@cse.nagoya-u.ac.jp 13305 received 6 Jun 2006}, author={Tetsu Iwata}, year=2006 }	2021-10-21 15:51:17	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5635450482368469, "perplexity": 12721.170244839983}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585424.97/warc/CC-MAIN-20211021133500-20211021163500-00062.warc.gz"}
https://planetmath.org/levycollapse	# Levy collapse Given any cardinals $\kappa$ and $\lambda$ in $\mathfrak{M}$, we can use the Levy collapse to give a new model $\mathfrak{M}[G]$ where $\lambda=\kappa$. Let $P=\operatorname{Levy}(\kappa,\lambda)$ be the set of partial functions $f:\kappa\rightarrow\lambda$ with $\|\operatorname{dom}(f)\|<\kappa$. These functions each give partial information about a function $F$ which collapses $\lambda$ onto $\kappa$. Given any generic subset $G$ of $P$, $\mathfrak{M}[G]$ has a set $G$, so let $F=\bigcup G$. Each element of $G$ is a partial function, and they are all compatible, so $F$ is a function. $\operatorname{dom}(G)=\kappa$ since for each $\alpha<\kappa$ the set of $f\in P$ such that $\alpha\in\operatorname{dom}(f)$ is dense (given any function without $\alpha$, it is trivial to add $(\alpha,0)$, giving a stronger function which includes $\alpha$). Also $\operatorname{range}(G)=\lambda$ since the set of $f\in P$ such that $\alpha<\lambda$ is in the range of $f$ is again dense (the domain of each $f$ is bounded, so if $\beta$ is larger than any element of $\operatorname{dom}(f)$, $f\cup\{(\beta,\alpha)\}$ is stronger than $f$ and includes $\lambda$ in its domain). So $F$ is a surjective function from $\kappa$ to $\lambda$, and $\lambda$ is collapsed in $\mathfrak{M}[G]$. In addition, $\|\operatorname{Levy}(\kappa,\lambda)\|=\lambda$, so it satisfies the $\lambda^{+}$ chain condition, and therefore $\lambda^{+}$ is not collapsed, and becomes $\kappa^{+}$ (since for any ordinal between $\lambda$ and $\lambda^{+}$ there is already a surjective function to it from $\lambda$). We can generalize this by forcing with $P=\operatorname{Levy}(\kappa,<\lambda)$ with $\kappa$ regular, the set of partial functions $f:\lambda\times\kappa\rightarrow\lambda$ such that $f(0,\alpha)=0$, $\|\operatorname{dom}(f)\|<\kappa$ and if $\alpha>0$ then $f(\alpha,i)<\alpha$. In essence, this is the product of $\operatorname{Levy}(\kappa,\eta)$ for each $\eta<\lambda$. In $\mathfrak{M}[G]$, define $F=\bigcup G$ and $F_{\alpha}(\beta)=F(\alpha,\beta)$. Each $F_{\alpha}$ is a function from $\kappa$ to $\alpha$, and by the same argument as above $F_{\alpha}$ is both total and surjective. Moreover, it can be shown that $P$ satisfies the $\lambda$ chain condition, so $\lambda$ does not collapse and $\lambda=\kappa^{+}$. Title Levy collapse LevyCollapse 2013-04-16 22:08:32 2013-04-16 22:08:32 ratboy (4018) e1568582 (1000182) 9 ratboy (1000182) Example msc 03E45	2019-09-20 18:02:12	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 67, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9685267210006714, "perplexity": 95.95166915987649}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514574058.75/warc/CC-MAIN-20190920175834-20190920201834-00034.warc.gz"}
https://www.reddit.com/user/AseOdin	[–] 0 points1 point (0 children) Are you asking us to break someone's RSA key? Your number is 200 digits long (665 bits), and looking at the RSA Factoring Challenge, a number roughly this same size (RSA-200 at 663 bits) took the equivalent of 75 years of computing time on a 2.2 GHz Opteron processor back in 2005. I don't think anyone's going to solve that here. [–] 0 points1 point (0 children) Question 5. You want to find the values of x for which f(x) < g(x), so replacing f(x) and g(x) by their expressions, we want to find the values of x for which x(x-1) < x. By rearranging, we get x2-2x < 0. The left hand side is a quadratic curve with roots at x=0 and x=2. If we check at x=1, we see that (1)2-2·1 = -1 is negative, and so we know that the curve x2-2x goes from positive to negative when crossing x=0 and then from negative to positive when crossing x=2. Hence the solution is that f(x) < g(x) whenever 0 < x < 2. Question 16. Because the function is defined by parts, we will analyse the range over two intervals. First, when x < 0, f(x) = 3·sin(x), and since the range of sin(x) is [-1,1], the range of f(x) is going to be [-3,3] when x < 0. However, when x ≥ 0, f(x) = sqrt(x), and the range of sqrt(x) is [0,∞). If we combine these two results, f(x) has a range of [-3,∞) over its entire domain. Question 46. First let's factor the numerator: x2-5·x+6 = (x-2)(x-3). Therefore, f(x) = (x-2)(x-3)/(x-2), which means that f(x) = x-3 whenever x is different from 2, and f(2) is undefined. Since f(x) is linear, it only has one x-intercept. Now the graph of f(x) and of x-3 look a lot alike, but you have to remember that f(2) is not defined, and so their graphs are distinct at x=2. Further, because of that difference, (2-3) = -1 is not in the range of f(x), though every other real number is. Question 49. We know that f(x) = sin(x) is an odd function, which means that f(-x) = -f(x). So since we want to know when f(-x) = f(x), this is the same as asking when -f(x) = f(x), i.e. when is f(x) = 0? This, we know to be the integer multiples of pi. Question 50. We have an intersection of the x-axis wherever we have a root of x(x2-2)(x2+x+1). This is a matter of factor the expression. Note that x2-2 = (x-sqrt(2))(x+sqrt(2)), and that x2+x+1 does not factor any further (that is because b2-4·a·c = 1-4 = -3 < 0). For that reason, we have that x(x2-2)(x2+x+1) = (x-0)(x-sqrt(2))(x+sqrt(2))(x2+x+1) = 0, which then clearly has 3 solutions: x = 0, x = -sqrt(2) and x = sqrt(2). [–] 0 points1 point (0 children) I've seen Z_p used for the p-adic integer (for p prime), so I usually tend not to use this notation, but in most cases there is no confusion to be had between the two objects anyway. [–] 0 points1 point (0 children) I'm not sure if this one tends too much towards the textbook kind, but From Fermat to Minkowski (original Von Fermat bis Minkowski) by Scharlau & Opolka gives a very nice historical introduction to number theory. The book is aimed at last-year undergrad student, so some knowledge from the reader is assumed (some field theory and ring theory). [–] 1 point2 points (0 children) I'd also be interested in joining this group. I don't know any of the proposed platforms (besides reddit), but Slack doesn't seem to fit the "classroom" model based on its introductory video. [–] 0 points1 point (0 children) Well I was assuming f was a function from R2 to R, and the question specifies that f(x,x) ∈ Z iff x ∈ Z, so the sines are to make sure that f(x,x) ∈ (0,1) for x ∈ R\Z. I might be assuming too many restrictions, though... [–] 0 points1 point (0 children) The real numbers are usually considered to be the “∞-adics”, i.e. the completion of Q with respect to the “∞-adic” metric \|·\|_∞ = \|·\|. The reason for this convention arises from an analogy that is nicely explained here. The wiki page is also not bad. The reason why Conway refers to those as the (−1)-adics is explained in the preface of the book. He writes: This reminds me of the fact that some people always smile indulgently when I mention “the prime −1”, and continue to use what they presume to be the grown up name “∞”. But consider: Every nonzero rational number is uniquely a product of powers of prime numbers p. For distinct odd primes (p/q) and (q/p) differ just when p ≡ q ≡ −1 (mod 4). There is an invariant called the p-signature whose definition involves summing p-parts of numbers. If there are p-adically integral root vectors of norm k and kp, then p is in the spinor kernel. Each of these statements includes the case p = −1, but none of them is even meaningful when we use the silly name “∞”. In the future, I shall smile indulgently back! As for your follow-up question, I have no idea. Sorry. [–] 2 points3 points (0 children) [–] 2 points3 points (0 children) If I'm correct, the "sum" is defined by x + y + z = (xy)::(xz)::(xy+xz-y), where :: denotes concatenation. Based on this, the solution is 8 + 3 + 9 = (83)::(89)::(83+89-3) = 24::72::(24+72-3) = 247293. It's possible however for there to be other rules. In particular, for any number α, you can define polynomial in three variables such that the two triples (5,3,2) and (7,2,5) give the correct answer, and the last triple (8,3,9) give α. In that sense, you could say that there are uncountably many solutions. [–] 4 points5 points (0 children) I think you are right, and that there is a slight error. That being said, the proof is still essentially correct: if you look at the mean value theorem, it only requires for F to be continuous on the closed interval [a,b] and differentiable on the open interval (a,b), which, as you've mentioned, is given by assumption. Similarly, F is continuous on each closed [x_i-1, x_i] and differentiable on each open (x_i-1, x_i). Good catch! Someone should probably fix it. I know Wikipedia's rules are sometimes very stringent, so I won't dare do it myself. [–] 0 points1 point (0 children) First of all, you want to only consider the relevant resistances. You don't really care for the flow going from the brewery to the first bottling facility, since it does not vary depending on the location of the junction point (its value is always k·4/(0.8)4). The only flow that matters is the one from the junction to the second bottling facility. If d is the distance from the brewery to J, then you can easily find the distance between J and the second bottling facility, and consequently you can also find a nice equation for the resistance in that pipe. Since it's the only resistance that matters, you can minimise its value for d between 0 and 4, and get your answer. [–] 3 points4 points (0 children) The reason why no one has studied those problems in details is simply because no one has deemed them worthy of research. Of course, whether a problem is interesting or not is subjective, but I can try to give my point of view. I would say that mathematicians usually tend to study problems that either: 1. Arise as sub-problems of larger problems that are already considered to be interesting, or; 2. Appear naturally from studying a structure or an object (e.g. when trying to generalise a result). (I'm sure there are other reasons, but I think those two motivate most problems.) The problems you are interested in, however, seem to be no more than brain-teasers. They do not appear as sub-problems of larger problems, and there does not seem to be any structure to their solutions. This could also simply mean that we have not yet found the underlying concept, but I doubt it. Personally, I don't believe that there is a deeper structure here. For example, dealing with digits (which depend on the base ten representation) rather than integers indicates that a solution would not be algebraic. (For example, 13 = (2+63)/5 depends on the fact that the concatenation of the digits 6 and 3 represents the integer 63.) On the other hand, your solution also depends on the behaviour of algebraic operations which care not about your choice of base. This leads to a "contradiction" in the sense that you should expect a general solution to be both algebraic and not algebraic at the same time... TL;DR: The problem is not useful and there's no nice structure to its solutions. This makes it uninteresting to study. [–][S] 0 points1 point (0 children) Thank you so much! Every single concept from that book is completely new to me, so it's very helpful to have feedback to make sure I understand the material well. [–] 0 points1 point (0 children) The set {z : \|z\| = 2} denotes the set of points with norm 2, which represents a circle centred at the origin of radius 2. The obvious parametrization is then simply 2e for 0 ≤ ϑ < 2π. We can then use this parametrization to also find γ' = {f(2e) : 0 ≤ ϑ < 2π} = {(1+2e)2/4 : 0 ≤ ϑ < 2π} = {1/4+e+e2iϑ : 0 ≤ ϑ < 2π} which looks like this. Note that f(γ) and γ' are used in part (c) of the question. [–] 0 points1 point (0 children) As you said, this really depends on the definitions you are using. If this is in the context of a course, you should simply use the definition provided in class. The definition of integral domains I'm used to is the one provided on Wikipedia and doesn't mention anything about the existence of an identity. For this definition, the statement is certainly true. [–] 1 point2 points (0 children) You have the right idea on how to do this, so here is a detailed solution. If at some point you feel like you can finish the problem by yourself, please try to do it. You want to find for which x does [; \frac{x^2+\frac{1}{x}}{x+\frac{1}{x}-1} = 1 ;] so you can multiply the numerator and the denominator by [; x ;] to get [; \frac{x^3+1}{x^2+1-x} = 1. ;] Then, multiply both sides by the denominator [; x^3+1 = x^2-x+1 ;] and bring back everything on one side [; x^3 - x^2 + x = 0. ;] You can then factor the left-hand side by pulling out a factor of [; x ;] to get [; x\cdot(x^2 - x + 1) = 0. ;] The quadratic equation itself has no real roots, so you can see then that [; f(x) = 1 ;] only when [; x = 0 ;]. I want to note that, technically, [; f ;] is not defined at [; x = 0 ;] (because of the [; 1/x ;] terms) which means that [; f^{-1}(1) ;] is not defined either. The correct way to define [; f ;] for every real number is [; f(x) = \frac{x^3+1}{x^2-x+1}. ;] [–][S] 0 points1 point (0 children) This clears things up, especially with the "I look down upon you" example. If I understand correctly, one way to avoid ending the original sentence with a preposition would be with "This is the sort of English with which I will not put up," but then we lose the idiomatic meaning of "put up with" (as we did in your example with "look down upon"). Thank you for the clarification! [–] 0 points1 point (0 children) There is a symmetry to these equations, so it's not too weird to try to find a solution while assuming that x, y and z are all equal. In this case, you get x = y = z = (1+sqrt(1+4a))/2 or x = y = z = (1-sqrt(1+4a))/2. You can check that these work as solutions, though they might not be the only ones. Finding sum of a series by in cheatatmathhomework [–] 1 point2 points (0 children) Okay, first note that Sum (5n-2)/((n-1)n(n+2)) = 5 * Sum 1/((n-1)(n+2)) - 2 * Sum 1/((n-1)n(n+2)), where both series on the right are converging (from the comparison test with the p-series). So we'll evaluate the two series individually. We start with the first series. By partial fraction decomposition, we find that 1/((n-1)(n+2)) = 1/(3(n-1)) - 1/(3(n+2)). What you must see here is that this gives telescoping partial sums. To be more precise, the terms with denominator greater than or equal to 12 cancel out. You are left with Sum 1/((n-1)(n+2)) = 1/3 + 1/6 + 1/9 = 11/18. Now for the second one. We use the same technique and find that 1/((n-1)n(n+2)) = 2/(6(n-1)) - 3/(6n) + 1/(6(n+2)). This again gives telescoping partial sums where all terms with denominator greater than or equal to 24 cancel out. You are left with Sum 1/((n-1)n(n+2)) = (2/6 - 3/12) + (2/12 - 3/18) + (2/18) = 7/36. Hence the series is equal to Sum (5n-2)/((n-1)n(n+2)) = 5(11/18) - 2(7/36) = 8/3. [–] 0 points1 point (0 children) So your average is the random variable X ~ N(21.1, 6.82/80). If we normalize this variable, we get Z = (X - 21.1)/(6.8/sqrt(80)) ~ N(0,1) and from a table for a normal variable, we find that P(Z ∈ [-1.96, 1.96]) = 0.95. Now if we simply rewrite the set inside the probability in terms of X we see that Z ∈ [-1.96, 1.96] iff (X - 21.1)/(6.8/sqrt(80)) ∈ [-1.96, 1.96] iff X - 21.1 ∈ [-1.966.8/sqrt(80), 1.966.8/sqrt(80)] iff X ∈ [21.1 - 1.966.8/sqrt(80), 21.1 + 1.966.8/sqrt(80)]. From this, you find that the constant you are looking for is a = 1.96*6.8/sqrt(80) ≈ 1.49012. In particular, 17 is outside of this interval since 17 < 21.1-1.5. [–] 1 point2 points (0 children) Well the sample mean X_bar is normally distributed and both the mean and the standard deviation are known. This is true regardless of the sample size. Since you know the exact distribution, there is no reason to look for an approximation via the t-distribution. The t-distribution will be useful when you can only approximate your standard deviation with your sample. You can then just calculate the probability of a normally distributed random variable with mean 98.2 and standard deviation 0.62/sqrt(19) to be less than 98.5. This gives you 0.982534.	2017-04-30 13:01:16	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8265580534934998, "perplexity": 820.6140294704684}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917125532.90/warc/CC-MAIN-20170423031205-00047-ip-10-145-167-34.ec2.internal.warc.gz"}
https://www.proofwiki.org/wiki/Category:Definitions/Complex_Multiplication	# Category:Definitions/Complex Multiplication This category contains definitions related to Complex Multiplication. Related results can be found in Category:Complex Multiplication. The multiplication operation in the domain of complex numbers $\C$ is written $\times$. Let $z = a + i b, w = c + i d$ where $a, b, c, d \in \R$. Then $z \times w$ is defined as: $\paren {a + i b} \times \paren {c + i d} = \paren {a c - b d} + i \paren {a d + b c}$ ## Pages in category "Definitions/Complex Multiplication" The following 5 pages are in this category, out of 5 total.	2023-03-28 21:58:56	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9162834286689758, "perplexity": 1297.3085238138697}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948871.42/warc/CC-MAIN-20230328201715-20230328231715-00653.warc.gz"}
https://tex.stackexchange.com/questions/648463/latex-latex-error-missing-inserted	# Latex LaTex error : "Missing } inserted" I have written this: but latex gives me the error "Missing } inserted", i tried some stuff but everytime latex gives me this. Can you help me please ? • Welcome to TSE. Please post a Minimal Working Example, instead of a code snippet. Jun 21 at 9:17 • \begin{array}{2} contains a syntax error. Please tell us how your array environment is supposed to be organized: how many columns, and how should the columns be aligned? – Mico Jun 21 at 9:27 • Move \end{document} up until you find the exact line that causes the problem. Then you can create a MINIMAL example showing the problem. Until then, the best we can say is the you are missing a }. Jun 21 at 9:31 • Off-topic: Don't write $G^'$ and $L^'$, respetively. Instead, just write $G'$ and $L'$, i.e., omit the ^ (caret, "exponentiation") symbols. – Mico Jun 21 at 9:39 • Please post the code itself, not a screenshot of the code. In general, I'd say, you're using far to many curly braces. For instance, there's is absolutely no advantage in writing ${C_G}$. Indeed, by writing $C_G$ you would do yourself a big favor as you'd have to deal with much less code clutter. Similarly, do replace ({C_{{G_{end}}(V,E}}) with (C_{G_{\mathrm{end}}(V,E)}). – Mico Jun 21 at 10:00	2022-06-29 06:02:52	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8460111618041992, "perplexity": 1406.5179525457893}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103624904.34/warc/CC-MAIN-20220629054527-20220629084527-00744.warc.gz"}
http://mathhelpforum.com/trigonometry/57509-sine-angle.html	# Math Help - sine angle 1. ## sine angle How do I solve this? What is the answer? sin x/2= (square root 3)/2 , -pi<x<pi (this is the range given) (I had no idea how to put a squre root symbol of the 3...) 2. Originally Posted by jcfulton How do I solve this? What is the answer? sin x/2= (square root 3)/2 , -pi<x<pi (this is the range given) (I had no idea how to put a squre root symbol of the 3...) hint: $\sin (?) = \frac {\sqrt{3}}2$, what is $?$? 3. sin 60 degrees is (square root 3)/2, therefore the answer is 120 degrees, since the angle is x/2. 4. sin of 60 degrees is (square root 3)/2..........or pi/3........where do I go from here? 5. sin x = (square root 3)/2 means that x is 60 degrees. sin x/2 = (square root 3)/2 means that x is 120....120/2 = 60	2014-08-01 10:26:44	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9520092606544495, "perplexity": 2888.9448336028504}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510274967.3/warc/CC-MAIN-20140728011754-00239-ip-10-146-231-18.ec2.internal.warc.gz"}
https://papers.nips.cc/paper/2015/file/c9e1074f5b3f9fc8ea15d152add07294-Reviews.html	Paper ID: 104 Title: On the Limitation of Spectral Methods: From the Gaussian Hidden Clique Problem to Rank-One Perturbations of Gaussian Tensors Current Reviews Submitted by Assigned_Reviewer_1 Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http://nips.cc/PaperInformation/ReviewerInstructions) This paper is about hypothesis testing of Gaussian Hidden Clique problem. It shows that, for any \epsilon, the case where there exists a planted clique of size L = (1-\epsilon) \sqrt{n} cannot be distinguished against the case where there is no hidden clique, using spectral methods. This complements the positive result of [12], where it is possible to distinguish the two cases using a spectral method, when L > (1+\epsilon) \sqrt{n}. The main technical contribution is a general analysis based on rank-1 perturbation of order-k symmetric Gaussian tensors, which may be of independent interest. Quality: The result is technically sound as far as I have checked. Clarity: The paper is well-written overall; I am confused about Equation (24) -- why does it follow directly from Lemma 5? Originality: Although the proof technique based on Lemma 2 is quite interesting, to be honest, I am not able to evaluate the novelty since I am not familiar with the relevant literature of proving such lower bound. Significance: the paper establishes an interesting result, which nicely complements the previous positive result on spectral methods. But the downside is that this is only a lower bound for spectral methods", which makes the usage rather restrictive. It seems that extending this lower bound to any efficient detection algorithm is not easy, since the technique heavily relies on the fact that the eigenvalues are conjugation-invariant. This paper is about hypothesis testing of Gaussian Hidden Clique problem using spectral methods. The techniques are general, the results are interesting and nicely complements the existing lower bound. Submitted by Assigned_Reviewer_2 Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http://nips.cc/PaperInformation/ReviewerInstructions) The authors consider the following problem: Given a symmetric matrix X of dimension n, design a spectral test (a statistical test that depends only on the eigenvalues of X) to distinguish between the following two hypotheses: H_0 = all elements are drawn from N(0,1) H_{1,L} = there exists a submatrix that is drawn from N(1,1) It is known that when L >= (1-\epsilon)\sqrt(n) there is a simple test (involving checking the top eigenvalue) to distinguish H_0 and H_{1,L}. What the authors show is that the result is tight and that no spectral test is reliable for L \leq (1-\epsilon)\sqrt(n). The authors also prove results regarding the analogous tensor variant in particular distinguishing among H_0: X = Z H_1: X = \beta v^{\otimes k} + Z The papers' contribution is primarily theoretical and there are no experiments. Questions/Concerns: (1) It's not entirely clear to me what the machine learning applications are for such a problem. (2) Could the authors explain why the eigenvalue distribution of \beta * vv^T + Z is the same as Z if the norm of v is fixed? (3) Is there some intuition to why examining the top eigenvalue is sufficient for the gaussian clique problem (if L is large enough?) (3) I don't have an intuitive understanding of Theorem 1. In particular, why spectral contiguity of H_{1,L} with respect to H_0 implies that no spectral test is reliable for L <= (1-\epsilon)\sqrt(n). The authors prove some theoretical lower bounds for the gaussian hidden clique problem in both matrices and tensors. The contributions of the paper are primarily theoretical (i.e. no experiments). I am not familiar with the related work and have some questions about some of the concepts that I detail below. Submitted by Assigned_Reviewer_3 Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http://nips.cc/PaperInformation/ReviewerInstructions) The paper provides a result on how the eigenvalues of a random matrix can be exploited to answer the detection problem "Gaussian hidden clique problem". The authors provide a tight result on how for small size of planted clique, the eigenvalues are not enough for the detection problem. Revise the first sentence in the abstract. It is a very long sentence, and it is hard to read and understand it. I also suggest to move some of the related works discussion provided at the end of paper to the beginning in the introduction section. The paper has a reasonable contribution, and it is clearly written. Author Feedback Author Feedback Q1:Author rebuttal: Please respond to any concerns raised in the reviews. There are no constraints on how you want to argue your case, except for the fact that your text should be limited to a maximum of 5000 characters. Note however, that reviewers and area chairs are busy and may not read long vague rebuttals. It is in your own interest to be concise and to the point. We thank the reviewers for their useful and insightful comments. Our responses to some of the issues are summarized below: 1) Reviewer 2 asks for further pointers of applications of our result to the NIPS community and notes that perturbations of matrices of rank greater than one is of interest to researchers in machine learning. Of course, spectral clustering provides another broad set of application for spectral methods. Our approach may provide insights into the limitations of spectral clustering. The main technical is to deal with higher-rank structures, which we believe to be feasible. 2) Reviewer 2 correctly observes that our results are asymptotic, hence the question arises to what extent they apply to finite instances that are encountered in practice. All calculations are explicit, and in principle we can derive explicit bounds on the total variation distance for each fixed k. For instance in Eq. (27) the integral yields E\Lambda^2 \le 1 + C'/n^{(k/2)-1} for k\ge 3 Hence allready for $k=4$ the total variation decays at a linear rate. 3) Reviewer 2, raises the issue of spectral algorithms that use eigenvectors in addition to eigenvalues. We believe that establishing the limitations of such methods is of great interest. However, It should be noted that proving unconditional hardness results for detection problems which consider both eigenvectors and eigenvalues is unlikely, as the whole matrix is determined by its eigenvalues and eigenvectors (up to permutations). 4) We thank reviewer 3 for his suggestion regarding the first sentence. We will make the first sentence shorter and easier to parse.	2021-04-16 01:58:21	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8749339580535889, "perplexity": 599.7858378448817}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038088471.40/warc/CC-MAIN-20210416012946-20210416042946-00160.warc.gz"}
https://zbmath.org/?q=an:1066.60058	## Exponentially stable stationary solutions for stochastic evolution equations and their perturbation.(English)Zbl 1066.60058 For the abstract evolution equation $$dX=\bigl(AX+f(X)\bigr)\,dt+B(X)\,dW_t$$, where $$A$$ generates a strongly continuous semigroup with growth bound $$a$$ on a separable Hilbert space $$H$$, $$f:H\to H$$ is globally Lipschitz with Lipschitz constant $$L_f$$, $$W$$ is a trace class Wiener process on some Hilbert space $$U$$, and $$B$$ is globally Lipschitz with Lipschitz constant $$L_B$$ from $$H$$ to the space of linear operators from $$U$$ to $$H$$, it is assumed that $a+L_f+L_B/2<0.\tag{1}$ Under this condition, existence of an exponentially attracting stationary solution is proved. Assuming a finite-dimensional Wiener process with $$B$$ linear and commuting with $$A$$, it is shown that exponential attraction holds uniformly in bounded sets of initial conditions. Finally, it is shown that the stationary solution depends continuously on perturbations of the nonlinearity $$f$$, provided (1) holds uniformly. ### MSC: 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 37L55 Infinite-dimensional random dynamical systems; stochastic equations 93E15 Stochastic stability in control theory ### Keywords: random dynamical systems; random attractors Full Text:	2022-12-05 00:06:36	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7913769483566284, "perplexity": 214.80380369654253}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711001.28/warc/CC-MAIN-20221205000525-20221205030525-00565.warc.gz"}
https://math.stackexchange.com/questions/2824392/choosing-10-cards-randomly-from-a-52-card-deck/2824414	# Choosing 10 cards randomly from a 52 card deck Given a 52-card deck, if we pick 10 cards, what's the probability of having all four aces among the 10 cards we picked? My attempt was defining $\Omega = \{(1,2,...,52)^{10}\}$. Now $\|\Omega\|=\binom{52}{10}$ Let $A$ be the event that we're looking for. $A=\{\{1,2,3,4\}\cup(5,6,...,52)^{6}\}$. $\|A\|=\binom{48}{6}$ Now $Pr(A)=\dfrac{\|A\|}{\|\Omega\|}=\dfrac{\binom{48}{6}}{\binom{52}{10}}$ I don't have the answer but I noticed I got a really small number ($0.07\%$), so I don't think I solved it right. • I dont see a probem with your solution. Getting 4 specific cards in a 10 card draw is going to be a low probability event. – user625 Jun 19 '18 at 1:18 • I got an even lower answer: $\frac{6}{7735}$ – Joseph Eck Jun 19 '18 at 1:27 • @JosephEck, what do you mean by "lower"? $6/7735=0.00077569489\approx0.0776\%$. – Barry Cipra Jun 19 '18 at 1:47 • @BarryCipra Lol, brain fart, forgot to convert my decimal to a percent – Joseph Eck Jun 19 '18 at 1:52 For any 4 cards from 10 there are $^{10}C_4 = 210$ combinations. For any 4 cards from 52 there are $^{52}C_4 = 270725$ combinations. $P(4A) = \frac{^{10}C_4}{^{52}C_4} = \frac{210}{270725} = .0007757$ In other words, your $10$ cards only comprises of $210$ four card combinations from a possible $270725.$ • Ah ha! It took me a moment to realise what you were counting here. Rather than "selecting 4 cards from 10", I would say "selecting 4 places from 10 (when shuffling the deck)". – Graham Kemp Jun 19 '18 at 2:27 Comment: You have several correct answers already--including your own! I just wanted to show the connection with the hypergeometric distribution. The number $X$ of Aces among ten cards chosen at random without replacement from a 52 card deck has a hypergeometric distribution. There are four favorable cards (Aces) and 48 unfavorable cards (non-Aces). In R statistical software, one can compute the probabilities $P(X = k)$ for $k = 0, 1, 2, 3, 4.$ cbind(k, pdf) k pdf 0 0.4134453782 1 0.4240465417 2 0.1431157078 3 0.0186166774 4 0.0007756949 You seek $$P(X=4) = \frac{{4 \choose 4}{48 \choose 6}}{{52 \choose 10}} = 0.0007756949.$$ You are correct that getting all four Aces, even among ten cards, has a very small probability. The figure below shows a bar chart of the hypergeometric distribution of $X.$ Your probability corresponds to the very short bar at the far right. Notes: (1) As an experiment, I tried doing ten million draws of 10 cards and counting the Aces each time. The expected number of aces is $10^7 \times 0.0007756949 \approx 7757.$ In my simulation I got 7591, which is fewer than 7757, but within the margin of simulation error. (2) In many versions of poker each player's hand consists of five supposedly randomly chosen cards. A poker player would be delighted to find all four Aces in his/her hand. But the probability of that is even smaller than your probability: $1.847 \times 10^{-5}.$ dhyper(4, 4, 48, 5) ## 1.846893e-05 (3) If you sample the ten cards with replacement, then the number $Y$ of Aces in ten independent draws has $Y \sim \mathsf{Binom}(10, 1/13).$ Then $P(Y = 4) = 0.004548553.$ This is larger than your probability because the Aces don't get "used up" as they get chosen. (Each Ace could possibly be chosen more than once.) dbinom(4, 10, 1/13) ## 0.004548553 • So the chance of getting at least 1 ace is higher than having no aces at all? – steenbergh Jun 19 '18 at 7:47 • Seems so...by a smidge. That's when 10 cards are drawn. But that's not true for a 5-card poker hand: dhyper(0:1, 4, 48, 5) returns 0.6588420 for no Ace and 0.2994736 for one Ace. – BruceET Jun 19 '18 at 7:58 Note that we pick the ten cards without "putting them back", so the space $\Omega$ is the set of all subsets with ten elements of $\Omega_0=\{1,2,\dots,52\}$. This is not the cartesian product $\Omega_0^{\times 10}$. (Which allows repetitions, and also knows the order the ten cards came to the hand.) Our convention is to always sort the hand. (First w.r.t values, than w.r.t. colors.) And the aces are last after sorting. The number of all "possible cases" is thus $$\|\Omega\|=\binom{52}{10}\ .$$ Now let us consider the "favorable cases". Yes, the aces are at the end, for the remained $6$ places we have the remained $48$ cards, there are $$\binom{48}6$$ good cases. This is the posted solution. The quotient of the above two binomial coefficients is the searched probability. We may think alternatively as follows. The cards are extracted one by one, so the new modelling space is $\Omega'$ the space of all ordered tuples with different elements from $\Omega_0$. There are $$\|\Omega'\|=(52-0)(52-1)\cdots(52-9)=\frac {52!}{(52-10)!}$$ possible cases. Out of them, come cases are favorable. Let us count them. We have $\binom{10}4$ choices for the positions of the aces among the ten places of a "good ordered (tuple) hand". We have $4!$ possibilities to insert into these places the four aces. The remained positions can be filled with the remained cards in $$(48-0)(48-1)\cdots(48-5)=\frac {48!}{(48-6)!}$$ different ways. So there are totally $$\binom{10}4\cdot 4!\cdot \frac {48!}{(48-6)!}= \frac{10!}{6!}\cdot \frac {48!}{(48-6)!}$$ "good cases". The probability, computed in this model is: $$\frac {\frac{10!}{6!}\cdot \frac {48!}{(48-6)!}} {\frac{52!}{42!}} = \frac {\frac{48!}{42!6!}} {\frac{52!}{42!10!}} = \frac{\binom{48}6}{\binom{52}{10}}\ ,$$ same as in the first modelling. The probability is approximatively: sage: binomial(48,6) / binomial(52,10) 6/7735 sage: _.n() 0.000775694893341952 Let us simulate some cases to convince ourselves, sage. N = 10**5 # trials C = Combinations( [1..52], 10 ) hitcounter = 0 # so far for trial in xrange(N): hand = C.random_element() # it is already sorted if hand[:4] == [1,2,3,4]: hitcounter += 1 print ( "N=%s trials, %s hits, quote = %s = %f" % (N, hitcounter, hitcounter/N, hitcounter/float(N)) ) This gives this time: N=100000 trials, 84 hits, quote = 21/25000 = 0.000840 Here is the list of probabilities to get four aces (4A) if we extract $k$ cards, sage: for k in [4..52]: ....: print ( "4A in a hand with %2s cards come with probability ~ %f" ....: % (k, binomial(48,k-4)/binomial(52,k)) ) ....: 4A in a hand with 4 cards come with probability ~ 0.000004 4A in a hand with 5 cards come with probability ~ 0.000018 4A in a hand with 6 cards come with probability ~ 0.000055 4A in a hand with 7 cards come with probability ~ 0.000129 4A in a hand with 8 cards come with probability ~ 0.000259 4A in a hand with 9 cards come with probability ~ 0.000465 4A in a hand with 10 cards come with probability ~ 0.000776 4A in a hand with 11 cards come with probability ~ 0.001219 4A in a hand with 12 cards come with probability ~ 0.001828 4A in a hand with 13 cards come with probability ~ 0.002641 4A in a hand with 14 cards come with probability ~ 0.003697 4A in a hand with 15 cards come with probability ~ 0.005042 4A in a hand with 16 cards come with probability ~ 0.006723 4A in a hand with 17 cards come with probability ~ 0.008791 4A in a hand with 18 cards come with probability ~ 0.011303 4A in a hand with 19 cards come with probability ~ 0.014317 4A in a hand with 20 cards come with probability ~ 0.017896 4A in a hand with 21 cards come with probability ~ 0.022107 4A in a hand with 22 cards come with probability ~ 0.027020 4A in a hand with 23 cards come with probability ~ 0.032708 4A in a hand with 24 cards come with probability ~ 0.039250 4A in a hand with 25 cards come with probability ~ 0.046726 4A in a hand with 26 cards come with probability ~ 0.055222 4A in a hand with 27 cards come with probability ~ 0.064826 4A in a hand with 28 cards come with probability ~ 0.075630 4A in a hand with 29 cards come with probability ~ 0.087731 4A in a hand with 30 cards come with probability ~ 0.101228 4A in a hand with 31 cards come with probability ~ 0.116225 4A in a hand with 32 cards come with probability ~ 0.132829 4A in a hand with 33 cards come with probability ~ 0.151150 4A in a hand with 34 cards come with probability ~ 0.171303 4A in a hand with 35 cards come with probability ~ 0.193407 4A in a hand with 36 cards come with probability ~ 0.217582 4A in a hand with 37 cards come with probability ~ 0.243956 4A in a hand with 38 cards come with probability ~ 0.272657 4A in a hand with 39 cards come with probability ~ 0.303818 4A in a hand with 40 cards come with probability ~ 0.337575 4A in a hand with 41 cards come with probability ~ 0.374070 4A in a hand with 42 cards come with probability ~ 0.413445 4A in a hand with 43 cards come with probability ~ 0.455850 4A in a hand with 44 cards come with probability ~ 0.501435 4A in a hand with 45 cards come with probability ~ 0.550356 4A in a hand with 46 cards come with probability ~ 0.602770 4A in a hand with 47 cards come with probability ~ 0.658842 4A in a hand with 48 cards come with probability ~ 0.718737 4A in a hand with 49 cards come with probability ~ 0.782624 4A in a hand with 50 cards come with probability ~ 0.850679 4A in a hand with 51 cards come with probability ~ 0.923077 4A in a hand with 52 cards come with probability ~ 1.000000 and note that even if we have a hand with $51$ cards, there is still a $4/52=1/13$ probability to miss an $A$... Yes, the probability for selecting four specific cards (and six others) when selecting ten cards from a standard 52 card deck (without replacement) is $${\left.{\dbinom 44}\dbinom{48}6\middle/\dbinom{52}{10}\right.}$$ The probability that those four cards shall be placed among the top ten places in the deck is $${\left.\dbinom{10}4\middle/\dbinom{52}{4}\right.}$$ Which is the same small value $6/7735$. The chance that A♣️ is in your hand is $10/52$. Given that, the chance the A♦️ also is is then $9/51$, etc, so the answer is $\dfrac{10\cdot9\cdot8\cdot7}{52\cdot51\cdot50\cdot49}$.	2019-05-20 22:53:45	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.931822657585144, "perplexity": 813.3111594185553}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232256163.40/warc/CC-MAIN-20190520222102-20190521004102-00267.warc.gz"}
https://dylaniki.org/Baby,%20I'm%20in%20the%20Mood%20for%20You	Written by Bob Dylan Recorded 9 Jul, 1962 during the Freewheelin’ sessions, released on Biograph (1985) Tabbed by Eyolf Østrem B7/f# = 22120x G Sometimes I'm in the mood, I wanna leave my lonesome home C G And sometimes I'm in the mood, I wanna hear my milk cow moan B7/f# C/g And sometimes I'm in the mood, I wanna hit the highway road G D G C But then again, but then again, I said oh, I said oh, I said G D G Oh babe, I'm in the mood for you. Sometimes I'm in the mood, Lord, I had my overflowin' fill Sometimes I'm in the mood, I'm gonna make out my final will Sometimes I'm in the mood, I'm gonna head for the walkin' hill But then again, but then again, I said oh, I said oh, I said Oh babe, I'm in the mood for you. Sometimes I'm in the mood, I wanna lay right down and die Sometimes I'm in the mood, I wanna climb up to the sky Sometimes I'm in the mood, I'm gonna laugh until I cry But then again, I said again, I said again, I said Oh babe, I'm in the mood for you. Sometimes I'm in the mood, I'm gonna sleep in my pony's stall Sometimes I'm in the mood, I ain't gonna do nothin' at all Sometimes I'm in the mood, I wanna fly like a cannon ball But then again, but then again, I said oh, I said oh, I said Oh babe, I'm in the mood for you. Sometimes I'm in the mood, I wanna back up against the wall Sometimes I'm in the mood, I wanna run till I have to crawl Sometimes I'm in the mood, I ain't gonna do nothin' at all But then again, but then again, I said oh, I said oh, I said Oh babe, I'm in the mood for you. Sometimes I'm in the mood, I wanna change my house around Sometimes I'm in the mood, I'm gonna make a change in this here town Sometimes I'm in the mood, I'm gonna change the world around But then again, but then again, I said oh, I said oh, I said Oh babe, I'm in the mood for you.	2022-08-12 14:29:08	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6947536468505859, "perplexity": 6440.668768247935}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571719.48/warc/CC-MAIN-20220812140019-20220812170019-00354.warc.gz"}
http://mkyongtutorial.com/advance-loan-virginia-pertaining-to-pay-day-loans-10	# Advance Loan Virginia. PERTAINING TO PAY DAY LOANS AND PAY time LOANS IN Virginia Advance Loan Virginia. PERTAINING TO PAY DAY LOANS AND PAY time LOANS IN Virginia Advance Loan Virginia. PERTAINING TO PAYDAY ADVANCES AND PAY time LOANS IN Virginia KNOW THE ADVANCED RULES AND REGULATIONS* You may think it is one of several states having A apr that is 36-percent once you very very very first glance at VirginiaвЂ™s cash loan laws and regulations. Virginia includes a 36-percent limitation on yearly interest, but that’s totally different from APR. APR includes the complete finance cost, not just the interest that is yearly. Brand groundbreaking Hampshire and Montana both capped APR at 36-percent, but Virginia enables for just about any other finance fees with the 36-percent interest that is yearly. ItвЂ™s important to not ever confuse both these acutely different instructions. ThatвЂ™s why weвЂ™ll dig a little much deeper and appearance at a number of the particulars that are key VirginiaвЂ™s advance loan recommendations. To begin with with your loan need at the moment, simultaneously as much as our protected loan need sort. ## Virginia Advance Loan Regulations. Your loan term has to be at the very least as long as two for the pay durations. In Virginia, the many loan volume is $500. Consequently, if you have actually paid every or two, your loan term will have to be at the least 28 times very very very long week. The utmost finance cost comprises of a 36-percent annual interest,$5 verification price, and 20% for the loan volume. Which means that for the 14-day, $100 loan, youвЂ™d pay a$26.38 finance price. in cases like this, your 14-day APR could possibly be 687.76-percent. Perhaps you are simply allowed to eliminate one cash that is outstanding at a quantity of the time in Virginia. YouвЂ™re not permitted to grow your loan. This implies no rollovers. Your financial institution may charge you the after collection costs: a $25 NSF cost, reasonable attorneyвЂ™s costs, and court costs. Your loan company simply is not allowed to follow action that is criminal you. Virginia Advance Loan Regulations ## It doesn’t matter what crisis that is monetary one selection for short-term, small-dollar funding many customers go on to may be the loan that is payday. This financial product is just like an advance loan. ItвЂ™s managed as a result of the state legislation that is exact same. • So just how Payday Loans Perform вЂ“ a loan company supplies a loan that is smallвЂ™s likely to be reimbursed within a period that is brief of time, frequently across the date the debtor expects become paid. • Optimum Amount вЂ“ In Virginia, financial institutions can offer at the most$500 through this form of cash. Virginia State Economy Prior to taking away an advance loan, it is advisable to produce re payment plan consequently youвЂ™re better ready to invest through the loan. DonвЂ™t just look into your finances that are personal additionally consider very carefully your stateвЂ™s economy. In-may 2015, the jobless cost in Virginia finished up being 4.9-percent. ThatвЂ™s a little a lot more than nearby states, like brand name completely new Hampshire (3.8-percent) and Massachusetts (4.6-percent). VirginiaвЂ™s unemployment price that is greatest wound up being 7.9-percent in December 1982. The lowest priced finished up being 2.1-percent in November 2000. While Virginia caps the yearly interest at 36-percent for pay day loans, there are many different other costs related to getting this kind of loan. Make certain you figure out what your unique finance that is total should really be before you consent into the home loan. ## Requesting Title Loans in Virginia. Virginia residents that will be struggling to produce ends meet can think about trying additionally to have a car or truck payday loans Arizona title loan. Title loans are short-term, small-dollar loans and that can be requested against a borrowerвЂ™s automobile title. Loan amounts can cover any such thing from $100 to$5,500, or 25% to 50per cent about the worth associated with car being borrowed against. Title loans in Virginia have in fact actually comparable guidelines that are legal pay day loans and loans which are payday their state. Title loans are capped at just as much as 50percent concerning the market that is reasonable connected with automobile being borrowed against. Pertaining to cost restrictions: there clearly was a cost restriction of 22per cent every month of outstanding balances most of the option to $700, 18% for degrees of$701 to $1400, and 15% for levels of$1,401 or higher, as well as a lien price. a lien price is a charge for a title to ensure the safe re re payment connected with economic responsibility owed. ## Why Clients Ask for Title Loans? Virginia residents can put on for the true title loan on the web. While requirement shall vary regarding the loan company, prospects are required to provide the following that is immediate Potential borrowers should keep in your mind that financial institutions might also typically check an applicantвЂ™s credit score to make sure they shall find a way to repay their loan right appropriate straight right straight back on some time satisfy other fine print. Shorter-duration funding brings relief for Virginia residents who might be struggling to help with making ends satisfy. Title loans can be used for the following that is immediate Clients should simply subscribe to short-term loans if they’re able to handle them, as these loans normally have high-interest costs and costs	2021-01-25 09:38:39	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.21774278581142426, "perplexity": 6823.173507699051}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703565541.79/warc/CC-MAIN-20210125092143-20210125122143-00525.warc.gz"}
http://docs.plasmapy.org/en/stable/api/plasmapy.atomic.integer_charge.html	# integer_charge¶ plasmapy.atomic.integer_charge(particle: plasmapy.atomic.particle_class.Particle) → int Return the integer charge of a particle. Parameters: particle (str) – String representing a particle. Z – The charge as a multiple of the elementary charge. int InvalidParticleError – If the argument does not correspond to a valid particle or contradictory information is provided. ChargeError – If charge information for the particle is not available. AtomicWarning – If the input represents an ion with an integer charge that is less than or equal to -3, which is unlikely to occur in nature. Notes This function supports two formats for integer charge information. The first format is a string that has information for the element or isotope at the beginning, a space in between, and the integer charge information in the form of an integer followed by a plus or minus sign, or a plus or minus sign followed by an integer. The second format is a string containing element information at the beginning, following by one or more plus or minus signs. Examples >>> integer_charge('Fe-56 2+') 2 >>> integer_charge('He -2') -2 >>> integer_charge('H+') 1 >>> integer_charge('N-14++') 2	2019-02-21 17:31:08	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.43349090218544006, "perplexity": 917.7143433492569}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247506094.64/warc/CC-MAIN-20190221172909-20190221194909-00629.warc.gz"}
https://www.zbmath.org/serials/?q=se%3A9088	# zbMATH — the first resource for mathematics ## Modern Birkhäuser Classics Short Title: Mod. Birkhäuser Classics Publisher: Birkhäuser, Cham ISSN: 2197-1803 Online: https://link.springer.com/bookseries/7588 Comments: Book series Documents Indexed: 106 Publications (since 2007) all top 5 #### Authors / Editors 3 Cartier, Pierre 3 Gromov, Mikhael Leonidovich 3 Illusie, Luc 3 Katz, Nicholas Michael 3 Laumon, Gérard 3 Manin, Yuri Ivanovich 3 Mumford, David Bryant 3 Ribet, Kenneth Alan 3 Rota, Gian-Carlo 3 Triebel, Hans 2 Alon, Noga M. 2 Arnol’d, Vladimir Igorevich 2 Aubin, Jean-Pierre 2 Bourgain, Jean 2 Connes, Alain 2 Duistermaat, Johannes Jisse 2 Falb, Peter L. 2 Gusein-Zade, Sabir M. 2 Jardine, John Frederick 2 Milman, Vitali D. 2 Varchenko, Alexander Nikolaevich 1 Amrein, Werner O. 1 Bardi, Martino 1 Başar, Tamer 1 Berndt, Rolf 1 Bernhard, Pierre 1 Bethuel, Fabrice 1 Bichteler, Klaus 1 Borel, Armand 1 Boutet de Monvel, Anne Marie 1 Brézis, Haïm 1 Brieskorn, Egbert 1 Bruggeman, Roelof Wichert 1 Brylinski, Jean-Luc 1 Buser, Peter 1 Capuzzo Dolcetta, Italo 1 Cherix, Pierre-Alain 1 Cherkaev, Andrej V. 1 Chung, Kai Lai 1 Collet, Pierre 1 Conlon, Lawrence 1 Cowling, Michael G. 1 Davis, Philip J. 1 Denker, Manfred 1 Dieudonné, Jean Alexandre 1 Drucker, Thomas L. 1 Eckmann, Jean-Pierre 1 Edwards, Harold Mortimer jun. 1 Elias, Juan 1 Facchini, Alberto 1 Filippov, Alexandre Tikhonovich 1 Fomin, Vasiliĭ Mikhaĭlovich 1 Frankowska, Hélène 1 Freeman, Randy A. 1 Frenkel, Victor Ya. 1 Gel’fand, Israil’ Moiseevich 1 Georgescu, Vladimir 1 Gessel, Ira Martin 1 Giral, Jose M. 1 Goerss, Paul G. 1 Gorelik, Gennady E. 1 Gray, Jeremy John 1 Greene, Daniel H. 1 Grosswald, Emil 1 Hélein, Frédéric 1 Hentschel, Klaus 1 Hersh, Reuben 1 Hofer, Helmut H. W. 1 Holz, Michael 1 Hörmander, Lars Valter 1 Jolissaint, Paul 1 Julg, Pierre 1 Kac, Mark 1 Kaiser, Gerald 1 Kanwal, Ram Prakash 1 Kapovich, Michael 1 Kapranov, Mikhail M. 1 Karloff, Howard J. 1 Kimura, Hidenori 1 Kiselev, Sergey P. 1 Knörrer, Horst 1 Knuth, Donald Ervin 1 Kohn, Robert Vita 1 Kokotovic, Petar V. 1 Krantz, Steven George 1 Kunz, Ernst 1 Laugwitz, Detlef 1 Lawler, Gregory Francis 1 Lubotzky, Alexander 1 Lunardi, Alessandra 1 Lusztig, George 1 Madras, Neal 1 Marchisotto, Elena Anne Corie 1 Mirò-Roig, Rosa Maria 1 Monastyrskiĭ, Mikhail Il’ich 1 Monk, James Donald 1 Morales-Ruiz, Juan José 1 Murasugi, Kunio 1 Murty, Maruti Ram 1 Murty, Vijaya Kumar ...and 45 more Authors all top 5 #### Fields 26 History and biography (01-XX) 16 Algebraic geometry (14-XX) 13 General and overarching topics; collections (00-XX) 12 Global analysis, analysis on manifolds (58-XX) 11 Partial differential equations (35-XX) 10 Number theory (11-XX) 9 Operator theory (47-XX) 8 Differential geometry (53-XX) 8 Manifolds and cell complexes (57-XX) 8 Systems theory; control (93-XX) 7 Dynamical systems and ergodic theory (37-XX) 6 Harmonic analysis on Euclidean spaces (42-XX) 6 Calculus of variations and optimal control; optimization (49-XX) 6 Algebraic topology (55-XX) 6 Probability theory and stochastic processes (60-XX) 5 Group theory and generalizations (20-XX) 5 Functions of a complex variable (30-XX) 5 General topology (54-XX) 5 Computer science (68-XX) 4 Mathematical logic and foundations (03-XX) 4 Combinatorics (05-XX) 4 Commutative algebra (13-XX) 4 Topological groups, Lie groups (22-XX) 4 Real functions (26-XX) 4 Measure and integration (28-XX) 4 Several complex variables and analytic spaces (32-XX) 4 Ordinary differential equations (34-XX) 4 Functional analysis (46-XX) 4 Geometry (51-XX) 4 Fluid mechanics (76-XX) 4 Quantum theory (81-XX) 3 Linear and multilinear algebra; matrix theory (15-XX) 3 Integral equations (45-XX) 3 Mechanics of deformable solids (74-XX) 3 Information and communication theory, circuits (94-XX) 3 Mathematics education (97-XX) 2 Field theory and polynomials (12-XX) 2 Associative rings and algebras (16-XX) 2 Nonassociative rings and algebras (17-XX) 2 Category theory; homological algebra (18-XX) 2 $$K$$-theory (19-XX) 2 Abstract harmonic analysis (43-XX) 2 Integral transforms, operational calculus (44-XX) 2 Numerical analysis (65-XX) 1 Order, lattices, ordered algebraic structures (06-XX) 1 Potential theory (31-XX) 1 Special functions (33-XX) 1 Approximations and expansions (41-XX) 1 Convex and discrete geometry (52-XX) 1 Statistics (62-XX) 1 Mechanics of particles and systems (70-XX) 1 Optics, electromagnetic theory (78-XX) 1 Statistical mechanics, structure of matter (82-XX) 1 Relativity and gravitational theory (83-XX) 1 Operations research, mathematical programming (90-XX) 1 Game theory, economics, finance, and other social and behavioral sciences (91-XX) #### Citations contained in zbMATH Open 79 Publications have been cited 2,113 times in 2,020 Documents Cited by Year Theory of function spaces. Reprint of the 1983 original. Zbl 1235.46002 Triebel, Hans 2010 Singularities of differentiable maps, Volume 2. Monodromy and asymptotics of integrals. Transl. from the Russian by Hugh Porteous and revised by the authors and James Montaldi. Reprint of the 1988 hardback edition. Zbl 1297.32001 Arnold, V. I.; Gusein-Zade, S. M.; Varchenko, A. N. 2012 Singularities of differentiable maps, Volume 1. Classification of critical points, caustics and wave fronts. Transl. from the Russian by Ian Porteous, edited by V. I. Arnol’d. Reprint of the 1985 hardback edition. Zbl 1290.58001 Arnold, V. I.; Gusein-Zade, S. M.; Varchenko, A. N. 2012 Theory of function spaces II. Reprint of the 1992 edition. Zbl 1235.46003 Triebel, Hans 2010 Discriminants, resultants, and multidimensional determinants. Reprint of the 1994 edition. Zbl 1138.14001 Gelfand, I. M.; Kapranov, M. M.; Zelevinsky, A. V. 2008 Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Reprint of the 1997 original. Zbl 1134.49022 Bardi, Martino; Capuzzo-Dolcetta, Italo 2008 Introduction to quantum groups. Reprint of the 1994 ed. Zbl 1246.17018 Lusztig, George 2010 Metric structures for Riemannian and non-Riemannian spaces. Transl. from the French by Sean Michael Bates. With appendices by M. Katz, P. Pansu, and S. Semmes. Edited by J. LaFontaine and P. Pansu. 3rd printing. Zbl 1113.53001 Gromov, Misha 2007 Set-valued analysis. Reprint of the 1990 original. Zbl 1168.49014 Aubin, Jean-Pierre; Frankowska, Hélène 2009 Discrete groups, expanding graphs and invariant measures. With an appendix by Jonathan D. Rogawski. Reprint of the 1994 original. Zbl 1183.22001 Lubotzky, Alexander 2010 Analytic semigroups and optimal regularity in parabolic problems. Reprint of the 1995 hardback ed. Zbl 1261.35001 Lunardi, Alessandra 2013 Linear algebraic groups. Reprint of the 1998 2nd ed. Zbl 1202.20048 Springer, T. A. 2009 Vector bundles on complex projective spaces. With an appendix by S. I. Gelfand. Corrected reprint of the 1988 edition. Zbl 1237.14003 Okonek, Christian; Schneider, Michael; Spindler, Heinz 2011 Geometry and spectra of compact Riemann surfaces. Reprint of the 1992 original. Zbl 1239.32001 Buser, Peter 2010 Optimal control. Reprint of the 2000 ed. Zbl 1215.49002 Vinter, Richard 2010 Robust nonlinear control design. State-space and Lyapunov techniques. Softcover reprint of the 1996 ed. Zbl 1130.93005 Freeman, Randy A.; Kokotović, Petar V. 2008 Loop spaces, characteristic classes and geometric quantization. Reprint of the 1993 edition. Zbl 1136.55001 Brylinski, Jean-Luc 2008 Simplicial homotopy theory. Reprint of the 1999 original. Zbl 1195.55001 Goerss, Paul G.; Jardine, John F. 2010 An introduction to quantum stochastic calculus. Reprint of the 1992 ed. Zbl 1338.81010 Parthasarathy, K. R. 2013 Hyperbolic manifolds and discrete groups. Reprint of the 2001 hardback edition. Zbl 1180.57001 Kapovich, Michael 2009 Viability theory. 2nd printing. Zbl 1179.93001 Aubin, Jean-Pierre 2009 The implicit function theorem. History, theory, and applications. Reprint of the 2003 hardback edition. Zbl 1269.58003 Krantz, Steven G.; Parks, Harold R. 2013 Notions of convexity. Reprint of the 1994 edition. Zbl 1108.32001 Hörmander, Lars 2007 The self-avoiding walk. Reprint of the 1996 edition. Zbl 1254.01051 2013 Mathematical control theory: an introduction. Reprint of the 2nd corrected printing 1995. Zbl 1123.93003 Zabczyk, Jerzy 2008 Intersections of random walks. Reprint of the 1996 ed. Zbl 1253.60003 Lawler, Gregory F. 2013 $$H^ \infty$$-optimal control and related minimax design problems. A dynamic game approach. Softcover reprint of the 1995 2nd ed. Zbl 1130.93002 Başar, Tamer; Bernhard, Pierre 2008 Tata lectures on theta. II: Jacobian theta functions and differential equations. With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman, and H. Umemura. Reprint of the 1984 edition. Zbl 1112.14003 Mumford, David 2007 Non-vanishing of $$L$$-functions and applications. Reprint of the 1997 edition. Zbl 1235.11086 Murty, M. Ram; Murty, V. Kumar 2012 Dynamical systems of algebraic origin. Reprint of the 1995 edition. Zbl 1248.37004 Schmidt, Klaus 2011 Iterated maps on the interval as dynamical systems. Reprint of the 1980 original. Zbl 1192.37003 Collet, Pierre; Eckmann, Jean-Pierre 2009 Fourier integral operators. Reprint of the 1996 ed. Zbl 1214.35092 Duistermaat, J. J. 2011 Tata lectures on theta. I. With the collaboration of C. Musili, M. Nori, E. Previato, and M. Stillman. Reprint of the 1983 edition. Zbl 1112.14002 Mumford, David 2007 Elements of the representation theory of the Jacobi group. Reprint of the 1998 original. Zbl 1235.11046 Berndt, Rolf; Schmidt, Ralf 2012 Module theory. Endomorphism rings and direct sum decompositions in some classes of modules. Reprint of the 1998 original. Zbl 1273.16007 Facchini, Alberto 2012 Symplectic invariants and Hamiltonian dynamics. Reprint of the 1994 original. Zbl 1223.37001 Hofer, Helmut; Zehnder, Eduard 2011 Mathematics for the analysis of algorithms. Reprint of the 1990 ed. Zbl 1151.68750 Greene, Daniel; Knuth, Donald E. 2008 A friendly guide to wavelets. Reprint of the 1994 original. Zbl 1230.42001 Kaiser, Gerald 2011 Differential Galois theory and non-integrability of Hamiltonian systems. Reprint of the 1999 original. Zbl 1295.37015 Morales Ruiz, Juan J. 2013 The Navier-Stokes equations. An elementary functional analytic approach. Reprint of the 2001 hardback ed. Zbl 1388.35001 Sohr, Hermann 2013 Algebraic $$K$$-theory. Paperback reprint of the 1996 2nd edition. Zbl 1125.19300 Srinivas, V. 2008 A history of algebraic and differential topology 1900–1960. 2nd printing. Zbl 1180.55001 Dieudonné, Jean 2009 Tata lectures on theta III. In collaboration with Madhav Nori and Peter Norman. Reprint of the 1991 edition. Zbl 1124.14043 Mumford, David 2007 Generalized polygons. Reprint of the 1998 edition. Zbl 1235.51002 Van Maldeghem, Hendrik 2011 Generalized étale cohomology theories. Reprint of the 1997 original. Zbl 1207.19001 Jardine, J. F. 2010 Evolutionary integral equations and applications. Reprint of the 1993 original. Zbl 1258.45008 Prüss, Jan 2012 The structure of functions. Reprint of the 2001 hardback ed. Zbl 1282.46002 Triebel, Hans 2013 Differentiable manifolds. Reprint of the 2nd ed. 2001. Zbl 1137.57001 Conlon, Lawrence 2008 Ginzburg-Landau vortices. Reprint of the 1994 hardback edition. Zbl 1372.35002 Bethuel, Fabrice; Brezis, Haïm; Hélein, Frédéric 2017 Knot theory and its applications. Transl. from the Japanese by Bohdan Kurpita. Reprint of the 1996 translated edition. Zbl 1138.57001 Murasugi, Kunio 2008 Plane algebraic curves. Transl. from the German by John Stillwell. Reprint of the hardback ed. 1986. Zbl 1254.14002 Brieskorn, Egbert; Knörrer, Horst 2012 Convex integration theory. Solutions to the $$h$$-principle in geometry and topology. Reprint of the 1998 original. Zbl 1223.58007 Spring, David 2010 Linear representations of groups. Transl. from the Russian by A. Iacob. Reprint of the 1989 original. Zbl 1206.20008 Vinberg, Ernest B. 2010 Introduction to stochastic integration. Reprint of the 1990 2nd edition. Zbl 1279.60002 Chung, Kai Lai; Williams, Ruth J. 2014 A probability path. Reprint of the 2005 edition. Zbl 1280.60001 Resnick, Sidney I. 2014 Number theory. An approach through history from Hammurapi to Legendre. Reprint of the 1984 edition. Zbl 1149.01013 Weil, André 2007 Groups with the Haagerup property. Gromov’s a-T-menability. Reprint of the 2001 edition. Zbl 1308.43001 Cherix, Pierre-Alain; Cowling, Michael; Jolissaint, Paul; Julg, Pierre; Valette, Alain 2015 Introduction to commutative algebra and algebraic geometry. Transl. from the German by Michael Ackerman. With a preface by David Mumford. Reprint of the 1985 edition. Zbl 1263.13001 Kunz, Ernst 2013 Linear differential equations and group theory from Riemann to Poincaré. Reprint of the 2nd edition 2000. Zbl 1141.01010 Gray, Jeremy J. 2008 Introduction to cardinal arithmetic. Reprint of the 1999 original. Zbl 1187.03037 Holz, M.; Steffens, K.; Weitz, E. 2010 The versatile soliton. Transl. from the Russian. Reprint of the 2000 edition. Zbl 1205.35001 Filippov, Alexandre T. 2010 Topics from the theory of numbers. Reprint of the 1984 2nd ed. Zbl 1247.11002 Grosswald, Emil 2009 The heat kernel Lefschetz fixed point formula for the spin-c Dirac operator. Reprint of the 1996 original. Zbl 1222.58014 Duistermaat, J. J. 2011 An introduction to the mechanics of fluids. 2nd printing. Zbl 1153.76005 Truesdell, C.; Rajagopal, K. R. 2009 Families of automorphic forms. Reprint of the 1994 orginial. Zbl 1184.11016 Bruggeman, Roelof W. 2010 Logic for computer scientists. Reprint of the 1989 hardback ed. Zbl 1206.03001 Schöning, Uwe 2008 Linear integral equations. Theory and technique. Reprint of the 2nd edition 1997. Zbl 1259.45001 Kanwal, Ram P. 2013 Bernhard Riemann 1826-1866: Turning points in the conception of mathematics. Transl. from the German by Abe Shenitzer. With the editorial assistance of the author, Hardy Grant, and Sarah Shenitzer. Reprint of the 1999 original. Zbl 1140.01019 Laugwitz, Detlef 2008 Riemann, topology and physics. Transl. from the Russian by Roger Cooke, James King and Victoria King. With a foreword by Freeman J. Dyson. Reprint of the 2nd ed. 1999. Zbl 1141.01019 Monastyrsky, Michael 2008 Indiscrete thoughts. With forewords by Reuben Hersh and Robert Sokolowski. Edited, with notes and an epilogue by Fabrizio Palombi. Reprint of the 1997 original. Zbl 1131.00005 Rota, Gian-Carlo 2008 Bifurcations of planar vector fields and Hilbert’s sixteenth problem. Reprint of the 1998 edition. Zbl 1278.37002 Roussarie, Robert 2013 $$C_0$$-groups, commutator methods and spectral theory of $$N$$-body Hamiltonians. Reprint of the 1996 original. Zbl 1278.47001 Amrein, Werner O.; Boutet de Monvel, Anne; Georgescu, Vladimir 2013 Classic papers in combinatorics. Reprint of the 1987 original. Zbl 1154.05001 Gessel, Ira (ed.); Rota, Gian-Carlo (ed.) 2009 Beyond the quartic equation. Paperback reprint of the 1996 edition. Zbl 1177.12001 King, R. Bruce 2008 Schrödinger equations and diffusion theory. Reprint of the 1993 edition. Zbl 1375.60012 Nagasawa, Masao 2013 The non-Euclidean revolution. With an introduction by H. S. M. Coxeter. Reprint of the 1987 original. Zbl 1133.51001 Trudeau, Richard J. 2008 The mathematical experience. Study edition. With an introduction by Gian-Carlo Rota. Reprint of the 1995 edition, updated with epilogues by the authors. Zbl 1230.00003 Davis, Philip J.; Hersh, Reuben; Marchisotto, Elena Anne 2012 Visions in mathematics. GAFA 2000 special volume, Part II. Proceedings of the meeting “Visions in Mathematics – Towards 2000, Tel Aviv, Israel, August 25–September 3, 1999. Special volume of the journal Geometric and Functional Analysis – GAFA. Reprint of the 2000 original. Zbl 1185.00038 Alon, Noga (ed.); Bourgain, Jean (ed.); Connes, Alain (ed.); Gromov, Mikhael (ed.); Milman, Vitali D. (ed.) 2010 Advanced calculus. A differential forms approach. Reprint of the 1994 edition. Zbl 1279.26002 Edwards, Harold M. 2014 Ginzburg-Landau vortices. Reprint of the 1994 hardback edition. Zbl 1372.35002 Bethuel, Fabrice; Brezis, Haïm; Hélein, Frédéric 2017 Groups with the Haagerup property. Gromov’s a-T-menability. Reprint of the 2001 edition. Zbl 1308.43001 Cherix, Pierre-Alain; Cowling, Michael; Jolissaint, Paul; Julg, Pierre; Valette, Alain 2015 Introduction to stochastic integration. Reprint of the 1990 2nd edition. Zbl 1279.60002 Chung, Kai Lai; Williams, Ruth J. 2014 A probability path. Reprint of the 2005 edition. Zbl 1280.60001 Resnick, Sidney I. 2014 Advanced calculus. A differential forms approach. Reprint of the 1994 edition. Zbl 1279.26002 Edwards, Harold M. 2014 Analytic semigroups and optimal regularity in parabolic problems. Reprint of the 1995 hardback ed. Zbl 1261.35001 Lunardi, Alessandra 2013 An introduction to quantum stochastic calculus. Reprint of the 1992 ed. Zbl 1338.81010 Parthasarathy, K. R. 2013 The implicit function theorem. History, theory, and applications. Reprint of the 2003 hardback edition. Zbl 1269.58003 Krantz, Steven G.; Parks, Harold R. 2013 The self-avoiding walk. Reprint of the 1996 edition. Zbl 1254.01051 2013 Intersections of random walks. Reprint of the 1996 ed. Zbl 1253.60003 Lawler, Gregory F. 2013 Differential Galois theory and non-integrability of Hamiltonian systems. Reprint of the 1999 original. Zbl 1295.37015 Morales Ruiz, Juan J. 2013 The Navier-Stokes equations. An elementary functional analytic approach. Reprint of the 2001 hardback ed. Zbl 1388.35001 Sohr, Hermann 2013 The structure of functions. Reprint of the 2001 hardback ed. Zbl 1282.46002 Triebel, Hans 2013 Introduction to commutative algebra and algebraic geometry. Transl. from the German by Michael Ackerman. With a preface by David Mumford. Reprint of the 1985 edition. Zbl 1263.13001 Kunz, Ernst 2013 Linear integral equations. Theory and technique. Reprint of the 2nd edition 1997. Zbl 1259.45001 Kanwal, Ram P. 2013 Bifurcations of planar vector fields and Hilbert’s sixteenth problem. Reprint of the 1998 edition. Zbl 1278.37002 Roussarie, Robert 2013 $$C_0$$-groups, commutator methods and spectral theory of $$N$$-body Hamiltonians. Reprint of the 1996 original. Zbl 1278.47001 Amrein, Werner O.; Boutet de Monvel, Anne; Georgescu, Vladimir 2013 Schrödinger equations and diffusion theory. Reprint of the 1993 edition. Zbl 1375.60012 Nagasawa, Masao 2013 Singularities of differentiable maps, Volume 2. Monodromy and asymptotics of integrals. Transl. from the Russian by Hugh Porteous and revised by the authors and James Montaldi. Reprint of the 1988 hardback edition. Zbl 1297.32001 Arnold, V. I.; Gusein-Zade, S. M.; Varchenko, A. N. 2012 Singularities of differentiable maps, Volume 1. Classification of critical points, caustics and wave fronts. Transl. from the Russian by Ian Porteous, edited by V. I. Arnol’d. Reprint of the 1985 hardback edition. Zbl 1290.58001 Arnold, V. I.; Gusein-Zade, S. M.; Varchenko, A. N. 2012 Non-vanishing of $$L$$-functions and applications. Reprint of the 1997 edition. Zbl 1235.11086 Murty, M. Ram; Murty, V. Kumar 2012 Elements of the representation theory of the Jacobi group. Reprint of the 1998 original. Zbl 1235.11046 Berndt, Rolf; Schmidt, Ralf 2012 Module theory. Endomorphism rings and direct sum decompositions in some classes of modules. Reprint of the 1998 original. Zbl 1273.16007 Facchini, Alberto 2012 Evolutionary integral equations and applications. Reprint of the 1993 original. Zbl 1258.45008 Prüss, Jan 2012 Plane algebraic curves. Transl. from the German by John Stillwell. Reprint of the hardback ed. 1986. Zbl 1254.14002 Brieskorn, Egbert; Knörrer, Horst 2012 The mathematical experience. Study edition. With an introduction by Gian-Carlo Rota. Reprint of the 1995 edition, updated with epilogues by the authors. Zbl 1230.00003 Davis, Philip J.; Hersh, Reuben; Marchisotto, Elena Anne 2012 Vector bundles on complex projective spaces. With an appendix by S. I. Gelfand. Corrected reprint of the 1988 edition. Zbl 1237.14003 Okonek, Christian; Schneider, Michael; Spindler, Heinz 2011 Dynamical systems of algebraic origin. Reprint of the 1995 edition. Zbl 1248.37004 Schmidt, Klaus 2011 Fourier integral operators. Reprint of the 1996 ed. Zbl 1214.35092 Duistermaat, J. J. 2011 Symplectic invariants and Hamiltonian dynamics. Reprint of the 1994 original. Zbl 1223.37001 Hofer, Helmut; Zehnder, Eduard 2011 A friendly guide to wavelets. Reprint of the 1994 original. Zbl 1230.42001 Kaiser, Gerald 2011 Generalized polygons. Reprint of the 1998 edition. Zbl 1235.51002 Van Maldeghem, Hendrik 2011 The heat kernel Lefschetz fixed point formula for the spin-c Dirac operator. Reprint of the 1996 original. Zbl 1222.58014 Duistermaat, J. J. 2011 Theory of function spaces. Reprint of the 1983 original. Zbl 1235.46002 Triebel, Hans 2010 Theory of function spaces II. Reprint of the 1992 edition. Zbl 1235.46003 Triebel, Hans 2010 Introduction to quantum groups. Reprint of the 1994 ed. Zbl 1246.17018 Lusztig, George 2010 Discrete groups, expanding graphs and invariant measures. With an appendix by Jonathan D. Rogawski. Reprint of the 1994 original. Zbl 1183.22001 Lubotzky, Alexander 2010 Geometry and spectra of compact Riemann surfaces. Reprint of the 1992 original. Zbl 1239.32001 Buser, Peter 2010 Optimal control. Reprint of the 2000 ed. Zbl 1215.49002 Vinter, Richard 2010 Simplicial homotopy theory. Reprint of the 1999 original. Zbl 1195.55001 Goerss, Paul G.; Jardine, John F. 2010 Generalized étale cohomology theories. Reprint of the 1997 original. Zbl 1207.19001 Jardine, J. F. 2010 Convex integration theory. Solutions to the $$h$$-principle in geometry and topology. Reprint of the 1998 original. Zbl 1223.58007 Spring, David 2010 Linear representations of groups. Transl. from the Russian by A. Iacob. Reprint of the 1989 original. Zbl 1206.20008 Vinberg, Ernest B. 2010 Introduction to cardinal arithmetic. Reprint of the 1999 original. Zbl 1187.03037 Holz, M.; Steffens, K.; Weitz, E. 2010 The versatile soliton. Transl. from the Russian. Reprint of the 2000 edition. Zbl 1205.35001 Filippov, Alexandre T. 2010 Families of automorphic forms. Reprint of the 1994 orginial. Zbl 1184.11016 Bruggeman, Roelof W. 2010 Visions in mathematics. GAFA 2000 special volume, Part II. Proceedings of the meeting “Visions in Mathematics – Towards 2000, Tel Aviv, Israel, August 25–September 3, 1999. Special volume of the journal Geometric and Functional Analysis – GAFA. Reprint of the 2000 original. Zbl 1185.00038 Alon, Noga; Bourgain, Jean; Connes, Alain; Gromov, Mikhael; Milman, Vitali D. 2010 Set-valued analysis. Reprint of the 1990 original. Zbl 1168.49014 Aubin, Jean-Pierre; Frankowska, Hélène 2009 Linear algebraic groups. Reprint of the 1998 2nd ed. Zbl 1202.20048 Springer, T. A. 2009 Hyperbolic manifolds and discrete groups. Reprint of the 2001 hardback edition. Zbl 1180.57001 Kapovich, Michael 2009 Viability theory. 2nd printing. Zbl 1179.93001 Aubin, Jean-Pierre 2009 Iterated maps on the interval as dynamical systems. Reprint of the 1980 original. Zbl 1192.37003 Collet, Pierre; Eckmann, Jean-Pierre 2009 A history of algebraic and differential topology 1900–1960. 2nd printing. Zbl 1180.55001 Dieudonné, Jean 2009 Topics from the theory of numbers. Reprint of the 1984 2nd ed. Zbl 1247.11002 Grosswald, Emil 2009 An introduction to the mechanics of fluids. 2nd printing. Zbl 1153.76005 Truesdell, C.; Rajagopal, K. R. 2009 Classic papers in combinatorics. Reprint of the 1987 original. Zbl 1154.05001 Gessel, Ira; Rota, Gian-Carlo 2009 Discriminants, resultants, and multidimensional determinants. Reprint of the 1994 edition. Zbl 1138.14001 Gelfand, I. M.; Kapranov, M. M.; Zelevinsky, A. V. 2008 Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Reprint of the 1997 original. Zbl 1134.49022 Bardi, Martino; Capuzzo-Dolcetta, Italo 2008 Robust nonlinear control design. State-space and Lyapunov techniques. Softcover reprint of the 1996 ed. Zbl 1130.93005 Freeman, Randy A.; Kokotović, Petar V. 2008 Loop spaces, characteristic classes and geometric quantization. Reprint of the 1993 edition. Zbl 1136.55001 Brylinski, Jean-Luc 2008 Mathematical control theory: an introduction. Reprint of the 2nd corrected printing 1995. Zbl 1123.93003 Zabczyk, Jerzy 2008 $$H^ \infty$$-optimal control and related minimax design problems. A dynamic game approach. Softcover reprint of the 1995 2nd ed. Zbl 1130.93002 Başar, Tamer; Bernhard, Pierre 2008 Mathematics for the analysis of algorithms. Reprint of the 1990 ed. Zbl 1151.68750 Greene, Daniel; Knuth, Donald E. 2008 Algebraic $$K$$-theory. Paperback reprint of the 1996 2nd edition. Zbl 1125.19300 Srinivas, V. 2008 Differentiable manifolds. Reprint of the 2nd ed. 2001. Zbl 1137.57001 Conlon, Lawrence 2008 Knot theory and its applications. Transl. from the Japanese by Bohdan Kurpita. Reprint of the 1996 translated edition. Zbl 1138.57001 Murasugi, Kunio 2008 Linear differential equations and group theory from Riemann to Poincaré. Reprint of the 2nd edition 2000. Zbl 1141.01010 Gray, Jeremy J. 2008 Logic for computer scientists. Reprint of the 1989 hardback ed. Zbl 1206.03001 Schöning, Uwe 2008 Bernhard Riemann 1826-1866: Turning points in the conception of mathematics. Transl. from the German by Abe Shenitzer. With the editorial assistance of the author, Hardy Grant, and Sarah Shenitzer. Reprint of the 1999 original. Zbl 1140.01019 Laugwitz, Detlef 2008 Riemann, topology and physics. Transl. from the Russian by Roger Cooke, James King and Victoria King. With a foreword by Freeman J. Dyson. Reprint of the 2nd ed. 1999. Zbl 1141.01019 Monastyrsky, Michael 2008 Indiscrete thoughts. With forewords by Reuben Hersh and Robert Sokolowski. Edited, with notes and an epilogue by Fabrizio Palombi. Reprint of the 1997 original. Zbl 1131.00005 Rota, Gian-Carlo 2008 Beyond the quartic equation. Paperback reprint of the 1996 edition. Zbl 1177.12001 King, R. Bruce 2008 The non-Euclidean revolution. With an introduction by H. S. M. Coxeter. Reprint of the 1987 original. Zbl 1133.51001 Trudeau, Richard J. 2008 Metric structures for Riemannian and non-Riemannian spaces. Transl. from the French by Sean Michael Bates. With appendices by M. Katz, P. Pansu, and S. Semmes. Edited by J. LaFontaine and P. Pansu. 3rd printing. Zbl 1113.53001 Gromov, Misha 2007 Notions of convexity. Reprint of the 1994 edition. Zbl 1108.32001 Hörmander, Lars 2007 Tata lectures on theta. II: Jacobian theta functions and differential equations. With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman, and H. Umemura. Reprint of the 1984 edition. Zbl 1112.14003 Mumford, David 2007 Tata lectures on theta. I. With the collaboration of C. Musili, M. Nori, E. Previato, and M. Stillman. Reprint of the 1983 edition. Zbl 1112.14002 Mumford, David 2007 Tata lectures on theta III. In collaboration with Madhav Nori and Peter Norman. Reprint of the 1991 edition. Zbl 1124.14043 Mumford, David 2007 Number theory. An approach through history from Hammurapi to Legendre. Reprint of the 1984 edition. Zbl 1149.01013 Weil, André 2007 all top 5 #### Cited by 3,155 Authors 13 Sickel, Winfried 10 Natroshvili, David 9 Escher, Joachim 9 Haroske, Dorothee D. 9 Sawano, Yoshihiro 9 Yang, Dachun 8 Gusein-Zade, Sabir M. 8 Skrzypczak, Leszek 8 Triebel, Hans 8 Yuan, Wen 7 Fan, Jishan 7 Georgiadis, Athanasios G. 7 Guidetti, Davide 7 Isaev, Alexander 7 Janeczko, Stanisław 7 Mironchenko, Andrii 6 Berceanu, Stefan 6 Hieber, Matthias 6 Petrushev, Pencho P. 6 Ragusa, Maria Alessandra 6 Ruzhansky, Michael V. 6 Seeger, Andreas 6 Shakhmurov, Veli B. 6 Wang, Yinxia 5 Averboukh, Yuriĭ Vladimirovich 5 Bachmayr, Annette 5 Davydov, Alekseĭ Aleksandrovich 5 Duduchava, Roland 5 Fang, Daoyuan 5 Gala, Sadek 5 Gough, John E. 5 Greenblatt, Michael 5 Gupta, Subhojoy 5 Katzarkov, Ludmil 5 Liu, Zhengrong 5 Pilaud, Vincent 5 Schmeisser, Hans-Jürgen 5 Tang, Hao 5 Vasil’ev, Viktor Anatol’evich 5 Wang, Yuzhu 5 Wu, Jiahong 4 Albeverio, Sergio A. 4 Bao, Huanchen 4 Boutry, Nicolas 4 Chae, Dongho 4 Chaubey, Yogendra P. 4 Chen, Jiecheng 4 Cleanthous, Galatia 4 Colding, Tobias Holck 4 Favero, David 4 Frankowska, Hélène 4 Frauenfelder, Urs Adrian 4 Géraud, Thierry 4 Giga, Yoshikazu 4 Hermosilla, Cristopher 4 Hussein, Amru 4 Izumiya, Shyuichi 4 Johnsen, Jon 4 Kim, KyeongHun 4 Krishna, Amalendu 4 Kröner, Axel 4 Maciejewski, Andrzej J. 4 Marigonda, Antonio 4 Matioc, Bogdan-Vasile 4 Moussai, Madani 4 Nielsen, Morten 4 Pattanayak, Santosha Kumar 4 Prömel, David J. 4 Schneider, Cornelia 4 Taylor, Jay 4 Tomita, Naohito 4 Trabs, Mathias 4 Ullrich, Tino 4 Wang, Weiqiang 4 Zakalyukin, Vladimir M. 4 Zhou, Yong 3 Alimohammady, Mohsen 3 Alkauskas, Giedrius 3 Astashov, Evgeniĭ Aleksandrovich 3 Ballard, Matthew Robert 3 Bettiol, Piernicola 3 Bhat, B. V. Rajarama 3 Binyamini, Gal 3 Bivià-Ausina, Carles 3 Borodzik, Maciej 3 Bringmann, Kathrin 3 Caetano, António M. 3 Cattani, Carlo 3 Chen, Zhimin 3 Colaneri, Patrizio 3 Colombo, Fabrizio 3 Delgado, Julio 3 Ding, Yong 3 Doosti, Hassan 3 Ebeling, Wolfgang 3 Fan, Zhaobing 3 Favini, Angelo 3 Feleqi, Ermal 3 Ferrara, Sergio 3 Fu, Zunwei ...and 3,055 more Authors all top 5 #### Cited in 453 Journals 82 Mathematische Nachrichten 55 Transactions of the American Mathematical Society 39 Journal of Mathematical Physics 39 Advances in Mathematics 34 Mathematical Methods in the Applied Sciences 28 Communications in Partial Differential Equations 26 Communications in Mathematical Physics 24 Journal of Functional Analysis 23 Journal of Algebra 23 Journal of Pure and Applied Algebra 22 Proceedings of the American Mathematical Society 21 Automatica 20 Mathematische Annalen 19 SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 18 Journal of Geometry and Physics 18 Annales de l’Institut Fourier 17 Journal of Mathematical Analysis and Applications 17 Functional Analysis and its Applications 17 Journal of Differential Equations 16 Ergodic Theory and Dynamical Systems 16 International Journal of Mathematics 16 Journal of High Energy Physics 15 Topology and its Applications 15 Journal of Mathematical Sciences (New York) 14 Applicable Analysis 14 Communications in Algebra 14 Mathematische Zeitschrift 14 The Journal of Fourier Analysis and Applications 13 Reviews in Mathematical Physics 13 Linear Algebra and its Applications 13 Selecta Mathematica. New Series 13 Proceedings of the Steklov Institute of Mathematics 12 Israel Journal of Mathematics 12 Numerical Functional Analysis and Optimization 12 SIAM Journal on Control and Optimization 11 Journal of Optimization Theory and Applications 11 Abstract and Applied Analysis 11 Boundary Value Problems 10 Compositio Mathematica 10 Geometriae Dedicata 10 Inventiones Mathematicae 10 Discrete and Continuous Dynamical Systems 10 Journal of Inequalities and Applications 10 Geometry & Topology 10 Dynamic Games and Applications 9 The Annals of Probability 9 Systems & Control Letters 9 Revista Matemática Complutense 8 Duke Mathematical Journal 8 Manuscripta Mathematica 8 Communications in Statistics. Theory and Methods 8 Journal de Mathématiques Pures et Appliquées. Neuvième Série 8 Potential Analysis 8 Transformation Groups 8 Communications in Contemporary Mathematics 8 Algebraic & Geometric Topology 8 Journal of the Australian Mathematical Society 8 Communications on Pure and Applied Analysis 8 Complex Variables and Elliptic Equations 7 Bulletin of the Australian Mathematical Society 7 Nuclear Physics. B 7 Annali di Matematica Pura ed Applicata. Serie Quarta 7 Journal of Symbolic Computation 7 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 7 Geometric and Functional Analysis. GAFA 7 Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI 7 Calculus of Variations and Partial Differential Equations 7 European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations 7 Journal of Mathematical Fluid Mechanics 7 Acta Mathematica Sinica. English Series 7 Journal of Algebra and its Applications 7 Analysis and Applications (Singapore) 7 Set-Valued and Variational Analysis 7 Science China. Mathematics 7 Evolution Equations and Control Theory 7 Arnold Mathematical Journal 7 Journal de l’École Polytechnique – Mathématiques 6 Journal of the Franklin Institute 6 Mathematics of Computation 6 Applied Mathematics and Computation 6 Glasgow Mathematical Journal 6 Memoirs of the American Mathematical Society 6 Acta Applicandae Mathematicae 6 Physica D 6 Discrete & Computational Geometry 6 The Journal of Geometric Analysis 6 Journal of Mathematical Imaging and Vision 6 Experimental Mathematics 6 NoDEA. Nonlinear Differential Equations and Applications 6 Chaos 6 Journal of Dynamical and Control Systems 6 International Journal of Wavelets, Multiresolution and Information Processing 6 Journal of Physics A: Mathematical and Theoretical 6 Journal of Function Spaces 5 Computers & Mathematics with Applications 5 Communications on Pure and Applied Mathematics 5 Journal d’Analyse Mathématique 5 Letters in Mathematical Physics 5 Mathematical Notes 5 Nonlinearity ...and 353 more Journals all top 5 #### Cited in 62 Fields 427 Partial differential equations (35-XX) 273 Algebraic geometry (14-XX) 250 Functional analysis (46-XX) 174 Harmonic analysis on Euclidean spaces (42-XX) 174 Differential geometry (53-XX) 147 Dynamical systems and ergodic theory (37-XX) 142 Global analysis, analysis on manifolds (58-XX) 136 Several complex variables and analytic spaces (32-XX) 136 Operator theory (47-XX) 130 Calculus of variations and optimal control; optimization (49-XX) 126 Systems theory; control (93-XX) 125 Fluid mechanics (76-XX) 113 Group theory and generalizations (20-XX) 112 Quantum theory (81-XX) 111 Probability theory and stochastic processes (60-XX) 103 Manifolds and cell complexes (57-XX) 87 Number theory (11-XX) 87 Numerical analysis (65-XX) 82 Ordinary differential equations (34-XX) 76 Nonassociative rings and algebras (17-XX) 70 Combinatorics (05-XX) 68 Functions of a complex variable (30-XX) 55 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 53 Algebraic topology (55-XX) 50 Associative rings and algebras (16-XX) 49 Computer science (68-XX) 48 Statistics (62-XX) 48 Operations research, mathematical programming (90-XX) 45 Commutative algebra (13-XX) 44 Convex and discrete geometry (52-XX) 44 Mechanics of deformable solids (74-XX) 43 Statistical mechanics, structure of matter (82-XX) 42 Topological groups, Lie groups (22-XX) 40 Category theory; homological algebra (18-XX) 38 Real functions (26-XX) 32 Biology and other natural sciences (92-XX) 29 Measure and integration (28-XX) 28 Abstract harmonic analysis (43-XX) 27 Linear and multilinear algebra; matrix theory (15-XX) 27 Approximations and expansions (41-XX) 26 Mechanics of particles and systems (70-XX) 24 $$K$$-theory (19-XX) 22 Potential theory (31-XX) 21 Relativity and gravitational theory (83-XX) 21 Information and communication theory, circuits (94-XX) 20 Geometry (51-XX) 19 Field theory and polynomials (12-XX) 18 Mathematical logic and foundations (03-XX) 17 Optics, electromagnetic theory (78-XX) 15 Integral equations (45-XX) 15 General topology (54-XX) 13 History and biography (01-XX) 12 Integral transforms, operational calculus (44-XX) 11 Special functions (33-XX) 10 Difference and functional equations (39-XX) 10 Classical thermodynamics, heat transfer (80-XX) 8 Geophysics (86-XX) 6 General and overarching topics; collections (00-XX) 6 Order, lattices, ordered algebraic structures (06-XX) 2 Sequences, series, summability (40-XX) 2 Mathematics education (97-XX) 1 Astronomy and astrophysics (85-XX)	2021-09-25 18:43:59	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5114467740058899, "perplexity": 4243.603362791858}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057733.53/warc/CC-MAIN-20210925172649-20210925202649-00335.warc.gz"}
http://docs.scipy.org/doc/numpy-1.4.x/reference/generated/numpy.vdot.html	# numpy.vdot¶ numpy.vdot(a, b) Return the dot product of two vectors. The vdot(a, b) function handles complex numbers differently than dot(a, b). If the first argument is complex the complex conjugate of the first argument is used for the calculation of the dot product. For 2-D arrays it is equivalent to matrix multiplication, and for 1-D arrays to inner product of vectors (with complex conjugation of a). For N dimensions it is a sum product over the last axis of a and the second-to-last of b: dot(a, b)[i,j,k,m] = sum(a[i,j,:] * b[k,:,m]) Parameters: a : array_like If a is complex the complex conjugate is taken before calculation of the dot product. b : array_like Second argument to the dot product. output : ndarray Returns dot product of a and b. Can be an int, float, or complex depending on the types of a and b. See also dot Return the dot product without using the complex conjugate of the first argument. Notes The dot product is the summation of element wise multiplication. Examples >>> a = np.array([1+2j,3+4j]) >>> b = np.array([5+6j,7+8j]) >>> np.vdot(a, b) (70-8j) >>> np.vdot(b, a) (70+8j) >>> a = np.array([[1, 4], [5, 6]]) >>> b = np.array([[4, 1], [2, 2]]) >>> np.vdot(a, b) 30 >>> np.vdot(b, a) 30 numpy.dot numpy.inner	2015-03-28 14:21:51	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7669293284416199, "perplexity": 1409.0755118608497}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-14/segments/1427131297587.67/warc/CC-MAIN-20150323172137-00227-ip-10-168-14-71.ec2.internal.warc.gz"}
https://www.proofwiki.org/wiki/Definition:Normal_Operator	# Definition:Normal Operator ## Definition Let $\HH$ be a Hilbert space. Let $\mathbf T: \HH \to \HH$ be a bounded linear operator. Then $\mathbf T$ is said to be normal if and only if: $\mathbf T^* \mathbf T = \mathbf T \mathbf T^$ where $\mathbf T^$ denotes the adjoint of $\mathbf T$.	2023-02-06 09:24:56	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9909980893135071, "perplexity": 83.23678187394374}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500334.35/warc/CC-MAIN-20230206082428-20230206112428-00696.warc.gz"}
http://cloud.originlab.com/doc/Quick-Help/calculate-half-life	# 3.116 FAQ-676 We are trying to use a single exponential decay equation to determine the half-life of a compound, but your equation is slightly different than the standard form. How do we calculate the half-life? Last Update: 2/3/2015 Typically, the standard form of the single exponential decay function is $A(t) = A_0e^{-kt}$ where $A_0$ is the initial population, $k$ is the decay constant, and $t$ is time. In this case the formula for $t$ is $\frac {\ln(2)} {k}$. In Origin's case, one of its single exponential decay equations (ExpDecay1) is described as: $y = y_0 + Ae^{\frac {-(x-x0)} {t}}$ Suppose $y_0 = 0$. The equation then becomes $y = Ae^{\frac {-(x-x0)} {t}}$. If the equations are then set equal to each other and solved for $k$ one finds that $k=\frac {-(x-x_0)} {t^2}$. Since this is the case, the equation for a half-life becomes $t(\frac {1}{2}) = x_0 + t\ln(2)$ Keywords:Exponential Fit	2019-03-24 00:58:34	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 12, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9542933106422424, "perplexity": 196.27787527690248}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912203123.91/warc/CC-MAIN-20190324002035-20190324024035-00135.warc.gz"}
https://hal.inria.fr/inria-00000373v2	# An Experimental Assessment of the 2D Visibility Complex 1 VEGAS - Effective Geometric Algorithms for Surfaces and Visibility INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications Abstract : We make an experimental assessment of the size of the 2D visibility complex of disjoint unit discs randomly distributed in the plane with density $\mu$. We observe that the number of free bitangents is asymptotically linear in the number of discs and we study the dependence of the linear asymptote in terms of the density of the scene. Specifically, for a particular range of scene densities $\mu$, we exhibit an approximation of the number of free bitangents in terms of $\mu$ and the number $n$ of discs, for $n$ larger than some function of $\mu$. We also notice how our approximation gained for rather large densities can be used to guess the onset of the linear behavior for small densities. Document type : Conference papers https://hal.inria.fr/inria-00000373 Contributor : Sylvain Lazard <> Submitted on : Tuesday, October 4, 2005 - 12:12:25 PM Last modification on : Friday, February 26, 2021 - 3:28:08 PM Long-term archiving on: : Monday, September 20, 2010 - 11:39:23 AM ### Identifiers • HAL Id : inria-00000373, version 2 ### Citation Hazel Everett, Sylvain Lazard, Sylvain Petitjean, Linqiao Zhang. An Experimental Assessment of the 2D Visibility Complex. 17th Canadian Conference on Computational Geometry - CCCG'2005, Aug 2005, Windsor, Canada. ⟨inria-00000373v2⟩ Record views	2021-09-27 11:18:21	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.44956618547439575, "perplexity": 2069.0297878354604}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780058415.93/warc/CC-MAIN-20210927090448-20210927120448-00082.warc.gz"}
https://www.numerade.com/questions/find-the-exact-length-of-the-curve-y-ln-1-x2-0-le-x-le-frac12/	💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here! # Find the exact length of the curve.$y = \ln (1 - x^2)$ , $0 \le x \le \frac{1}{2}$ ## $-\frac{1}{2}+\ln 3$ #### Topics Applications of Integration ### Discussion You must be signed in to discuss. Lectures Join Bootcamp ### Video Transcript it's Clara. So when you read here, So here, we're gonna find that exactly. We're gonna first start with the derivative. And we got negative two acts over one minus x square. We're gonna use this on plug it into our our Klink equation Square root of one plus negative to X over one minus x square square. When we factor this out we got four x square all over one minus two X square less x to the fourth when we're gonna make one bye. Using a common denominator two X square, it was four x because X to the fore. Excuse me one minus two X square plus X to the fourth. And when we add and simplify, this part becomes from zero to wouldn't have one plus x square over one minus x squared DX. Because we're taking this square root off the integral. We're gonna defy the top and bottom. Using long division in this equals from 0 to 1/2 negative one plus two over one minus x squared d x and we're going to use partial fractions. So a over one plus tax must be over one minus X or finding A and B and we get a to be one be to be one. So it becomes negative X plus from 0 to 1/2 one over one plus x plus one over one minus x the ex. When we just integrate this, we got negative one have plus Helton of three. #### Topics Applications of Integration Lectures Join Bootcamp	2021-09-23 05:29:08	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6112711429595947, "perplexity": 1400.9247212756757}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057417.10/warc/CC-MAIN-20210923044248-20210923074248-00153.warc.gz"}
https://physics.com.hk/2021/02/12/problem-2-6/	# Problem 2.6 A First Course in String Theory . 2.5 Constructing $\displaystyle{T^2/\mathbb{Z}_3}$ orbifold (a) A fundamental domain, with its boundary, is the parallelogram with corners at $\displaystyle{z = 0, 1}$ and $\displaystyle{e^{i \pi/3}}$. Where is the fourth corner? Make a sketch and indicate the identifications on the boundary. The resulting space is an oblique torus. (b) Consider now an additional $\displaystyle{\mathbb{Z}_3}$ identification $\displaystyle{z \sim R(z) = e^{2 \pi i/3} z}$ To understand how this identification acts on the oblique torus, draw the short diagonal that divides the torus into two equilateral triangles. Describe carefully the $\displaystyle{{Z}_3}$ action on each of the two triangles (recall that the action of $\displaystyle{R}$ can be followed by arbitrary action with $\displaystyle{T_1}$, $\displaystyle{T_2}$, and their inverses). [guess] (a) $\displaystyle{z = 1 + e^{\frac{i \pi}{3}}}$ (b) [guess] — Me@2021-02-11 06:03:36 PM . .	2021-09-26 21:54:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 10, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9316924810409546, "perplexity": 637.4268618349868}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057973.90/warc/CC-MAIN-20210926205414-20210926235414-00237.warc.gz"}
http://conversion.org/speed/kilometre-per-second/inches-per-minute	# kilometre per second to inches per minute conversion Conversion number between kilometre per second [km/s] and inches per minute [ipm] is 2362204.7244095. This means, that kilometre per second is bigger unit than inches per minute. ### Contents [show][hide] Switch to reverse conversion: from inches per minute to kilometre per second conversion ### Enter the number in kilometre per second: Decimal Fraction Exponential Expression [km/s] eg.: 10.12345 or 1.123e5 Result in inches per minute ? precision 0 1 2 3 4 5 6 7 8 9 [info] Decimal: Exponential: ### Calculation process of conversion value • 1 kilometre per second = (exactly) (1000) / ((0.0254/60)) = 2362204.7244095 inches per minute • 1 inches per minute = (exactly) ((0.0254/60)) / (1000) = 4.2333333333333 × 10-7 kilometre per second • ? kilometre per second × (1000 ("m/s"/"kilometre per second")) / ((0.0254/60) ("m/s"/"inches per minute")) = ? inches per minute ### High precision conversion If conversion between kilometre per second to metre-per-second and metre-per-second to inches per minute is exactly definied, high precision conversion from kilometre per second to inches per minute is enabled. Decimal places: (0-800) kilometre per second Result in inches per minute: ? ### kilometre per second to inches per minute conversion chart Start value: [kilometre per second] Step size [kilometre per second] How many lines? (max 100) visual: kilometre per secondinches per minute 00 1023622047.244095 2047244094.488189 3070866141.732284 4094488188.976378 50118110236.22047 60141732283.46457 70165354330.70866 80188976377.95276 90212598425.19685 100236220472.44095 110259842519.68504 Copy to Excel ## Multiple conversion Enter numbers in kilometre per second and click convert button. One number per line. Converted numbers in inches per minute: Click to select all ## Details about kilometre per second and inches per minute units: Convert Kilometre per second to other unit: ### kilometre per second Definition of kilometre per second unit: ≡ 1 km / 1 s. The speed with which the body moves 1 km in 1 second. The Earth's circulation speed is about 30 km/s. Convert Inches per minute to other unit: ### inches per minute Definition of inches per minute unit: ≡ 1 in / 60 s = 2.54 cm / 60 s. The speed with which the body moves 1 inch (or 2.54 centimetres) in 1 minute. ← Back to Speed units	2022-01-20 16:32:09	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.18724820017814636, "perplexity": 11439.07901936377}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320302355.97/warc/CC-MAIN-20220120160411-20220120190411-00647.warc.gz"}
https://www.gradesaver.com/textbooks/math/other-math/thinking-mathematically-6th-edition/chapter-3-logic-3-2-compound-statements-and-connectives-exercise-set-3-2-page-134/100	## Thinking Mathematically (6th Edition) The dominance of connectives is used to clarify the means of the statement. The ascending order in which dominance of connectives is used is: (1) Negation, $\sim$ (2) Disjunction, $\vee$ Conjunction, $\wedge$ (3) Conditional, $\to$ (4) Biconditional, $\leftrightarrow$ Biconditional statement is most dominant and Negation statement is least dominant and conjunction, disjunction has the same level of dominance. Use this order of dominance of connectives then the statement in the clarify form is: $\left( p\to p \right)\leftrightarrow \left[ \left( p\vee p \right)\to \sim p \right]$. The statement is biconditional because biconditional is most dominant. Hence, the statement in the clarify form is$\left( p\to p \right)\leftrightarrow \left[ \left( p\vee p \right)\to \sim p \right]$.	2019-11-20 07:52:13	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6645389795303345, "perplexity": 1775.743904408998}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496670512.94/warc/CC-MAIN-20191120060344-20191120084344-00301.warc.gz"}
https://tug.org/pipermail/xetex/2011-November/021965.html	[XeTeX] Wacky behavior of XeLaTeX in TeXLive 2011 Vafa Khalighi vafaklg at gmail.com Mon Nov 7 13:14:27 CET 2011 ```I uploaded a minimal working example, it's a stripped down version of a > book, text removed apart from a stub using greek text (Unicode > capabilities), the prologue is intact, the default font is Tinos, a freely > downloadable font, but if you want you can substitute it with > LiberationSerif or any other font that supports polytonic Greek. > Thanks for this. I used LiberationSerif font which is by default shipped with Ubuntu. These are the warnings, I get: LaTeX Font Warning: Font shape `EU1/LiberationSerif(0)/m/sl' undefined (Font) using `EU1/LiberationSerif(0)/m/n' instead on input line 43 . LaTeX Font Warning: Some font shapes were not available, defaults substituted. Which I believe is perfectly normal and you ought to get the same warnings with TeXLive 2010. This happens because Italic or oblique version of LiberationSerif does not exist and so fontspec uses the regular shape of the font for slanted shape (produced by \textsl, etc). Please provide me with an example for pdflatex, I don't know how to use it > :) > What I meant was that you produce a similar example with pdflatex (no polyglossia, no fontspec, no xelatex specific packages) and see if the same issue happens.	2023-03-22 00:39:50	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9734845161437988, "perplexity": 11748.768139208512}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296943747.51/warc/CC-MAIN-20230321225117-20230322015117-00342.warc.gz"}
http://aas.org/archives/BAAS/v30n3/dps98/318.htm	DPS Meeting, Madison, October 1998 Session 31P. Comets II Contributed Poster Session, Wednesday, October 14, 1998, 5:10-6:10pm, Hall of Ideas ## [31P.15] Molecular Line Mapping of Comets Hyakutake and Hale-Bopp Amy J. Lovell, F. Peter Schloerb, James E. Dickens, Christopher H. DeVries, William M. Irvine, Matthew C. Senay (FCRAO) We present a comprehensive set of spectral line maps of the molecular emission from Comets C/1996 B2 Hyakutake and C/1995 O1 Hale-Bopp, obtained with the Five College Radio Astronomy Observatory 14-m telescope. The comets were observed with the QUARRY focal plane array receiver, and the spectral line were assembled into a final data cube'' consisting of spectra at different positions within the coma. This final data set has a spatial resolution of approximately 80" and velocity resolution of approximately 0.1 km s-1. The HCN spectral lines in Comet Hyakutake during the near-perigee period revealed a systematic progression from asymmetric, single-peaked lines, to symmetric double-peaked lines. The median line velocity is blue-shifted approximately 0.1 km s-1, and the line width increases as expected with heliocentric distance. During the same time, the maps of the inner 20,000 km in the coma initially revealed strong, asymmetric emission, but evolved to weaker, azimuthally symmetric distributions. In Comet Hale-Bopp, which was observed over a greater span of time, the spectral lines were also generally asymmetric, or weakly double-peaked. The median line velocity was generally blueshifted, particularly at larger heliocentric distances, and the observed line widths are consistent with the outflow velocities derived by Biver {\it et al.} ({\it Earth, Moon & Planets}, 1998, in press). The emission maps reveal some asymmetries over the observed 200,000 km region. [Previous] \| [Session 31P] \| [Next]	2015-07-29 22:10:26	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8442447185516357, "perplexity": 9011.929143036332}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042986646.29/warc/CC-MAIN-20150728002306-00341-ip-10-236-191-2.ec2.internal.warc.gz"}
https://deepai.org/publication/lower-bounds-for-symbolic-computation-on-graphs-strongly-connected-components-liveness-safety-and-diameter	# Lower Bounds for Symbolic Computation on Graphs: Strongly Connected Components, Liveness, Safety, and Diameter A model of computation that is widely used in the formal analysis of reactive systems is symbolic algorithms. In this model the access to the input graph is restricted to consist of symbolic operations, which are expensive in comparison to the standard RAM operations. We give lower bounds on the number of symbolic operations for basic graph problems such as the computation of the strongly connected components and of the approximate diameter as well as for fundamental problems in model checking such as safety, liveness, and co-liveness. Our lower bounds are linear in the number of vertices of the graph, even for constant-diameter graphs. For none of these problems lower bounds on the number of symbolic operations were known before. The lower bounds show an interesting separation of these problems from the reachability problem, which can be solved with O(D) symbolic operations, where D is the diameter of the graph. Additionally we present an approximation algorithm for the graph diameter which requires Õ(n √(D)) symbolic steps to achieve a (1+ϵ)-approximation for any constant ϵ > 0. This compares to O(n · D) symbolic steps for the (naive) exact algorithm and O(D) symbolic steps for a 2-approximation. Finally we also give a refined analysis of the strongly connected components algorithms of Gentilini et al., showing that it uses an optimal number of symbolic steps that is proportional to the sum of the diameters of the strongly connected components. • 61 publications • 14 publications • 51 publications • 2 publications 11/08/2020 ### Tight Conditional Lower Bounds for Approximating Diameter in Directed Graphs Among the most fundamental graph parameters is the Diameter, the largest... 08/12/2020 ### Improved SETH-hardness of unweighted Diameter We prove that, assuming the Strong Exponential Time Hypothesis, for any ... 07/06/2021 ### Noisy Boolean Hidden Matching with Applications The Boolean Hidden Matching (BHM) problem, introduced in a seminal paper... 08/30/2021 ### BDD-Based Algorithm for SCC Decomposition of Edge-Coloured Graphs Edge-coloured directed graphs provide an essential structure for modelli... 11/06/2017 ### Nearly Work-Efficient Parallel Algorithm for Digraph Reachability One of the simplest problems on directed graphs is that of identifying t... 09/11/2019 ### Quasipolynomial Set-Based Symbolic Algorithms for Parity Games Solving parity games, which are equivalent to modal μ-calculus model che... 05/20/2019 ### Subcubic Equivalences Between Graph Centrality Measures and Complementary Problems Despite persistent efforts, there is no known technique for obtaining un... ## 1 Introduction Graph algorithms are central in the formal analysis of reactive systems. A reactive system consists of a set of variables and a state of the system corresponds to a set of valuations, one for each of these variables. This naturally induces a directed graph: Each vertex represents a state of the system and each directed edge represents a state transition that is possible in the system. As the number of vertices is exponential in the number of variables of the system, these graphs are huge and, thus, they are usually not explicitly represented during their analysis. Instead they are implicitly represented using e.g., binary-decision diagrams (BDDs) [Bryant86, Bryant92]. To avoid considering specifics of the implicit representation and their manipulation, an elegant theoretical model for algorithms that work on this implicit representation has been developed, called symbolic algorithms (see e.g. [BurchCMDH90, ClarkeMCH96, Somenzi99, ClarkeGP99, ClarkeGJLV03, GentiliniPP08, ChatterjeeHJS13]). In this paper we will give novel upper and (unconditional) lower bounds on the number of operations required by a symbolic algorithm for solving classic graph-algorithmic questions, such as computing the strongly connected components and the (approximate) diameter, as well as for graph-algorithmic questions that are important in the analysis of reactive systems, such as safety, liveness, and co-liveness objectives. Our lower bounds are based on new reductions of problems from communication complexity to symbolic algorithms. Symbolic algorithms. A symbolic algorithm is allowed to use the same mathematical, logical, and memory access operations as a regular RAM algorithm, except for the access to the input graph: It is not given access to the input graph through an adjacency list or adjacency matrix representation but instead only through two types of symbolic operations: 1. One-step operations Pre and Post: Each predecessor (Pre) (resp., successor (Post)) operation is given a set of vertices and returns the set of vertices with an edge to (resp., edge from) some vertex of . 2. Basic set operations: Each basic set operation is given one or two sets of vertices and performs a union, intersection, or complement on these sets. An initial set of vertices is given as part of the input, often consisting of a single vertex. Symbolic operations are more expensive than the non-symbolic operations and thus one is mainly interested in the number of symbolic operations of such an algorithm (and the exact number of non-symbolic operations is often neglected). Moreover, as the symbolic model is motivated by the compact representation of huge graphs, we aim for symbolic algorithms that only store or many sets of vertices as otherwise algorithms become impractical due to the huge space requirements. Additionally, every computable graph-algorithmic question can be solved with symbolic one-step operations when storing many sets (and allowing an unbounded number of non-symbolic operations): For every vertex perform a Pre and a Post operation, store the results, which represent the full graph, and then compute the solution on this graph, using only non-symbolic operations. Note, however, that our lower bounds do not depend on this requirement, i.e., they also apply to symbolic algorithms that store an arbitrary number of sets. Furthermore the basic set operations (that only require vertices, i.e., the current state variables) are computationally much less expensive than the one-step operations (that involve both variables of the current and of the next state). Thus, to simplify the analysis of symbolic algorithms, we only analyze the number of one-step operations in the lower bounds that we present. For all upper bounds in prior work and in our work the number of basic-set operations is at most linear in the number of one-step operations. There is an interesting relationship between the two types of symbolic operations and Boolean matrix-vector operations: Interpreting the edge relationship as a Boolean matrix and a vertex set as a Boolean vector, the one-step operations correspond to (left- and right-sided) matrix-vector multiplication, where the matrix is the adjacency matrix, and basic set operations correspond to basic vector manipulations. Thus, an equivalent way of interpreting symbolic algorithms is by saying that the access to the graph is only allowed by performing a Boolean matrix-vector multiplication, where the vector represents a set of vertices and the matrix is the adjacency matrix. Note also that there is a similarity to the CONGEST and the LOCAL model in synchronous distributed computation, as in these models each vertex in a synchronous network knows all its neighbors and can communicate with all of them in one round (exchanging bits in the CONGEST model), and the algorithmic complexity is measured in rounds of communication. While in these models all neighbors of every individual vertex, i.e., all edges of the graph, can be determined in one round of communication, in the symbolic model this might require Pre and Post operations, each on an singleton set. Thus, determining (and storing) all edges of a symbolically represented graph is expensive and we would ideally like to have algorithms that use sub-linear (in the number of vertices) many symbolic one-step operations. Objectives. First we formally introduce the most relevant graph-algorithmic questions from the analysis of reactive systems [MannaP92]. Given a graph and a starting vertex , let be the set of infinite paths in starting from . Each objective corresponds to a set of requirements on an infinite path and the question that needs to be decided by the algorithm is whether there is a path in that satisfies these requirements, in which case we say the path satisfies the objective. An objective is the dual of an objective if a path satisfies iff it does not satisfy . Let be a set of target vertices given as input. The most basic objective is reachability where an infinite path satisfies the objective if the path visits a vertex of at least once. The dual safety objective is satisfied by infinite paths that only visit vertices of . The next interesting objective is the liveness (aka Büchi) objective that requires an infinite path to visit some vertex of infinitely often. The dual co-liveness (aka co-Büchi) objective requires an infinite path to eventually only visit vertices in . Verifying these objectives are the most fundamental graph-algorithmic questions in the analysis of reactive systems. Computing the strongly connected components (SCCs) is at the heart of the fastest algorithms for liveness and co-liveness: For example, there is a reduction from liveness to the computation of SCCs that takes symbolic steps in the order of the diameter of the graph. Thus, determining the symbolic complexity of SCCs also settles the symbolic complexity of liveness. Furthermore, the diameter computation plays a crucial role in applications such as bounded model-checking [BiereCCSZ03], where the goal is to analyze the system for a bounded number of steps, and it suffices to choose the diameter of the graph as bound. Second, in many scenarios, such as in hardware verification, the graphs have small diameter, and hence algorithms that can detect if this is the case and then exploit the small diameter are relevant [BiereCCSZ03]. Motivated by these applications, we define the diameter of a graph as the largest finite distance in the graph, which coincides with the usual graph-theoretic definition on strongly connected graphs and is more general otherwise. Note that linear lower bounds for the number of symbolic operations are non-trivial, since a one-step operation can involve all edges. For example, to determine all the neighbors of a given vertex takes one symbolic operation, while it takes many operations in the classic setting. In the following we use to denote the number of vertices of a graph and to denote its diameter. Previous results. To the best of our knowledge, no previous work has established lower bounds for symbolic computation. There is some prior work on establishing upper bounds on the number of symbolic operations: In [GentiliniPP08] a symbolic algorithm that computes the SCCs with symbolic operations is presented. This algorithm leads to an algorithm for liveness and co-liveness with symbolic operations and improves on earlier work by [BloemGS06], which requires symbolic operations. Note that for the reachability objective the straightforward algorithm requires symbolic operations: Starting from the set containing only the start vertex , repeatedly perform a Post-operation until is reached. For safety the straightforward algorithm takes symbolic operations: Iteratively remove from vertices that do not have an outgoing edge to another vertex of , i.e., vertices of , until a fixed point is reached. Finally, there is a trivial algorithm for computing the diameter of the graph: Simply determine the depth of a breadth-first search from every vertex and output the maximum over all depths. Computing the depth of a breadth-first search can be done with many symbolic steps, thus this requires many symbolic steps in total. In a strongly connected graph a 2-approximation of the diameter of the graph can be obtained by computing one breadth-first search from and one to some arbitrary vertex and output the sum of the depths. This takes symbolic steps. Our contributions. Our main contributions are novel lower bounds for the number of symbolic operations for many of the above graph-algorithmic questions, leading to an interesting separation between seemingly similar problems. 1. For reachability objectives, the basic symbolic algorithm requires symbolic operations. Quite surprisingly, we show that such diameter-based upper bounds are not possible for its dual problem, namely safety, and are also not possible for liveness and co-liveness objectives. Specifically, we present tight lower bounds to show that, even for constant-diameter graphs, one-step symbolic operations are required for safety, liveness, and co-liveness objectives. In addition we establish tight bounds for symbolic operations required for the computation of SCCs, showing a lower bound of for constant-diameter graphs. See Table 1 for a summary of these results. 2. We show that even for strongly-connected constant-diameter graphs approximating the diameter requires symbolic steps. More precisely, any -approximation algorithm requires symbolic one-step operations, even for undirected and connected graphs with constant diameter. We also give a novel upper bound: We present a -approximation algorithm for any constant that takes symbolic steps. This can be compared to the trivial 2-approximation algorithm and the exact algorithm. Notice that for explicitly represented graphs the approximation of the diameter is already hard for constant-diameter graphs while in the symbolic model there exists a trivial upper bound in this case, thus showing a lower bound of is non-trivial. See Table 2 for a summary of these results. 3. Finally we give a refined analysis of the number of symbolic steps required for computing strongly connected components based on a different problem parameter. Let be the set of all SCCs of and the diameter of the strongly connected component . We give matching upper and lower bounds showing that the SCCs can be computed with symbolic steps. Note that can be a factor larger than . Key technical contribution. Our key technical contribution is based on the novel insight that lower bounds for communication complexity can be used to establish lower bounds for symbolic computation. We feel that this connection is of interest by itself and might lead to further lower bounds for symbolic algorithms. Our lower bounds are by two kinds of reductions, both from the communication complexity problem of Set Disjointness with elements. First, we give reductions that construct graphs such that one-step operations can be computed with bits of communication between Alice and Bob and thus allow for linear lower bounds on the number symbolic operations. Second, we give a reduction that constructs a graph with only many vertices, i.e., , but allows one-step operations to require bits of communication. This again results in linear lower bounds on the number of symbolic operations. ## 2 Preliminaries Symbolic Computation. We consider symbolic computation on graphs. Given an input graph and a set of vertices , the graph can be accessed only by the following two types of operations: 1. Basic set operations like , , , , and ; 2. One-step operations to obtain the predecessors or successors of the vertices of in . In particular we define the operations Pre(S)={v∈V∣∃s∈S:(v,s)∈E} % and Post(S)={v∈V∣∃s∈S:(s,v)∈E}. In the applications the basic set operations are much cheaper as compared to the one-step operations. Thus we aim for lower bounds on the number of one-step operations, while not accounting for set operations. In all our upper bounds the number of set operations is at most of the same order as the number of one-step operations. Note that there is a one-to-one correspondence between a one-step operation and a Boolean matrix-vector multiplication with the adjacency matrix and that for undirected graphs and are equivalent. Communication Complexity Lower Bound for Set Disjointness. Our lower bounds are based on the known lower bounds for the communication complexity of the Set Disjointness problem. The classical symmetric two-party communication complexity model is as follows [KushilevitzN97]. There are three finite sets , the former two are possible inputs for a function , where the actual input is only known by Alice, and the actual input is only known by Bob. Alice and Bob want to evaluate a function while sending as few bits as possible to each other. The communication happens according to a fixed protocol, known to both players beforehand, that determines which player sends which bits when, and when to stop. Set Disjointness. In the Set Disjointness problem we have a universe of elements and both sets , contain all bit vectors of length , i.e., they represent all possible subsets of and are of size . Alice has a vector and Bob has a vector , and the function is defined as if for all either or , and otherwise. We will sometimes use and to denote the sets corresponding to the vectors and , i.e., and and iff . We next state a fundamental lower bound for the communication complexity of the Set Disjointness problem which will serve as basis for our lower bounds on the number of symbolic operations. ###### Theorem 2.1 ([KalyanasundaramS92, Razborov92, Bar-YossefJKS04, HastadW07, KushilevitzN97]). Any (probabilistic bounded error or deterministic) protocol for the Set Disjointness problem sends bits in the worst case over all inputs. ## 3 Lower Bounds In this section we present our lower bounds, which are the main results of the paper. ### 3.1 Lower Bounds for Computing Strongly Connected Components We first consider the problem of computing the strongly connected components (SCCs) of a symbolically represented graph. The best known symbolic algorithm is by Gentilini et al. [GentiliniPP08] and computes the SCCs of a Graph G with symbolic one-step operations and thus matches the linear running time of the famous Tarjan algorithm [Tarjan72] in the non-symbolic world. We provide lower bounds showing that the algorithm is essentially optimal, in particular we show that algorithms are impossible. These lower bounds are by reductions from the communication complexity problem of Set Disjointness to computing SCCs in a specific graph. In particular, we show that any algorithm that computes SCCs with symbolic one-step operations would imply a communication protocol for the Set Disjointness problem with communication. ###### Reduction 3.1. Let be an instance of Set Disjointness and let w.l.o.g. for some integers . We construct a directed graph with vertices and edges as follows. (1) The vertices are given by with . (2) There is an edge from to if either or and . (3) For , there is an edge from to iff or . In our communication protocol both Alice and Bob compute the number of SCCs on the graph from Reduction 3.1, according to a given algorithm. While both know all the vertices of the graph, they do not know all the edges (some depend on both and ) and thus whenever such an edge is relevant for the algorithm, Alice and Bob have to communicate with each other. We show that the graph is constructed such that for each subset the operations and can be computed with only four bits of communication between Alice and Bob. ###### Theorem 3.2. Any (probabilistic bounded error or deterministic) symbolic algorithm that computes the SCCs of graphs with vertices needs symbolic one-step operations. Moreover, for a graph with the set of SCCs and diameter any algorithm needs symbolic one-step operations. We first show that Reduction 3.1 is a valid reduction from the Set Disjointness problem to an SCC problem. The missing proofs are given in Section 6.1. ###### Lemma 3.3. iff the graph constructed in Reduction 3.1 has exactly SCCs. The critical observation for the proof of Theorem 3.2 is that any algorithm that computes SCCs with many symbolic one-step operations implies the existence of a communication protocol for Set Disjointness that only requires communication. ###### Lemma 3.4. For any algorithm that computes SCCs with symbolic one-step operations there is a communication protocol for Set Disjointness that requires communication. ###### Proof. In our communication protocol both Alice and Bob consider the graph from Reduction 3.1. We call edges of the graph that are present independently of and definite edges and edges whose presence depends on and possible edges. Both Alice and Bob execute the given symbolic algorithm to decide whether the graph has SCCs (cf. Lemma 3.3). As both know all the vertices, they can execute set operations without communicating. Communication is only needed when executing symbolic one-step operations, since for these some of the possible edges might affect the outcome. We next argue that each symbolic one-step operations can be executed with a constant number of bits of communication. First notice that as both Alice and Bob execute the same algorithm simultaneously, they both know the input set to an operation and they only need communication about the possible edges that can change the output. Both can independently identify these possible edges and they can decide whether such an edge exists by sending one bit each. We next argue that for each one-step operation we need to consider at most two possible edges. For this we consider the vertices in their linear ordering given by , e.g., and . operation: Let be the input set and let the vertex with the minimum index in . Then we have and potentially also and can be in , but no other vertices. That is, we have . To decide whether is in , we first check whether and if so we check whether the edge is present. To decide , we check whether the edge is present. That is, to compute we only access two possible edges. operation: Let be the input set and let the vertex with the maximum index in . Then we have and potentially also and can be in , but no other vertices. That is, we have . To decide whether is in , we first check whether and if so we check whether the edge is present. To decide if , we check whether the edge is present. That is, we can compute with accessing only two possible edges. By the above we have that a symbolic algorithm with one-step operations gives rise to a communication protocol for Set Disjointness with bits of communication. ∎ By Lemma 3.4 we have that any algorithm computing SCCs with symbolic one-step operations would contradict Theorem 2.1. Now inspecting the graph of Reduction 3.1, we observe that its diameter is equal to , which leads to the following lower bounds. For the graph has diameter and thus the holds even for graphs of constant diameter. On the other side, for disjoint sets and correspond to strongly connected graphs and thus the lower bounds also holds for graphs with a bounded number of SCCs, i.e., there are no symbolic algorithms. Finally for we obtain a lower bound. ###### Remark 3.5. By the above no algorithm can compute SCCs with or symbolic one-step operations for any function . In contrast, if we consider both parameters simultaneously, there is an symbolic algorithm. The above lower bounds for computing SCCs match the bound by the algorithm of Gentilini et al. [GentiliniPP08]. One way to further improve the algorithm is to not consider the diameter of the whole graph but the diameter of each single SCC . In that direction the previous reduction already gives us an lower bound and we will next improve it to an lower bound (i.e., it is even if ). These two bounds differ if the graph has a large number of trivial SCCs. Thus we next give a reduction that constructs a graph that has only trivial SCCs if and are disjoint. ###### Reduction 3.6. Given an instance of Set Disjointness, we construct a directed graph with vertices and edges as follows. (1) The vertices are given by . (2) There is an edge from to for . (3) For there is an edge from to iff and . ###### Theorem 3.7. Any (probabilistic bounded error or deterministic) symbolic algorithm that computes the SCCs needs symbolic one-step operations. ### 3.2 Lower Bounds for Liveness, Reachability, and Safety Objectives In this section we extend our lower bounds for SCC computation to Liveness, Reachability, and Safety Objectives on graphs. Lower Bounds for Reachability. The lower bounds for Reachability are an immediate consequence from our lower bounds for SCC computation in Theorem 3.2. When setting in Reduction 3.1 then the vertex is reachable from all vertices iff the graph is strongly connected iff the sets and are disjoint. ###### Theorem 3.8. Any (probabilistic bounded error or deterministic) symbolic algorithm that solves Reachability in graphs with diameter requires symbolic one-step operations. Lower Bounds for Liveness. To show an lower bound for Liveness objectives which holds even for graphs of bounded diameter, we introduce another reduction. This reduction is again from the Set Disjointness Problem and also constructs a graph such that and operations can be executed with a constant number of bits of communication between Alice and Bob. ###### Reduction 3.9. Given an instance of Set Disjointness, we construct a directed graph with vertices and edges as follows. (1) The vertices are given by . (2) There is an edge from to for and there is a loop edge . (3) For there is a loop edge iff and . Notice that the graph in Reduction 3.9 has diameter and thus allows to show the lower bounds stated in Theorem 3.10 when considering , with the exception of (2) which is by Reduction 3.1 and . ###### Theorem 3.10. For any (probabilistic bounded error or deterministic) symbolic algorithm that solves the following lower bounds on the required number of symbolic one-step operations hold: (1) even for instances with constant ; (2) even for instances with ; (3) even for instances with constant ; (4) ; and (5) . Lower Bounds for co-Liveness and Safety. The following lower bounds are by Reduction 3.9 (and variations of it) and the set of safe vertices . ###### Theorem 3.11. For any (probabilistic bounded error or deterministic) symbolic algorithm that solves or the following lower bounds on the required number of symbolic one-step operations hold: (1) even for constant diameter graphs; (2) even for constant diameter graphs; and (3) even for constant diameter graphs. Notice that the parameters diameter, number of SCCs, or diameters of SCCs do not help in the case of Safety. This is because every graph can be reduced to a strongly connected graph with diameter without changing the winning set as follows: Add a new vertex that has an edge to and an edge from all original vertices but do not add to the safe vertices . We complete this section with a lower bound for which is by a variant of Reduction 3.1. ###### Proposition 3.12. Any (probabilistic bounded error or deterministic) symbolic algorithm that solves on graphs with diameter needs symbolic one-step operations. ### 3.3 Lower Bound for Approximate Diameter The Approximate Diameter Problem. Let be a directed graph with vertices and edges . Let denote the shortest distance from to in , i.e., the smallest number of edges of any path from to in . Recall that we define the diameter of as the maximum of over all pairs for which can reach in . 111Usually the diameter is defined over all pairs of vertices, not just the reachable ones, and is therefore if is not strongly connected. Our definition is more general since determining whether the graph is strongly connected takes only symbolic steps and additionally our definition is more natural in the symbolic setting as it provides an upper bound on the number of one-step operations needed until a fixed point is reached, which is an essential primitive in symbolic graph algorithms. We consider the problem of approximating the diameter of a graph by a factor , where the goal is to compute an estimate such that . As undirected graphs are special cases of directed graphs, the lower bound is presented for undirected graphs and the upper bound for directed graphs (see Section 4), i.e., both hold for undirected and directed graphs. Result. We show a lower bound of on the number of symbolic steps needed to distinguish between a diameter of 2 and a diameter of 3, even in an undirected connected graph. The basic symbolic algorithm for computing the diameter exactly takes many symbolic steps. Thus our lower bound is tight for constant-diameter graphs. Outline Lower Bound. We show how to encode an instance of the Set Disjointness Problem with a universe of size in an (undirected, connected) graph with vertices and edges such that 1) in a communication protocol any symbolic one-step operation can be simulated with bits and 2) the graph has diameter 2 if the two sets are disjoint and diameter 3 otherwise. Thus the communication complexity lower bound of for Set Disjointness implies a lower bound of for the number of symbolic one-step operations to compute a -approximation of the diameter of a graph with vertices. ###### Reduction 3.13. Let be an instance of the Set Disjointness problem of size and let . We construct an undirected graph with vertices and edges as follows. (1) There are three sets with vertices each and two auxiliary vertices and . We denote the -th vertex of each of with a lowercase letter indicating the set and subscript . (2) There is an edge between and and between and each vertex of and and between and each vertex of . (3) For each there is an edge between and . (4) For let be such that . There is an edge between and iff and there is an edge between and iff . We first show that this graph has diameter if and are disjoint and diameter otherwise and then show how Alice can obtain a communication protocol for the Set Disjointness problem from any symbolic algorithm that can distinguish these two cases. ###### Lemma 3.14. Let be the graph given by Reduction 3.13 and let denote its diameter. If , then , otherwise . In the graph given by Reduction 3.13 Alice knows all the vertices of the graph and all the edges except those who are constructed based on , i.e., Alice does not know the edges between and . To take into account the edges between and , Alice has to communicate with Bob. To show a lower bound on the number of symbolic steps, we show next an upper bound on the number of bits of communication between Alice and Bob needed to simulate a symbolic one-step operation on . With the simulation of the one-step operations, the symbolic algorithm can be used as a communication protocol for distinguishing whether has diameter or and thus by Lemma 3.14 to solve the Set Disjointness problem. Whenever the symbolic algorithm performs a or operation (which are equivalent on undirected graphs) for a set that contains vertices of or , then Alice can simulate this one-step operation by specifying the vertices of and that are in with a bit vector of size , where Bob answers with a bit vector, again of size , that indicates all vertices that are adjacent to . Thus the communication protocol can simulate a symbolic algorithm that performs one-step operations with at most bits of communication. Hence we have by Theorem 2.1 that and thus . Together with Lemma 3.14, this proves the following theorem. Note that any -approximation algorithm for the diameter of a graph can distinguish between diameter 2 and 3. ###### Theorem 3.15. Any (probabilistic bounded error or deterministic) symbolic algorithm that computes a -approximation of the diameter of an undirected connected graph with vertices needs symbolic one-step operations. ## 4 Upper Bounds In this work we present the following upper bounds. ### 4.1 Upper Bounds for Computing Strongly Connected Components We revisit the symbolic algorithm of Gentilini et al. [GentiliniPP08] that computes the SCCs and present a refined analysis to show that it only requires symbolic operations, improving the previously known bound and matching the lower bound of Theorem 3.7 (details in Section 6.4). ###### Theorem 4.1. The algorithm of Gentilini et al. [GentiliniPP08] computes the SCCs of a graph with symbolic operations. ### 4.2 Upper Bounds for Liveness, Reachability, and Safety Objectives The upper bounds for Reachability, Safety, Liveness, and co-Liveness, are summarized in the following proposition, which is straightforward to obtain as discussed below. ###### Proposition 4.2. Let be the number of symbolic steps required to compute the SCCs of the graph and let be the set of target/safe vertices. Then can be solved with symbolic operations; can be solved with symbolic operations; can be solved with symbolic operations; and can be solved with symbolic operations. Algorithm for Reachability. Given a target set , we can easily compute the vertices that can reach by iteratively applying operations until a fixed-point is reached. By the definition of diameter, this requires only symbolic operations. Algorithms for Liveness. A simple algorithm for Liveness first starts an algorithm for computing SCCs and whenever an SCC is reported it tests whether the SCC contains one of the target vertices and if so adds all vertices of the SCC to the winning set. Finally, after all SCCs have been processed, the algorithm computes all vertices that can reach the current winning set and adds them to the winning set. That is, in total the algorithm only needs many symbolic operations where is the number of symbolic operations required by the algorithm. An alternative algorithm for Liveness with symbolic operations is as follows. For each check with symbolic one-step operations whether the vertex can reach itself and if not remove the vertex from . Then do a standard reachability with the updated set as target, again with symbolic one-step operations. Algorithm for co-Liveness. Given a set of safe vertices, an algorithm for co-Liveness first restricts the graph to the vertices of ; in the symbolic model this can be done by intersecting the outcome of each and operation with . One then uses an SCC algorithm and whenever a non-trivial SCC is reported, all its vertices are added to the winning set. Finally, after all SCCs have been processed, all vertices that can reach the current winning set in the original graph are added to the winning set. That is, in total the algorithm only needs many symbolic operations, where comes from the linear number of symbolic operations required by the algorithm for the modified graph. Algorithm for Safety. Given a set of safe vertices, an algorithm for safety first restricts the graph to the vertices of . One then uses an SCC algorithm and whenever a non-trivial SCCs is reported, all its vertices are added to the winning set. Finally, after all SCCs have been processed, all vertices that can reach, within , the current winning set are added to the winning set. That is, in total the algorithm only needs symbolic operations as both the algorithm for the modified graph and reachability in the modified graph are in . Also notice that reachability is not bounded by as restricting the graph to vertices of can change the diameter. Notice that none of the above algorithms stores all the SCCs, but processes one SCC at a time. That is, the algorithms themselves only need a constant number of sets plus the sets stored in the algorithm for computing SCCs (which can be done with many sets). ### 4.3 Upper Bounds for Approximate Diameter We present a -approximation algorithm for the diameter (for any constant ) that takes symbolic operations (the -notation hides logarithmic factors). ###### Theorem 4.3. A -approximation of the diameter of a directed graph for any constant can be obtained with symbolic operations, using many sets. Technical Overview -Approximation Algorithm. The symbolic algorithm is based on the -approximation algorithm by Aingworth et al. [AingworthCIM99] for explicit graphs, we give a high-level overview of the differences here. An expensive step in the algorithm of [AingworthCIM99] is the computation of -partial BFS trees that contain vertices closest to a root vertex and can be determined with explicit operations (this part was later replaced and improved upon by [RodittyW13, ChechikLRSTW14]). In the symbolic model computing -partial BFS trees would be less efficient, however, we can compute all vertices with distance at most from with only many symbolic operations. The limitation of the approximation ratio to in the algorithm of [AingworthCIM99] comes from having to deal with vertices for which less than vertices are within distance at most . In the symbolic model we do not have to consider this case since with a budget of operations we can always reach at least vertices (assuming for now that the graph is strongly connected and ). Thus the algorithm simplifies to the second part of their algorithm, whose core part is to find a vertex within distance at most for each vertex of the graph by using a greedy approximation algorithm for dominating set. However, in the symbolic model storing a linear number of sets is too costly, hence we inherently use that we can recompute vertices at distance at most efficiently when needed. Details are presented in Section 6.5. ## 5 Discussion ### 5.1 SCCs and Verification Objectives First, our results show that the symbolic SCC algorithm by Gentilini et al. [GentiliniPP08] is essentially optimal. That is, we have the three upper bounds of , and and matching lower bounds of (Theorem 3.2), (Theorem 3.2), and (Theorem 3.7). Our results for the different kinds of verification objectives are summarized in Table 4. We have an interesting separation between the reachability objective and the other objectives in terms of the diameter . While reachability can be solved with symbolic operations, all the other objectives require symbolic one-step operation on graphs of constant diameter. When considering the diameters of the SCCs we get another separation. There we have that Liveness and Reachability can be solved with many symbolic operations, while Safety and co-Liveness requires symbolic one-step operations on strongly connected graphs with constant diameter. This reflects the fact that in the standard algorithm for Safety and co-Liveness the SCC computation is performed on a modified graph. ### 5.2 Approximate Diameter For explicitly represented graphs a -approximation of the diameter can be computed in time [RodittyW13, ChechikLRSTW14], while under the strong exponential time hypothesis no time algorithm exists to distinguish graphs of diameter 2 and 3 (i.e., no -approximation can be obtained) [RodittyW13]. The fastest exact algorithms take time. While for explicitly represented graphs small, constant diameters are a hard case, the current results suggest that in the symbolic model the diameter of graphs with constant diameter can be determined more efficiently than for large diameters, as both the upper bound for exact and approximate computation of the diameter depend on the diameter of the graph and are linear when the diameter is constant. While the threshold of an approximation ratio of appears in our (linear) lower bound, the current symbolic upper bounds do not show this behavior. Several interesting open questions remain: Is there a c-approximation algorithm when ? Is there a linear -approximation algorithm for graphs with super-constant diameter? Or are there better lower bounds? ## 6 Detailed Proofs ### 6.1 Proofs of Section 3.1 ###### Proof of Lemma 3.3.. We have to show that iff the graph constructed in Reduction 3.1 has exactly SCCs. First notice that there are no edges from a set to a set if and thus there are at least SCCs, independently of the actual values of and . If then all possible edges exists and it is easy to verify that the SCCs of the graphs are exactly the sets for . If then there are , such that there is no edge from to . Now the set splits up in at least two SCCs and thus there are at least SCCs. ∎ ###### Proof of Theorem 3.7.. We have to show that any (probabilistic bounded error or deterministic) symbolic algorithm that computes the SCCs needs symbolic one-step operations. First consider Reduction 3.1 and notice that for the constructed graph . Then by Theorem 3.2 we already have a bound. Now consider Reduction 3.6 and notice that the constructed graph has SCCS iff and are disjoint. Now we can use the same argument as in the proof of Theorem 3.2 that each symbolic one-step operations just needs constant communication. The instances where and are disjoint have SCCs and . Hence an algorithm with symbolic one-step operations would imply a communication protocol with communication, a contradiction to Theorem 2.1. By combining the two lower bounds we get the desired bound. ∎ ### 6.2 Proofs of Section 3.2 ###### Proof of Theorem 3.8.. We have to show that any (probabilistic bounded error or deterministic) symbolic algorithm that solves Reachability in graphs with diameter requires symbolic one-step operations. Consider the graph from Reduction 3.1 with parameter . We have that is reachable from all vertices iff the graph is strongly connected. From the proof of Theorem 3.2 we have that testing whether the graph is strongly connected requires symbolic one-step operations. Now notice that (a) the graph is strongly connected iff can reach and that (b) if the graph is strongly connected then . ∎ ###### Proof of Theorem 3.10.. For (1) & (3) consider the graph constructed in Reduction 3.9 and the target set . We have a valid reduction from the Set Disjointness problem as the vertex is winning for iff there is a loop for one of the vertices in iff . By the same argument as in the proof of Theorem 3.2 we have that each symbolic one-step operations just needs constant communication. Hence an algorithm with , or symbolic one-step operations would imply a communication protocol with communication, a contradiction. For (2) consider the graph constructed in Reduction 3.1 with and the target set . It is easy to verify that the vertex is winning if it can reach . Thus the lower bound for reachability also applies here. Notice that this also gives a lower bound. The lower bound in (4) is a direct consequence of the lower bound for instances with constant size target sets , and the lower bound for instances with constant diameter . Finally, (5) is by the bound from Reduction 3.9 and the bound by Reduction 3.1 with . ∎ ###### Proof of Theorem 3.11.. 1) & 2) Consider the graph constructed in Reduction 3.9 and the set of safe vertices . We have a valid reduction from the Set Disjointness problem as the vertex is winning for iff there is a loop for one of the vertices in iff . By the same argument as in the proof of Theorem 3.2 we have that each symbolic one-step operations just needs constant communication. Thus an algorithm with or symbolic one-step operations would imply a communication protocol with communication, a contradiction. 3) Consider the graph constructed in Reduction 3.9 but replace the edge by the edge . When considering the set of safe vertices the same arguments as above apply and thus we get a lower bound. However, the graph is strongly connected and has diameter and thus . ∎ ###### Proof of Proposition 3.12.. Consider the graph constructed in Reduction 3.1 with , add an additional loop edge , and consider the set of safe vertices, i.e., is the only safe vertex. It is easy to verify that the vertex is winning in if it can reach . Thus the lower bound for reachability also applies here. ∎ ### 6.3 Proofs of Section 3.3 ###### Proof of Lemma 3.14.. Let be the graph given by Reduction 3.13 and let denote its diameter. We have to show that if , then , otherwise . First note that through the edges adjacent to the auxiliary vertices the diameter of is at most . Furthermore we have for all that , , and , for all additionally , and for all it holds that and . Thus whether is or depends only on the maximum over all of and . If , then for each pair of indices at least one of the edges and exists. Since , we have for all and all that and hence . If , let and let be such that . Then neither the edge nor the edge exists. Thus the vertex , and analogously the vertex , has edges to the following vertices only: the auxiliary vertex , the vertex , and vertices with . The vertex has edges only to the auxiliary vertex and to vertices and with . Hence none of the vertices adjacent to , and respectively for , is adjacent to and thus we have . ∎ ### 6.4 An Improved Upper Bound for Strongly Connected Components Result. Gentilini et al. [GentiliniPP08] provide a symbolic algorithm for computing the strongly connected components (SCCs) and show a bound of on the number of its symbolic operations for a directed graph with vertices, diameter , and many SCCs. Let be the diameter of an SCC . We give a tighter analysis of the algorithm of [GentiliniPP08] that shows an upper bound of symbolic operations that matches our lower bound (Theorem 3.7). We have both and and thus our upper bound is always at most the previous one. We additionally observe that the algorithm can be implemented with many sets (when the SCCs are output immediately and not stored). We first explain the intuition behind the algorithm of [GentiliniPP08] and then present the improved analysis of its number of symbolic steps. Symbolic Breadth-First Search. While explicit algorithms for SCCs are based on depth-first search (DFS), DFS is impractical in the symbolic model. However, breadth-first search (BFS) from a set can be performed efficiently symbolically, namely proportional to its depth, as defined below. ###### Definition 6.1 (Symbolic BFS). A forward search from a set of vertices is given by a sequence of operations such that for until we have . We call the -th level of the forward search and the index of the last non-empty level the depth of the forward search. Let be the forward set, which is equal to the vertices reachable from . Analogously we define the backward search of depth and the backward set for operations. We denote a singleton set by . There is a simple algorithm for computing the SCCs symbolically with BFS that takes many symbolic steps: Start with an arbitrary vertex . Compute the SCC containing by taking the intersection of and , remove the obtained SCC from the graph, and repeat. Skeleton-based Ordering. The importance of DFS for SCCs lies in the order in which the SCCs are computed. Starting from a vertex that lies in an SCC without outgoing edges (i.e. a sink in the DAG of SCCs of the graph), the forward search does not leave the SCC and for computing SCCs the backward search can be restricted to the vertices of , i.e., the SCC of can be determined with proportional to the diameter of the SCC many symbolic steps. The DFS-based SCC algorithm of [Tarjan72] finds such an SCC first. The algorithm of [GentiliniPP08] is based on an ordering obtained via BFS that achieves a DFS-like ordering suitable for computing SCCs symbolically. Our tighter analysis essentially shows that their approach achieves the best ordering we can hope for. The ordering is given by so-called skeletons. ###### Definition 6.2. A pair with and is a skeleton of for if has maximum distance from and the vertices of form a shortest path from to . Let denote the SCC containing . The SCCs of the vertices of will be computed in the following order: First is computed by performing a backward search from within . The remaining SCCs of of the vertices of are then computed in the reverse order of the path induced by , starting with . We now describe the overall algorithm, where in addition to the SCCs of are computed (potentially using a different, previously computed skeleton). The Algorithm. The pseudo-code of the algorithm is given in Algorithm LABEL:alg:SymbolicSCC, the pseudo-code for the sub-procedure for computing a forward set including a skeleton in Algorithm LABEL:alg:Skeleton. Processing a graph , the algorithm proceeds as follows: it starts from some vertex and computes the set of reachable vertices, i.e, the forward set , including a skeleton that includes exactly one vertex of each level of the forward search and forms a shortest path in . It then starts a backward search starting from in the subgraph induced by . Clearly the SCC of is given by the vertices that are reached by the backward search. The algorithm returns this SCC as an SCC of and recurses on (a) the subgraph induced by the vertices of and (b) the subgraph induced by the vertices of . For the recursion on (a) we update a potentially already existing skeleton by removing all vertices that are in the current SCC (initially we have an empty skeleton) while for the recursion on (b) we use the skeleton computed by the forward search (but also remove vertices of the current SCC). The skeleton is then used to select the starting vertex in the consecutive steps of the algorithm: when the algorithm is called with skeleton , then the forward search is started from ; when the skeleton was computed in a forward search from a vertex , then this corresponds to the vertex of the skeleton that is furthest away from and contained in this recursive call. algocf[t] algocf[t] A Refined Analysis. The correctness of the algorithm is by [GentiliniPP08]. Notice that the algorithm would be correct even without the usage of skeletons but the skeletons are necessary to make it efficient, i.e., to avoid unnecessarily long forward searches. We show the following theorem. ###### Theorem 6.3 (Restatement of Theorem 4.1). With input Algorithm LABEL:alg:SymbolicSCC computes the SCCs of and requires symbolic operations. The analysis of [GentiliniPP08] of the number of symbolic steps of Algorithm LABEL:alg:SymbolicSCC uses that (1) each vertex is added to at most two skeletons, (2) the steps of the forward searches can be charged to the vertices in the skeletons, and (3) backward searches are only performed to immediately identify an SCC and thus can be charged to the vertices of the SCC; hence both the steps of the forward and of the backward searches can be bounded with . For the backward searches it can easily be seen that the number of operations to identify the SCC is also bounded by . For the forward searches we show that each part of the skeleton (that in turn is charged for the forward search) can be charged to for some SCC ; this in particular exploits that skeletons are shortest paths (in the graph in which they are computed). ###### Lemma 6.4 ([GentiliniPP08]). For each recursive call of LABEL:alg:SymbolicSCC with input we have that is a set of vertices that induces a shortest path in the graph and is the last vertex of this path. We first recall the result from [GentiliniPP08] that shows that the number of symbolic operations in an execution of LABEL:alg:Skeleton is proportional to the size of the computed skeleton. ###### Lemma 6.5 ([GentiliniPP08]). LABEL:alg:Skeleton only requires symbolic operations, i.e., is linear in the output, and can be implemented using only constantly many sets. ###### Proof. For each level of the forward search we need one operation in the first while loop and one operation in the second while loop, and the number of set operations is in the order of one-step operations. As for each level we add one vertex to , the result follows. Moreover, there is no need to explicitly store all the levels as they can be easily recomputed from the next level when needed, increasing the number of symbolic operations only by a constant factor. ∎ ###### Remark 6.6. Given Lemma 6.5 we can implement LABEL:alg:SymbolicSCC using many sets at a time by recursing on the smaller of the two sub-graphs and first. We split the cost of symbolic operations for LABEL:alg:Skeleton into two parts: the part where is the SCC identified in this level of recursion and the part that is passed to one of the recursive calls. The following lemma shows that the first part and the subsequent backward search can be charged to , using that is a shortest path in . ###### Lemma 6.7. Without accounting for the recursive calls, each call of LABEL:alg:SymbolicSCC takes	2022-08-16 01:01:15	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8898815512657166, "perplexity": 510.9189016457704}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572215.27/warc/CC-MAIN-20220815235954-20220816025954-00705.warc.gz"}
http://love2d.org/forums/viewtopic.php?f=4&t=3380	## Weird love.graphics.line behaviour Questions about the LÖVE API, installing LÖVE and other support related questions go here. Forum rules Before you make a thread asking for help, read this. Zaphio Prole Posts: 2 Joined: Thu Jun 02, 2011 1:49 pm ### Weird love.graphics.line behaviour Code: Select all function love.load() love.graphics.setMode(500, 500) love.graphics.setLineWidth(100000) ------------------------ -- EDIT THIS VARIABLE -- interval = 1 -- Try out: .9, 1, 2, 5, 10, 50 ------------------------ end function love.draw() love.graphics.setLine(25) drawRainbow({255, 0, 0, 255}, 10) drawRainbow({255, 127, 0, 255}, 20) drawRainbow({127, 255, 0, 255}, 30) drawRainbow({ 0, 255, 0, 255}, 40) drawRainbow({ 0, 255, 127, 255}, 50) drawRainbow({ 0, 127, 255, 255}, 60) drawRainbow({ 0, 0, 255, 255}, 70) drawRainbow({127, 0, 255, 255}, 80) drawRainbow({255, 0, 127, 255}, 90) end local function f(x) return 250+10*math.sin(x) end function drawRainbow(col, offset) love.graphics.setColor(unpack(col)) local ly, t = f(0), love.timer.getTime() for x = 1, 500, interval do local y = f(x/30+t) + offset love.graphics.line(x-1, ly, x, y) ly = y end end I'm trying to draw a rainbow, but I'm getting skewed vertical lines that turn instead of a smooth horizontal line. Why is this happening? Have I hit a bug? Or am I just plain dumb? Also note setLineWidth doesn't seem to affect anything. vrld Party member Posts: 917 Joined: Sun Apr 04, 2010 9:14 pm Location: Germany Contact: ### Re: Weird love.graphics.line behaviour Zaphio wrote:I'm trying to draw a rainbow, but I'm getting skewed vertical lines that turn instead of a smooth horizontal line. It's because of how you loop over x. Replacing 1 with interval and adjusting the vertical starting position should fix it: Code: Select all local t = love.timer.getTime() local ly = f(t) + offset for x = interval, 500, interval do local y = f(x/30+t) + offset love.graphics.line(x-interval, ly, x, y) ly = y end Zaphio wrote:Also note setLineWidth doesn't seem to affect anything. That's a feature of how OpenGL defines line drawings and how graphic card vendors implement it: There is an arbitrary upper (and lower) limit of the line width, which depends on the type of graphics card you have installed. I can get a line width up to 10, but yours seems to be limited to about 2. The line drawing method will be replaced in the next LÖVE version. I have come here to chew bubblegum and kick ass... and I'm all out of bubblegum. hump \| HC \| SUIT \| moonshine kikito Inner party member Posts: 3153 Joined: Sat Oct 03, 2009 5:22 pm Contact: ### Re: Weird love.graphics.line behaviour I must point out that the current result looks more interesting than the "plain sinusoidal curves" to me. When I write def I mean function.	2020-09-30 00:01:07	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2793234586715698, "perplexity": 7841.1684472616735}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600402093104.90/warc/CC-MAIN-20200929221433-20200930011433-00750.warc.gz"}
https://bryanavery.co.uk/embed-picasa-slideshows-into-blogengine-net/	# Embed Picasa slideshows into BlogEngine.net I was having a problem embedding a slide show from Picasa into my BlogEngine.net editor. Basically, the editor kept stripping the embed html. The trick is to edit \admin\tinyMCE.ascx file. You need to modify the extended_valid_elements entry. Apparently this is used to define valid html tags for the tinyMCE editor. Here are my additions in red:	2021-07-28 18:15:45	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9205002784729004, "perplexity": 8839.529371164743}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046153739.28/warc/CC-MAIN-20210728154442-20210728184442-00038.warc.gz"}
http://www.bobbycapri.com/physics-techniques-projectile-action-one-four-years-old/	# Physics Techniques Projectile Action (One of four years old) y Means directory place (m) Little diversions inside the sensitive mouse movement and also velocity have been to determine uncertainness. All projectile action occurs in some sort of bilaterally shaped way, assuming that the aim of screening machine as well as go back appear on the similar outside area. For instance period of flight as well as maximum level, all the different your projectile is often a purpose of 1st rate. Projectile motion: Projectile moving after a parabola.Initial release direction is $\alpha$, and also the speed is actually $\text_0$. ### Maximum Height The hearth will be 4.Double zero m off the incline, and also 1.50 m higher in comparison to the height on the bring. There isn’t a acceleration in this way given that seriousness solely works top to bottom. To the left over difficulties, the word appears appropriate due to the fact holistic remedy within mathematics is certainly one which also includes the actual particular circumstances, but I’m just fewer adamant concerning this term. The actual velocity on the projectile is consequently entirely motivated the moment them matches the meaning of some sort of projectile. Because of this a x posture is definitely x = Several.00 m. ## RELATED CONTENT Range with Trajectory: The plethora of the flight is demonstrated in this amount. Projectiles at the Angle: That online video provides for a clear and simple justification of precisely how to eliminate a dilemma in Projectiles Brought out within an Position. A 50-gram Disc throwing will be unveiled within an very first swiftness connected with Something like 20 meters every secondly sheer with its ripped side parallel to the ground. The war previously was clearly progressively more sectarian and Islamic-the velocity ended up being evident. 2019 Inspite of the most up-to-date downbeat assurance survey, unemployment boasts, a major signal of your economy’s potential trajectory, ticked straight down yesterday inspite of rising economic anxiety. If there is a clear distance, deborah, that you want the object to get so you be aware of preliminary swiftness when it can be brought out, the initial start perspective necessary to understand which length is termed your point of view associated with attain. #### Maximum Height A new cast helium-filled go up isn’t ordinarily deemed the projectile since the exhaust plus buoyant allows about it usually are because significant as the load. The trail the object follows is named their velocity. Range of Trajectory: The plethora of a new flight can be proven with this number. (It is deemed an simple meaning.) The path of any projectile is named the . From the displacement scenario we are able to discover the highest possible height Now our normal projectile stops being a thing that has a unveiling factor along with a attaining point but help me do my assignment it commences as a , forever circling the world, constantly changing course thereby quickening ingesting severity, nevertheless certainly not landing at any place. As a result, garden $\text$ (inside the side path) has since: Bilateral evenness shows that your movement is actually symmetrical inside the usable planes. – Ben Taylor, Harper’s magazine, “Exit Ghost,” Twelve Ruin. #### Maximum Height Because the projectile techniques up wards it goes towards seriousness, hence the velocity starts to decrease. Here, $\text$ would be the time period of the air travel prior to a target it is the surface. Cantor’s supple muscles. Your message can be used in most cases within physics and also anatomist, although not generally; we can also declare, such as, the flight of a whole living may very well be occur an individual’s junior, and also a fresh book remnants a extensive trajectory of the French enterprise. Seriousness isn’t regular to start with, although the alternative isn’t visible in excess of on a daily basis amounts around height. 2018 The important thing to the summary derives from an study of Oumuamua’s trajectory while measured by a assortment of telescopes, such as the Hubble Living space Telescope, through Marco Micheli with the Western Living space Agency’s SSA-NEO Control Heart, around Frascati, Tuscany. By using we can arrange the speed formula to uncover the it can take to the resist accomplish maximum height In ballistics, the simplest way to describe a new velocity is as simple as their x– in addition to z-components, while using z aspect suffering from nearby gravitational pressure. Many of us change $\theta$ using $\theta -- \alpha$ plus $\text$ along with $\text \cdot \cos where [latex]\text_\text$ signifies the amount of time it takes to achieve utmost level. By presuming a consistent worth with the development as a result of gravitational pressure, most of us increase the risk for trouble much easier to resolve and also (oftentimes) don't genuinely drop that much with respect to accuracy. ### Пожаловаться на видео? Because the object travels length $\text$ inside the usable direction prior to this traffic the floor, we could utilize kinematic picture for that up and down activity: A 50-gram Frisbee will be unveiled with an 1st rate associated with Something like 20 feets a second upright which consists of toned side simultaneous to the floor. The devices of vertical and horizontal posture usually are measures (m). Cookbook article author, ‘Extra' coordinator, diamond jewelry range, ‘Basketball Wives'-we are all aware the particular velocity intended for celebrity couples as well as exes. ## Did You are aware of? - David Grossman, Popular Mechanics, "What Can be a Complete Vortex? This Arctic Conditions Style Explained," Sixteen February. We change in the ideal specifics: Since the activity is due to the parabolic shape, this can appear two times: one time while traveling upwards, in addition to once again when the subject is traveling lower. $\text_0 Equals 0$ plus $\text_0 = 0$ ). By way of computing the issue of gravitational forces as well as other makes, this flight of any subject released in space for a acknowledged swiftness is often worked out specifically. After a Disc throwing will be unveiled, not one but two pushes act on the item, each pointing lower: Essentially beforehand, $\text$ depends upon your initial swiftness specifications and the perspective on the projectile: Explain the relationship involving the range as well as duration of flight Use the particular top to bottom course to find the amount of your airвЂ¦ Because of this we could utilize formula involving displacement inside the top to bottom direction, $\text-\text_0$: Mail you suggestions. We have outlined the negative impacts of numerous kick off perspectives upon variety, top, and also use of airfare. The regular sums of drag and buoyancy just will not be big enough to save lots of your passengers using a destined airfare through an unfortunate finish. It might adhere to all of them, in ever before greater amount of training, right until the trajectory could deliver the item plunging homeward. • The it takes via an item being projected plus acreage is referred to as any time with journey. Depends to the 1st acceleration of your projectile and also the direction regarding screening machine. • Пожаловаться • If an item will be projected within the exact same original quickness, yet a pair of secondary perspectives of screening machine, all the different this projectile could be the exact. • The duration of flight of the thing, with the preliminary introduction position as well as 1st rate is found together with: $\text=\frac your flat route, the object travels at a continual pace v0 throughout the trip. The number R (while in the horizontal way) has while: [latex]\text= \text_0 \cdot \text = \text_0 \sqrt side to side displacement of the projectile known as all the various a projectile, along with relies on the initial speed in the subject. How To Solve Any Projectile Motion Challenge (This Tool resource Strategy): Presenting the particular “Toolbox” approach to handling projectile movement problems! Take a look at apply kinematic equations along with change with very first disorders to create a “toolbox” of equations in which to unravel a vintage three-part projectile movements challenge. The vast geographic range in addition to the wide old range of these matters we all phone projectiles elevates some damage to the common student associated with physics. Projectile movements is a kind of movement exactly where something steps in a very bilaterally shaped, parabolic route. In a prior atom many of us reviewed just what the many components of something in projectile motions are usually. #### Пожаловаться на видео? The time of airfare on the projectile movement 's what it appears including. Complications of all sorts in science tend to be easier to remedy if you collection the things that you know (a “givens”). There isn't any top to bottom component within the initial acceleration ([latex]\text_0$ ) considering that the item can be released flat in a trench. Using details, we are able to solve many troubles affecting projectile movement. Using data, we are able to clear up lots of complications regarding projectile movement. Projectile motions solely happens when there is a single push utilized in the beginning to the flight, after which it the only interference comes from gravity. y Equates to usable position (m) ### Displacement Some sort of chucked helium-filled device isn't ordinarily viewed as the projectile as being the drag and also buoyant aids into it are while crucial as extra weight. $\text_0 Equals 0$ plus $\text_0 Means 0$ ). Projectile action is actually a sort of action wherever an object movements inside parabolic direction. Purchaser Self-confidence Dies out in 12," Twenty seven 12. Now that we understand just how the kick off viewpoint represents a significant purpose in lots of alternative elements of this flight of any item inside projectile movement, you can submit an application this information for you to make an item area where by we'd like that. Your is actually virtually any target by having an 1st non-zero, side velocity as their velocity is a result of gravitational pressure alone. Cookbook author, ‘Extra' sponsor, rings collection, ‘Basketball Wives'-we truly realize the particular trajectory to get celeb couples and exes. The maximum height of the target in a projectile trajectory occurs when the directory piece of pace, $\text_\text$, equals nil. x = x0 + v0xt + ?axt 2 x = 0 + (v cos ?) t + 0 xfinal = (v cos ?) tfinal A baseball will be placed in the oxygen, attaining various yards out. #### Displacement A projectile and also a satellite tv both are controlled by the identical actual physical rules whilst they have various labels. Construct a model with projectile motions through which includes amount of airline flight, utmost top, and also range Therefore the particular x place will be x = Several.00 m. Because of this you can take advantage of the scenario of displacement from the straight direction, $\text-\text_0$:	2019-10-22 23:49:38	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 18, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5925856232643127, "perplexity": 2471.847153740224}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987826436.88/warc/CC-MAIN-20191022232751-20191023020251-00191.warc.gz"}
http://www.mathemafrica.org/?p=14874	Recall the definition of a group: A set G is “upgraded” into a group if it satisfied the following axioms under one binary operation () : 1. Closure: $\forall x, y \in G, xy \in G$ 2. Associativity: $\forall x, y, z \in G, (xy)z = x(yz)$ 3. Identity: $\exists e \in G, \text{ called the identity element such that } \forall x \in G, xe = ex = x$ 4. Inverse: $\exists y \in G, \text{ called the inverse of x, with } xy = yx = e \forall x \in G$ An Abelian group is a group that is follows the axioms 1 – 4 with the addition of one property: 1. Commutativity: $\forall x, y \in G, xy = yx$ In addition to the axioms, the following properties of groups are important to note: 1. Uniqueness of the identity element 2. Uniqueness of the inverse element 3. Cancellation law 4. Inverse property (extended) Uniqueness of an element in mathematics means there exists only one such element with that property. We prove uniqueness by making an assumption that there are two elements in the set that satisfy the property, and show that if such a situation holds, then the two elements must be equal! We use * to denote the binary operation between elements and “QED” to signal the end of the proof. The remainder of the post aims to go through the proofs of these properties! uniqueness of the identity element proof We want to show that the identity element of a group is unique. Let’s assume the opposite. Proof: Suppose for group G, there are two identity elements, $e_1$ and $e_2.$ Then for any element, $x \in G, xe_1 = e_1x = x \text{ and } xe_2 = e_2x = x.$ So $e_1 = e_1e_2 = e_2e_1 = e_2$ This shows that $e_1 = e_2$ (hence if there are two identity elements in a set, then those elements are the same thing just with different labelling)! QED To break this proof, look at the following bullet points: • we can say $e_1 = e_1e_2$ because here $e_2$ acts as the identity element, and $e_1$ acts as a regular element of the set • $e_1e_2 = e_2e_1$ holds since identity elements are commutative, by definition (RECALL) Identity: $\exists e \in G, \text{ called the identity element such that } \forall x \in G, xe = ex = x$. This shows the commutative nature of identity elements • finally we conclude $e_2e_1 = e_2$ since we use the fact that $e_2$ can also act as a regular element and $e_1$ can act as the identity element uniqueness of inverse element proof We want to show that the inverse element is unique. Suppose the opposite Proof: Suppose there exists two inverse elements, $h, k \in G \text{ for } g \in G.$ Then $hg = gh = e \text{ and } kg = gk = e$ where $e$ is the identity element of G. Then $h = he = h(gk) = (hg)k = ek = k.$ Hence, $h=k.$ QED Again, I’ll breakdown the proof below: • using the identity property, we can say $h = he$ • since we defined the inverse $h$ of $g$ as: $hg = gh = e,$ then we can substitute $he$ for $h(gk)$ • by associativity of G, $h(gk) = (hg)k$ • and by assumption $kg = gk = e,$ we can substitute $hg$ for $e$ • and finally, we use the identity property again to conclude $h = he$ cancellation laws proof In a group G, if $ag = bg$ then $a=b$ for $a, b, g \in G$ Proof: Suppose $ag = bg$ Let h be the inverse of g, then $hg = gh = e$ for identity element, $e$ Then $a = ae = a(gh) = (ag)h = (bg)h = b(gh) = be = b,$ as required QED The breakdown is as follows: • $a = ae$ is possible using the identity property • $ae = a(gh)$ is possible since $gh = e,$ hence used substitution • $a(gh) = (ag)h$ by associativity • $(ag)h = (bg)h$ since we assumed $ag = bg$ • $(bg)h = b(gh)$ by associativity • $b(gh) = be$ by inverse property • $be = b$ by identity property inverse of $gh \text{ is } h^{-1} g^{-1}$ Proof: Let $g, h \in G,$ then we know $\exists g^{-1}, h^{-1} \in G$ satisfying $gg^{-1} = g^{-1}g = e$ and $hh^{-1} = h^{-1}h = e$ We claim $h^{-1} * g^{-1}$ is the inverse of $gh$. To show this, we want to show that $(gh) * (h^{-1} * g^{-1}) = (h^{-1} * g^{-1})(gh)$ Since $(gh) * (h^{-1} * g^{-1}) = g(h h^{-1}) * g^{-1} = ge g^{-1} = gg^{-1} = e$ And $(h^{-1} g^{-1})(gh) = h^{-1} (g^{-1}g)h = h^{-1} eh = h^{-1}h = e$ Then $(gh) * (h^{-1} * g^{-1}) = (h^{-1} * g^{-1})(gh)$ QED The breakdown: • $(gh) * (h^{-1} * g^{-1}) = g(h h^{-1}) * g^{-1}$ by associativity • $g(h h^{-1}) * g^{-1} = ge g^{-1}$ by inverse property • $ge g^{-1} = gg^{-1}$ by identity property • $gg^{-1} = e$ by identity property • $(gh) (h^{-1} * g^{-1}) = (h^{-1} * g^{-1})*(gh)$ since both equal $e$ How clear is this post?	2019-09-22 23:24:50	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 67, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9452285766601562, "perplexity": 546.0768495373135}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514575751.84/warc/CC-MAIN-20190922221623-20190923003623-00173.warc.gz"}
https://math.sciences.ncsu.edu/events/	August 2017 ## Daniel Scofield, NC State, Patterns in Khovanov homology Khovanov homology is a recent link invariant that lifts the Jones polynomial. We analyze torsion in Khovanov homology by describing a related homology theory that lifts the chromatic polynomial. In particular, we describe torsion in Khovanov homology of several link families… Find out more » ## Jichun Li, University of Nevada Las Vegas, Electromagnetic cloaking: mathematical analysis and simulation In June 23, 2006's "Science" magazine, Pendry et al and Leonhardt independently published their papers on electromagnetic cloaking. In Nov.10, 2006's Science magazine, Pendry et al demonstrated the first practical realization of such a cloak with the use of artificially constructed… Find out more » September 2017 ## Joey Hart, NCSU, SIAM Tutorial Series: Introduction to Monte Carlo Methods In this lecture we will present basis theoretical and algorithmic properties of Monte Carlo methods. In particular, their convergence properties and implementational simplicity will be highlighted. There are a variety of Monte Carlo methods but we will focus on two,… Find out more » ## Adam Levine, Duke, Heegaard Floer invariants for homology $S^1 \times S^3$s Using Heegaard Floer homology, we construct a numerical invariant for any smooth, oriented 4-manifold X with the homology of $S^1 \times S^3$. Specifically, we show that for any smoothly embedded 3-manifold Y representing a generator of H_3(X), a suitable version… Find out more » October 2017 November 2017	2017-08-21 17:56:36	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7373175024986267, "perplexity": 3658.2533373863384}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886109470.15/warc/CC-MAIN-20170821172333-20170821192333-00079.warc.gz"}
https://quant.stackexchange.com/questions/44895/pricing-in-the-heston-model	# Pricing in the Heston Model The dynamics of the Heston Model is \begin{align} \frac{dS}{S} & = \lambda \sqrt{\nu} d W^S \\[0.5em] d \nu & = k (1- \nu )dt + \epsilon \sqrt{\nu} dW^\sigma \end{align} where $$\lambda$$ is the instantaneous volatility. Let $$\nu_0 = 1$$. The Brownian motions are correlated with $$\rho dt$$. Now I want to use this online pricer: https://kluge.in-chemnitz.de/tools/pricer/heston_price.php to determine the prices. How would I go about this without $$\lambda$$ being specified in the model of the pricer? Say I want to find the price for $$S_0=100$$, $$K=90$$, $$\epsilon = 0.3$$, $$\kappa = 0.05$$, $$\rho = 0.5$$ and $$\lambda = 0.2$$. I know that by using Ito on the spot I get $$d \log S_t = \lambda \sqrt{\nu_t} d W_t^S - \frac{1}{2} \lambda^2 \nu_t dt$$ Do I somehow need to use the relationship between $$\lambda^2 \nu_t$$? If so, how? This is not the typical Heston stochastic differential equation (SDE). In the original Heston paper, the SDE is defined without $$\lambda$$, that is $$\lambda=1$$ and $$v(0)=v_0$$ not necessarily 1. In your case you have to do the change of variable $$y= \lambda^2 v$$ which leads to $$dS/S = \sqrt{y}dW_S$$ $$dy = k(\lambda^2 - y) + \epsilon\lambda\sqrt{y} dW_y$$ and $$y(0)=\lambda^2$$. the initial vol is $$\sqrt{y(0)}$$ and the vol of vol (actually really a vol of var) is $$\xi=\epsilon\lambda$$.	2019-10-15 18:09:17	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 22, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.927861213684082, "perplexity": 436.6302156902732}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986660067.26/warc/CC-MAIN-20191015155056-20191015182556-00189.warc.gz"}
https://www.nature.com/articles/s41598-017-17599-1?error=cookies_not_supported&code=fbaa775d-4dcc-4e32-baf4-2adb4e0c6fd9	Article \| Open \| Published: # Aerosols cause intraseasonal short-term suppression of Indian monsoon rainfall ## Abstract Aerosol abundance over South Asia during the summer monsoon season, includes dust and sea-salt, as well as, anthropogenic pollution particles. Using observations during 2000–2009, here we uncover repeated short-term rainfall suppression caused by coincident aerosols, acting through atmospheric stabilization, reduction in convection and increased moisture divergence, leading to the aggravation of monsoon break conditions. In high aerosol-low rainfall regions extending across India, both in deficient and normal monsoon years, enhancements in aerosols levels, estimated as aerosol optical depth and absorbing aerosol index, acted to suppress daily rainfall anomaly, several times in a season, with lags of a few days. A higher frequency of prolonged rainfall breaks, longer than seven days, occurred in these regions. Previous studies point to monsoon rainfall weakening linked to an asymmetric inter-hemispheric energy balance change attributed to aerosols, and short-term rainfall enhancement from radiative effects of aerosols. In contrast, this study uncovers intraseasonal short-term rainfall suppression, from coincident aerosol forcing over the monsoon region, leading to aggravation of monsoon break spells. Prolonged and intense breaks in the monsoon in India are associated with rainfall deficits, which have been linked to reduced food grain production in the latter half of the twentieth century. ## Introduction The Indian summer monsoon affects water availability and therefore water management related to rain-fed agricultural practices1,2. Long-term changes in Indian monsoon precipitation that are linked to aerosol radiative forcing, termed slow-responses, have been caused by thermodynamic adjustments of mean temperature and moisture content3,4, reductions in land–sea temperature differences and zonal winds, and dynamic circulation adjustments to regional energy imbalances5,6,7. These, in turn, have been associated with precipitation deficits on multi-decadal time scales. Short-term changes in Indian monsoon precipitation that are linked to aerosols, termed fast-responses, occur through the enhancement of meridional surface temperature or pressure gradients6,8 and mid-tropospheric diabatic heating9, which cause an increase in the northward transport of moisture, leading to the onset and enhancement of precipitation on time scales of days to a month. The importance of synoptic, large-scale convection in supporting vertically integrated moisture transport in monsoon systems is well established4,10,11. Aerosols were linked to significant decreases in convective instability over India, inferred from modeled lower atmosphere warming4 and increased tropospheric temperature trends, in agreement with microwave sounder measurements during 1979–2003. In studies not specifically related to the Indian monsoon, aerosols have been linked to the inhibition of cloud and precipitation development, by altering the vertical profile of heating rate, inducing stabilization12 and suppressing mesoscale convective motion. Aerosol abundance is persistent over the Indian subcontinent during the summer monsoon season13,14,15. Aerosol-induced effects on cloud microphysical properties have been linked to both precipitation suppression16 and invigoration17, at different aerosol levels. These occur through changes in cloud droplet size distributions, redistribution of precipitable water, and latent heat changes associated with condensation and evaporation12. Reduction in the median size and width of cloud droplet distributions reduces the efficiency of droplet growth18. Fast microphysical effects have been linked to precipitation shut-off in ship tracks for pristine marine clouds19. Recently, influences of aerosols in the monsoon region have been reported, suggestive of inhibition or invigoration of clouds and rainfall20,21. To our knowledge, the causal effects of coincident aerosols on the changes in Indian monsoon precipitation have not been investigated through observational analysis. Here we find causal relationships between aerosol enhancement and suppression of lagged daily precipitation and mean cloud drop sizes, through atmospheric stabilization, increased moisture divergence and reduced convection, leading to a higher frequency of prolonged rainfall breaks. ## Results Aerosol and cloud properties from satellite observations22,23,24, precipitation from ground based measurements25, and meteorological variables from European Centre for Medium-Range Weather Forecasts (ECMWF) re-analysis (ERA)-interim reanalysis data26 from 2000–2009 over the Indian subcontinent (6.5–40°N and 66.5–100°E), at 1 × 1° resolution for the monsoon months of June to September (JJAS), were used for the analysis (for more information see Methods). Precipitation data used were from 1803 irregularly located meteorological stations over India25, reported to be gridded to 1 × 1o, using Shepard’s interpolation methodology, yielding approximately 350 pixels over the Indian domain. Normalized daily anomaly was calculated for each variable, as deviation for a specific day and pixel from its mean (calculated across years), normalized by its standard deviation (for more information see Methods). The data were clustered for each year using hierarchical clustering27, using season average of normalized anomaly of AOD and precipitation, to identify clusters of (a) high AOD-low precipitation (HL), (b) low AOD-low precipitation (LL), (c) high AOD-high precipitation (HH), and (d) low AOD-high precipitation (LH). Cluster average time series of normalized anomalies of AOD and precipitation were used to detect Granger causality28 which tests statistically significant improvement in the prediction of precipitation, using past information of AOD, as compared to only past information of precipitation (for more information see Methods). In case of causal association, path analysis29 was used to enable identification of mechanisms through which AOD influenced precipitation (for more information see Methods). ### Causal Influence of Aerosols on Short-term Precipitation Suppression It was found that high aerosol-low precipitation (HL) clusters extended over large parts of India in 2004, 2005 and 2009 (Supplementary Fig. 2). In HL clusters, a leading positive AOD anomaly caused a negative precipitation anomaly, with a lag time of 1–5 days, during the JJAS monsoon season in 2004, 2005, and 2009 (Fig. 1a,b). However, no causal influence was found in the corresponding LL clusters, characterized by low aerosol abundance (Supplementary Figs. 3,4). Precipitation suppression lagged aerosol enhancement and lasted one to five days (2004, 2–5 days; 2005, 1–2 days; 2009, 2–5 days), with the maximum influence, in terms of correlation coefficient magnitude, occurring on different days (2004, day 5; 2005, day 1; 2009, day 3). The corresponding lagged time series between AOD anomaly and precipitation anomaly (Fig. 1c), corresponding to maximum correlation magnitude, showed several intra-seasonal periods of high AOD anomaly followed by periods of low precipitation. These negative causal relationships did not manifest in 2000, 2001, and 2002; 2006 and 2008 lacked sufficient data (<50 pixels in HL and LL clusters) for conclusive results. A positive causal effect of daily AOD on daily precipitation was found in 2003 and 2007, identified as abundant monsoon years30. In addition to total aerosol abundance (measured by AOD), the effects of absorbing aerosols (measured by absorbing aerosol index, AAI) were examined in the HL and LL clusters. Again a positive anomaly in AAI exerted a strong causal influence on lagged negative precipitation anomaly (2004 and 2005, 2–5 days; 2009, 1 day), which lasted one to five days, with strongest causal influence occurring on different days (quantified as correlation coefficient) in different years (2004, day 5; 2005, day 5; 2009, day 1). This behavior also manifested in LL clusters (with negative AOD anomalies) in 2005 and 2009. Since clustering was performed with AOD, the LL clusters contained many (30–40%) positive AAI anomaly values and thus included several days of high absorbing aerosol levels. Overall, enhanced levels of aerosols caused suppression of lagged daily precipitation, 3–5 times during the monsoon season, in high aerosol-low precipitation regions (HL). Other studies found positive correlations between AOD and cloud properties20 and daily precipitation over the monsoon region31, suggesting a cloud invigoration effect, however, did not test for causation. The aerosol-cloud invigoration effect is beyond the scope of the present study, which focuses only on variables linked to aerosol-induced suppression of precipitation. ### Cause–Effect Model Development and Validation The physical mechanisms (causality) underlying the observed aerosol caused suppression of precipitation are studied using cause-effect model and path-analysis. To unravel the mechanism of AOD- or AAI-induced precipitation suppression, it was postulated that a microphysical pathway linked AOD with precipitation through the mean cloud droplet effective radius (CDER) (AOD–CDER–PRECIP), while a radiative pathway linked aerosols (AOD or AAI) with precipitation through the lapse rate (defined as AOD–lapse rate–PRECIP and AAI–lapse rate–PRECIP). Causality was first tested between AOD and CDER, AOD and lapse rate, and AAI and lapse rate, and path analysis was used to segregate and quantify the effects of the two mediating pathways. These pathways were compared with pathways that directly linked column water vapor (CWV) with both precipitation (PRECIP) and CDER, to evaluate their respective strengths32. In high aerosol-low precipitation (HL) regions, enhancement of AOD caused a reduction in lapse rate (2004) and in CDER (2005), extending to five days (panels 1 and 3, Fig. 2a,b). No significant effect of AOD on CDER was found in 2004 or 2009. Here we found aerosol induced increase in static stability and decrease in moisture availability, with short time-lags of 1-5 days, subsequently, causing reduction in cloud droplet size (CDER). This effect of reduction in CDER via divergence of moisture takes place over a period of days as compared to microphysical effects, where increases in aerosols within clouds leads to formation of larger number of smaller drops on time scales of minutes to hours. Enhancement of AAI exerted suppression of lapse rate (stabilization) at shorter lag times of one day (panel 2, Fig. 2a,c), indicating that the radiative effects on lapse rate changes were largely influenced by absorbing aerosols in all three years. In contrast, no causal influences were found in low aerosol-low precipitation (LL) regions, implying absence of these effects. ### Cloud Microphysical Pathway The cause-effect model and lagged correlation coefficients were used as input in the path analysis; the overall correlation coefficients were segregated into path coefficients whose sign and magnitude indicated the direction and strength of the causal influence (Fig. 3a,b). The cloud microphysical pathway (Fig. 3a) showed inverse effects of AOD on CDER (blue color; arrow direction) or mean drop size, but direct effects of CDER on PRECIP (red color; arrow direction), indicating increases in AOD causing lagged decreases in CDER and rainfall. A positive causal influence of water vapor availability (CWV) on both PRECIP and CDER showed the expected relation of increased moisture availability to cloud and rainfall development. The CWV–PRECIP direct positive pathway was significantly stronger than the negative AOD–CDER–PRECIP pathway, based on the magnitudes of the overall path effects (Supplementary Table 1), indicating that rainfall suppression through changes in cloud drop size was not a strong pathway. In the low aerosol-low precipitation (LL) regions no causal influence of aerosol level on CDER was found (missing causal lines between AOD and CDER in Fig. 3b), however, moisture availability exerted positive causal effects on rainfall, both directly (CWV–PRECIP, Fig. 3b) and indirectly through CDER (CWV–CDER–PRECIP). Thus, the aerosol effects acted only in the high-aerosol regions, but moisture effects acted in both high and low aerosol regions. Cloud microphysical processes, which influence rainfall suppression, result from direct increases in aerosols at cloud level, leading to the redistribution of moisture to a larger number of smaller drops. This reduces coalescence efficiency, slowing down the conversion of cloud drops to raindrops or graupel18. Raindrop formation is initiated from vapor condensation and collision/coalescence processes, on time scales of a minute, but it subsequently intensifies from scavenging of small cloud drops by gravitational settling of larger drops (termed autoconversion). This leads to precipitation onset on time scales of about 15–20 minutes33. Such fast microphysical effects have been linked to precipitation shut-off in ship tracks in pristine marine clouds19. Microphysical and radiative processes are largely independent, occurring at different ranges of AOD values; the time responses of the microphysical processes are much shorter than those of the radiative processes34. The presence of 1–5 day lag times between aerosol enhancement and CDER or precipitation suppression indicates that these causal relationships might not be a direct microphysical effect, which typically acts on time scales of minutes to hours. In regions of high-aerosol and low-precipitation, the radiative pathway (Fig. 3a) showed inverse effects of AOD on lapse rate (blue color; arrow direction) calculated as the slope of potential temperature, with a lower magnitude of lapse rate, indicating higher atmospheric stability. Positive effects of lapse rate on PRECIP (red color; arrow direction), indicated increases in AOD causing lagged decreases in lapse rate and rainfall. The causal influence between AAI and lapse rate was especially strong in 2004, 2005 and 2009, substantiated by larger magnitudes of negative path coefficients (Supplementary Table 1), compared to that of AOD which acted only in 2004. The radiative pathway of absorbing aerosols (AAI–lapse rate–PRECIP), equaled the moisture-rainfall (CWV-PRECIP) effect in strength in some years, indicating the potential for strong aerosol-induced rainfall suppression. A stronger influence of the radiative pathway, than the cloud microphysical pathway, was found (larger negative values of path coefficients; Supplementary Table 1) on rainfall suppression. The radiative pathway showed significant effects even in the low-aerosol regions (LL clusters which were based on AOD) indicating overall more spatially widespread effects of aerosols on rainfall suppression. The radiative and cloud microphysical pathways, along with CWV, together explained approximately half of total precipitation variability (PRECIP R 2 > 0.50; Supplementary Table 1). Exclusion of the radiative pathway from the model resulted in a significant drop in PRECIP R 2 for the HL and LL clusters (Supplementary Table 2), highlighting the strong effects of aerosol-induced atmospheric stabilization on precipitation suppression. The distributions of anomaly values of CWV and cloud fraction (CF), which could influence rainfall development, were not statistically different (Supplementary Fig. 5a,b) between the high- and low-aerosol regions, further supporting aerosol-induced stabilization as the primary cause of the observed rainfall suppression. Short-term radiative effects can manifest within a day of an increase in aerosol levels and last for two or more days35,36,37,38. The absorption of solar radiation by aerosols has been linked to local atmospheric heating, with cooling of the surface leading to a reduction in the atmospheric lapse rate or a consequent increase in stability36,39; this is consistent with the radiative pathway seen in this study. The effect is further linked to a suppression of moisture and heat fluxes from the surface34, a reduction of convection12, and the vertical mixing of moisture. Aerosol levels were linked to a reduction in cloud fraction and drop sizes of shallow continental clouds34,40. Higher atmospheric stabilization by aerosols was linked to modeled decreases in monsoon precipitation on multi-decadal time scales4, while it was suggested as significant on short time scales in another study5. An increase in black carbon aerosols increased stability of the boundary layer and reduced convection, which further reduced cumulus precipitation in large eddy simulations (LES)12 and general circulation model simulations41. To our knowledge, the findings here are the first demonstration of causality of aerosol-induced atmospheric stabilization on Indian monsoon precipitation suppression. In contrast with the aerosol-induced suppression of precipitation found in this study, Sarangi et al.31 found an increase in daily precipitation intensity with increased aerosol loading over the core monsoon zone. Differences in the studies include the use of clustering into HL and LL regions here, in contrast with the use of non-segregated data in the other study. Further, cloud fraction in the present study ranged from 0.5 to 1 in the HL and 0.3 to 1 in the LL regions, in contrast with larger cloud fractions considered in the other study. Aerosol-induced invigoration of rainfall acts at different levels of AOD on different regimes of cloud fraction, in comparison with the suppression mechanism investigated here, and manifests in changes in micro- and macro-physical cloud properties, the analysis of which is beyond the scope of the present study. ### Mechanisms of Short-term Precipitation Suppression Further analysis was performed to link the observed causality between enhanced aerosol levels and short-term precipitation suppression to factors typically used to explain monsoon variability, such as, vertical integral of divergence of moisture flux (VIDMF), vertical wind (ω850) and surface pressure10,42,43. Increases in VIDMF and ω850 have been linked to suppression in precipitation, while an increase in surface pressure has been associated with short monsoon break spells44. In periods of high aerosol levels (AOD anomaly >0.7 for at least three consecutive days) in the high-aerosol regions, VIDMF and ω850 anomalies (upward wind being negative) were found to be positive (Fig. 4a,c,e), indicating higher moisture flux divergence and a simultaneous reduction in convective activity in the column, in contrast to that in low aerosol periods (Fig. 4b,d,f). These effects were more pronounced during periods of higher aerosol levels (at different AOD anomaly thresholds; Supplementary Fig. 6) from an increase in gross moist stability, defined as the ratio of vertically integrated horizontal divergence of moisture to vertical convection, consistent with earlier studies45, associated with subsequently reduced precipitation. The role of common moderating meteorological variables is often debated in relation to aerosol–cloud–precipitation interactions. Surface pressure is typically identified as one of these variables, whose positive anomalies have been linked to break spells in monsoon precipitation44. Increases in AOD anomalies had causal effects on increases in anomalies of surface pressure and VIDMF, with lags of 1–5 days (Supplementary Table 3). In HL regions, a greater frequency of positive surface pressure anomalies was found (Supplementary Fig. 7). However, there was no causality between increased anomalies in surface pressure and those in VIDMF (Supplementary Table 3), ruling out the role of surface pressure in the observed effects. The monsoon region is reported to experience the persistence of both dust aerosols13,46 and build-up of anthropogenic fine particles47,48,49, the combined effects of which can diminish surface reaching radiation and can cool the surface, subsequently increasing the atmospheric stability. Further, transient occurrence of black carbon aerosols has been measured both at surface and at elevations of 1 to 3 km over India and adjoining oceans50,51. Convection is reported to accommodate positively, on daily to monthly timescales, to radiative effects of absorbing aerosols like black carbon leading to short-term increases in precipitation8,9. Specifically, active phases following monsoon break periods, were linked to a build-up of aerosols which caused aerosols moisture convergence and onset of rainfall47,52, on time-scales of about 20 days. However, reduction in convection with increases in black carbon aerosols is reported, both in observation studies of biomass burning40 and modelling studies12. Modeling study conducted by Das et al.53 reported increase in moisture divergence with increased absorbing aerosols. Guo et al.54 point out that a significant threshold of black carbon loading is necessary to induce convection, which was found only in simulations using 5xBC of present day emissions. The overall mesoscale mechanism observed here (Fig. 5), is enhancement in aerosol levels causing atmospheric stabilization, with increased horizontal moisture divergence, reduced convection and vertical velocity and subsequent suppression of precipitation. These two effects of limited moisture availability and restricted convection conflate and contribute to the suppression of precipitation. ### Implications for Monsoon Break Spells Break spells are inherent to the Indian monsoon. However, their intensity and duration plays an important role in determining deficient precipitation or drought conditions2,55. While several definitions are used to identify monsoon break spells, a widely accepted one is based on a normalized anomaly threshold of one standard deviation below the mean2,55,56, which occurs for at least three consecutive days (Fig. S7). The possible influence of aerosols on mediating break spells was examined through comparing break occurrence in regions of higher (HL) and lower (LL) aerosols (Fig. 6). A larger number of total break days and increased frequency of break spells and prolonged break spells (lasting seven days or longer) was found in HL regions (3–4 times in a season) than in LL regions (1–2 times in a season). The chosen threshold of precipitation anomalies corresponded to the negative one standard deviation and more stringent decreases (Supplementary Fig. 8) and results were averaged over all threshold values. It is accepted that prolonged or extended breaks (lasting seven days or more) often result in droughts2,55. Recent studies examining monsoon variability over India during the last 50 years found an increased duration57 and frequency58 of break spells. However, explicit attribution to aerosol effects was not investigated in the earlier studies. The widespread nature of aerosol-induced rainfall suppression observed here, is evidenced by their occurrence not only in 2004 and 2009, widely reported as deficient monsoon years on sub-continental scales14,35, but also in 2005, acknowledged as a normal monsoon year. It was suggested that when the Indian summer monsoon anomaly is large (>15%), it is often uniform across the country, but when it is within a few percent of the mean, several regions could have deficit or excess precipitation1. This suggests that the aerosol-induced causal mechanisms uncovered here could aggravate break spells and precipitation deficits in normal monsoon years with modest monsoon precipitation anomalies. In other years, where the analysis was inconclusive due to lack of data, the use of a finer resolution or larger dataset could yield better insights. Intra-seasonal oscillations of the Indian monsoon typically occur on cycles of 6–9 days, 10–20 days, and 30–60 days59. Monsoon break periods are generally associated with increased surface pressure anomalies, weaker moisture-laden low-level flows from the southern Indian Ocean, decreased cyclonic vorticity over the monsoon region, and positive anomalies in outgoing longwave radiation associated with the scarcity of clouds44,55,60. The aerosol-induced suppression of precipitation seen in this study, which results from mesoscale atmospheric stabilization on shorter time scales of 3–5 days (Fig. 1c), is therefore a distinct phenomenon, whose interplay with monsoon break dynamics warrants further investigation. ## Summary and Discussion Anthropogenic pollution particles over South Asia comprise a mixture of light scattering species (including sulfate, nitrate, organic carbon and others) and light absorbing black carbon (or soot), the latter emitted primarily by traditional technologies burning biomass fuels61. This study uncovered a causal influence of aerosols on repeated meso-scale suppression of precipitation, which occurred throughout the monsoon season. The suppression mechanism was mediated primarily through a radiative pathway acting to increase atmospheric stability and horizontal divergence of moisture. Consensus in previous studies points to monsoon rainfall weakening linked to an asymmetric inter-hemispheric energy balance change attributed to aerosols5,7 and short-term rainfall enhancement linked to radiative effects of non-local absorbing aerosols8,9,47,52. In this study, short-term rainfall suppression is linked to radiative effects of coincident aerosols, acting through repeated atmospheric stabilization, reduction in convection and increased moisture divergence, leading to aggravation of monsoon break conditions. Interestingly, in addition to being manifested in deficient monsoon years, causal influences of aerosols on precipitation suppression also occurred in a normal monsoon year, indicating the possibility of a widespread occurrence of this phenomenon. The causal influence of aerosols on precipitation suppression is relevant to the inter-annual variability of monsoon precipitation and the timing of monsoon break spells. Prolonged and intense breaks in the monsoon were associated with rainfall deficits55, which have been linked to reduced food grain production during latter half of the twentieth century1. Thus, aerosol-induced precipitation suppression and aggravation of break spells, uncovered here, could influence future rainfall deficits and agricultural vulnerability in India. ## Methods ### Data set Aerosol and cloud properties from satellite observations22,23,24, precipitation from ground based measurements25, and meteorological variables from European Centre for Medium-Range Weather Forecasts (ECMWF) re-analysis (ERA)-interim reanalysis data26 from 2000–2009 over the Indian subcontinent (6.5–40° N and 66.5–100° E), at 1 × 1° resolution for the monsoon months of June to September (JJAS), were used for the analysis. Level-3 (L3) atmospheric aerosol data, retrieved from the moderate resolution imaging spectroradiometer (MODIS) on board the Earth Observing System’s (EOS) Terra (MOD08_D3v6) and Aqua (MYD08_D3v6) satellites, made available through the National Aeronautics and Space Administration (NASA) Deep blue (Collection-6) algorithm24, were used for aerosol optical depth (AOD). The data includes AOD retrievals over regions with high reflectance. Positive values of absorbing aerosol index (AAI) from the total ozone mapping spectrometers (TOMS) on board the Earth Probe satellites and the ozone monitoring instrument (OMI) sensor on board the EOS Aura satellite, were derived using the Earth Probe TOMS22 and OMAERUV62 algorithms. These measure absorbing aerosols that are within the range of 0.5 and 3.5 km63 and neglect the residues related to purely scattering aerosols62. MODIS L3 cloud droplet effective radius (CDER) data, obtained from both EOS Terra (MOD08_v3) and Aqua (MYD08_v3) satellites, were used. Lapse rate was calculated using the temperature of nine layers of atmosphere between 1000 and 750 hPa, corresponding to the lower troposphere in the ERA-interim dataset26. Precipitation data, as a gridded product, were obtained through the interpolation of data from 1803 irregularly located meteorological stations over India, with approximately 350 pixels over the Indian domain25. ### Data Processing The description and source of the data is listed in Table S4 for years 2000–2009 at resolution of 1 × 1o. The spatial coverage for the study region was 6.5–40 °N to 66.5–100 °E. AOD values less than 0.8 were retained for the analysis as higher values may be due to misclassification of clouds as aerosols40. Daily pixel-wise absolute value for each variable was transformed into normalized anomaly. To achieve this, at a given pixel, deviation of daily absolute value from its mean daily value (across years) was divided with standard deviation of daily absolute value (across years). This was repeated for all the pixels. Further, season average anomaly was calculated by taking mean of normalized anomaly for each year and every pixel. Normalized daily anomaly (Δx tiy ) for each variable was calculated as described below: The normalized anomaly (Δx tiy ) was the deviation of variable for a specific day (t) and pixel (i) from the mean (calculated across years), normalized by its standard deviation. Further, seasonal average anomaly $$(\overline{{\rm{\Delta }}{x}_{iy}})$$ was calculated for each year and each pixel. Depending upon the value of $$\overline{{\rm{\Delta }}{x}_{iy}}$$ the complete 122-day temporal series of anomaly (Δx tiy ) was assigned to either low value cluster or high value cluster. $$\overline{{x}_{ti}}=\frac{\sum _{y=1}^{Y}{x}_{tiy}}{Y},\{i=1,2,\mathrm{...}.,N;t=1,2,\mathrm{...}.,{\rm{T}}\}$$ (1) $$\sigma ({x}_{ti})=\sqrt{\frac{\sum _{y=1}^{Y}{({x}_{tiy}-\overline{{x}_{ti}})}^{2}}{Y-1}},\{i=1,2,\mathrm{...}.,N;t=1,2,\mathrm{...}.,{\rm{T}}\}$$ (2) $${\rm{\Delta }}{x}_{tiy}=\frac{{x}_{tiy}-\overline{{x}_{ti}}}{\sigma ({x}_{ti})},\{i=1,2,\mathrm{...}.,N;t=1,2,\mathrm{...}.,{\rm{T}};y=1,2,\mathrm{...}.,{\rm{Y}}\}$$ (3) $${\overline{{\rm{\Delta }}x}}_{iy}=\frac{\sum _{t=1}^{T}{\rm{\Delta }}{x}_{tiy}}{T},\{i=1,2,\mathrm{...}.,N;y=1,2,\mathrm{...}.,{\rm{Y}}\}$$ (4) $${\overline{{\rm{\Delta }}x}}_{iy}=\{\begin{array}{c} > 0,\,i\,{\rm{is}}\,{\rm{an}}\,{\rm{high}}\,{\rm{value}}\,{\rm{pixel}}\,{\rm{in}}\,{\rm{year}}\,y\\ < 0,\,i\,{\rm{is}}\,{\rm{a}}\,{\rm{low}}\,{\rm{value}}\,{\rm{pixel}}\,{\rm{in}}\,{\rm{year}}\,y\end{array},\{i=1,2,\mathrm{...}.,N;y=1,2,\mathrm{...}.,{\rm{Y}}\}$$ (5) here t denotes a day in June-September (JJAS) (t = 1, 2, …, T). Pixels are denoted as i (i = 1, 2, …, N) and y denotes years (y = 1, 2, …, Y). T is total number of days in JJAS, N is total number of pixels and Y is total number of years. The season averaged pixel-level AOD and precipitation anomalies were subjected to hierarchical clustering for each of the years (2000–2009). Pixels were assigned to (a) high AOD-low precipitation (HL), (b) low AOD-low precipitation (LL), (c) high AOD-high precipitation (HH), and (d) low AOD-high precipitation (LH) clusters. The high AOD-low precipitation and low AOD-low precipitation clusters were selected to investigate possible effects of different levels of aerosols on precipitation suppression. Clustering process is described below: Pixels were first clustered into high and low AOD anomaly pixels depending upon $$\overline{{\rm{\Delta }}{x}_{iy}}$$ AOD values. These pixels were further clustered into high and low precipitation anomaly pixels using $$\overline{{\rm{\Delta }}{x}_{iy}}$$ precipitation values. Thus, pixels were clustered into four clusters i.e. (a) high AOD-low precipitation (HL), (b) low AOD-low precipitation (LL), (c) high AOD-high precipitation (HH), and (d) low AOD-high precipitation (LL). Once the pixels were assigned to respective clusters, cluster average of all the pixels was taken to get an average temporal series Δx ty for each year which was used for causality analysis. $${\rm{\Delta }}{x}_{ty}=\frac{\sum _{i=1}^{{\rm{M}}(y)}{\rm{\Delta }}{x}_{tiy}}{M(y)},$$ (6) where M(y) is the number of pixels in a given cluster, t is the day and y is the year. HL clusters for years 2004, 2005 and 2009 are shown in Supplementary Fig. 2 while LL clusters are shown in Supplementary Figs 3,4. For year 2004, HL cluster had 113 pixels while LL cluster had 98 pixels while year 2009 had 156 and 169 pixels respectively for HL and LL clusters. HL cluster corresponded to highly deficient region for year 2004 and peninsular regions for 2009, as reported in literature35,64. Though 2005 was normal year the HL pixels (belonging to north-eastern region of India) received less precipitation compared to other years65. The above clustering and analysis was based on normalized anomaly values. The definition of anomaly ensured that regional effects were excluded while establishing causality. The individual time series were found to be stationary using Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test or were made stationary by first order differencing, if the original time series was non-stationary. In the cause-effect model, Granger causality (GC) was tested pair-wise both ways and once the causality was established, the lagged correlation coefficient (lag obtained from GC analysis as discussed next) was calculated and provided as input to path analysis. Lags with washout (i.e. precipitation causing AOD, feedback (i.e. AOD causing precipitation and precipitation causing AOD), and lags associated with statistically not significant correlations were excluded from the analysis. The statistical significance tests were performed at α = 0.1 throughout the study. ### Granger causality The notion of Granger causality (GC) was first introduced by Granger28. It relies on the principle that the causal event leads its effect and has unique information about the future. A variable Y is said to Granger cause variable X, if inclusion of past information of both Y and X gives statistically significant improvement in the prediction of X as compared to only inclusion of past information of X. GC has been applied in climate domain for causal attribution66,67,68. Consider stationary time series of Y t and X t to test the null hypothesis of no Granger causality. Towards this end, an auto-regressive model of X t is compared with auto- and cross-regressive model of X t involving Y t as, $${X}_{t}={\alpha }_{0}+{\alpha }_{1}{X}_{t-1}+\mathrm{.....}.+{\alpha }_{n}{X}_{t-n}+{\varepsilon }_{x}$$ (7) $${X}_{t}={\tilde{\alpha }}_{0}+{\tilde{\alpha }}_{1}{X}_{t-1}+\mathrm{.....}.+{\tilde{\alpha }}_{n}{X}_{t-n}+{\beta }_{1}{Y}_{t-1}+\mathrm{.....}.+{\beta }_{n}{Y}_{t-n}+{\varepsilon }_{xy}$$ (8) If the variance of the residual in the second model, labelled $${\sigma }_{{\varepsilon }_{xy}}^{2}$$, is significantly less than the variance of the residual in the first model, labelled $${\sigma }_{{\varepsilon }_{x}}^{2}$$, then the inclusion of information of Y is improving the prediction of X t implying that Y is Granger causing X. GC was tested at varying lags, and lags with statistically significant causality were retained for further analysis. First order difference was performed for all the variables to ensure stationarity before performing GC test for years 2004, 2005 and 2009. Along with this, causality of a particular day was also tested by comparing the regression model with and without inclusion of a particular day value. $${X}_{t}={\alpha }_{0}+{\alpha }_{1}{X}_{t-1}+\mathrm{.....}.+{\alpha }_{n}{X}_{t-n}+{\tilde{\varepsilon }}_{x}$$ (9) $${X}_{t}={\tilde{\alpha }}_{0}+{\tilde{\alpha }}_{1}{X}_{t-1}+\mathrm{.....}.+{\tilde{\alpha }}_{n}{X}_{t-n}+{\beta }_{n}{Y}_{t-n}+{\tilde{\varepsilon }}_{xy}$$ (10) If $${{\sigma }^{2}}_{{\tilde{\varepsilon }}_{xy}}$$ is significantly less than $${{\sigma }^{2}}_{{\tilde{\varepsilon }}_{x}}$$ the inclusion of information of Y t−n is improving the prediction of X t implying that Y t−n (i.e. a particular nth day in the past) is Granger causing X. The results found were same as with conventional causality methods. ### Path Analysis Proposed by Wright29, path analysis enables splitting of the net effect of one variable on other variables into direct and indirect effects. The direct effect is the path coefficient of the directed edge between two variables under consideration. The indirect effect is the sum of product of path coefficients for all paths, other than the direct edge, connecting the variables under consideration. To illustrate, consider the example in Supplementary Fig. 969 where two exogenous variables Z1 and Z2 are affecting a common endogenous variable Y. The variables Z1 and Z2 are correlated with Y. The correlation coefficients ρ12, ρ1y and ρ2y are available for a given data set. In Supplementary Fig. 9 the double headed arrow between the exogenous variables Z1 and Z2 represents the correlation between exogenous variables. The single headed arrow from exogenous variables Z1 to Y and Z2 to Y signifies that Y is dependent on Z1 and Z2. The self-loop on variable Y represents the error term and coefficient p represents the error path coefficient. The regression model of Y, in standardized form, can be written as: $$Y={p}_{1y}{Z}_{1}+{p}_{2y}{Z}_{2}+{p}_{y\varepsilon }\varepsilon$$ (11) Path coefficients p 1y , p 2y and p are calculated using: $${\rho }_{1y}={p}_{1y}(1)+{p}_{2y}{\rho }_{12}$$ (12) $${\rho }_{2y}={p}_{1y}{\rho }_{12}+{p}_{2y}(1)$$ (13) $$1={\sigma }_{Y}={p}_{1y}^{2}+{p}_{2y}^{2}+{p}_{y\varepsilon }^{2}+2{\rho }_{12}{p}_{1y}{p}_{2y}$$ (14) Once the path coefficients are calculated the total effect can be segregated into direct and indirect effects. The direct and indirect effects of exogenous variables Z1 and Z2 on Y are shown in Supplementary Table 5. 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Clim. 19, 1182–1194 (2006). 68. 68. Mokhov, I. I. & Smirnov, D. A. Diagnostics of a cause-effect relation between solar activity and the Earth’s global surface temperature. Izv. Atmos. Ocean. Phys. 44, 263–272 (2008). 69. 69. Johnson, R. A. & Wichern, D. W. Applied Multivariate Statistical Analysis. (Prentice-Hall, Inc., 1988). ## Acknowledgements This study was supported by the Indian Institute of Technology Bombay, Centre of Excellence in Climate Studies (IITB-CECS) project of the Department of Science and Technology (DST), New Delhi, India. The author(s) wish to acknowledge use of the Ferret program for analysis and graphics in this paper. Ferret is a product of NOAA’s Pacific Marine Environmental Laboratory (http://ferret.pmel.noaa.gov/Ferret/). We would also like to acknowledge R Core Team (2013): A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria (http://www.R-project.org/). ## Author information ### Affiliations 1. #### Interdisciplinary Programme in Climate Studies, Indian Institute of Technology Bombay, Mumbai, 400076, India • Prashant Dave • , Mani Bhushan • & Chandra Venkataraman 2. #### Department of Chemical Engineering, Indian Institute of Technology Bombay, Mumbai, 400076, India • Mani Bhushan • & Chandra Venkataraman ### Contributions M.B. and C.V. provided the study concepts and interpretation of the results; P.D. carried out the data analysis, with guidance from M.B. and C.V.; M.B., C.V. and P.D. wrote the manuscript. ### Competing Interests The authors declare that they have no competing interests. ### Corresponding author Correspondence to Chandra Venkataraman.	2019-02-23 16:43:17	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6557101607322693, "perplexity": 7774.557765378196}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550249508792.98/warc/CC-MAIN-20190223162938-20190223184938-00339.warc.gz"}
https://xtremepape.rs/threads/igcse-economics-coursebook-2nd-edtion-cambridge-textbook-answers.109496/page-2	# IGCSE Economics (Coursebook 2nd edtion, Cambridge) - Textbook Answers #### raghav4igcse Hi - which textbook do these chapter answers relate to? Thanks Of course, "Cambridge IGCSE & O level Economics, Coursebook 2nd edition" by Susan Grant. #### Shamar Hi - which textbook do these chapter answers relate to? Thanks Yes thank you so much - I could not find which textbook because I have a few with the same title! Thankfully, I have managed to find my relevant textbook. Sorry for the confusion.... thank you again for posting the answer chapters, I really appreciate it. #### Rayyan Sohail Thank You sooooo much, This helped me a lot!!! #### raghav4igcse Thank You sooooo much, This helped me a lot!! No problem at all ! I hope it did. Yes thank you so much - I could not find which textbook because I have a few with the same title! Thankfully, I have managed to find my relevant textbook. Sorry for the confusion.... thank you again for posting the answer chapters, I really appreciate it.	2022-07-07 17:02:48	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8643089532852173, "perplexity": 3383.390423127877}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104495692.77/warc/CC-MAIN-20220707154329-20220707184329-00389.warc.gz"}
https://physics.stackexchange.com/questions/665113/the-same-frequency-sounds-sound-as-different-sounds	# The same frequency sounds sound as different sounds Imagine, the piano is sounded $$500 \mathrm{Hz}$$ sound and the same frequency is sounded in a violin. We always observe their sounds are different even though they're harmonic. I think there must be some other parameters that provide the same frequency sounds differently. Why it's sounded differently?	2022-07-03 13:09:58	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 1, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.48505473136901855, "perplexity": 717.3470241603019}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104240553.67/warc/CC-MAIN-20220703104037-20220703134037-00609.warc.gz"}
https://learnopengl.com/In-Practice/2D-Game/Collisions/Collision-resolution	If you're running AdBlock, please consider whitelisting this site if you'd like to support LearnOpenGL; and no worries, I won't be mad if you don't :) Collision resolution At the end of the last tutorial we had a working collision detection scheme. However, the ball does not react in any way to the detected collisions; it just moves straight through all the bricks. We want the ball to bounce of the collided bricks. This tutorial discusses how we can accomplish this so called collision resolution within the AABB - circle collision detection scheme. Whenever a collision occurs we want two things to happen: we want to reposition the ball so it is no longer inside the other object and second, we want to change the direction of the ball's velocity so it looks like its bouncing of the object. Collision repositioning To position the ball object outside the collided AABB we have to figure out the distance the ball penetrated the bounding box. For this we'll revisit the diagrams from the previous tutorial: Here the ball moved slightly into the AABB and a collision was detected. We now want to move the ball out of the shape so that it merely touches the AABB as if no collision occurred. To figure out how much we need to move the ball out of the AABB we need to retrieve the vector $$\color{brown}{\bar{R}}$$ which is the level of penetration into the AABB. To get this vector $$\color{brown}{\bar{R}}$$ we subtract $$\color{green}{\bar{V}}$$ from the ball's radius. Vector $$\color{green}{\bar{V}}$$ is the difference between closest point $$\color{red}{\bar{P}}$$ and the ball's center $$\color{blue}{\bar{C}}$$. Knowing $$\color{brown}{\bar{R}}$$ we offset the ball's position by $$\color{brown}{\bar{R}}$$ and position the ball directly alongside the AABB; the ball is now properly positioned. Collision direction Next we need to figure out how to update the ball's velocity after a collision. For Breakout we use the following rules to change the ball's velocity: 1. If the ball collides with the right or left side of an AABB, its horizontal velocity (x) is reversed. 2. If the ball collides with the bottom or top side of an AABB, its vertical velocity (y) is reversed. But how do we figure out the direction the ball hit the AABB? There are several approaches to this problem and one of them is that instead of 1 AABB we use 4 AABBs for each brick that we position each at one of its edges. This way we can determine which AABB and thus which edge was hit. However, a simpler approach exists with the help of the dot product. You probably still remember from the transformations tutorial that the dot product gives us the angle between two normalized vectors. What if we were to define four vectors pointing north, south, west or east and calculate the dot product between them and a given vector? The resulting dot product between these four direction vectors and the given vector that is highest (dot product's maximum value is 1.0f which represents a 0 degree angle) is then the direction of the vector. This procedure looks as follows in code: Direction VectorDirection(glm::vec2 target) { glm::vec2 compass[] = { glm::vec2(0.0f, 1.0f), // up glm::vec2(1.0f, 0.0f), // right glm::vec2(0.0f, -1.0f), // down glm::vec2(-1.0f, 0.0f) // left }; GLfloat max = 0.0f; GLuint best_match = -1; for (GLuint i = 0; i < 4; i++) { GLfloat dot_product = glm::dot(glm::normalize(target), compass[i]); if (dot_product > max) { max = dot_product; best_match = i; } } return (Direction)best_match; } The function compares target to each of the direction vectors in the compass array. The compass vector target is closest to in angle, is the direction returned to the function caller. Here Direction is part of an enum defined in the game class's header file: enum Direction { UP, RIGHT, DOWN, LEFT }; Now that we know how to get vector $$\color{brown}{\bar{R}}$$ and how to determine the direction the ball hit the AABB we can start writing the collision resolution code. AABB - Circle collision resolution To calculate the required values for collision resolution we need a bit more information from the collision function(s) than just a true or false so we're going to return a tuple of information, namely if a collision occurred, what direction it occurred and the difference vector ($$\color{brown}{\bar{R}}$$). You can find the tuple container in the <tuple> header. To keep the code slightly more organized we'll typedef the collision relevant data as Collision: typedef std::tuple<GLboolean, Direction, glm::vec2> Collision; Then we also have to change the code of the CheckCollision function to not only return true or false, but also the direction and difference vector: Collision CheckCollision(BallObject &one, GameObject &two) // AABB - AABB collision { [...] return std::make_tuple(GL_TRUE, VectorDirection(difference), difference); else return std::make_tuple(GL_FALSE, UP, glm::vec2(0, 0)); } The game's DoCollision function now doesn't just check if a collision occurred, but also acts appropriately whenever a collision did occur. The function now calculates the level of penetration (as shown in the diagram at the start of this tutorial) and adds or subtracts it from the ball's position based on the direction of the collision. void Game::DoCollisions() { for (GameObject &box : this->Levels[this->Level].Bricks) { if (!box.Destroyed) { Collision collision = CheckCollision(Ball, box); if (std::get<0>(collision)) // If collision is true { // Destroy block if not solid if (!box.IsSolid) box.Destroyed = GL_TRUE; // Collision resolution Direction dir = std::get<1>(collision); glm::vec2 diff_vector = std::get<2>(collision); if (dir == LEFT \|\| dir == RIGHT) // Horizontal collision { Ball->Velocity.x = -Ball->Velocity.x; // Reverse horizontal velocity // Relocate GLfloat penetration = Ball->Radius - std::abs(diff_vector.x); if (dir == LEFT) Ball->Position.x += penetration; // Move ball to right else Ball->Position.x -= penetration; // Move ball to left; } else // Vertical collision { Ball->Velocity.y = -Ball->Velocity.y; // Reverse vertical velocity // Relocate GLfloat penetration = Ball->Radius - std::abs(diff_vector.y); if (dir == UP) Ball->Position.y -= penetration; // Move ball back up else Ball->Position.y += penetration; // Move ball back down } } } } } Don't get too scared by the function's complexity since it is basically a direct translation of the concepts introduced so far. First we check for a collision and if so we destroy the block if it is non-solid. Then we obtain the collision direction dir and the vector $$\color{green}{\bar{V}}$$ as diff_vector from the tuple and finally do the collision resolution. We first check if the collision direction is either horizontal or vertical and then reverse the velocity accordingly. If horizontal, we calculate the penetration value $$\color{brown}R$$ from the diff_vector's x component and either add or subtract this from the ball's position based on its direction. The same applies to the vertical collisions, but this time we operate on the y component of all the vectors. Running your application should now give you a working collision scheme, but it's probably difficult to really see its effect since the ball will bounce towards the bottom edge as soon as you hit a single block and be lost forever. We can fix this by also handling player paddle collisions. Player - ball collisions Collisions between the ball and the player are slightly different than what we've previously discussed since this time the ball's horizontal velocity should be updated based on how far it hit the paddle from its center. The further the ball hits the paddle from its center, the stronger its horizontal velocity should be. void Game::DoCollisions() { [...] Collision result = CheckCollision(Ball, Player); if (!Ball->Stuck && std::get<0>(result)) { // Check where it hit the board, and change velocity based on where it hit the board GLfloat centerBoard = Player->Position.x + Player->Size.x / 2; GLfloat distance = (Ball->Position.x + Ball->Radius) - centerBoard; GLfloat percentage = distance / (Player->Size.x / 2); // Then move accordingly GLfloat strength = 2.0f; glm::vec2 oldVelocity = Ball->Velocity; Ball->Velocity.x = INITIAL_BALL_VELOCITY.x percentage * strength; Ball->Velocity.y = -Ball->Velocity.y; Ball->Velocity = glm::normalize(Ball->Velocity) * glm::length(oldVelocity); } } After we checked collisions between the ball and each brick, we'll check if the ball collided with the player paddle. If so (and the ball is not stuck to the paddle) we calculate the percentage of how far the ball's center is removed from the paddle's center compared to the half-extent of the paddle. The horizontal velocity of the ball is then updated based on the distance it hit the paddle from its center. Aside from updating the horizontal velocity we also have to reverse the y velocity. Note that the old velocity is stored as oldVelocity. The reason for storing the old velocity is that we only update the horizontal velocity of the ball's velocity vector while keeping its y velocity constant. This would mean that the length of the vector constantly changes which has the effect that the ball's velocity vector is much larger (and thus stronger) if the ball hit the edge of the paddle compared to if the ball would hit the center of the paddle. For this reason the new velocity vector is normalized and multiplied by the length of the old velocity vector. This way, the strength and thus the velocity of the ball is always consistent, regardless of where it hits the paddle. You may or may not have noticed it when you ran the code, but there is still a large issue with the player and ball collision resolution. The following video clearly shows what might happen: This issue is called the sticky paddle issue which happens because the player paddle moves with a high velocity towards the ball that results in the ball's center ending up inside the player paddle. Since we did not account for the case where the ball's center is inside an AABB the game tries to continuously react to all the collisions and once it finally breaks free it will have reversed its y velocity so much that it's unsure whether it goes up or down after breaking free. We can easily fix this behavior by introducing a small hack which is possible due to the fact that the we can assume we always have a collision at the top of the paddle. Instead of reversing the y velocity we simply always return a positive y direction so whenever it does get stuck, it will immediately break free. //Ball->Velocity.y = -Ball->Velocity.y; Ball->Velocity.y = -1 * abs(Ball->Velocity.y); If you try hard enough the effect is still noticeable, but I personally find it an acceptable trade-off. The bottom edge The only thing that is still missing from the classic Breakout recipe is some loss condition that resets the level and the player. Within the game class's Update function we want to check if the ball reached the bottom edge, and if so, reset the game. void Game::Update(GLfloat dt) { [...] if (Ball->Position.y >= this->Height) // Did ball reach bottom edge? { this->ResetLevel(); this->ResetPlayer(); } } The ResetLevel and ResetPlayer functions simply re-load the level and reset the objects' values to their original starting values. The game should now look a bit like this: And there you have it, we just finished creating a clone of the classical Breakout game with similar mechanics. You can find the game class' source code here: header, code. A few notes Collision detection is a difficult topic of video game development and possibly its most challenging. Most collision detection and resolution schemes are combined with physics engines as found in most modern-day games. The collision scheme we used for the Breakout game is a very simple scheme and one specialized specifically for this type of game. It should be stressed that this type of collision detection and resolution is not perfect. It calculates possible collisions only per frame and only for the positions exactly as they are at that timestep; this means that if an object would have such a velocity that it would pass over another object within a single frame, it would look like it never collided with this object. So if there are framedrops or you reach high enough velocities, this collision detection scheme will not hold. Several of the issues that can still occur: • If the ball goes too fast, it might skip over an object entirely within a single frame, not detecting any collisions. • If the ball hits more than one object within a single frame, it will have detected two collisions and reverse its velocity twice; not affecting its original velocity. • Hitting a corner of a brick could reverse the ball's velocity in the wrong direction since the distance it travels in a single frame could make the difference between VectorDirection returning a vertical or horizontal direction. These tutorials are however aimed to teach the readers the basics of several aspects of graphics and game-development. For this reason, this collision scheme serves its purpose; its understandable and works quite well in normal scenarios. Just keep in mind that there exist better (more complicated) collision schemes that work quite well in almost all scenarios (including movable objects) like the separating axis theorem. Thankfully, there exist large, practical and often quite efficient physics engines (with timestep-independent collision schemes) for use in your own games. If you wish to delve further into such systems or need more advanced physics and have trouble figuring out the mathematics, Box2D is a perfect 2D physics library for implementing physics and collision detection in your applications. HI	2019-02-18 16:34:06	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.45595622062683105, "perplexity": 1063.8778131688598}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247487595.4/warc/CC-MAIN-20190218155520-20190218181520-00526.warc.gz"}
https://techwhiff.com/learn/meadow-hills-hospital-is-a-211-bed-acute-care/130186	# Meadow Hills Hospital is a 211-bed acute care hospital w its medical staff. Meadow serves a... ##### Help on 13,14,15,16 standard deviation to find the Be sure to note for questions 13-16, you... help on 13,14,15,16 standard deviation to find the Be sure to note for questions 13-16, you do not need the mean or Z-score. 13. Find the Z-score that describes the following proportions under the normal curve (within 0.01 points): Top 38% of scores,38 14. Find the Z-score that describes the fol... ##### Consco, a U.S. company, hires Tekco to act on its behalf in doing business in Kenya.... Consco, a U.S. company, hires Tekco to act on its behalf in doing business in Kenya. Tekco has the authority to sign contracts and negotiate with government officials on behalf of Consco. Consco's activity can best be described as: a. indirect exporting. b. direct exporting. c. licensing. d. man... ##### 4. Compute the following determinants: 230 0 6 0 3 2 0 4 0 0 (a)... 4. Compute the following determinants: 230 0 6 0 3 2 0 4 0 0 (a) b b a b 5 -3 0 0 0 0 6 0 6 0 0 0... ##### How do we account for unusual experimental results? How do we account for unusual experimental results?... ##### Middlefield Motors is evaluating project Z. The project would require an initial investment of 76,000 dollars... Middlefield Motors is evaluating project Z. The project would require an initial investment of 76,000 dollars that would be depreciated to 6,000 dollars over 6 years using straight-line depreciation. The first annual operating cash flow of 28,000 dollars is expected in 1 year, and annual operating c... On March 1, 2021, Stratford Lighting issued 12% bonds, dated March 1, with a face amount of $720,000. The bonds sold for$714,000 and mature on February 28, 2041 (20 years). Interest is paid semiannually on August 31 and February 28. Stratford uses the straight-line method and its fiscal year ends D... ##### Suppose that the following equations characterize the economy; C=$160+0.80Y_D I=$300 G=$200 T=$200 Please answer the following... Suppose that the following equations characterize the economy; C=$160+0.80Y_D I=$300 G=$200 T=$200 Please answer the following questions; Write down the equilibrium condition (please use Z for the total demand for goods and Y for production). Write down an equation for Z. Calculate GDP (Y=?) Calcula... ##### Kaler Company purchased a building and land with a fair market value of $575,000 (building,$325,000... Kaler Company purchased a building and land with a fair market value of $575,000 (building,$325,000 and land, $250,000) on January 1, 2018. Kaler signed a 25-year, 15% mortgage payable. Kaler will make monthly payments of$7,364.78. Round to two decimal places. Explanations are not required for jou...	2023-03-30 05:14:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.30102092027664185, "perplexity": 3321.2093888963386}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296949097.61/warc/CC-MAIN-20230330035241-20230330065241-00067.warc.gz"}
http://tex.stackexchange.com/questions/95899/longtable-and-rowcolors	# longtable and rowcolors I tried to use the longtable with the row color and I just simply can get it to work. When I try to do: \documentclass[10pt, landscape]{report} %Packages \usepackage[utf8]{inputenc} \usepackage[english, ngerman]{babel} \usepackage{booktabs} \usepackage{array} \newcounter{rowno} \setcounter{rowno}{0} \usepackage{longtable} \usepackage[table]{xcolor} \begin{document} \rowcolors{1}{gray!40!white}{blue!10!white} \begin{longtable}{>{\stepcounter{rowno}\therowno.}c c l c l l l l } \hiderowcolors \multicolumn{1}{c}{No.} & Article & Word & Type & 3. Person & Präteritum & Perfect & Translation \\ \toprule \showrowcolors \hiderowcolors \multicolumn{1}{c}{No.} & Article & Word & Type & 3. Person & Präteritum & Perfect & Translation \\ \toprule \showrowcolors & --- & abdecken & verb & deckt ab & deckte ab & abgedeckt & to cover, \\ \midrule & --- & abholen & verb & holt ab & holte ab & abgeholte & to collect or pick up \\ \midrule \end{longtable} \end{document} I am getting every row to be the same color - gray. I tried doing different things with it but to no avail. Did anyone dealt with something like that and have a solution? - give a complete example which shows what you see – Herbert Jan 29 '13 at 18:34 What do want to do with {\stepcounter{rowno}\therowno.}? Please make your given code compilable, so add used packages, document class and \end{document}. – Kurt Jan 29 '13 at 18:59 The \stepcounter{rowno}\therowno gives an automatic count to the rows in the table (there are ~240 entries in it) – ulek Jan 29 '13 at 19:06 it is a problem with your setting of \show/hiderowcolors. This works: \documentclass[10pt, landscape]{report} %Packages \usepackage[utf8]{inputenc} \usepackage[english, ngerman]{babel} \usepackage{booktabs} \usepackage{array} \newcounter{rowno} \setcounter{rowno}{0} \usepackage{longtable} \usepackage[table]{xcolor} \begin{document} \rowcolors{3}{gray!40!white}{blue!40!white!80} \begin{longtable}{>{\stepcounter{rowno}\therowno.}c c l c l l l l } \multicolumn{1}{c}{No.} & Article & Word & Type & 3. Person & Präteritum & Perfect & Translation \\\toprule \multicolumn{1}{c}{No.} & Article & Word & Type & 3. Person & Präteritum & Perfect & Translation \\ \toprule & --- & abdecken & verb & deckt ab & deckte ab & abgedeckt & to cover, \\ & --- & abholen & verb & holt ab & holte ab & abgeholte & to collect or pick up\\ & --- & abholen & verb & holt ab & holte ab & abgeholte & to collect or pick up\\ & --- & abholen & verb & holt ab & holte ab & abgeholte & to collect or pick up\\ \bottomrule \end{longtable} \end{document} - Actually it appears that \midrule was the problem. When I removed the \show/hidecolors I got nowhere fast, however after removing the \midrule' everything works fine – ulek Jan 29 '13 at 19:26 Actually it appears that \midrule was the problem. When I removed the \show/hidecolors I got nowhere fast, however after removing the \midrule` everything works fine	2014-04-24 09:51:51	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.821324348449707, "perplexity": 5902.47667919368}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00179-ip-10-147-4-33.ec2.internal.warc.gz"}
http://clay6.com/qa/138224/torque-acting-on-a-electric-dipole-in-a-uniform-electric-fiels-hat-e	Answer Comment Share Q) # Torque acting on a electric dipole in a uniform electric fiels $\hat E$ $\begin{array}{1 1} (a)\; Torque=\overrightarrow{p}.\hat E \\ (b)\;Torque=\|\overrightarrow{p}\|\|\hat E\| \cos \theta\\ (c) Torque =\overrightarrow{P} \times \overrightarrow{E} \\ (d) Torque =\large\frac{\overrightarrow{P} \times \overrightarrow{E}}{2} \end{array}$ ## 1 Answer Comment A) Solution : $(c) Torque =\overrightarrow{P} \times \overrightarrow{E}$	2019-09-16 02:20:10	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9940485954284668, "perplexity": 4861.242795797185}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514572471.35/warc/CC-MAIN-20190916015552-20190916041552-00214.warc.gz"}
http://mydisneymania.blogspot.com/2015/08/wdw-picture-of-day_20.html	## Thursday, August 20, 2015 ### WDW Picture of the Day Entrance to Spaceship Earth at Epcot. So, if you've been a long-time WDW fan, did you ever remember Spaceship Earth having this long of a line LATE in the day? Unfortunately, this is a direct consequence of the Fastpass+. Attractions that normally do not have a long line late in the day are now getting these long lines. And fastpass lines that used to move fast are now crawling. In fact, in some cases, there's a line trying to get into the fastpass line! See Kilimanjaro Safari! Zz.	2021-10-17 07:32:14	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8064866065979004, "perplexity": 3211.821877474326}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585121.30/warc/CC-MAIN-20211017052025-20211017082025-00489.warc.gz"}
http://mathhelpforum.com/differential-equations/186209-finding-particular-solution-differential-equations.html	# Math Help - Finding particular solution from differential equations 1. ## Finding particular solution from differential equations I am stuck on how to solve the following, i don't weather my arithmetic is wrong or what i am doing. So could someone please do each step. Find the particular solution to a) dh/dt=2h(3-h) where y=1 and t=0 b) dy/dx= e^y +1/e^y where y=0 when x=0. 2. ## Re: Finding particular solution from differential equations First of all try to find the general solution of the DE (it's a separable one), for example the first one: $\frac{dh}{dt}=2h(3-h)$ Calculating general solution, first rewrite the DE: $\frac{dh}{2h(3-h)}=dt$ Take the integral of both sides: $\int \frac{dh}{2h(3-h)}=\int dt$ If you split the integrand in partial fractions (in LHS) you'll get: $\frac{1}{6}\int \frac{dh}{h} - \frac{1}{6}\int \frac{dh}{h-3}=tt+C$ $=\frac{1}{6}\ln\|h\|-\frac{1}{6}\ln\|h-3\|=t+C$ $\Leftrightarrow \ln\left\|\frac{h}{h-3}\right\|=6t+6C$ $\Leftrightarrow e^{6t+6C}=\left\|\frac{h}{h-3}\right\|$ Now try to continue and try to get an expression $h(t)=...$ and afterwards substitute $h=1,t=0$ to find $C$. (Note: $\ln$ is the natural logarithm) 3. ## Re: Finding particular solution from differential equations Originally Posted by johnsy123 I am stuck on how to solve the following, i don't weather my arithmetic is wrong or what i am doing. So could someone please do each step. Find the particular solution to a) dh/dt=2h(3-h) where y=1 and t=0 b) dy/dx= e^y +1/e^y where y=0 when x=0. Hiya Johnsy, is the second one $\displaystyle \frac{dy}{dx} = e^y + \frac{1}{e^y}$ or $\displaystyle \frac{dy}{dx} = \frac{e^y + 1}{e^y}$? For the first... \displaystyle \begin{align} \frac{dh}{dt} &= 2h(3 - h) \\ \frac{dt}{dh} &= \frac{1}{2h(3-h)} \\ t &= \int{\frac{1}{2h(3 -h)}\,dh} \end{align} You will now need to apply partial fractions. 4. ## Re: Finding particular solution from differential equations It is (e^y +1)/(e^y) 5. ## Re: Finding particular solution from differential equations Originally Posted by johnsy123 It is (e^y +1)/(e^y) OK, so \displaystyle \begin{align} \frac{dy}{dx} &= \frac{e^y + 1}{e^y} \\ \frac{dx}{dy} &= \frac{e^y}{e^y + 1} \\ x &= \int{\frac{e^y}{e^y + 1}\,dy} \end{align} Now make the substitution $\displaystyle u = e^y + 1 \implies du = e^y\,dy$ and the integral becomes $\displaystyle x = \int{\frac{1}{u}\,du}$. I'm sure you can go from here 6. ## Be carefully using the results of Wolfram!... Originally Posted by johnsy123 It is (e^y +1)/(e^y) ... so that the DE is... $y^{'}= 1+e^{-y}\ ,\ y(0)=0$ (1) In (1) the variables are separable and the solution is found with the standard approach... $\int \frac{d y}{1+e^{-y}} = \int dx \implies y+ \ln (1+e^{-y}) = x + c$ (2) ... and the 'initial condition' means that is $c=0$. All right?... yes, of course, but it is interesting to spend a little of time about the details of the integration performed in (2). The integral is... $\int \frac{d y}{1+e^{-y}} = \int dy - \int \frac{e^{-y}}{1+e^{-y}}\ dy= y+\ln (1+e^{-y}) + c$ (3) The integral (3) is not 'trascendental' but may be that someone prefers to use Wolfram because 'it saves time'... well!... please observe the result supplied by Wolfram ... Wolfram Mathematica Online Integrator Kind regards $\chi$ $\sigma$ 7. ## Re: Be carefully using the results of Wolfram!... Just so everyone knows, in Australia Separation of Variables isn't taught until university. The OP's question is from Year 12 Specialist Mathematics, so the method I gave is the method that is expected. 8. ## Re: Be carefully using the results of Wolfram!... Originally Posted by Prove It Just so everyone knows, in Australia Separation of Variables isn't taught until university. The OP's question is from Year 12 Specialist Mathematics, so the method I gave is the method that is expected. Of course different methods can give the same result... in this case the 'Year 12 Specialist Mathematics method' gives the solution... $x= \ln (1+e^{y}) + c$ (1) ... and the 'chisigma method' gives the solution... $x= y + \ln (1+e^{-y}) + c = y + \ln (1+e^{-y}) + c$ (2) ... but considering the 'identity'... $\ln (1+e^{y}) = \ln e^{y} + \ln (1+e^{-y})= y + \ln (1+e^{-y})$ (3) ... it is obvious that (1) and (2) are exactly the same expression. Scope [a little polemical may be...] of my post was to do some considerations about the 'reliability' of some 'very popular tools' ... Kind regards $\chi$ $\sigma$	2014-12-20 10:10:11	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 28, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9981970191001892, "perplexity": 2508.951937017038}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802769642.136/warc/CC-MAIN-20141217075249-00088-ip-10-231-17-201.ec2.internal.warc.gz"}
https://www.springerprofessional.de/complex-analysis/13784754	main-content ## Über dieses Buch The main idea of this book is to present a good portion of the standard material on functions of a complex variable, as well as some new material, from the point of view of functional analysis. The main object of study is the algebra H(G) of all holomorphic functions on the open set G, with the topology on H(G) of uniform convergence on compact subsets of G. From this point of vie~, the main theorem of the theory is Theorem 9.5, which concretely identifies the dual of H(G) with the space of germs of holomorphic functions on the complement of G. From this result, for example, Runge's approximation theorem and the global Cauchy integral theorem follow in a few short steps. Other consequences of this duality theorem are the Germay interpolation theorem and the Mittag-Leffler Theorem. The approach via duality is entirely consistent with Cauchy's approach to complex variables, since curvilinear integrals are typical examples of linear functionals. The prerequisite for the book is a one-semester course in com plex variables at the undergraduate-graduate level, so that the elements of the local theory are supposed known. In particular, the Cauchy Theorem for the square and the circle are assumed, but not the global Cauchy Theorem in any of its forms. The second author has three times taught a graduate course based on this material at the University of Illinois, with good results. ## Inhaltsverzeichnis ### § 1. Preliminaries: Set Theory and Topology Abstract We assume familiarity with the rudiments of informal set theory including such notions as set, subset, superset, the null set Ɉ, the union or intersection of a family of sets, set difference (A\B), complement (compl A), Cartesian product; functions, domain, range, one-to-one, onto, image, inverse, restriction; partial ordering, linear (or total) ordering, and equivalence relation. D. H. Luecking, L. A. Rubel ### § 2. Preliminaries: Vector Spaces and Complex Variables Abstract Our goal is the treatment of complex analysis from the point of view of topological vector spaces. Here we present the basic definitions and some well-known facts. D. H. Luecking, L. A. Rubel ### § 3. Properties of C(G) and H(G) Abstract With little change we can study functions of several variables. To keep the notation simple, we will restrict ourselves to two variables. In that case, G is an open set in ℂ × ℂ = ℂ2. Then C(G) is defined just as in the one-variable case. (The distance in ℂ2 between two points (z,w) and (z’,w’) will be denoted d((z,w), (z’,w’)) = (\|z - z’\|2 + \|w - w’\|2)½. We also define H(G) as before, but must first define holomorphic. D. H. Luecking, L. A. Rubel ### § 4. More about C(G) and H(G) Abstract It is commonly prove in courses of advanced calculus that compact sets in $$\mathbb{R}$$ (or more generally $${\mathbb{R}^n}$$) are characterized by being closed and bounded. In a general topological vector space only one implication is correct. Here and in the future we will assume that our topological vector spaces are Hausdorff. D. H. Luecking, L. A. Rubel ### § 5. Duality Abstract The concept of duality is one of the most productive in analysis. One can very nearly characterize applications of functional analysis as applications of duality. Most of our goal in succeeding sections will be identifying the dual of H(G) and exploiting that identification. Toward this end, we begin with the definition. D. H. Luecking, L. A. Rubel ### § 6. Duality of H(G)—The Case of the Unit Disc Abstract We begin with a general result about linear functionals on a locally convex topological vector space. Let E have the topology generated by a family P of seminorms. For each non-empty finite set A = {‖•‖1, ‖•‖2,…, ‖•‖n} ⊂ P, define$${\left\\| x \right\\|_A} = \mathop{{\max }}\limits_{{l \leqslant j \leqslant n}} \,{\left\\| x \right\\|_j}$$, x ∈ E. Then ‖•‖A is a seminorm. Let = P ∪ {‖•‖A: A is a non empty finite subset of P}; then P and generate the same topology on E (Exercise 2). Consequently, we may assume P = in the following proposition. D. H. Luecking, L. A. Rubel ### § 7. The Hahn-Banach Theorem, and Applications Abstract A major tool in the application of duality results (in any locally convex topological vector space) is the Hahn-Banach Theorem. We state here one standard version (there are many equivalent versions) and two important corollaries. D. H. Luecking, L. A. Rubel ### § 8. More Applications Abstract In this section E will denote the space H(ℂ), i.e., the space of entire functions with the topology of uniform convergence on compact sets. D. H. Luecking, L. A. Rubel ### § 9. The Dual of H(G) Abstract We want to prove, as in the case of the disk, that H(G)* = H 0(ℂ \ G). We first study the dual of C(G). We change our notation here and write L(f) = ∫ fdμ when L ∈ C(G). (For the reader unfamiliar with integration theory this is simply a change in notation: The left-hand side defines the right-hand side. There are two advantages to this notation. First, it is the notation in which research papers are written. Second, the reader can call upon her experience with integration for intuition. For the mathematically advanced reader: we are invoking the Riesz Representation Theorem for C(G) .) We call μ the “measure” associated with L, and we may identify μ and L. The collection of all such μ is denoted M0(G), so that M0(G) = C(G)*. We also write L(f) = ∫ f(z)dμ(z) when it is necessary to indicate the independent variable. “Measures” have the same properties as continuous linear functionals (which is what they are); for reinforcement, we list them here. Given μ ∈ M0(G): i) ∫ (f + g)dμ = ∫ fdμ + ∫ gdμ, f, g ∈ C(G). ii) ∫ afdμ = a ∫ fdμ, f ∈ C(G), a ∈ ℂ. iii) If fn → f in C(G) then ∫ fndμ → ∫ fdμ. iv) There is a compact set K ⊆ G such that \| ∫ fdμ \| ≤ C‖f‖K for all f ∈ C(G). D. H. Luecking, L. A. Rubel ### § 10. Runge’s Theorem Abstract If f ∈ H(G), G a connected open set, it is a consequence of the power series expansion for holomorphic functions that if f(zn) = 0, zn → z0 ∈ G then f = 0 in G. It is also a consequence that if f(n)(z0) = 0 for n = 0,1,2,…, then f = 0 in G. We adopt conventions about “sets with multiplicity” that allow us to treat both cases as one. D. H. Luecking, L. A. Rubel ### § 11. The Cauchy Theorem Abstract Runge’s Theorem can be used to prove Cauchy’s Theorem. This will require the elements of integration theory described in Chapter 2. Recall that for a rectifiable curve γ: [0,1] → ℂ, we let ‖γ‖ denote its length, i.e. ‖γ‖ = <Inline>1</Inline> \|γ′(t)\|dt, so that $$\left\| {\int_{\gamma } {f(z)dz} } \right\| \leqslant \left\\| \gamma \right\\| \cdot \left\\| f \right\\|$$ where ‖f‖ = sup{\| f(z) \|: z ∈ γ}. The reader is reminded that γ^ denotes the “physical curve”, that is the image of γ. D. H. Luecking, L. A. Rubel ### § 12. Constructive Function Theory Abstract The goal of this section is the construction, by means of sums and products of simpler functions, of holomorphic functions with prescribed behavior. In what follows, when we speak of a sequence of complex numbers we ordinarily mean a sequence with multiplicity, so that a function taking some value at a point of the sequence must take that value with the appropriate multiplicity. We also exclude the trivial function f ≡ 0 unless otherwise noted. D. H. Luecking, L. A. Rubel ### § 13. Ideals in H(G) Abstract We use the results of the previous section to derive some descriptive results on ideals of holomorphic functions. D. H. Luecking, L. A. Rubel ### § 14. The Riemann Mapping Theorem Abstract The Riemann Mapping Theorem implies that, as far as H(G) can tell, all simply connected regions are the “same”. To clarify what this means we need the following notion of equivalence. D. H. Luecking, L. A. Rubel ### § 15. Carathéodory Kernels and Farrell’s Theorem Abstract Given a sequence {Gn} of regions and a region G ≠ Ɉ such that G ⊆ Gn+1 ⊆ Gn for n = 1,2,3,.., we say that a superset G’ of G is suitable if G’ is connected and G’ ⊆ ∩Gn. Then ker[Gn: G], the kernel of {Gn} with respect to G, is defined as the union of all suitable supersets of G. D. H. Luecking, L. A. Rubel ### § 16. Ring (not Algebra) Isomorphisms of H(G) Abstract We return here to the ring structure of H(G). A ring homomorphism of R1 to R2 is a function φ: R1 → R2 which preserves multiplication i.e. φ(rs) = φ(r)φ(s) and φ(r + s) = φ(r) + φ(s) for all r, s ∈ R1. A ring isomorphism is a ring homomorphism that is one-to-one and onto. If G and G’ are two conformally equivalent domains in ℂ then there is an algebra isomorphism from H(G) to H(G’) as we saw in Chapter 5. An algebra isomorphism will be a ring isomorphism which additionally preserves scalar multiplication. It follows from Proposition 5.4 that H(G) and H(G’) are isomorphic as algebras if and only if G and G’ are conformally equivalent. But a ring isomorphism can exist without conformal equivalence. D. H. Luecking, L. A. Rubel ### § 17. Dual Space Topologies Abstract This chapter is intended as a prerequisite for later chapters. In it we introduce a topology on the dual of a topological vector space. We present some of the standard results in the theory of Fréchet spaces and some additional results on topological vector spaces in general. D. H. Luecking, L. A. Rubel ### § 18. Interpolation Abstract In this chapter we study we study interpolation by holomorphic functions from a different point of view from that used in 12. In particular, the Germay Theorem (12.14) says that there exists an entire function that interpolates given values at a given sequence of points. There are three major approaches that can be taken to proving such a theorem. The first is via the Mittag-Leffler Theorem and Weierstrass products, which involves writing an explicit formula. The second is via solving infinitely many linear equations in infinitely many unknowns, the Taylor coefficients. (See [M. Eidelheit] and [P. J. Davis].) The third is via functional analysis—specifically the Banach-Dieudonné theorem. Kere we take the third route, obtaining in the process a functional analysis proof of Theorem 12.18. D. H. Luecking, L. A. Rubel ### § 19. Gap-Interpolation Theorems Abstract There are many theorems in classical analysis where gaps play a rôle. We take up now some considerations from [N. Kalton and L. A. Rubel] where gaps and interpolation are mixed. The idea is to take the Germay interpolation situation, where we want f(zn) = wn, n = 1,2,3,… for some entire function f but now require that f have the form $$f(z) = \mathop{\Sigma }\limits_{{\lambda \in \Lambda }} \,{a_{\lambda }}{z^{\lambda }}$$ where ⋀ is a given set of positive integers. For certain ⋀ (like ⋀ = $$\Lambda = \mathbb{N}$$, the set of all positive integers), this interpolation is always possible—provided we require \|zn\| → ∞ (and no zn = 0). D. H. Luecking, L. A. Rubel ### § 20. First-Order Conformal Invariants Abstract The Theme of this section is the following: Suppose you find yourself on a plane domain, with only a restricted logic at your disposal; how closely can you determine which domain you are on—up to conformal equivalence? This leads to a study of a system of conformal invariants, the first-order conformal invariants (FOCI), which are obtained from the elementary properties of the algebra (or ring) of analytic functions on plane domains. Although the formal definition of FOCI is given in the terminology of mathematical logic, these invariants are nonetheless all included within the framework of classical function theory. Each of the FOCI corresponds to an elementary assertion about analytic functions that can be understood without any knowledge of mathematical logic. D. H. Luecking, L. A. Rubel ### Backmatter Weitere Informationen	2020-06-07 00:38:24	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8621355891227722, "perplexity": 953.873240750678}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590348521325.84/warc/CC-MAIN-20200606222233-20200607012233-00115.warc.gz"}
https://mitgcm.readthedocs.io/en/latest/overview/finding_pressure.html	# 1.3.6. Finding the pressure field¶ Unlike the prognostic variables $$u$$, $$v$$, $$w$$, $$\theta$$ and $$S$$, the pressure field must be obtained diagnostically. We proceed, as before, by dividing the total (pressure/geo) potential in to three parts, a surface part, $$\phi _{s}(x,y)$$, a hydrostatic part $$\phi _{hyd}(x,y,r)$$ and a non-hydrostatic part $$\phi _{nh}(x,y,r)$$, as in (1.25), and writing the momentum equation as in (1.26). ## 1.3.6.1. Hydrostatic pressure¶ Hydrostatic pressure is obtained by integrating (1.27) vertically from $$r=R_{o}$$ where $$\phi _{hyd}(r=R_{o})=0$$, to yield: $\int_{r}^{R_{o}}\frac{\partial \phi _{hyd}}{\partial r}dr=\left[ \phi _{hyd} \right] _{r}^{R_{o}}=\int_{r}^{R_{o}}-bdr$ and so (1.33)$\phi _{hyd}(x,y,r)=\int_{r}^{R_{o}}bdr$ The model can be easily modified to accommodate a loading term (e.g atmospheric pressure pushing down on the ocean’s surface) by setting: (1.34)$\phi _{hyd}(r=R_{o})=loading$ ## 1.3.6.2. Surface pressure¶ The surface pressure equation can be obtained by integrating continuity, (1.3), vertically from $$r=R_{fixed}$$ to $$r=R_{moving}$$ $\int_{R_{fixed}}^{R_{moving}}\left( \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v} }_{h}+\partial _{r}\dot{r}\right) dr=0$ Thus: $\frac{\partial \eta }{\partial t}+\vec{\mathbf{v}}.\nabla \eta +\int_{R_{fixed}}^{R_{moving}}\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}} _{h}dr=0$ where $$\eta =R_{moving}-R_{o}$$ is the free-surface $$r$$-anomaly in units of $$r$$. The above can be rearranged to yield, using Leibnitz’s theorem: (1.35)$\frac{\partial \eta }{\partial t}+\mathbf{\nabla }_{h}\cdot \int_{R_{fixed}}^{R_{moving}}\vec{\mathbf{v}}_{h}dr=\text{source}$ where we have incorporated a source term. Whether $$\phi$$ is pressure (ocean model, $$p/\rho _{c}$$) or geopotential (atmospheric model), in (1.26), the horizontal gradient term can be written (1.36)$\mathbf{\nabla }_{h}\phi _{s}=\mathbf{\nabla }_{h}\left( b_{s}\eta \right)$ where $$b_{s}$$ is the buoyancy at the surface. In the hydrostatic limit ($$\epsilon _{nh}=0$$), equations (1.26), (1.35) and (1.36) can be solved by inverting a 2-d elliptic equation for $$\phi _{s}$$ as described in Chapter 2. Both ‘free surface’ and ‘rigid lid’ approaches are available. ## 1.3.6.3. Non-hydrostatic pressure¶ Taking the horizontal divergence of (1.26) and adding $$\frac{\partial }{\partial r}$$ of (1.28), invoking the continuity equation (1.3), we deduce that: (1.37)$\nabla _{3}^{2}\phi _{nh}=\nabla .\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{ \nabla }_{h}^{2}\phi _{s}+\mathbf{\nabla }^{2}\phi _{hyd}\right) =\nabla . \vec{\mathbf{F}}$ For a given rhs this 3-d elliptic equation must be inverted for $$\phi _{nh}$$ subject to appropriate choice of boundary conditions. This method is usually called The Pressure Method [Harlow and Welch (1965) [HW65]; Williams (1969) [Wil69]; Potter (1973) [Pot73]. In the hydrostatic primitive equations case (HPE), the 3-d problem does not need to be solved. ### Boundary Conditions¶ We apply the condition of no normal flow through all solid boundaries - the coasts (in the ocean) and the bottom: (1.38)$\vec{\mathbf{v}}.\widehat{n}=0$ where $$\widehat{n}$$ is a vector of unit length normal to the boundary. The kinematic condition (1.38) is also applied to the vertical velocity at $$r=R_{moving}$$. No-slip $$\left( v_{T}=0\right) \$$or slip $$\left( \partial v_{T}/\partial n=0\right) \$$conditions are employed on the tangential component of velocity, $$v_{T}$$, at all solid boundaries, depending on the form chosen for the dissipative terms in the momentum equations - see below. Eq. (1.38) implies, making use of (1.26), that: (1.39)$\widehat{n}.\nabla \phi _{nh}=\widehat{n}.\vec{\mathbf{F}}$ where $\vec{\mathbf{F}}=\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{\nabla }_{h}\phi_{s}+\mathbf{\nabla }\phi _{hyd}\right)$ presenting inhomogeneous Neumann boundary conditions to the Elliptic problem (1.37). As shown, for example, by Williams (1969) [Wil69], one can exploit classical 3D potential theory and, by introducing an appropriately chosen $$\delta$$-function sheet of ‘source-charge’, replace the inhomogeneous boundary condition on pressure by a homogeneous one. The source term $$rhs$$ in (1.37) is the divergence of the vector $$\vec{\mathbf{F}}.$$ By simultaneously setting $$\widehat{n}.\vec{\mathbf{F}}=0$$ and $$\widehat{n}.\nabla \phi _{nh}=0\$$on the boundary the following self-consistent but simpler homogenized Elliptic problem is obtained: $\nabla ^{2}\phi _{nh}=\nabla .\widetilde{\vec{\mathbf{F}}}\qquad$ where $$\widetilde{\vec{\mathbf{F}}}$$ is a modified $$\vec{\mathbf{F}}$$ such that $$\widetilde{\vec{\mathbf{F}}}.\widehat{n}=0$$. As is implied by (1.39) the modified boundary condition becomes: (1.40)$\widehat{n}.\nabla \phi _{nh}=0$ If the flow is ‘close’ to hydrostatic balance then the 3-d inversion converges rapidly because $$\phi _{nh}\$$is then only a small correction to the hydrostatic pressure field (see the discussion in Marshall et al. (1997a,b) [MHPA97] [MAH+97]. The solution $$\phi _{nh}\$$to (1.37) and (1.39) does not vanish at $$r=R_{moving}$$, and so refines the pressure there.	2019-08-25 13:35:01	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8561485409736633, "perplexity": 1076.24740442569}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027330233.1/warc/CC-MAIN-20190825130849-20190825152849-00550.warc.gz"}
https://www.transtutors.com/questions/activity-based-costing-p7-kall-company-produces-cellular-phones-it-has-just-complete-1355638.htm	# Activity-Based Costing p7. Kall Company produces cellular phones. It has just completed an order... Activity-Based Costing p7. Kall Company produces cellular phones. It has just completed an order for 10,000 phones placed by Connect, Ltd. Kall recently shifted to an activity-based costing sys- tem, and its controller is interested in the impact that the ABC system had on the Connect order. Data for that order are as follows: direct materials, $36,950; pur- chased parts,$21,100; direct labor hours, 220; and average direct labor pay rate per hour, $15. Under Kall’s traditional costing system, overhead costs were assigned at a rate of 270 percent of direct labor cost. Data for activity-based costing for the Connect order follow. Activity Cost Driver Activity Cost rate Activity Usage Electrical engineering design Engineering hours$19 per engineering hour 32 engineering hours Setup Number of setups $29 per setup 11 setups Parts production Machine hours$26 per machine hour 134 machine hours Product testing Number of tests $32 per test 52 tests Packaging Number of packages$0.0374 per package 10,000 packages Building occupancy Machine hours $9.80 per machine hour 134 machine hours Assembly Direct labor hours$15 per direct labor hour 220 direct labor hours reQUIreD 1. Use the traditional costing approach to compute the total cost and the product unit cost of the Connect order. (Round unit costs to the nearest cent.) 2. Using the cost hierarchy, identify each activity as unit level, batch level, product level, or facility level. 3. Prepare a bill of activities for the activity costs. 4. Use ABC to compute the total cost and product unit cost of the Connect order. (Round activity costs to the nearest dollar, and round unit costs to the nearest cent.) 5. aCCounting ConneCtion ▶ What is the difference between the product unit cost you computed using the traditional approach and the one you computed using ABC? Does the use of ABC guarantee cost reduction for every order?	2018-09-24 09:20:52	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.24120891094207764, "perplexity": 9826.577025379862}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267160337.78/warc/CC-MAIN-20180924090455-20180924110855-00530.warc.gz"}
http://mathoverflow.net/feeds/question/81721	A homotopy commutative diagram that cannot be strictified - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T23:56:14Z http://mathoverflow.net/feeds/question/81721 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/81721/a-homotopy-commutative-diagram-that-cannot-be-strictified A homotopy commutative diagram that cannot be strictified Akhil Mathew 2011-11-23T17:16:50Z 2011-12-30T04:03:11Z <p>By a "homotopy commutative diagram," I mean a functor $F: \mathcal{I} \to \mathrm{Ho}(\mathrm{Top})$ to the homotopy category of spaces. By a "strictification," I mean a lifting of such a functor to the category $\mathrm{Top}$ of topological spaces. I am curious about simple instances where such a "strictification" does not exist: that is, there are obstacles to making a diagram that commutes only up to homotopy strictly commute.</p> <p>For example, it is known that a homotopy coherent diagram in topological spaces can always be strictified. In other words, if one imposes homotopy commutativity, but also keeps track of all the homotopies and requires that they satisfy compatibility conditions with one another, then one can strictify. More precisely, a homotopy coherent diagram $\mathcal{I} \to \mathrm{Top}$ can be described as a simplicial functor $\mathfrak{C} \mathcal{I} \to \mathcal{Top}$, where $\mathfrak{C}\mathcal{I}$ (notation of HTT) is a "thickened" version of $\mathcal{I}$ where the usual relations in $\mathcal{I}$ only hold up to coherent homotopy. Then it is a theorem of Dwyer and Kan that the projective model structures on homotopy coherent diagrams (i.e. $\mathrm{Fun}(\mathfrak{C} \mathcal{I}, \mathrm{SSet})$) and on commutative diagrams (i.e. $\mathrm{Fun}(\mathcal{I}, \mathrm{SSet})$ are Quillen equivalent under some form of Kan extension.</p> <p>Part of this question is to help myself understand why homotopy coherence is more natural than homotopy commutativity. One example I have in mind is the following: let $X$ be a connected H space, and suppose $X$ is not weakly equivalent to a loop space. Then one can construct a simplicial diagram in the homotopy category given by the nerve construction applied to $X$ (as a group object in the homotopy category). This cannot be lifted to the category of spaces, because then Segal's de-looping machinery would enable us to show that $X$ was the loop space of the geometric realization of this simplicial space. </p> <p>However, this example is rather large. Is there a simple, toy example (preferably involving a finite category) of a homotopy commutative diagram that cannot be strictified?</p> http://mathoverflow.net/questions/81721/a-homotopy-commutative-diagram-that-cannot-be-strictified/81749#81749 Answer by Tyler Lawson for A homotopy commutative diagram that cannot be strictified Tyler Lawson 2011-11-23T20:02:18Z 2011-11-26T22:52:41Z <p>EDIT: Tom Goodwillie, in the comments, points out (interpreted in a quite charitable way) that there are two mistakes with the following argument. The <code>$\pi_1$</code>-obstruction does exist. However:</p> <ul> <li><p>It misreads the question and assumes that there is some portion of an actual diagram which is homotopy commutative.</p></li> <li><p>Even given that, it makes a mistake: it asserts that we can ignore the indeterminacy. This is wrong. For any space in the diagram, the space of maps to the terminal object $S^1$ is not simply-connected, but instead is homotopy equivalent to $S^1$. This allows you to simply erase the obstruction in <code>$\pi_1$</code> by simply making different choices of homotopies.</p></li> </ul> <p>As such, it's advisable that what's written below be demoted and I'll try to get a correct version later.</p> <hr> <p>One of the classical obstructions to realizability is for cubical diagrams. Here, the category $I$ is the poset of subsets of <code>$\{0,1,2\}$</code>. Given such a diagram which is homotopy commutative, you can get twelve maps (one per edge in the cube) and six homotopies (one per face, which are well-defined up to "multiplication by an element in <code>$\pi_1$</code>") and the collection of all possible ways to compose maps, or compose maps with homotopies, gives rise to a hexagon in the space $Map(X,Z)$. Here $X$ is the image of initial object and $Z$ is the image of the terminal object in the cube. If the diagram is actually commutative then you can choose your homotopies to be trivial, and get a trivial hexagon; if it is equivalent to an honestly commutative diagram then you can choose your six homotopies on faces so that the hexagon in $Map(X,Z)$ can be filled in with a disc, or equivalently "represents the trivial element in <code>$\pi_1$</code>".</p> <p>All scare quotes in the above are places where I'm neglecting basepoints. You have to be more careful in a real-world situation.</p> <p>Here's an example where all the spaces in the diagram are contractible, except for the terminal one. This makes it easy to ignore the indeterminacy.</p> <p>Let $Z$ be $S^1$, viewed as a quotient of $[0,1]$. We have six subsets of $Z$, which are the images of <code>$$[0,1/3], [1/3, 2/3], [2/3, 1], \{0\}, \{1/3\}, \{2/3\}$$</code> We get a corresponding cubical diagram in the homotopy category as follows. The space $S^1$ and its six subspaces define an honestly commutative diagram which is almost all of a commutative cube, except that it's missing the initial vertex. Let the initial vertex be a point $\ast$, which maps isomorphically to all three objects <code>$\{0\}, \{1/3\}, \{2/3\}$</code>.</p> <p>(Sorry, I'm not really up to TeXing up the commutative diagram on MO today, it would be much easier to grasp.)</p> <p>This diagram in the homotopy category is homotopy commutative. In fact, all the spaces are connected and the sources of nontrivial morphisms are contractible, so the diagram has no choice but to commute. The six homotopies all occur in contractible mapping spaces, so there is no <code>$\pi_1$</code>-indeterminacy. The hexagon maps to the mapping space $Map(\ast,S^1) \cong S^1$, and if you use the most obvious choices of homotopies then the hexagon maps to $S^1$ by a homotopy equivalence.</p> <p>In this instance you can choose a witness to each commutativity diagram. The failure to rectify homotopy commutativity occurs here because your witnesses aren't telling stories that are compatible.</p> http://mathoverflow.net/questions/81721/a-homotopy-commutative-diagram-that-cannot-be-strictified/81766#81766 Answer by Dylan Wilson for A homotopy commutative diagram that cannot be strictified Dylan Wilson 2011-11-24T01:01:29Z 2011-11-24T01:01:29Z <p>I'm not sure if this is what Tom was referring to, but here's an explicit example that Adams gave of a homotopy-associative H-space that couldn't be strictified, this is quoted from Stasheff's book on H-spaces:</p> <blockquote> <p>Let $Y$ be a Moore space, $Y(G, 2n-1)$ where divisibility by $2$ and $3$ but no other primes is possible. Thus $Y$ has cohomology only in dimensions $2n-1$ and for $i \ne n$, $\pi_iY \cong \sum_{p>3} \pi_i(S^{2n-1})_p$. Consider $Y \subset \Omega^2\Sigma^2Y$ which is an associative H-space. The obstructions to forming $Y \times Y \rightarrow \Omega^2\Sigma^2Y \times \Omega^2\Sigma^2Y \rightarrow \Omega^2Y$ into $Y$ lie in $H^i(Y \wedge Y; \pi_i(\Omega^2\Sigma^2Y,Y))$. The relative groups are trivial to at least $5(2n)-3$, so the obstructions vanish. Similarly, there are no obstructions to deforming an associating homotopy within $\Omega^2\Sigma^2Y$ to one in $Y$. That $Y$ is not a loop space follows from the decomposability of $P^nu_{2n} \cdot u_{2n}P$ for $p>3$ and $n \not\vert p-1$.</p> </blockquote> <p>I assume that he's referring to Steenrod operations at the end there... Anyway, I know there are probably fancier examples using Zabrodsky mixing, but I like this one because we see an explicit obstruction to straightening the homotopy associativity, via the Steenrod operations. </p> http://mathoverflow.net/questions/81721/a-homotopy-commutative-diagram-that-cannot-be-strictified/81962#81962 Answer by Tom Goodwillie for A homotopy commutative diagram that cannot be strictified Tom Goodwillie 2011-11-26T20:05:59Z 2011-11-27T13:41:07Z <p>One thing to be aware of is that, even when liftings to strict diagrams exist, their non-uniqueness is a serious matter. For example, consider the square pushout diagram in which $S^n$ maps to a point, and to $D^{n+1}$ by inclusion; the fourth space is $S^{n+1}$. Compare this with a different diagram in which the four spaces are the same and three of the maps are the same but the map $S^n\to D^{n+1}$ is replaced by a constant map (to some point in the boundary). This is again strictly commutative, and it yields an isomorphic diagram in $Ho(Top)$, but the one square is "highly connected" while the other is not: the first square gives a map from the homotopy fiber of $S^n\to \star$ (which is $S^n$) to the homotopy fiber of $D^{n+1}\to S^{n+1}$ (which is homotopy equivalent to $\Omega S^{n+1}$) that induces an isomorphism of $H_n$, while the second square induces a map between precisely the same two homotopy fibers which, being a constant map, cannot possibly induce an isomorphism.</p> <p>EDIT </p> <p>Of course there are similar examples involving chain complexes instead of spaces. </p> <p>Note that in the case of the category $Ch(K)$ of chain complexes over a field $K$ the existence of a lifting is automatic. In fact, let $Ch'(K)$ be the full subcategory consisting of chain complexes whose boundary operators are zero: the composed functor $Ch'(K)\to Ch(K)\to Ho(Ch(K))$ is an equivalence of categories, so every diagram (of any shape) in the homotopy category can be lifted. </p> <p>But of course this does not mean that we are silly to bother with chain complexes that have nonzero boundary operators. The point is that a commutative diagram in the homotopy category is not good for much. For example, Mayer-Vietoris sequences arise from certain square diagrams of chain complexes (say, those which are both pushouts and pullbacks), but two of these may be isomorphic as diagrams in the homotopy category and give different results.</p> http://mathoverflow.net/questions/81721/a-homotopy-commutative-diagram-that-cannot-be-strictified/82516#82516 Answer by Jeff Smith for A homotopy commutative diagram that cannot be strictified Jeff Smith 2011-12-02T22:43:41Z 2011-12-02T22:43:41Z <p>Generically speaking, the obstructions to rigidification are generalized Toda brackets.</p> <p>Let C be the partial order {0,1}x{0,1}x{0,1}. (C for cube) Let f:A->B, g:B->C and h:C->D be pointed maps such that gf and hg are null homotopic. Lay f, g and h along three edges of the cube with A at 000, B at 001, C at 011, D at 111. Put * (a point) at all other vertices of C and fill in the rest of the arrows in the only way possible. This gives a hunky dory diagram in the homotopy category. The diagram can be rigidified if and only if the Toda bracket contains 0. Jeff Smith</p> http://mathoverflow.net/questions/81721/a-homotopy-commutative-diagram-that-cannot-be-strictified/82958#82958 Answer by Jesper Grodal for A homotopy commutative diagram that cannot be strictified Jesper Grodal 2011-12-08T11:49:57Z 2011-12-30T04:03:11Z <p>A good place to look for finite toy examples is in the theory of <em>group extensions with non-abelian kernel</em>. </p> <p>A homomorphism $G \to Out(H)$ is the same as a functor <code>$G \to Ho(Top)$</code> sending <code>$$</code>to <code>$BH$</code>, where <code>$G$</code> is viewed as a category with one object <code>$$</code> and <code>$G$</code> as morphisms. By the Dwyer-Kan obstruction theory (noting that the diagram is <em>centric</em>) the obstructions to lifting this to a diagram in Top lies in $H^3(G;Z(H))$.</p> <p>A closer inspection shows that this obstruction class is the same as the obstruction in $H^3(G;Z(H))$ to constructing an extension</p> <p>$$1 \to H \to ? \to G \to 1$$</p> <p>It is possible to calculate this obstruction group in a number of cases:</p> <p>Eilenberg-MacLane show in the 1947 Annals paper "Cohomology Theory in Abstract Groups II" that given a group $G$ and an abelian group $C$ with an action of $C$, and a class $v$ in $H^3(G;C)$ then there exist a group $H$ with center $C$ and a morphism $G \to Out(H)$ realizing the class $v$. (A special case is proven in MacLane's book Homology, and there is an exposition on some of this in Brown's book on group cohomology.) Note that in this construction $H$ may be infinite even if $G$ and $C$ are not.</p> <p>It is also possible to give completely finite examples: Given a group H there is a <em>universal</em> obstruction in $H^3(Out(H);Z(H))$ associated to this group, determining whether the extension </p> <p>$$1 \to H \to ? \to Out(H) \to 1$$ </p> <p>exists. Other obstruction classes will be a pull-back of this one (in particular it has the best chance of being non-zero).</p> <p>The topological interpretation of this obstruction class is that it is the unique k-invariant of the space $BAut(BH)$, where Aut means the space of self-homotopy equivalences. (Recall that group extensions correspond to fibrations, and fibrations with fiber $F$ are classified by maps into $BAut(F)$, and this is a good way to understand group extensions with non-abelian kernel.) </p> <p>Now, it is possible to calculate this class explicitly for small groups $H$. It of course vanishes when H is abelian. It also turns out to vanish for $H = D_8$ or $Q_8$, but is actually <em>non-zero</em> for $D_{16}$ and $Q_{16}$. </p> <p>[As an aside I can say that the H^3 class always vanishes for compact connected Lie groups, which is a theorem of de Siebenthal from the 1950's. Kasper Andersen and I proved that this generalizes to p-compact groups (in a G&T paper from 2008), which was why Kasper and I examined the case where $H$ finite.]</p> <p>[As a further aside I can say that there are a lot of similar obstruction questions over relatively small categories such as the orbit category and corresponding centric diagrams, which occur in e.g., in the theory of p-compact groups and p-local finite groups, when addressing certain uniqueness questions -- here lucily the obstructions are usually zero, although this is usually not so easy to prove, and in particular don't seem to follow from "formal" arguments.]</p>	2013-05-21 23:56:19	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8807443380355835, "perplexity": 452.5882425591172}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368700871976/warc/CC-MAIN-20130516104111-00003-ip-10-60-113-184.ec2.internal.warc.gz"}
http://jeit.ie.ac.cn/article/current?pageType=en	## Current Articles Display Method: \| 2019, 41(12): 1-30. [Abstract](148) [PDF 620KB](35) Abstract: 2019, 41(12): 1-4. [Abstract](106) [PDF 357KB](25) Abstract: 2019, 41(12): 2795-2801. doi: 10.11999/JEIT181165 [Abstract](798) [FullText HTML] (568) [PDF 1167KB](59) Abstract: Non-Orthogonal Multiple Access (NOMA) serves multiple transmitters using the same resource block, and the receiver decodes the information from different transmitters through Successive Interference Cancellation (SIC). However, most of the researches on NOMA systems are based on perfect SIC assumption, in which the impact of imperfect SIC on NOMA system is not considered. Focusing on this problem, a framework is provided to analyze the performance of single-cell uplink NOMA system under the assumption of imperfect SIC. Firstly, the Binomial Point Process (BPP) is used to model the spatial distribution of base station and user equipment in uplink NOMA system. Based on this model, the interference cancellation order which is based on large-scale fading is adopted, and then the error of interference cancellation is analyzed. Then, based on stochastic geometry theory and order statistics theory, the expression of coverage probability of user equipment which is at rank k in terms of the distance from the base station is derived, besides, the average coverage probability is adopted to reflect the reliability of NOMA transmission system. The analytical and simulation results show the influence of system parameters such as distance order and base station radius on transmission reliability. Also, the validity of theoretical deduction is verified. 2019, 41(12): 2802-2809. doi: 10.11999/JEIT190001 [Abstract](1368) [FullText HTML] (758) [PDF 2287KB](73) Abstract: Long Range(LoRa) Backscattering Communication (BC) not only has the advantages of low cost and low power consumption, but also has a long communication distance. However, the existing LoRa BC scheme is complex and can not be applied to actual engineering. For this purpose, a new LoRa BC method is proposed. A Direct Digital frequency Synthesis (DDS) technique is used to generate a square wave with a linear frequency variation as a LoRa scattering modulation signal. For the first time, the prototype of LoRa BC system based on MCU is demonstrated. Experimental results show that design can successfully realize backscatter communication at any position between the station and the receiver which are 208 meters apart, while being compatible with commodity LoRa chipset. In addition, the method is also applicable to an Application Specific Integrated Circuit (ASIC) design, which enables the LoRa backscattering IC to have higher robustness, lower cost, and lower power consumption. 2019, 41(12): 2810-2816. doi: 10.11999/JEIT190171 [Abstract](885) [FullText HTML] (585) [PDF 1708KB](66) Abstract: In order to solve the problem of the Device to Device (D2D) multi-multiplex communication resource blocks allocation in a cell, the resource blocks allocation scheme about D2D multi-multiplex mode based on non-equilibrium solution is proposed after analyzing a D2D user to multiplex two and three cells respectively. The problem of resource blocks partitioning is transformed into the problem of solving the joint revenue maximum value of the multiplexed cellular user by using game theory. When the Nash equilibrium solution does not exist, the objective function is analyzed, the "optimal solution" is solved in the feasible domain and the optimality of unbalanced solution processing is guaranteed. When the equilibrium solution exists, it is rounded up and used as the basis of the resource allocation scheme to maintain its optimality. The theoretical analysis and simulation results show that the proposed algorithm enhances significantly the system performance and sum rate. 2019, 41(12): 2817-2825. doi: 10.11999/JEIT181168 [Abstract](440) [FullText HTML] (345) [PDF 1628KB](24) Abstract: Machine-to-Machine (M2M) and Device-to-Device (D2D) communications are both key technologies in the Fifth Generation (5G) mobile communication systems. In M2M communications, the Energy Efficiency (EE) especially needs to be improved to extend the life cycle of the M2M equipment. In this paper, the M2M and D2D technologies are combined and the D2D technology is used to realize M2M transmission. At the same time, M2M users are allowed to reuse spectrum resources with Human-to-Human (H2H) devices in the cellular networks. To guarantee the Quality of Service (QoS) of these two systems simultaneously, a Multi-Objective Optimization Problem (MOOP) is then formulated to maximize the sum throughput of H2H systems, and the sum EE of M2M systems and to minimize the interference from M2M communications to H2H networks. To solve this MOOP, the penalty function method is firstly adopted to relax the original binary variables, and then the ConCave-Convex Procedure (CCCP) method is used to convert the non-convex single-objective problems into convex problems. Finally, the weighted Tchebyshev algorithm is utilized to obtain the Pareto solution of the original MOOP. By comparing with the traditional weighted sum method, the effectiveness of the proposed method is proved by simulation results. 2019, 41(12): 2826-2835. doi: 10.11999/JEIT190173 [Abstract](797) [FullText HTML] (401) [PDF 4700KB](28) Abstract: For the problem of sparse feature enhancement in Synthetic Aperture Radar (SAR) imagery, conventional methods are difficult to achieve a preferable balance between accuracy and efficiency. In this paper, a robust and efficient SAR imaging algorithm based on Complex Alternating Direction Method of Multipliers(C-ADMM) is proposed for general SAR imaging feature enhancement within complex raw data domain. The problem is firstly imposed by an augmented Lagrange function, and the complex ${\ell _1}$-norm of the intended SAR image is jointly formulated within the C-ADMM framework. Then, the proximal mapping of the sparse feature is derived as a soft-thresholding operator. Further, an iterative processing procedure is designed according to Gaussian-Deidel principle, and the convergence of the proposed algorithm is analyzed. In the experiment, the performance of the proposed algorithm is firstly examined by the simulated data in terms of Phase Transition Diagram (PTD) under different under-sampling rate and degree of sparsity. Then, various raw SAR and Inverse SAR(ISAR) data, for both stationary ground scene and Ground Moving Target Imaging(CMTIm), are applied to further verifying the proposed C-ADMM, and comparisons with classical Convex(CVX) and Bayesian Compress Sensing(BCS) algorithms are performed, so that both the effectiveness and superiority of the C-ADMM algorithm can be verified. 2019, 41(12): 2836-2843. doi: 10.11999/JEIT190148 [Abstract](372) [FullText HTML] (193) [PDF 2715KB](17) Abstract: The generalized Pareto distributed sea clutter model, known as one of the compound-Gaussian models, is able to describe heavy-tailed characteristic of sea clutter under high-resolution and low grazing angle detection scene efficiently, and the accuracy of parameter estimation under this condition heavily impacts radar’s detection property. In this paper, Combined BiPercentile (CBiP) estimator is proposed to estimate the parameters. The CBiP estimator is realized based on the explicit roots of low-order polynomial equations and full application of sample information in returns, which provides a highly-accurate parameter estimation process. Besides, the CBiP estimator can maintain the robustness of estimation performance when outliers with extremely large power are existing in samples, while other estimators, including moment-based and Maximum Likelihood (ML) estimators, degrade extremely in estimation accuracy. Without outliers in samples, the combined bipercentile estimator shows similar accuracy with the ML estimator. With outliers, the combined percentile estimator is the only method with robustness in performance, compared with other estimators aforementioned. Moreover, the ability of the new estimator is verified by measured clutter data. 2019, 41(12): 2844-2851. doi: 10.11999/JEIT181133 [Abstract](524) [FullText HTML] (412) [PDF 1811KB](30) Abstract: The constraint conditions of target assignment model for phased array radar network are unreasonable and the performance of model solving algorithms are not good enough. To solve these problems, a target assignment model for radar network based on Quality of Service (QoS) is constructed in this paper, and a model solving algorithm based on strong concave function approximation is proposed. Through the establishment of resource space and environment space in QoS model, radar resource constraints as well as the visibility constraints between radars and targets are described accurately. Then, sufficient conditions for the optimal solution of QoS model are derived by Karush-Kuhn-Tucker(KKT) condition, and a two-dimensional fast traversal method is used to approximate the strong concave function curve. Finally, the optimal assignment scheme is obtained by the stepwise iteration of operation setting points on the strong concave curve of each target. The simulation results show that the proposed model can effectively accomplish the target assignment of radar network, and model solving algorithm has better performance than the typical intelligent search algorithms. 2019, 41(12): 2852-2858. doi: 10.11999/JEIT190096 [Abstract](573) [FullText HTML] (321) [PDF 2463KB](32) Abstract: Forward-looking Synthetic Aperture Radar (SAR) imaging has the problem of left-right Doppler ambiguity, so it is necessary to use spatial resources for ambiguity resolution. Due to the weight and size of Unmanned Aerial Vehicle (UAV), the receiving array is usually small, and the ability of spatial beam-forming for Doppler ambiguity resolution is insufficient. In addition, the small Doppler gradient and narrow bandwidth of forward-looking SAR echo make the receiving bandwidth underutilized. Based on the above problems, a Doppler diversity Multiple Input Multiple Output (MIMO) forward-looking SAR imaging method is proposed. Based on the forward-looking SAR imaging technology, the narrow-band forward-looking Doppler echo is modulated to different Doppler centers by using Doppler diversity MIMO technology to make full use of the Doppler receiving bandwidth. Furthermore, a virtual receiving array with several times the aperture of the real receiving array can be obtained, which expands greatly the receiving channel and improves effectively the performance of forward-looking SAR imaging in de-Doppler left-right ambiguity. 2019, 41(12): 2859-2864. doi: 10.11999/JEIT190136 [Abstract](578) [FullText HTML] (301) [PDF 851KB](54) Abstract: The performance of a Constant False Alarm Rate (CFAR) detector is often evaluated in three typical backgrounds - homogeneous environment, multiple targets situation and clutter edges described by Prof. Rohling. However, there is a lack of the analytic expression of the false alarm rate for the Rank Sum (RS) nonparametric detector at clutter boundaries, and lack of a comparison of the ability for the RS detector to control the rise of the false alarm rate at clutter edges to that of the conventional parametric CFAR schemes; which is incomplete and imperfect for the detection theory of nonparametric detectors. The analytic expression of the false alarm rate Pfa for the RS nonparametric detector at clutter edges is given in this paper, and the ability of the RS nonparametric detector to control the rise of the false alarm rate at clutter edges is compared to that of the Cell Averaing (CA) CFAR, the Greatest Of (GO) CFAR and the Ordered Statistic (OS) CFAR with incoherent integration. When both of the heavy and the weak clutters follow a Rayleigh distribution, it is shown that the rise of the false alarm rate for the RS detector at clutter edges lies between that of the CA-CFAR and that of the OS-CFAR with incoherent integration. If a non-Gaussian distributed clutter with a long tail moves into the reference window, the rise of the CA-CFAR, the GO-CFAR and the OS-CFAR with incoherent integration reaches a peak of more than 3 orders of magnitude, and can not return to the original pre-designed Pfa. However, the RS nonparametric detector exhibits its inherent advantage in such situation, it can maintain a constant false alarm rate even the distribution form of clutter becomes a different one. 2019, 41(12): 2865-2872. doi: 10.11999/JEIT180933 [Abstract](963) [FullText HTML] (583) [PDF 975KB](47) Abstract: Bistatic radar has the advantages of high concealment and strong anti-interference performance, and plays an important role in modern electronic warfare. Based on the principle of radar coincidence imaging, the problem of bistatic radar coincidence imaging of moving targets is studied. Firstly, based on the bistatic radar system that uses uniform linear array as the transmitting and receiving antenna, the characteristics of the moving target radar echo signal are analyzed under the condition of transmitting random frequency modulation signal, and a bistatic radar coincidence imaging parametric sparse representation model is established. Secondly, an iterative coincidence imaging algorithm based on sparse Bayesian learning is proposed for the parametric sparse representation model established. Based on the Bayesian model, the sparse reconstructed signal is obtained by Bayesian inference, so that the moving target imaging and accurate estimation of motion parameters can be achieved. Finally, the effectiveness of the proposed method is verified by simulation experiments. 2019, 41(12): 2873-2880. doi: 10.11999/JEIT180953 [Abstract](685) [FullText HTML] (500) [PDF 3288KB](47) Abstract: A modified SPECtral ANalysis (SPECAN) algorithm based on Doppler resampling is proposed to deal with the azimuth Space-Variant (SV) phase coefficients of the High Squint (HS) SAR data acquired from maneuvering platform. Firstly, for HS SAR with constant acceleration, an orthogonal coordinate slant range model is presented, which can handle the coordinate rotation caused by the traditional method of Range Walk Correction (RWC), and solve the mismatch between the range model and the signal after RWC. Then azimuth Doppler resampling is used to correct the SV phase coefficients. The focused image is achieved by SPECAN technique. Finally, the proposed algorithm is validated by processing of simulated SAR data, and has significant improvement on focusing quality over the reference one. 2019, 41(12): 2881-2888. doi: 10.11999/JEIT190108 [Abstract](628) [FullText HTML] (441) [PDF 5087KB](19) Abstract: The backscattering of the radar targets is sensitive to the relative geometry between orientations of the targets and the radar line of sight. When the orientations of the same target are different from the radar line of sight, the scattering characteristics are quite different. Targets such as inclined ground and inclined buildings may reverse the polarization base of the backscattered echo, which causes the cross-polarization component to be too high and the volume scattering component of the image is overestimated. In this paper, a polarimetric interferometric decomposition method based on polarimetric parameters ($H/{\alpha}$) and Polarimetric Interferometric Similarity Parameters (PISP) is proposed to solve the overestimation problem. The method makes full use of the scattering diversity of the scatterer in the radar line of sight. The cross-polarization components generated by targets such as inclined grounds and inclined buildings with different orientations are better adapted to obtain better decomposition results. Finally, the effectiveness of the proposed method in polarimetric interferometric decomposition is verified by the airborne C-band PolInSAR data obtained by the Institute of Electronics, Chinese Academy of Sciences. The experimental results show that the proposed improved algorithm can distinguish the scattering characteristics of terrain types effectively and correctly. 2019, 41(12): 2889-2895. doi: 10.11999/JEIT190033 [Abstract](621) [FullText HTML] (412) [PDF 1825KB](24) Abstract: To solve the problem that the traditional micro-Doppler feature extraction technologies are generally hard to achieve resolution and parameter estimation of multi-target, a novel curve overlap extrapolation algorithm for wide-band resolution of micro-motion multi-target is proposed. According to the relative distance between filtering data points and the historical slope information of each curve, the point trace behind the overlapping location can be extrapolated to realize data association of micor-motion curve for each signal component. On this basis, the multi-target resolution can be realized by analyzing the difference of micor-motion characteristics between each curve. Extensive simulation experiments are provided to illustrate the effectiveness and robustnees of the proposed algorithm. 2019, 41(12): 2896-2902. doi: 10.11999/JEIT181152 [Abstract](721) [FullText HTML] (491) [PDF 2206KB](35) Abstract: In the system of distributed radar array system using phase interference angle measurement, the phase center coordinate error of arrays and the phase difference error have relatively large influence on the angle measurement. The phase center position is often inconsistent with physical center position. Thus it is necessary to compensate these errors precisely. Far field radiation sources are often used to calibrate radar in traditional calibration methods. However, it is usually hard to achieve far field radiation sources for distributed radar array with large space between units surveilling space targets. In this paper, a calibration method based on the precise ephemeris of refined orbit satellites without measuring with special instruments is proposed. The phase error caused by coordinate error can be whitened by the precise ephemeris of multiple arcs, and the coordinate and phase difference will be searched out by matching the minimum variance. This method can get the errors easily. The simulation results and actual data verify that angle measurement accuracy gets large improvement by the method. 2019, 41(12): 2903-2910. doi: 10.11999/JEIT181102 [Abstract](1829) [FullText HTML] (545) [PDF 1997KB](54) Abstract: It makes the Pulse Doppler (PD) radar widely applied that the PD radar has the obvious advantages of detecting the Doppler frequency of the target and suppressing the clutter effectively. However, it is difficult for the PD radar to detect the target due to velocity ambiguity. Combining with the characteristic and stagger-period model of the PD radar, a Doppler frequency estimation method based on all phase DFT Closed-Form Robust Chinese Remainder Theorem (CFRCRT) with spectrum correction is proposed. Both theoretical analysis and simulation experiment demonstrate that the proposed method can satisfy the engineering demand in measure accuracy and real-time performance. 2019, 41(12): 2911-2918. doi: 10.11999/JEIT190142 [Abstract](471) [FullText HTML] (260) [PDF 3644KB](22) Abstract: In order to satisfy the requirement of elliptical beam antenna with low profile, a novel design technique of the hybrid-structural antenna with elliptical beam is proposed. The hybrid-structural antenna consists of the ring-focus elliptical antenna in inner-ring region and the Cassegrain elliptical antenna in outer-ring region. The design method, procedure and shaping formula are presented in detail. A 600 mm×1200 mm reflector antenna is designed and its tolerance analysis is also given. The results show that the novel structural antenna can operate in Ku/Ka dual bands, antenna efficiency is greater than 56% and Voltage Standing Wave Ratio (VSWR) is better than 1.27, and its side lobe levels in the EL and AZ planes are below –12.2 dB and –14.6 dB respectively. The simulated results of Grasp and CST software agree well, which verify the effectiveness of the design method. 2019, 41(12): 2919-2924. doi: 10.11999/JEIT181145 [Abstract](657) [FullText HTML] (369) [PDF 1078KB](20) Abstract: Based on the beam wave synchronization interaction in transverse and longitudinal directions at the same time and derived from Maxwell’s equation and linear Vlasov equation, the planar metallic grating beam-wave interaction " hot” dispersion equation considering both cyclotron resonance and Cherenkov resonance is deduced. Through the reasonable selection for geometric and electrical parameters, the numerical calculation and analysis of the " hot” dispersion equation show that the beam-wave interaction gain and frequency band with the cyclotron resonance enhancement effect are higher than those with only Cherenkov resonance radiation. 2019, 41(12): 2925-2931. doi: 10.11999/JEIT181049 [Abstract](742) [FullText HTML] (508) [PDF 3668KB](16) Abstract: A novel wideband low RCS new super-surface array based on three reflective cell shared aperture is designed, which is composed of three kinds of Artificial Magnetic Conductor (AMC). Compared with the traditional AMC array, the new array uses one of AMC as phasor interference unit. A new phase cancellation relation is presented, the new phase cancellation relation is used to extend the traditional array phase cancellation band. Then, the parameters of the cell structure are further optimized to realize the reduction of RCS and the improvement of bandwidth. The physical sample is processed and tested. The results of simulation and field test show that: the backward reduction of RCS in the range of 5.2～13.9 GHz reaches more than 10 dB, and the relative bandwidth reaches 91%. It is shown that the new array can overcome the defect of the discontinuous operating band of the traditional array and has broadband low scattering characteristics. 2019, 41(12): 2932-2938. doi: 10.11999/JEIT181127 [Abstract](671) [FullText HTML] (563) [PDF 560KB](22) Abstract: The performance of the existing target localization algorithms is not ideal in complex acoustic environment. In order to improve this problem, a novel target binaural sound localization algorithm is presented. First, the algorithm uses binaural spectral features as input of a time-frequency units selector based on deep learning. Then, to reduce the negative impact of the time-frequency unit belonging to noise on the localization accuracy, the selector is emploied to select the reliable time-frequency units from binaural input sound signal. At the same time, a Deep Neural Network (DNN)-based localization system maps the binaural cues of each time-frequency unit to the azimuth posterior probability. Finally, the target localization is completed according to the azimuth posterior probability belonging to the reliable time-frequency units. Experimental results show that the performance of the proposed algorithm is better than comparison algorithms and achieves a significant improvement in target localization accuracy in low Signal-to-Noise Ratio(SNR) and various reverberation environments, especially when there is noise similar to the target sound source. 2019, 41(12): 2939-2944. doi: 10.11999/JEIT181125 [Abstract](604) [FullText HTML] (430) [PDF 801KB](13) Abstract: Penalized programs are widely used to solve linear inverse problems in the presence of noise. For now, the study of the performance of panelized programs has two disadvantages. First, the results have some limitations on the tradeoff parameters. Second, the effect of the direction of the noise is not clear. This paper studies the performance of penalized programs when bounded noise is presented. A geometry condition which is used to study the noise-free problems and constrained problems is provided. Under this condition, an explicit error bound which guarantees stable recovery (i.e., the recovery error is bounded by the observation noise up to some constant factor) is proposed. The results are different from many previous studies in two folds. First, the results provide an explicit bound for all positive tradeoff parameters, while many previous studies require that the tradeoff parameter is sufficiently large. Second, the results clear the role of the direction of the observation noise playing in the recovery error, and reveal the relationship between the optimal tradeoff parameters and the noise direction. Furthermore, if the sensing matrix has independent standard normal entries, the above geometry condition can be studied using Gaussian process theory, and the measurement number needed to guarantee stable recovery with high probability is obtained. Simulations are provided to verify the theoretical results. 2019, 41(12): 2945-2950. doi: 10.11999/JEIT190012 [Abstract](718) [FullText HTML] (425) [PDF 1228KB](35) Abstract: To solve the problem of weak signals detection in non-Gaussian background, a method based on Sigmoid function is proposed which is named Sigmoid Function Detector (SFD). Firstly, the non-Gaussian background is modeled as a mixed Gaussian model. Based on this, the relationship between parameter k and SFD's performance and characteristics are systematically analyzed. It is pointed out that SFD will be a constant false alarm detector when its detection performance is optimal. Secondly, a new non-parametric detector is proposed via fixing the parameter k, which has significant improvement over matched filter. Finally, simulation analysis is carried out to verify the effectiveness and superiority of SFD. 2019, 41(12): 2951-2956. doi: 10.11999/JEIT190067 [Abstract](960) [FullText HTML] (554) [PDF 1141KB](19) Abstract: The propagation of acoustic signal in space has a strong multipath effect, and the receiver often overlaps in the form of convolution. Especially in strong reverberation conditions such as ocean and theatre, where the length of impulse response of hybrid filter increases significantly. In order to eliminate the problem that long impulse response leads to the failure of the frequency domain convolution blind separation algorithm, two Short-Time Fourier Transforms (STFT) are applied to the observed signal. The first STFT shortens the length of the hybrid filter. The second STFT converts the signal model into instantaneous blind separation. Finally, the separation matrix is estimated by Joint Diagonalization (JD) technique. Compared with the existing methods, this method solves the problem of model failure under deep convolution mixing, and can obtain better separation performance when the number of source signals is large or additive noise exists. The simulation results verify the effectiveness and performance advantages of the proposed method. 2019, 41(12): 2957-2964. doi: 10.11999/JEIT190144 [Abstract](743) [FullText HTML] (445) [PDF 1765KB](45) Abstract: Outliers are non-Gaussian measurement values far from the bulk of data. In practical transmission, the signals added with outlier often have the heavy-tailed property. Particle filter is based on the Bayesian framework and applicable to the non-linear and non-Gaussian system. However, measurement noise with outlier degrades the performance of particle filter. In this paper, student-t distribution is used to model the measurement noise, combined with Variational Bayes (VB), a novel particle filter Marginalized Particle Filter with VB Mean(MPF-VBM) is designed, which can estimate all parameters of t-distributed measurement distribution including mean parameter as well as state. Further, particle filter with noise correlation (MPF-VBM-COR) at the same epoch which is applicable to time variant measurement noise is developed. For verifying the performances of the proposed algorithms, the simulations on the typical univariate non-stationary growth model are performed under the different noise conditions in detail. The outcomes show that the proposed two algorithms of MPF-VBM and MPF-VBM-COR (MPF-VBM-Corrlation) have the superior performances to the compared ones. 2019, 41(12): 2965-2971. doi: 10.11999/JEIT180947 [Abstract](456) [FullText HTML] (160) [PDF 1088KB](5) Abstract: The transient signal without modulation information of the radiation source can characterize the unintentional modulation characteristics of the radiation source. The analysis of the transient signal can realize the radiation source identification. In the switching on and frequency conversion process of the frequency-hopping signal, there is a transient adjustment time without information transmission. In the transient adjustment moment, the signal transmitted by the transmitter is a non-linear, non-stationary and non-Gaussian signal without modulation information. This transient time series can reflect the device characteristics of the frequency-hopping transmitter, and the sequence often exhibits complex chaotic characteristics. Therefore, from the idea of chaotic time series analysis and Low-rank characteristics of transient signal, a frequency-hopping transmitter classification algorithm is proposed based on chaotic attractor reconstruction and Low-rank clustering. The experimental tests show that the transient signal of the frequency-hopping transmitter belongs to the chaotic time series. At the same time, the classification results of the frequency-hopping signals demonstrate the feasibility of the Low-rank clustering algorithm in frequency-hopping transmitter classification. 2019, 41(12): 2972-2979. doi: 10.11999/JEIT190032 [Abstract](578) [FullText HTML] (397) [PDF 5673KB](51) Abstract: In order to enhance the useful information in the image and improve the visual effect of the image, a Non-local Multi-scale Fractional Differential(NMFD) image enhancement operator is proposed. The operator divides the image into several sub-images and calculates the edge intensity coefficient, entropy value and roughness of each sub-image, and the obtained feature data are normalized in a unified scale in the global image range. Then, the normalized data are weighted to be the non-local eigenvalues of the image. Finally, an exponential function is used to establish the non-linear quantization relationship between image detail features and the value of fractional order. Thus, the fractional order of different scales can be determined in different image sub-block regions, so that the non-local multi-scale image enhancement model is realized. 2019, 41(12): 2980-2986. doi: 10.11999/JEIT190261 [Abstract](1270) [FullText HTML] (1063) [PDF 2241KB](27) Abstract: A kind of restoration method of BP neural network fuzzy image based on Optimized Brain Storming intelligent Optimized(OBSO-BP) algorithm is proposed in this paper. With the method of brain storming intelligent optimized algorithm which is optimized in both clustering and variation, issues of multi-peak high-dimensional function is easily solved. This method optimizes brain storming intelligence algorithm from two aspects of clustering and mutation. This method makes use of the characteristics of brain storming optimization algorithm, which is easy to solve multi-peak and high-dimensional function problems, to automatically search for better initial weights and thresholds of BP neural network, thus reducing the sensitivity of BP network to its initial weights and thresholds, avoiding the network falling into local optimal solution, increasing the convergence speed of the network and reducing the network error and improving the quality of image restoration. Twenty different images are adopted to the image restoration experiment of their fuzzy images with Wiener filtering restoration(Wiener), Wiener filtering restoration based on optimized Brain Storming intelligent Optimized algorithm(Wiener-BSO), BP neural network restoration and BP neural network restoration based on optimized Brain Storming intelligent Optimized algorithm(BSO-BP). Results show that a better effect of image restoration can be achieved with this method. 2019, 41(12): 2987-2994. doi: 10.11999/JEIT190043 [Abstract](2060) [FullText HTML] (990) [PDF 2601KB](85) Abstract: The image forgery detection algorithm based on convolutional neural network can implement the image forgery detection that does not depend on a single image attribute by using the learning ability of convolutional neural network, and make up for the defect that the previous image forgery detection algorithm relies on a single image attribute and has low applicability. Although the image forgery detection algorithm using a single network structure of deep layers and multiple neurons can learn more advanced semantic information, the result of detecting and locating forgery regions is not ideal. In this paper, an image forgery detection algorithm based on cascaded convolutional neural network is proposed. Based on the general characteristics exhibited by convolutional neural network, and then the deeper characteristics are further explored. The cascaded network structure of shallow layers and thin neurons figures out the defect of the single network structure of deep layers and multiple neurons in image forgery detection. The proposed detection algorithm in this paper consists of two parts: the cascade convolutional neural network and the adaptive filtering post-processing. The cascaded convolutional neural network realizes hierarchical forgery regions localization, and then the adaptive filtering post-processing further optimizes the detection result of the cascaded convolutional neural network. Through experimental comparison, the proposed detection algorithm shows better detection results and has higher robustness. 2019, 41(12): 2995-2999. doi: 10.11999/JEIT180939 [Abstract](553) [FullText HTML] (413) [PDF 342KB](17) Abstract: Due to the wide applications in association schemes, authentication codes and secret sharing schemes etc., construction of the linear codes with a few weights is an important research topic. A class of linear codes with four-weight and six-weight over finite field ${F_p}$ (p is an odd prime) is constructed by a proper selection of the defining set. The explicit weight distribution is obtained using Gauss sums, and some examples from Magma program to illustrate the validity of the conclusions are provided. The results show that these codes include almost optimal codes with respect to Singleton bound. 2019, 41(12): 3000-3005. doi: 10.11999/JEIT190071 [Abstract](682) [FullText HTML] (437) [PDF 364KB](37) Abstract: Families of pseudorandom sequences derived from Euler quotients modulo odd prime power possess sound cryptographic properties. In this paper, according to the theory of residue class ring, a new classes of binary sequences with period $2{p^{m + 1}}$ is constructed using Euler quotients modulo $2{p^m},$ where $p$ is an odd prime and integer $m \ge 1.$ Under the condition of ${2^{p - 1}}\not \equiv 1 ({od}\,{p^2})$, the linear complexity of the sequence is examined with the method of determining the roots of polynomial over finite field ${F_2}$. The results show that the linear complexity of the sequence takes the value $2({p^{m + 1}} - p)$ or $2({p^{m + 1}} - 1)$, which is larger than half of its period and can resist the attack of Berlekamp-Massey (B-M) algorithm. It is a good sequence from the viewpoint of cryptography. 2019, 41(12): 3006-3013. doi: 10.11999/JEIT190013 [Abstract](688) [FullText HTML] (454) [PDF 1066KB](26) Abstract: In the Network Function Virtualization (NFV) environment, for the reliability problem of Service Function Chain (SFC) deployment, a joint optimization method is proposed for backup Virtual Network Function (VNF) selection, backup instance placement and service function chain deployment. Firstly, the method defines a virtual network function measurement standard named the unit cost reliability improvement value to improve the backup virtual network function selection method. Secondly, the joint backup mode is used to adjust the placement strategy between adjacent backup instances to reduce bandwidth resources overhead. Finally, the reliability-guarantee problem of the whole service function chain deployment is modeled as integer linear programming, and a heuristic algorithm based on the shortest path is proposed to overcome the complexity of integer linear programming. The simulation results show that the method optimizes resource allocation while prioritizing the network service reliability requirements, and improves the request acceptance rate. 2019, 41(12): 3014-3021. doi: 10.11999/JEIT190016 [Abstract](645) [FullText HTML] (457) [PDF 1055KB](47) Abstract: The close relationship between resource deployment and specific tasks in traditional Wireless Sensor Network(WSN) leads to low resource utilization and revenue. According to the dynamic changes of Virtual Sensor Network Request(VSNR), the resource allocation strategy based on Semi-Markov Decision Process(SMDP) is proposed in Virtual Sensor Network(VSN). Then, difining the state, action, and transition probability of the VSN, the expected reward is given by considering the energy and time to complete the VSNR, and the model-free reinforcement learning approach is used to maximize the long-term reward of the network resource provider. The numerical results show that the resource allocation strategy of this paper can effectively improve the revenue of the sensor network resource providers. 2019, 41(12): 3022-3028. doi: 10.11999/JEIT181101 [Abstract](844) [FullText HTML] (834) [PDF 784KB](55) Abstract: A distributed algorithm based on modified Newton method is proposed to solve the nodes localization problem in large scale Wireless Sensor Network(WSN). The algorithm includes network partitioning and distributed algorithm. Firstly, the network is divided into several overlapping subregions according to the nodes positions and the distance information between the sensors. The localization problem of subregions is formulated into an unconstrained optimization problem and each subregion can be calculated independently. Then distributed algorithm is used to determine nodes positions in subregions and merge the subregions. Simulation results indicate that the proposed algorithm is superior to the existing algorithms in terms of accuracy in large scale network, which can meet the needs of nodes localization in large scale network. 2019, 41(12): 3029-3035. doi: 10.11999/JEIT190168 [Abstract](563) [FullText HTML] (555) [PDF 2071KB](38) Abstract: Considering the problem of scattered node mapping and more hops of link mapping in the traditional virtual network energy-saving embedding, the node and link are mapped simultaneously by using the minimum spanning tree topology of the virtual network request, and Energy-saving Virtual Network Embedding algorithm based on Sliding Region Particle Swarm (EVNE_SRPS) is proposed. When a virtual network request arrives, the minimum spanning tree topology is generated, the root node is the node with the shortest path length; Multiple regions are randomly selected as the particle object in the substrate network, and the minimum spanning tree topology of the virtual network request is mapped in the regional center; The fitness of the particles is calculated. The optimal solution of the group and the individual is finded, and the sliding direction and the location of the update region under the guidance of the optimal solution are determined. After the iteration, the mapping scheme of the virtual network is obtained. The experimental results show that compared with the existing algorithms, the network energy consumption is reduced, and the internet service providers revenue to cost ratio is improved. 2019, 41(12): 3036-3042. doi: 10.11999/JEIT181169 [Abstract](608) [FullText HTML] (506) [PDF 1614KB](13) Abstract: Facing changeable network environment, current Quality of Service (QoS)-aware flow aggregation scheme is lack of flexibility. A dynamic flow aggregation method to overcome present problems is proposed. An Enhanced Rough K-Means (ERKM) algorithm is used to aggregate network flows properly. Importantly, it is able to adjust degree of membership to face ever-changing internet environment to make algorithm more flexible. Internet scheduler experiment is carried out and a comparison is made with existing methods. Experimental results suggest that proposed method has advantages not only on flexibility of aggregation, but also on assurance of QoS of Internet flows. In addition, the consistency of QoS allocation under different network environment is investigated. 2019, 41(12): 3043-3050. doi: 10.11999/JEIT190004 [Abstract](743) [FullText HTML] (486) [PDF 2824KB](20) Abstract: Against the problem of low detection rate to detect small hardware Trojan by side-channel in physical environment, the Marginal Fisher Analysis (MFA) is introduced. On the basis, a hardware Trojan detection method based on Compression Marginal Fisher Analysis (CMFA) is proposed. The projection space is constructed by reducing the distance between the sample and its same neighbor samples, and the distance between the same neighbor samples and the center of the same kind, and increasing the distance between the same neighbor samples of the center and the sample in different kind. Thus, the difference in the original data is found without any assumptions about data distribution, and the detection of hardware Trojan is achieved. The hardware Trojan detection experiment in AES encryption circuit shows that this method can effectively distinguish the statistical difference in side-channel signal between reference chip and Trojan chip and detect the hardware Trojan whose scale is 0.04% of the original circuit. Monthly Journal Founded in 1979 The Source Journal of EI Compendex The Source Journal of ESCI Database Competent unit：Authorized by CAS Host unit：Hosted by IECAS，Department of Information Science of NNSFC Editor-in-Chief：Yirong Wu ISSN 1009-5896 CN 11-4494/TN News more > Conference more > Author Center Wechat	2020-01-19 13:20:53	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5290826559066772, "perplexity": 1420.1679280946523}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250594603.8/warc/CC-MAIN-20200119122744-20200119150744-00155.warc.gz"}
http://forum.sagittal.org/viewtopic.php?f=15&p=906	## Search times out cmloegcmluin Posts: 721 Joined: Tue Feb 11, 2020 3:10 pm Location: San Francisco, California, USA Real Name: Douglas Blumeyer Contact: ### Search times out The "Search..." bar on the forum doesn't work for me; it always times out. If it is of help to anyone else, I've found that I can use my browser's omnibar as an alternative: simply typing "forum.sagittal.org: thing i was searching for" there works. Dave Keenan Posts: 1024 Joined: Tue Sep 01, 2015 2:59 pm Location: Brisbane, Queensland, Australia Contact: ### Re: Search times out Thanks for that. Fixed now. I had noticed it too, and tried rebuilding the index, to no avail. Searching the PHP help forum told me that the likely culprit was: failing to install and enable the optional "mbstring" extension when I upgraded PHP to version 7.4 recently. I fixed it by logging into the server as root and issuing the following commands: apt-get install php7.4-mbstring service apache2 restart cmloegcmluin	2020-09-19 06:14:37	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.44726690649986267, "perplexity": 9671.921140980068}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400190270.10/warc/CC-MAIN-20200919044311-20200919074311-00562.warc.gz"}
https://gwct.github.io/referee/scores.html	# Referee's scores Referee's scores in output files will range from -2 to 91. Since Referee's score is a ratio of probabilities, theoretically any score from -infinity to +infinity is possible. However, in practice scores tend to be limited to -300 to +300. Positive scores indicate support for the called reference base while negative scores indicate more support for some other base. We cap Referee's scores from 0 to 90. This means that all sites that score negative are converted to a score of 0, and any sites that score higher than 90 simply get a score of 90. This scaling is practical because it makes scores easily interpratable and allows for the converstion to ASCII characters in the FASTQ output. However, here are several scenarios where the scoring calculation is impossible. We have reserved some scores for these situations. ScenarioScore The sum of the genotype likelihoods that do not contain the reference base is 0 91 The reference base is called as N -1 No reads mapped to this site -2 For the case of the reference base being called as N, Referee can calculate the highest scoring base and report it with the --correct option. ### Referee's scores are not Phred-scaled error probabilities Rather, they are a a ratio of probabilities. While Referee's scores are said to be Phred-like because higher scores indicate more confidence in the called base, they do not represent a probability of error as traditional Phred scores do, and as such they are scaled differently. Put succinctly: A Phred score of 10 corresponds to a probability of error of 0.1. A Referee score of 10 means that the called base is $$10^{10}$$ times more likely than another base.	2023-04-02 05:50:27	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6018077731132507, "perplexity": 1503.3146059888502}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296950383.8/warc/CC-MAIN-20230402043600-20230402073600-00686.warc.gz"}
https://newey.me/the-phantom-email-j/	Since I started working at my internship placement this year, I’ve noticed a fairly frequent occurrence of a phenomenon where for some odd reason (and seemingly out of context) a phantom “J” character would appear in an email at the end of a sentence. Take this (fabricated, but representative) example: Hi all, There is cake downstairs if anyone wants some. J Bob Anyhoo, with a bit of Googling, I found this blog post which explains the reason behind the phantom J. When someone types a smiley face (i.e. :)) in Outlook on Windows, it’s automatically “corrected” to the smiley face character in the Wingdings font (Microsoft’s proprietary Dingbats font). The smiley face character in Wingdings just so happens to map to the ASCII character code “J”. So, if someone types a smiley face in an Outlook email, and then you view that email on another platform (OS X, Linux, Android, etc) - then chances are, you’ll just see a phantom “J”. You can check this by looking at the HTML source of an offending email: <font face="Wingdings">J</font> This is completely redundant - there’s actually a Unicode character for a smiley face. Unicode is supported across almost all platforms created in the last… 20 years, and isn’t reliant on an obscure proprietary font. The consequence of this is that anyone who doesn’t use Windows now has to read emails with phantom “J” characters in…	2018-03-22 04:11:55	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.20694571733474731, "perplexity": 2904.3681111239343}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647768.45/warc/CC-MAIN-20180322034041-20180322054041-00511.warc.gz"}
https://math.stackexchange.com/questions/1343452/any-shortcut-to-calculate-factorial-of-a-number-without-calculator-or-n-to-1?noredirect=1	# Any shortcut to calculate factorial of a number (Without calculator or n to 1)? I've been searching the internet for quite a while now to find anything useful that could help me to figure out how to calculate factorial of a certain number without using calculator but no luck whatsoever. I'm well aware of the fact that there is a way to calculate any number of sigma (summation notation sigma) but haven't figured out anything for factorials yet. Could you please show me any method that should do the trick. E.g. factorial of 10! is 3628800 but how do I calculate it without using any sorts of calculator or calculate the numbers from 10 to 1? • Multiply 10 by 9 by 8....by 2. It's a lot of work, but that is how you would do it by "pen-and-paper" – Zach466920 Jun 29 '15 at 16:14 • Thanks @Zach466920 but as I wrote in my last sentence, without any of these, no calculation from n to 1. So I assume there is not a possible way to do that? – Teo Carter Jun 29 '15 at 16:16 • You could calculate $n! = \exp(\log(1) + \log(2) + \ldots + \log(n))$ using a log table. – Jair Taylor Jun 29 '15 at 16:18 • @JairTaylor you could calculate $n!=123...n$ using a multiplication table ;) – Zach466920 Jun 29 '15 at 16:20 • There is no simpler exact formula, as far as I know. – user228113 Jun 29 '15 at 16:26 Rewriting the factorial as the Gamma function and Stirling's approximation we get what I think is the closest possible approximation that you could do by hand: $$n! \approx \sqrt{2 \pi n} \cdot \left( \frac{n}{e} \right)^n$$ Where $e = 2.71828\dots$. Unfortunately, this might not be quicker than multiplying all the numbers together by hand, but it's certainly the only shortcut I can think of that could be done by hand. • That approximation actually involves $n$ multiplications....so it's equal to or less efficient than doing it by brute force. – Zach466920 Jun 29 '15 at 16:21 • @Zach466920 not true - exponentiation can be done in $\log_2 (n)$ multiplications :) – lisyarus Jun 29 '15 at 16:22 • @lisyarus I'll assume your joking, since this is pencil and paper we are talking about... ;) – Zach466920 Jun 29 '15 at 16:23 • @Zach466920 actually, I'm not. It's very easy to put the power into binary form and reduce the multiplications to at most $\log_2 n$ multiplications plus at most $\log_2 n$ squarings. – lisyarus Jun 29 '15 at 16:26 • @lisyarus Ok, yes. There are exceptions to exceptions, I get it...I just assumed since its used in many numerical systems, it might be the most efficient. – Zach466920 Jun 29 '15 at 17:12 I see that this is an old question and the conversation thread is most likely completely dead. However, there is a partial shortcut to calculating fairly length factorials. Being that multiplication is commutative, the order in which multiplication is done can be rearranged... Take, for example, 10! which equals 12345678910, this can be rearranged as (101)(92)(83)(74)(65) → 1018242830 One thing that you will (or should) notice about the delta between the product of each pairing is that it always decreases by a value of 2, and always begins with a value of n-2. For n!, where n=10, the delta between the first two products will be n-2, or 8. The delta between the products of successive pairs will always decrease by 2. In other words, when the pairing is done as illustrated above (pairing the highest value number with the lowest, second highest with the second lowest, etc), then the product of each pairing will increase by a value that decreases by an amount of 2 for each successive pairing. This makes calculating the products of a long list of number pairs relatively easy. For n!, where n=10, we know that 110=10, we also know that the next pairing will result in 10+8, or 18... the next paring will result in a product of 18+6=24, followed by 24+4=28, and finally by 28+2=30. Without having to calculate each product, we can quickly predict what they will be. This works for factorials of odd numbers as well, except that there will be one number at the end that will not have a pairing. 7!, for example, could be written as (17)(26)(35)4 → 712154. The initial product is 7, and the product of the next pairing is 7+(n-2) or 7+5, which of course equals 12. The next product is 12+3, or 15. So the solution to 7! would be 712154. As a generalized rule, it can be said that the initial delta between the products of the first and second pairings will be n-2, this delta will always decrease by a value of 2 for each successive pairing, and the final delta will always be either 2 or 3 (depending on whether we are dealing with the factorial of an even number or an odd number [2 or greater]). Again, this is only a partial shortcut. It essentially reduces the number of calculations you need to perform by half... but if you absolutely need to do calculate 100! by hand, performing approximately half of the necessary steps would make a significant difference. :) No, but you can if you accept approximations. Since the factorial function is defined recursively, $(n+1)!=n! \cdot (n+1)$, your question boils down to whether or not the recurrence relation has a closed form solution, which it doesn't have. You want to be able to skip around calculating $1!$ through $9!$. However $10!$ is defined by $9!$, so there isn't a way of skipping the intermediate steps. • Ah.. well approximations is the best thing we can get I suppose since it involves multiple level of multiplications. Thanks @Zach466920 :) – Teo Carter Jun 29 '15 at 16:26 • @Zach466920 I think there is a typo.$$(n+1)!=(n+1).n!$$ – Integrator Jun 29 '15 at 16:28 • @Integrator thanks :) – Zach466920 Jun 29 '15 at 16:31	2020-01-26 03:13:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6973588466644287, "perplexity": 308.26157797920405}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251684146.65/warc/CC-MAIN-20200126013015-20200126043015-00377.warc.gz"}
http://openstudy.com/updates/4d9eb1d88f378b0bacc2e317	anonymous 5 years ago dy/dx=cos^2(x)cos^2(2y), y=? 1. anonymous This is separable. You just need to rewrite it as$\frac{dy}{\cos^2 (2y)}=\cos^2 (2x) dx \rightarrow \int\limits_{}{} \frac{dy} {\cos^2 (2y)}=\int\limits_{}{} \cos^2 (2x) dx$ 2. anonymous Use the fact that $\cos 2\theta =\cos^2 \theta - \sin^2 \theta = 2 \cos^2 \theta -1 \rightarrow \cos^2\theta =\frac{\cos 2 \theta +1}{2}$ 3. anonymous Hello dichalao 4. anonymous Hello loki~ haha 5. anonymous Isn't it ~1am there? 6. anonymous yep :) 7. anonymous i dont want to end my day as usual :( 8. anonymous Why's that? 9. anonymous find the exact solution to this bvp u''-u=0 u(0)=0 and u(1)=1 how do you solve this problem..i was trying to help the guy, but failed 10. anonymous I don know.. i just dont want to go to bed 11. anonymous It should be a exponential function right 12. anonymous Lol...this is a linear homogeneous second order differential equation with constant coefficients. We assume a solution of the form $y=e^{\lambda x}$, sub it in and solve the characteristic equation${\lambda}^2-1=0 \rightarrow \lambda = \pm 1$The solution's then$y=c_1 e^x + c_2 e^{-x}$ 13. anonymous how you got λ2−1=0→λ=±1 14. anonymous $(e^{\lambda x})''-e^{\lambda x}=0 \rightarrow {\lambda}^2e^{\lambda x}-e^{\lambda x}=0 \rightarrow ({\lambda}^2-1)e^{\lambda x}=0$In the last product, it can only be the case that${\lambda}^2-1=0$since$e^{\lambda x} \neq0$for all x. 15. anonymous because λ=+-1 you got ce^x+c2e^(-x)? 16. anonymous Yep, the aim is to find lambdas that will allow you to make the assumption true. There's a theorem in differential equations that tells us how many solutions there will be in a diff. equation and that those solutions are unique. For second order homogeneous (what we have here) there will be two solutions, and since we've found two independent solutions (that technically needs to be checked with something called the Wronskian, but no-one usually cares with these types), and since we know a set of solutions is unique, we've found the solutions to the equation. 17. anonymous Last bit wasn't explained all that well. 18. anonymous LOL it is good enough at my level 19. anonymous Happy now? You can hit the sheets! 20. anonymous LOL i am always happy when you are on :) flattery gets me to anywehre 21. anonymous So do you just have to tell that guy the solution? 22. anonymous you can do it :) i dont want to take any credits for it.. it is 10 thread up 23. anonymous You can just screen shot it and send as an attachment. 24. anonymous He just has to solve for the boundary conditions u(0)=0 and u(1)=1. 25. anonymous LOL can you please do that for me :P 26. anonymous so you will earn your 133th fan 27. anonymous Okay, I'll go over to him. I think nikvist might be dealing with it...	2016-10-28 07:09:31	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6828553676605225, "perplexity": 2207.9696988625547}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988721558.87/warc/CC-MAIN-20161020183841-00474-ip-10-171-6-4.ec2.internal.warc.gz"}
https://math.stackexchange.com/questions/2480567/proof-q-is-symmetric-and-reflexive-on-a	# Proof: Q is symmetric and reflexive on A Suppose that $\mathscr{P}$ is a partition of A and suppose $xQy$ if there exists $C \in \mathscr{P}$ such that $x \in C$ and $y \in C$. Prove that Q is symmetric and Q is reflexive on A. Proof. Let $x \in A$ and $y \in A$. Assume that $xQy$. By definition, for all $x \in A$ and $y \in A$, if $xQy$, then $yQx$. Therefore, $\exists C \in \mathscr{P}$ such that $x \in C$ and $y \in C$. Therefore $yQx$. Hence Q is symmetric. Let $x \in A$. By definition, the partition of A is represented as $$A =\bigcup_{c\in \mathscr{P}}C.$$ Since $x \in A$, it follows that $x \in \bigcup_{c \in \mathscr{P}}C$. So $\exists C \in \mathscr{P}$ such that $x \in C$. Hence $xQx$. Q is reflexive. These are my proofs. If needed, can someone correct them? • The rest of the problem is to show that Q is transitive, ie an equivalence relation, and that the set of equivalence classes of Q is the partition P. – William Elliot Oct 20 '17 at 2:05 Let $x \in A$ and $y \in A$. Assume that $xQy$. By definition, for all $x \in A$ and $y \in A$, if $xQy$, then $yQx$. There's an issue here; this is not "by definition," and is in fact precisely what you're trying to conclude, so it's not OK to claim that $y Q x$ yet. Therefore, $\exists C \in \mathscr{P}$ such that $x \in C$ and $y \in C$. Therefore $yQx$. It might be slightly better to explicitly restate that we can conclude for this $C$ that "$y \in C$ and $x \in C$", in that order. This is obviously true, but it's the crux of why you can claim that $y Q x$. Whether or not this is imperative depends on how picky your teacher is, most likely. Hence Q is symmetric. Let $x \in A$. By definition, the partition of A is represented as $$A =\bigcup_{c\in \mathscr{P}}C.$$ Since $x \in A$, it follows that $x \in \bigcup_{c \in \mathscr{P}}C$. So $\exists C \in \mathscr{P}$ such that $x \in C$. Again, it may be worth explicitly saying that "for this $C$, we can say '$x \in C$ and $x \in C$.'" But this feels even more nitpicky than the last thing and may be unnecessary. Hence $xQx$. Q is reflexive. (This whole thing mostly looks good, by the way! The only serious problem is the one about prematurely claiming that $y Q x$.)	2020-04-10 14:00:24	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9019327759742737, "perplexity": 181.6320425042303}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371896913.98/warc/CC-MAIN-20200410110538-20200410141038-00130.warc.gz"}
https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Book%3A_Mathematical_Methods_in_Chemistry_(Levitus)/10%3A_Plane_Polar_and_Spherical_Coordinates/10.04%3A_A_Brief_Introduction_to_Probability	# 10.4: A Brief Introduction to Probability We have talked about the fact that the wavefunction can be interpreted as a probability, but this is a good time to formalize some concepts and understand what we really mean by that. Let’s start by reviewing (or learning) a few concepts in probability theory. First, a random variable is a variable whose value is subject to variations due to chance. For example, we could define a variable that equals the number of days that it rains in Phoenix every month, or the outcome of throwing a die (the number of dots facing up), or the time it takes for the next bus to arrive to the bus station, or the waiting time we will have to endure next time we call a customer service phone line. Some of these random variables are discrete; the number of rain days or the number of dots facing up in a die can take on only a countable number of distinct values. For the case of the die, the outcome can only be $$\{1,2,3,4,5,6\}$$. In contrast, a waiting time is a continuous random variable. If you could measure with enough precision, the random variable can take any positive real value. Coming back to physical chemistry, the position of an electron in an atom or molecule is a good example of a continuous random variable. Imagine a (admittedly silly) game that involves flipping two coins. You get one point for each tail, and two points for each head. The game has three possible outcomes: 2 points (if you get two tails), 3 points (if you get one tail and one head) and 4 points (if you get two heads). The outcomes do not have the same probability. The probability of getting two heads or two tails is 1/4, while the probability that you get one head and one tail is 1/2. If we define a random variable ($$x$$) that equals the number of points you get in one round of the game, we can represent the probabilities of getting each possible outcome as: $$x$$ $$P(x)$$ 2 3 4 1/4 1/2 1/4 The collection of outcomes is called the sample space. In this case, the sample space is $$\{2,3,4\}$$. If we add $$P(x)$$ over the sample space we get, as expected, 1. In other words, the probability that you get an outcome that belongs to the sample space needs to be 1, which makes sense because we defined the sample space as the collection of all possible outcomes. If you think about an electron in an atom, and define the position in polar coordinates, $$r$$ (the distance from the nucleus of the atom) is a random variable that can take any value from 0 to $$\infty$$. The sample space for the random variable $$r$$ is the set of positive real numbers. Coming back to our discrete variable $$x$$, our previous argument translates into $\sum\limits_{sample\; space}P(x)=1 \nonumber$ Can we measure probabilities? Not exactly, but we can measure the frequency of each outcome if we repeat the experiment a large number of times. For example, if we play this game three times, we do not know how many times we’ll get 2, 3 or 4 points. But if we play the game a very large number of times, we know that half the time we will get 3 points, a quarter of the time will get 2 points, and another quarter 4 points. The probability is the frequency of an outcome in the limit of an infinite number of trials. Formally, the frequency is defined as the number of times you obtain a given outcome divided for the total number of trials. Now, even if we do not have any way of predicting the outcome of our random experiment (the silly game we described above), if you had to bet, you would not think it twice and bet on $$x=3$$ (one head and one tail). The fact that a random variable does not have a predictable outcome does not mean we do not have information about the distribution of probabilities. Coming back to our atom, we will be able to predict the value of $$r$$ at which the probability of finding the electron is highest, the average value of $$r$$, etc. Even if we know that $$r$$ can take values up to $$\infty$$, we know that it is much more likely to find it very close to the nucleus (e.g. within an angstrom) than far away (e.g. one inch). No physical law forbids the electron to be 1 inch from the nucleus, but the probability of that happening is so tiny, that we do not even think about this possibility. ## The Mean of a Discrete Distribution Let’s talk about the mean (or average) some more. What is exactly the average of a random variable? Coming back to our “game”, it would be the average value of $$x$$ you would get if you could play the game an infinite number of times. You could also ask the whole planet to play the game once, and that would accomplish the same. The planet does not have an infinite number of people, but the average you get with several billion trials of the random experiment (throwing two coins) should be pretty close to the real average. We will denote the average (also called the mean) with angular brackets: $$\left \langle x \right \rangle$$. Let’s say that we play this game $$10^9$$ times. We expect 3 points half the time (frequency = $$1/2$$), or in this case, $$5\times 10^8$$ times. We also expect 2 points or 4 points with a frequency of $$1/4$$, so in this case, $$2.5 \times 10^8$$ times. What is the average? $\left \langle x \right \rangle=\dfrac{1}{4}\times 2+\dfrac{1}{2}\times 3+\dfrac{1}{4}\times 4=3 \nonumber$ On average, the billion people playing the game (or you playing it a billion times) should get 3 points. This happens to be the most probable outcome, but it does not need to be the case. For instance, if you just flip one coin, you can get 1 point or 2 points with equal probability, and the average will be 1.5, which is not the most probable outcome. In fact, it is not even a possible outcome!. In general, it should make sense that for a discrete variable: $\label{eq:mean_discrete} \left \langle x \right \rangle=\sum\limits_{i=1}^k P(x_i)x_i$ where the sum is performed over the whole sample space, which contains $$k$$ elements. Here, $$x_i$$ is each possible outcome, and $$P(x_i)$$ is the probability of obtaining that outcome (or the fraction of times you would obtain it if you were to perform a huge number of trials). ## Continuous Variables How do we translate everything we just said to a continuous variable? As an example, let’s come back to the random variable $$r$$, which is defined as the distance of the electron in the hydrogen atom from its nucleus. As we will see shortly, the 1s electron in the hydrogen atom spends most of its time within a couple of angstroms from the nucleus. We may ask ourselves, what is the probability that the electron will be found exactly at 1Å from the nucleus? Mathematically, what is $$P(r=1$$ Å)? The answer will disappoint you, but this probability is zero, and it is zero for any value of $$r$$. The electron needs to be somewhere, but the probability of finding it an any particular value of $$r$$ is zero? Yes, that is precisely the case, and it is a consequence of $$r$$ being a continuous variable. Imagine that you get a random real number in the interval [0,1] (you could do this even in your calculator), and I ask you what is the probability that you get exactly $$\pi/4$$. There are infinite real numbers in this interval, and all of them are equally probable, so the probability of getting each of them is $$1/\infty=0$$. Talking about the probabilities of particular outcomes is not very useful in the context of continuous variables. All the outcomes have a probability of zero, even if we intuitively know that the probability of finding the electron within 1Å is much larger than the probability of finding it at several miles. Instead, we will talk about the density of probability ($$p(r)$$). If you are confused about why the probability of a particular outcome is zero check the video listed at the end of this section. A plot of $$p(r)$$ is shown in Figure $$\PageIndex{1}$$ for the case of the 1s orbital of the hydrogen atom. Again, we stress that $$p(r)$$ does not measure the probability corresponding to each value of $$r$$ (which is zero for all values of $$r$$), but instead, it measures a probability density. We already introduced this idea in page Formally, the probability density function ($$p(r)$$), is defined in this way: $\label{eq:coordinates_pdf} P(a\leq r\leq b)=\int\limits_{a}^{b}p(r)dr$ This means that the probability that the random variable $$r$$ takes a value in the interval $$[a,b]$$ is the integral of the probability density function from $$a$$ to $$b$$. For a very small interval: $\label{eq:coordinates_pdf2} P(a\leq r\leq a+dr)=p(a)dr$ In conclusion, although $$p(r)$$ alone does not mean anything physically, $$p(r)dr$$ is the probability that the variable $$r$$ takes a value in the interval between $$r$$ and $$r+dr$$. For example, coming back to Figure $$\PageIndex{1}$$, $$p(1$$ Å) = 0.62, which does not mean at all that 62% of the time we’ll find the electron at exactly 1Å from the nucleus. Instead, we can use it to calculate the probability that the electron is found in a very narrow region around 1Å . For example, $$P(1\leq r\leq 1.001)\approx 0.62\times 0.001=6.2\times 10^{-4}$$. This is only an approximation because the number 0.001, although much smaller than 1, is not an infinitesimal. In general, the concept of probability density function is easier to understand in the context of Equation \ref{eq:coordinates_pdf}. You can calculate the probability that the electron is found at a distance shorter than 1Å as: $P(0\leq r\leq 1)=\int\limits_{0}^{1}p(r)dr \nonumber$ and at a distance larger than 1Å but shorter than 2Å as $P(1\leq r\leq 2)=\int\limits_{1}^{2}p(r)dr \nonumber$ Of course the probability that the electron is somewhere in the universe is 1, so: $P(0\leq r\leq \infty)=\int\limits_{0}^{\infty}p(r)dr=1 \nonumber$ We haven’t written $$p(r)$$ explicitly yet, but we will do so shortly so we can perform all these integrations and get the probabilities discussed above. Confused about continuous probability density functions? This video may help! http: //tinyurl.com/m6tgoap ## The Mean of a Continuous Distribution For a continuous random variable $$x$$, Equation \ref{eq:mean_discrete} becomes: $\label{eq:mean_continuous} \left \langle x \right \rangle = \int\limits_{all\,outcomes}p(x) x \;dx$ Coming back to our atom: $\label{eq:mean_r} \left \langle r \right \rangle = \int\limits_{0}^{\infty}p(r) r \;dr$ Again, we will come back to this equation once we obtain the expression for $$p(r)$$ we need. But before doing so, let’s expand this discussion to more variables. So far, we have limited our discussion to one coordinate, so the quantity $$P(a\leq r\leq b)=\int\limits_{a}^{b}p(r)dr$$ represents the probability that the coordinate $$r$$ takes a value between $$a$$ and $$b$$, independently of the values of $$\theta$$ and $$\phi$$. This region of space is the spherical shell represented in Figure $$\PageIndex{2}$$ in light blue. The spheres in the figure are cut for clarity, but of course we refer to the whole shell that is defined as the region between two concentric spheres of radii $$a$$ and $$b$$. What if we are interested in the angles as well? Let’s say that we want the probability that the electron is found between $$r_1$$ and $$r_2$$, $$\theta_1$$ and $$\theta_2$$, and $$\phi_1$$ and $$\phi_2$$. This volume is shown in Figure $$\PageIndex{3}$$. The probability we are interested in is: $P(r_1\leq r\leq r_2, \theta_1\leq \theta \leq \theta_2, \phi_1 \leq \phi \leq\phi_2)=\int\limits_{\phi_1}^{\phi_2}\int\limits_{\theta_1}^{\theta_2}\int\limits_{r_1}^{r_2}p(r,\theta,\phi)\;r^2 \sin\theta dr d\theta d\phi \nonumber$ Notice that we are integrating in spherical coordinates, so we need to use the corresponding differential of volume. This probability density function, $$p(r,\theta,\phi)$$, is exactly what $$\|\psi(r,\theta,\phi)\|^2$$ represents! This is why we’ve been saying that $$\|\psi(r,\theta,\phi)\|^2$$ is a probability density. The function $$\|\psi(r,\theta,\phi)\|^2$$ does not represent a probability in itself, but it does when integrated between the limits of interest. Suppose we want to know the probability that the electron in the 1s orbital of the hydrogen atom is found between $$r_1$$ and $$r_2$$, $$\theta_1$$ and $$\theta_2$$, and $$\phi_1$$ and $$\phi_2$$. The answer to this question is: $\int\limits_{\phi_1}^{\phi_2}\int\limits_{\theta_1}^{\theta_2}\int\limits_{r_1}^{r_2}\|\psi_{1s}\|^2\;r^2 \sin\theta dr d\theta d\phi \nonumber$ Coming back to the case shown in Figure $$\PageIndex{2}$$, the probability that $$r$$ takes a value between $$a$$ and $$b$$ independently of the values of the angles, is the probability that $$r$$ lies between $$a$$ and $$b$$, and $$\theta$$ takes a value between 0 and $$\pi$$, and $$\phi$$ takes a value between 0 and $$2\pi$$: $\label{eq:coordinates9} P(a\leq r\leq b)=P(a\leq r\leq b,0 \leq \theta \leq \pi, 0 \leq \phi \leq 2\pi)=\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}\int\limits_{a}^{b}\|\psi_{1s}\|^2\;r^2 \sin\theta dr d\theta d\phi$ So far we established that $$\|\psi(r,\theta,\phi)\|^2$$ is a probability density function in spherical coordinates. We can perform triple integrals to calculate the probability of finding the electron in different regions of space (but not in a particular point!). It is often useful to know the likelihood of finding the electron in an orbital at any given distance away from the nucleus. This enables us to say at what distance from the nucleus the electron is most likely to be found, and also how tightly or loosely the electron is bound in a particular atom. This is expressed by the radial distribution function, $$p(r)$$, which is plotted in Figure $$\PageIndex{1}$$ for the 1s orbital of the hydrogen atom. In other words, we want a version of $$\|\psi(r,\theta,\phi)\|^2$$ that is independent of the angles. This new function will be a function of $$r$$ only, and can be used, among other things, to calculate the mean of $$r$$, the most probable value of $$r$$, the probability that $$r$$ lies in a given range of distances, etc. We already introduced this function in Equation \ref{eq:coordinates_pdf}. The question now is, how do we obtain $$p(r)$$ from $$\|\psi(r,\theta,\phi)\|^2$$? Let’s compare Equation \ref{eq:coordinates_pdf} with Equation \ref{eq:coordinates9}: $P(a\leq r\leq b)=\int\limits_{a}^{b}p(r)dr \nonumber$ $P(a\leq r\leq b)=P(a\leq r\leq b,0 \leq \theta \leq \pi, 0 \leq \phi \leq 2\pi)=\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}\int\limits_{a}^{b}\|\psi(r,\theta,\phi)\|^2\;r^2 \sin\theta dr d\theta d\phi \nonumber$ We conclude that $\int\limits_{a}^{b}p(r)dr=\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}\int\limits_{a}^{b}\|\psi(r,\theta,\phi)\|^2\;r^2 \sin\theta dr d\theta d\phi \nonumber$ All $$s$$ orbitals are real functions of $$r$$ only, so $$\|\psi(r,\theta,\phi)\|^2$$ does not depend on $$\theta$$ or $$\phi$$. In this case: $\int\limits_{a}^{b}p(r)dr=\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}\int\limits_{a}^{b}\psi^2(r)\;r^2 \sin\theta dr d\theta d\phi=\int\limits_{0}^{2\pi}d\phi \int\limits_{0}^{\pi}\sin\theta \;d\theta \int\limits_{a}^{b}\psi^2(r)\;r^2 dr=4\pi\int\limits_{a}^{b}\psi^2(r)\;r^2 dr \nonumber$ Therefore, for an $$s$$ orbital, $p(r)=4\pi\psi^2(r)\;r^2 \nonumber$ For example, the normalized wavefunction of the 1s orbital is the solution of Example $$10.1$$: $$\dfrac{1}{\sqrt{\pi a_0^3}}e^{-r/a_0}$$. Therefore, for the 1s orbital: $\label{eq:coordinates10} p(r)=\dfrac{4}{a_0^3}r^2e^{-2r/a_0}$ Equation \ref{eq:coordinates10} is plotted in Figure $$\PageIndex{1}$$. In order to create this plot, we need the value of $$a_0$$, which is a constant known as Bohr radius, and equals $$5.29\times10^{-11}m$$ (or 0.526 Å). Look at the position of the maximum of $$p(r)$$; it is slightly above 0.5Å and more precisely, exactly at $$r=a_0$$! Now it is clear why $$a_0$$ is known as a radius: it is the distance from the nucleus at which finding the only electron of the hydrogen atom is greatest. In a way, $$a_0$$ is the radius of the atom, although we know this is not strictly true because the electron is not orbiting at a fixed $$r$$ as scientists believed a long time ago. In general, for any type of orbital, $\int\limits_{a}^{b}p(r)dr=\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}\int\limits_{a}^{b}\|\psi(r,\theta,\phi)\|^2\;r^2 \sin\theta dr d\theta d\phi=\int\limits_{a}^{b}{\color{Red}\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}\|\psi(r,\theta,\phi)\|^2\;r^2 \sin\theta\; d\theta d\phi} dr \nonumber$ in the right side of the equation, we just changed the order of integration to have $$dr$$ last, and color coded the expression so we can easily identify $$p(r)$$ as: $\label{eq:p(r)} p(r)=\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}\|\psi(r,\theta,\phi)\|^2\;r^2 \sin\theta\; d\theta d\phi$ Equation \ref{eq:p(r)} is the mathematical formulation of what we wanted: a probability density function that does not depend on the angles. We integrate $$\phi$$ and $$\theta$$ so what we are left with represents the dependence with $$r$$. We can multiply both sides by $$r$$: $rp(r)=r\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}{\color{Red}\|\psi(r,\theta,\phi)\|^2}{\color{OliveGreen}r^2 \sin\theta\; d\theta d\phi}=\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}{\color{Red}\|\psi(r,\theta,\phi)\|^2}r{\color{OliveGreen}r^2 \sin\theta\; d\theta d\phi} \nonumber$ and use Equation \ref{eq:mean_r} to calculate $$\left \langle r \right \rangle$$ $\label{eq:coordinates12} \left \langle r \right \rangle = \int\limits_{0}^{\infty}{\color{Magenta}p(r) r} \;dr=\int\limits_{0}^{\infty}{\color{Magenta}\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}\|\psi(r,\theta,\phi)\|^2r\;r^2 \sin\theta\; d\theta d\phi}dr$ The colors in these expressions are aimed to help you track where the different terms come from. Let’s look at equation \ref{eq:coordinates12} more closely. Basically, we just concluded that: $\label{eq:coordinates13} \left \langle r \right \rangle = \int\limits_{all\;space}\|\psi\|^2r\;dV$ where $$dV$$ is the differential of volume in spherical coordinates. We know that $$\psi$$ is normalized, so $\int\limits_{all\;space}\|\psi\|^2\;dV=1 \nonumber$ If we multiply the integrand by $$r$$, we get $$\left \langle r \right \rangle$$. We will discuss an extension of this idea when we talk about operators. For now, let’s use Equation \ref{eq:coordinates13} to calculate $$\left \langle r \right \rangle$$ for the 1s orbital. Example $$\PageIndex{1}$$ Calculate the average value of $$r$$, $$\left \langle r \right \rangle$$, for an electron in the 1s orbital of the hydrogen atom. The normalized wavefunction of the 1s orbital, expressed in spherical coordinates, is: $\psi_{1s}=\dfrac{1}{\sqrt{\pi a_0^3}}e^{-r/a_0} \nonumber$ Solution The average value of $$r$$ is: $\left \langle r \right \rangle=\int\limits_{0}^{\infty}p(r)r\;dr \nonumber$ or $\left \langle r \right \rangle = \int\limits_{all\;space}\|\psi\|^2r\;dV \nonumber$ The difference between the first expression and the second expression is that in the first case, we already integrated over the angles $$\theta$$ and $$\phi$$. The second expression is a triple integral because $$\|\psi\|^2$$ still retains the angular information. We do not have $$p(r)$$, so either we obtain it first from $$\|\psi\|^2$$, or directly use $$\|\psi\|^2$$ and perform the triple integration: $\left \langle r \right \rangle = \int\limits_{0}^{\infty}\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}\|\psi(r,\theta,\phi)\|^2r\;{\color{Red}r^2 \sin\theta\; d\theta d\phi dr} \nonumber$ The expression highlighted in red is the differential of volume. For this orbital, $\|\psi(r,\theta,\phi)\|^2=\dfrac{1}{\pi a_0^3}e^{-2r/a_0} \nonumber$ and then, $\left \langle r \right \rangle = \dfrac{1}{\pi a_0^3}\int\limits_{0}^{\infty}e^{-2r/a_0}r^3\;dr\int\limits_{0}^{2\pi}d\phi\int\limits_{0}^{\pi}\sin\theta\; d\theta=\dfrac{4}{a_0^3}\int\limits_{0}^{\infty}e^{-2r/a_0}r^3\;dr \nonumber$ From the formula sheet: $$\int_{0}^{\infty}x^ne^{-ax}dx=\dfrac{n!}{a^{n+1}},\; a>0, n$$ is a positive integer. Here, $$n=3$$ and $$a= 2/a_0$$. $\dfrac{4}{a_0^3}\int\limits_{0}^{\infty}r^3e^{-2r/a_0}\;dr=\dfrac{4}{a_0^3}\times \dfrac{3!}{(2/a_0)^4}=\dfrac{3}{2}a_0 \nonumber$ $\displaystyle{\color{Maroon}\left \langle r \right \rangle=\dfrac{3}{2}a_0} \nonumber$ From Example $$\PageIndex{1}$$, we notice that on average we expect to see the electron at a distance from the nucleus equal to 1.5 times $$a_0$$. This means that if you could measure $$r$$, and you perform this measurement on a large number of atoms of hydrogen, or on the same atom many times, you would, on average, see the electron at a distance from the nucleus $$r=1.5 a_0$$. However, the probability of seeing the electron is greatest at $$r=a_0$$ (page ). We see that the average of a distribution does not necessarily need to equal the value at which the probability is highest2.	2021-10-25 13:36:21	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8775587677955627, "perplexity": 113.7696684606553}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323587711.69/warc/CC-MAIN-20211025123123-20211025153123-00226.warc.gz"}
https://www.vedantu.com/question-answer/the-normal-to-the-curve-x2-+-2xy-3y2-0-at-left-class-10-maths-cbse-5ee480fc6c74206b34365e84	Courses Courses for Kids Free study material Free LIVE classes More # The normal to the curve, ${x^2} + 2xy - 3{y^2} = 0$, at $\left[ {1,1} \right]$${\text{A}}{\text{.}} does not meet the curve again{\text{B}}{\text{.}} meets the curve again in the second quadrant{\text{C}}{\text{.}} meets the curve again in the third quadrant{\text{D}}{\text{.}} meets the curve again in the fourth quadrant Last updated date: 20th Mar 2023 Total views: 305.4k Views today: 8.84k Answer Verified 305.4k+ views Hint: Here, we will be using the concept of finding the slope of the normal to any curve and the formula for equation of any line which slope m and passing through a point. Given equation of the curve is {x^2} + 2xy - 3{y^2} = 0{\text{ }} \to {\text{(1)}} Complete step-by-step answer: Here, we need to first of all find the equation of the normal to the given curve. As, we know that the slope of normal to any curve is - \dfrac{{dx}}{{dy}} . Let us differentiate the given equation of curve both sides with respect to x, we get $\dfrac{{d\left( {{x^2} + 2xy - 3{y^2}} \right)}}{{dx}} = 0 \Rightarrow \dfrac{{d\left( {{x^2}} \right)}}{{dx}} + 2\left[ {\dfrac{{d\left( {xy} \right)}}{{dx}}} \right] - 3\left[ {\dfrac{{d\left( {{y^2}} \right)}}{{dx}}} \right] = 0 \Rightarrow 2x + 2\left[ {y\left( {\dfrac{{dx}}{{dx}}} \right) + x\left( {\dfrac{{dy}}{{dx}}} \right)} \right] - 3 \times 2y\left[ {\dfrac{{dy}}{{dx}}} \right] = 0 \\ \Rightarrow 2x + 2y + 2x\left( {\dfrac{{dy}}{{dx}}} \right) - 6y\left( {\dfrac{{dy}}{{dx}}} \right) = 0 \Rightarrow 2\left( {x + y} \right) + 2\left( {\dfrac{{dy}}{{dx}}} \right)\left( {x - 3y} \right) = 0 \Rightarrow 2\left( {\dfrac{{dy}}{{dx}}} \right)\left( {x - 3y} \right) = - 2\left( {x + y} \right) \\ \Rightarrow \left( {\dfrac{{dy}}{{dx}}} \right) = - \dfrac{{2\left( {x + y} \right)}}{{2\left( {x - 3y} \right)}} \Rightarrow \left( {\dfrac{{dy}}{{dx}}} \right) = - \dfrac{{\left( {x + y} \right)}}{{\left( {x - 3y} \right)}} \\$ Slope of normal to the given curve = - \dfrac{{dx}}{{dy}} = \dfrac{{ - 1}}{{\dfrac{{dy}}{{dx}}}} = \dfrac{{ - 1}}{{\left[ { - \dfrac{{\left( {x + y} \right)}}{{\left( {x - 3y} \right)}}} \right]}} = \dfrac{{\left( {x - 3y} \right)}}{{\left( {x + y} \right)}} In order to find the slope of the normal to the given curve at a point \left[ {1,1} \right], let us substitute x=1 and y=1 in the above expression of slope. Slope of normal to the given curve at \left[ {1,1} \right]$$ = - \dfrac{{dx}}{{dy}} = \dfrac{{\left( {1 - 3 \times 1} \right)}}{{\left( {1 + 1} \right)}} = \dfrac{{1 - 3}}{2} = - 1$ Also, we know that equation of the line with slope m and passing through any point (${x_1},{y_1}$) is given by $y - {y_1} = m\left( {x - {x_1}} \right)$ Now, equation of the normal with slope of -1 and passing through point $\left[ {1,1} \right]$ is given by $y - 1 = - 1\left( {x - 1} \right) \Rightarrow y - 1 = - x + 1 \Rightarrow x + y - 1 - 1 = 0 \Rightarrow x + y - 2 = 0{\text{ }} \to {\text{(2)}}$ Now, in order to find out whether this normal meets the given curve at any point other than the given point $\left[ {1,1} \right]$. Let us solve for the common solutions between the curve and the normal. Equation (2) can be written as $\Rightarrow x + y - 2 = 0 \Rightarrow x = 2 - y{\text{ }} \to {\text{(3)}}$ Substitute equation (3) in equation (1), we get ${\left( {2 - y} \right)^2} + 2y\left( {2 - y} \right) - 3{y^2} = 0 \Rightarrow 4 + {y^2} - 4y + 4y - 2{y^2} - 3{y^2} = 0 \Rightarrow 4 - 4{y^2} = 0 \Rightarrow {y^2} = 1 \Rightarrow y = \pm 1$ When y=1, equation (3) reduces to $\Rightarrow x = 2 - y = 2 - 1 \Rightarrow x = 1$ When y=-1, equation (3) reduces to $\Rightarrow x = 2 - y = 2 - \left( { - 1} \right) \Rightarrow x = 3$ So, the points where the given curve meets with the normal to this curve are (1,1) and (3,-1). Clearly, apart from point (1,1) the given curve meets the normal to this curve at (3,-1) which lies in the fourth quadrant because in the fourth quadrant x-coordinate is positive and y-coordinate is negative. Hence, option D is correct. Note: In these types of problems, we will simply find the equation for the normal to the given curve at the given point by finding the slope of the normal. Then, we will find the points of intersection of the given curve and normal to the given curve and if any other point appears except the given point that means that the normal meets the curve again else it does not.	2023-03-25 02:02:24	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8625494241714478, "perplexity": 371.2818564469907}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945292.83/warc/CC-MAIN-20230325002113-20230325032113-00422.warc.gz"}
https://www.dmgt.uz.zgora.pl/publish/view_short.php?ID=-20719	DMGT ISSN 1234-3099 (print version) ISSN 2083-5892 (electronic version) IMPACT FACTOR 2018: 0.741 SCImago Journal Rank (SJR) 2018: 0.763 Rejection Rate (2017-2018): c. 84% Discussiones Mathematicae Graph Theory THE EDGE GEODETIC NUMBER AND CARTESIAN PRODUCT OF GRAPHS A.P. Santhakumaran and S.V. Ullas Chandran Department of Mathematics St. Xavier's College (Autonomous) Palayamkottai - 627 002, India e-mail: apskumar1953@yahoo.co.in e-mail: ullaschandra01@yahoo.co.in Abstract For a nontrivial connected graph G = (V(G),E(G)), a set S⊆ V(G) is called an edge geodetic set of G if every edge of G is contained in a geodesic joining some pair of vertices in S. The edge geodetic number g1(G) of G is the minimum order of its edge geodetic sets. Bounds for the edge geodetic number of Cartesian product graphs are proved and improved upper bounds are determined for a special class of graphs. Exact values of the edge geodetic number of Cartesian product are obtained for several classes of graphs. Also we obtain a necessary condition of G for which g1(G[¯] K2) = g1(G). Keywords: geodetic number, edge geodetic number, linear edge geodetic set, perfect edge geodetic set, (edge, vertex)-geodetic set, superior edge geodetic set. 2010 Mathematics Subject Classification: 05C12. References [1] B. Bresar, S. Klavžar and A.T. Horvat, On the geodetic number and related metric sets in Cartesian product graphs, (2007), Discrete Math. 308 (2008) 5555-5561, doi: 10.1016/j.disc.2007.10.007. [2] F. Buckley and F. Harary, Distance in Graphs (Addison-Wesley, Redwood City, CA, 1990). [3] G. Chartrand, F. Harary and P. Zhang, On the geodetic number of a graph, Networks 39 (2002) 1-6, doi: 10.1002/net.10007. [4] G. Chartrand and P. Zhang, Introduction to Graph Theory (Tata McGraw-Hill Edition, New Delhi, 2006). [5] F. Harary, E. Loukakis and C. Tsouros, The geodetic number of a graph, Math. Comput. Modeling 17 (1993) 89-95, doi: 10.1016/0895-7177(93)90259-2. [6] W. Imrich and S. Klavžar, Product Graphs: Structure and Recognition (Wiley-Interscience, New York, 2000). [7] A.P. Santhakumaran and J. John, Edge geodetic number of a graph, J. Discrete Math. Sciences & Cryptography 10 (2007) 415-432.	2020-02-28 22:52:01	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.866355299949646, "perplexity": 6144.750337967577}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875147647.2/warc/CC-MAIN-20200228200903-20200228230903-00257.warc.gz"}
https://solvedlib.com/n/a-mass-weighting-40-lbs-stretches-a-spring-6-inches-the,6136238	# A mass weighting 40 lbs stretches a spring 6 inches. The mass is in a medium that exerts a viscous ###### Question: A mass weighting 40 lbs stretches a spring 6 inches. The mass is in a medium that exerts a viscous resistance of 90 lbs when the mass has velocity of 6 ftlsec Suppose the object is displaced an additional 6 inches and released. Find an equation for the object's displacement, u(t) , in feet after t seconds_ u(t) Preview Get help: Video #### Similar Solved Questions ##### Choose the best answer from the available descriptions about the following circuit diagram. A B 4... Choose the best answer from the available descriptions about the following circuit diagram. A B 4 V 6 V 10 V 5 V The circuit does not adhere to Kirchoffs Voltage Law. The circuit does adhere to Kirchoffs Voltage Law. The circuit does not adhere to Kirchoffs Current Law The circuit does adhere to Kir... ##### Use Karnaugh maps to simplify the following Boolean functions ex minterms 1. a) fx,y,z)-ml +m2+ m5+m6+... Use Karnaugh maps to simplify the following Boolean functions ex minterms 1. a) fx,y,z)-ml +m2+ m5+m6+ m7 xy b) f(w, x y,z) -2(0,2,4,5,6,7,12,13) c) f(w, x, y, z) Σ(3, 4, 5, 6, 7, 9, 12, 13, 14, 15) wx... ##### If I add water to 200 mL of a 0.15 M NaOH solution until the final volume is 250 mL; what will the molarity of the diluted solution be? Point)0.12 M0.03 M0.24 M0.19 M If I add water to 200 mL of a 0.15 M NaOH solution until the final volume is 250 mL; what will the molarity of the diluted solution be? Point) 0.12 M 0.03 M 0.24 M 0.19 M... ##### Jukanc DHOH_mbl DGUWH(LK 'up 824.2 742.2 7.40 FezO, 30,7 Hz (9 27.3 Ee 24L,82 128.21 138.8 Hz0l)) Jukanc DHOH_mbl DGUWH(LK 'up 824.2 742.2 7.40 FezO, 30,7 Hz (9 27.3 Ee 24L,82 128.21 138.8 Hz0l))... ##### Combining 0.204 mol FezO3 with excess carbon produced 12.4 g FeFezOz + 3C _> 2Fe + 3C0What is the actual yield of iron in moles?actual yield:molWhat is the theoretical yield of iron in moles?theoretical yield:molWhat is the percent yield?percent yield: Combining 0.204 mol FezO3 with excess carbon produced 12.4 g Fe FezOz + 3C _> 2Fe + 3C0 What is the actual yield of iron in moles? actual yield: mol What is the theoretical yield of iron in moles? theoretical yield: mol What is the percent yield? percent yield:... ##### 110 PST PLS. LUI 7. Find each of the following exactly (no decimal approximations!) Show all... 110 PST PLS. LUI 7. Find each of the following exactly (no decimal approximations!) 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Please write your answer in interval notation_... ##### WKOD-13% WCLM-24% WANR-22% WWCN-41% this is a circle graph of 400 people were asked to identify th TV channel they watch the evening News WKOD-13%WCLM-24%WANR-22%WWCN-41%this is a circle graph of 400 people were asked to identify th TV channel they watch the evening News.1.how many people preferred WCLM?2.how many more people preferred WWCN then WANR3.ratio express the people who preferred WANR to WCLMi want to see if i got them rig... ##### Question 3Find the mean, mode,median , for the followIng data 12 15 18 18,15,22 1530,12DCAcst Onc decual placeMean =2) Medlane3) Moda Question 3 Find the mean, mode,median , for the followIng data 12 15 18 18,15,22 1530,12 DCAcst Onc decual place Mean = 2) Medlane 3) Moda... ##### The Toyota Camry is one of the best-selling cars in NorthAmerica. The cost of apreviously owned Camry depends upon many factors, including themodel year, mileage,and condition. Our interest is to estimate sales price using thecar's mileage. The le Toy-otaCamry" in d2l contains data for sales price and the car'smileage for a 2007 model yearCamry for 19 sales (PriceHub website, February 24, 2012). Asdemonstrated in the lecture,please create a subset data of size 16 and perform yours The Toyota Camry is one of the best-selling cars in North America. The cost of a previously owned Camry depends upon many factors, including the model year, mileage, and condition. Our interest is to estimate sales price using the car's mileage. The le Toy- otaCamry" in d2l contains data f... ##### Consider the reaction:2 SOzlg) Oz(g) 2SOglg)The cquilibrium constant (Kz) for this reaction is 2.80 x 102at J given temperature: If the concentration of SOz is 2.56 x 10-? Mand the concentration of 0z is 8.06 x 10-2 Mat equilibrium; what is the concentration of SO3 at equilibrium? Express vour answer inM using at least three significant figures Do not use scientifc notation: Consider the reaction: 2 SOzlg) Oz(g) 2SOglg) The cquilibrium constant (Kz) for this reaction is 2.80 x 102at J given temperature: If the concentration of SOz is 2.56 x 10-? Mand the concentration of 0z is 8.06 x 10-2 Mat equilibrium; what is the concentration of SO3 at equilibrium? Express vour ans...	2023-03-23 02:43:07	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.46263229846954346, "perplexity": 6087.463290636495}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296944606.5/warc/CC-MAIN-20230323003026-20230323033026-00188.warc.gz"}
https://tarungehlots.blogspot.com/2014/08/chapter-7-worked-out-examples_9.html	tgt ## Saturday, 9 August 2014 ### CHAPTER 7 - Worked Out Examples Example: 7 Find the sum $S$ given by $S = {1^2} \cdot {C_1} + {2^2} \cdot {C_2} + {3^2} \cdot {C_3} + \ldots + {n^2} \cdot {C_n}$ Solution: 7 We have to plan an approach wherein we are able to generate ${r^2}$ with ${C_r}$. We can generate one $r$ with every ${C_r}$, as we did earlier, and which is now repeated here: ${(1 + x)^n} = {C_0} + {C_1}x + {C_2}{x^2} + \ldots + {C_n}{x^n}$ Differentiating both sides with respect to $x$, we have $n{(1 + x)^{n - 1}} = {C_1} + 2{C_2}x + 3{C_3}{x^2} + \ldots + n{C_n}{x^{n - 1}}$ Now we have reached the stage where we have an $r$ with every ${C_r}$. We need to think how to get the other $r$. If we differentiate once again, we’ll have $r(r - 1)$ with every ${C_r}$ instead of ${r^2}$ (understand this point carefully). To ‘make-up’ for the power that falls one short of the required value, we simply multiply by $x$ on both sides of the relation above to obtain: $nx{(1 + x)^{n - 1}} = {C_1}x + 2{C_2}{x^2} + 3{C_3}{x^3} + \ldots + n{C_n}{x^n}$ It should be evident now that the next step is differentiation: $n(n - 1)x{(1 + x)^{n - 2}} + n{(1 + x)^{n - 1}} = {C_1} + {2^2} \cdot {C_2}x + {3^2}$ $\cdot {C_3}{x^2} +\ldots + {n^2} \cdot {C_n}{x^{n - 1}}$ Now we simply substitute $x = 1$ to obtain $n(n - 1) \cdot {2^{n - 2}} + n \cdot {2^{n - 1}} = {C_1} + {2^2} \cdot {C_2} + {3^2} \cdot {C_3} +\ldots + {n^2} \cdot {C_n}$ The required sum $S$ is thus $S\;\; = \;n(n - 1) \cdot {2^{n - 2}} + n \cdot {2^{n - 1}}$ $= n \cdot {2^{n - 2}}\left\{ {(n - 1) + 2} \right\}$ $= n(n + 1) \cdot {2^{n - 2}}$ Example: 8 Evaluate the following sums: (a) ${S_1} = \dfrac{{{C_0}}}{1} + \dfrac{{{C_2}}}{3} + \dfrac{{{C_4}}}{5} +\ldots$ (b) ${S_2} = \dfrac{{{C_1}}}{2} + \dfrac{{{C_3}}}{4} + \dfrac{{{C_5}}}{6} +\ldots$ Solution: 8-(a) The first sum contains only the even-numbered binomial coefficients, while the second contains only odd-numbered ones. Recall that we have already evaluated the sum $S$ given by $S = {C_0} + \dfrac{{{C_1}}}{2} + \dfrac{{{C_2}}}{3} + \ldots + \dfrac{{{C_n}}}{{n + 1}} = \dfrac{{{2^{n + 1}} - 1}}{{n + 1}}$ Note that $S$ is the sum of ${S_1}$ and ${S_2}$, i.e., ${S_1} + {S_2} = \dfrac{{{2^{n + 1}} - 1}}{{n + 1}}$ Thus, if we determine ${S_1}$ , ${S_2}$ is automatically determined, and vice-versa. Let us try to determine ${S_1}$ first. Consider again the general expansion ${(1 + x)^n} = {C_0} + {C_1}x + {C_2}{x^2} +\ldots + {C_n}{x^n}$ Integrating with respect to $x$, we have (we have not yet decided the limits) $\left. {\dfrac{{{{(1 + x)}^{n + 1}}}}{{n + 1}}} \right\|_a^b = \left. {{C_0}x} \right\|_a^b + \left. {{C_1}\dfrac{{{x^2}}}{2}} \right\|_a^b + \left. {{C_2}\dfrac{{{x^3}}}{3}} \right\|_a^b +\ldots + \left. {{C_n}\dfrac{{{x^{n + 1}}}}{{n + 1}}} \right\|_a^b$ Since we are trying to determine ${S_1}$ which contains only the even-numbered terms, we have to choose the limits of integration such that the odd-numbered terms vanish. This is easily achievable by setting $a = - 1$ and $b = 1$ (understand this carefully). Thus, we have $\dfrac{{{2^{n + 1}}}}{{n + 1}} = 2\left( {{C_0} + \dfrac{{{C_2}}}{3} + \dfrac{{{C_4}}}{5} + \ldots } \right)$ which implies that ${S_1} = \dfrac{{{2^n}}}{{n + 1}}$ Solution: 8-(b) ${S_2}$ is now simply given by ${S_2}\;\; = S - {S_1}$ $= \dfrac{{{2^{n + 1}} - 1}}{{n + 1}} - \dfrac{{{2^n}}}{{n + 1}}$ $= \dfrac{{{2^n} - 1}}{{n + 1}}$	2017-05-29 01:55:23	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 48, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8664478659629822, "perplexity": 386.2979504693569}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463612008.48/warc/CC-MAIN-20170529014619-20170529034619-00369.warc.gz"}
http://umj.imath.kiev.ua/article/?lang=en&article=9524	2018 Том 70 № 12 Topological stability of the averagings of functions Abstract We present sufficient conditions for the topological stability of the averagings of piecewise smooth functions $f : R \rightarrow R$ with finitely many extrema with respect to discrete measures with finite supports. Citation Example: Maksimenko S. I., Marunkevych O. V. Topological stability of the averagings of functions // Ukr. Mat. Zh. - 2016. - 68, № 5. - pp. 625-633.	2019-01-17 17:03:44	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.22268490493297577, "perplexity": 2847.86000244743}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583659056.44/warc/CC-MAIN-20190117163938-20190117185938-00203.warc.gz"}
https://math.stackexchange.com/questions/2973554/definition-of-adjoint-operator-for-quantum-mechanics	# Definition of Adjoint Operator for Quantum Mechanics While learning about adjoint operators for quantum mechanics, I encountered two definitions. The first definition is given by Shankar in The Principle of Quantum Mechanics: Given a ket $$A\lvert V \rangle = \lvert A V \rangle$$ the correspoding bra is $$\langle AV \rvert=\langle V \rvert A^\dagger$$ which defines the operator $$A^\dagger$$. The second definition is: For a linear operator $$A$$, its adjoint is defined so that $$\langle u \rvert A^\dagger \lvert v \rangle = {\langle v \rvert A \lvert u \rangle}^$$ where $$^$$ means to take the complex conjugate. From the first definition, it seems that the adjoint $$A^\dagger$$ should only act on bras. But from the second definition the adjoint $$A^\dagger$$ acts on a ket $$\lvert v \rangle$$. How does the two definition reconcile with each other? Is the first definition of the adjoint somewhat misleading, since it can also act on kets? • We have $\langle u\mid v\rangle =\langle v\mid u\rangle^$ for all vectors $u, v$. – Berci Oct 27 '18 at 15:48 • This is one reason Mathematicians do not use the bra-ket notation: you cannot tell what the operator in the middle acts on. If the operator is selfadjoint, then everything is okay, but it is useful to be able to study symmetric operators that aren't selfadjoint, in order to determine the deficiency spaces, which determine the boundary conditions for the problem. – DisintegratingByParts Oct 27 '18 at 16:14 • I have seen notation $\langle x \| A \} \| y\rangle$ and $\langle x \| \{ A \| y \rangle$. That fixes a lot of problems, but it sure is awkward. – DisintegratingByParts Oct 27 '18 at 16:42 • @DisintegratingByParts But $\left(\langle x \| A\right) \| y \rangle$ and $\langle x \| \left(A \| y \rangle\right)$ are exactly the same, aren't they? Isn't this the whole point of why this notation is used in the first place? – Joppy Oct 28 '18 at 10:45 • @Joppy : The notation $\langle x\| A\| y\rangle$ is not adequate. That's the point of my remark. And the substitute notation that you and I mentioned is needlessly complex and awkward. That's why I don't like bra-ket notation, and have no interest in using it. The usual inner product defined by von Neumann was perfectly adequate and was around well before Dirac devised his notation. – DisintegratingByParts Oct 28 '18 at 15:49 The underlying reason is that for a Hilbert space $$H$$, its dual is again $$H$$. What this means is that any bounded linear functional $$H\to\mathbb C$$ is given by the inner product against some fixed vector. So, any linear functional is given by choosing some $$v\in H$$ and then doing $$u\longmapsto \langle v\|u\rangle$$ (this is known to mathematicians as the Riesz Representation Theorem). That is, the "kets" are the elements of the space $$H$$, and the "bras" are the elements of the dual. Now, the adjoint $$A^\dagger$$ is defined precisely as the operator (on the dual $$H^$$, but for us it is again $$H$$) such that $$\tag1\langle A^\dagger v\|u\rangle=\langle v\|Au\rangle.$$ When $$A$$ is selfadjoint, you have $$\langle v\|Au\rangle=\langle Av\|u\rangle=\overline{\langle u\|Av\rangle}.$$ As a mathematician, it is hard for me to grasp any advantage from talking about bras and kets. It would make a lot more sense to write $$v^u$$ instead of $$\langle v\|u\rangle$$. Here, for a vector $$v$$, the vector $$v^$$ is the conjugate transpose, and then $$v^u$$ is precisely the matrix product. When there is an operator in the mix, $$v^(Au)=v^Au=(A^v)^u,$$ which is the same as $$(1)$$ but doesn't require any convention other than the mathematical properties of taking adjoints (namely, "conjugate and tranpose"). If you really care about the math behind, given an operator $$T:X\to Y$$, one can always defined an adjoint $$T^:Y^\to X^$$ (where $$X^$$ is the dual, i.e. the space of bounded linear functionals on $$X$$) by $$(T^g)(x)=g(Tx).$$ Because, as already mentioned, for a Hilbert $$H$$ the dual is again $$H$$, we can see the adjoint again as an operator on $$H$$. • The last $\langle Av \| u \rangle = \langle u \| Av \rangle$ does not hold - there should be a complex conjugation in there. – Joppy Oct 28 '18 at 11:55 • Nice catch, thanks. – Martin Argerami Oct 28 '18 at 14:10 Suppose the underlying (complex Hilbert) vector space is $$V$$, with inner product $$(-, -)_V$$. We then have bras and kets. Kets are the easiest to understand: $$\| v \rangle$$ is just the vector $$v$$. The bra $$\langle v \|$$ is a linear map $$V \to \mathbb{C}$$, given by $$\langle v \| (\| w \rangle ) = (v, w)_V$$. For aesthetic reasons, we leave out the parentheses and extra vertical line when feeding a ket into a bra, to get $$\langle v \| w \rangle = (v, w)_V$$. Any linear operator $$A: V \to V$$ acts naturally on both bras and kets. Since a ket is a vector, its action is the usual one: $$A \| v \rangle$$ means the vector $$Av \in V$$. Recall that a bra $$\langle v \|$$ is a map $$V \to \mathbb{C}$$, so the map $$\langle v \| A$$ is the map $$V \to \mathbb{C}$$ which first applies $$A$$, then applies $$\langle v\|$$. We get $$\langle v \| A \| w \rangle = \langle v \| (A (\| w \rangle)) = (v, Aw)_V = (\langle v\| A)(\| w\rangle)$$ i.e. the expression $$\langle v \| A \| w \rangle$$ is consistent with both these interpretations (of course it is, it's just composition of operators). So what is the adjoint of $$A$$? An interesting thing that happens because the inner product is conjugate-linear in the first argument is that if $$\|v\rangle + \lambda \|w \rangle$$ is a ket, its corresponding bra is $$\langle v \| + \lambda^* \langle w \|$$, where $$\lambda^*$$ means the complex conjugate. The adjoint is just the translation of this for operators: fix an operator $$A: V \to V$$. It can be shown that there exists an operator $$A^\dagger: V \to V$$ such that if $$A \| v \rangle$$ is a ket, then its corresponding bra is $$\langle v \| A^\dagger$$. This definition of adjoint was the first one you listed.	2019-08-25 12:06:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 63, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9686306118965149, "perplexity": 185.50007068640898}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027323328.16/warc/CC-MAIN-20190825105643-20190825131643-00537.warc.gz"}
http://mnwx.ihuq.pw/a-algorithm-c.html	# A Algorithm C An algorithm is a finite set of steps defining the solution of a particular problem. Panda seemed to crack down on thin content, content farms, sites with high ad-to-content ratios, and a number of other quality issues. Extended Description The use of a non-standard algorithm is dangerous because a determined attacker may be able to break the algorithm and compromise whatever data has been protected. , the ratio of the distance between A and C and the distance between A and B is u). Many C++11 features are used. Flowchart of an algorithm (Euclid's algorithm) for calculating the greatest common divisor (g. of Computer Science Garden City College-Bangalore. Uses elimination in order to cut down the running time substantially. Read more about C Programming Language. The one we will talk about is known as the Simple Genetic Algorithm and this one is fairly straightforward. The first line inside PathFinder. C program for drawing a circle using Midpoint Circle Algorithm /* Midpoint Circle Drawing Algorithm Created by: Pirate / #include #include #include Program to implement Deadlock Detection Algorithm in C. Bucket sort is a sorting algorithm that works by inserting the elements of the sorting array intro buckets, then, each bucket is sorted individually. The above applet demonstrates the circleSimple() algorithm. Algorithms is a featured book on Wikibooks because it contains substantial content, it is well-formatted, and the Wikibooks community has decided to feature it on the main page or in other places. It's also true that most people are afraid to open up the black box, and so they suffer from a complete lack of an appropriate diff algorithm, when actually it is quite easy to write a custom one that works on the problem at hand. An algorithm is a series of specific steps which solve a particular problem. Selection sort is notable for its programming simplicity and it can over perform other sorts in certain situations (see complexity analysis for more details). See execution policy for details. A is the most popular choice for pathfinding, because it's fairly flexible and can be used in a wide range of contexts. Like BFS, this famous graph searching algorithm is widely used in programming and problem solving, generally used to determine shortest tour in a weighted graph. My needs were complex (and I didn't want a C++ version), so I wrote this as generic as possible, and I'm releasing it for public consumption. The data structures that they can operate on include not only the C++ Standard Library container classes such as vector and list, but also program-defined data structures and arrays of elements that satisfy the requirements of a particular algorithm. A handful of algorithms are also in the header. Algorithm to Reverse String in C : Start Take 2 Subscript Variables 'i','j' 'j' is Positioned on Last Character 'i' is positioned on first character str[i] is interchanged with str[j] Increment 'i' Decrement 'j' If 'i' > 'j' then goto step 3 Stop Dry Run :. ; Andreas, A. Here the algorithms have been loosely translated into a real' programming language, i. C Programming: Data Structures and Algorithms is a ten week course, consisting of three hours per week lecture, plus assigned reading, weekly quizzes and five homework projects. The A* Algorithm # I will be focusing on the A* Algorithm [4]. Robin Friedman has written a Java class that implements the SpectralAlignment algorithm in section 8. Each data structure and each algorithm has costs and beneﬁts. Join over 5 million developers in solving code challenges on HackerRank, one of the best ways to prepare for programming interviews. I'm hoping to turn it into a book, but even in its incomplete state is should provide a good deal of useful algorithms for people working within the field of finance. The algorithm you use in C programming language is also the same algorithm you use in every other language. Data Structures and Algorithm Analysis in C++ is an advanced algorithms book that bridges the gap between traditional CS2 and Algorithms Analysis courses. Backpropagation Algorithm. p code} \noindent This algorithm is sometimes useful. 0003 degrees based on the date, time, and location on Earth. We use cookies to ensure you have the best browsing experience on our website. • The idea behind this model is that users will keep searching if they reach a dead end. 2 • In this graph, 1/4 of E’s page rank is distributed to pages A, B, C, and D. Extended Description The use of a non-standard algorithm is dangerous because a determined attacker may be able to break the algorithm and compromise whatever data has been protected. Learn how to apply them to optimize your C# developer skills and answer crucial interview questions. The reader might find it useful to work through an example by hand while following the discussion in this Appendix. My needs were complex (and I didn't want a C++ version), so I wrote this as generic as possible, and I'm releasing it for public consumption. Here the algorithms have been loosely translated into a real' programming language, i. In particular, it lacks implementations of many common data structures and algorithms. A major algorithm update hit sites hard, affecting up to 12% of search results (a number that came directly from Google). of Computer Science Garden City College-Bangalore. a set of mathematical instructions or rules that, especially if given to a computer, will help to calculate an answer to a problem: 2. Computer Programming - C++ Programming Language - Algorithms Sample Codes - Build a C++ Program with C++ Code Examples - Learn C++ Programming. I have the C++ equivalent on my shelf. In postfix or reverse polish notation, every operator follows all of its operands. It has similar properties and structure to DES with much smaller parameters. In April, 2011, MathWorks introduced MATLAB Coder as a stand-alone product to generate C code from MATLAB code. Read more: C. Tags for Bubble sort algorithm using function in C. As an example, for the array mentioned above - [5, 1, 4, 2, 3] we can see that 5 should not be on the left of 1 and so, we swap them to get: [1, 5, 4, 2, 3]. Evaluation of Postfix Expression in C [Algorithm and Program] Here you will get algorithm and program for evolution of postfix expression in C. The thing that you notice for the XOR operator is that x ^ k ^ k == x. net/problem/15596 15596번: 정수 N개의 합 정수 n개가 주어졌을 때, n개의 합을 구하는. We define complexity as a numerical function T(n) - time versus the input size n. Offers a variety of source code listings for the algorithm. A Linear Time Majority Vote Algorithm This algorithm, which Bob Boyer and I invented in 1980 decides which element of a sequence is in the majority, provided there is such an element. As a note: usually a good A* implementation does not use a standard ArrayList or List for the open nodes. Another reason would be to learn by example or hands-on experience. Use the directory structure of the repository. In a lot of cases, using STL algorithms in C++ code allows to make it more expressive. Creating a genetic algorithm for beginners Introduction A genetic algorithm (GA) is great for finding solutions to complex search problems. Given above is a sample source code for Dijkstra's algorithm, referred from Wikipedia. specific A step must NOT be replaced by a similar step. SCAN algorithm is sometimes called the elevator algorithm , since the disk arms behaves just like an elevator in a building, first servicing all the requests going. is ordered before) the second. Data Structures & Algorithm Analysis by Clifford A. Design goals. They're often used in fields such as engineering to create incredibly high quality products thanks to their ability to search a through a huge combination of parameters to find the best match. An algorithm is a procedure or step-by-step instruction for solving a problem. We've partnered with Dartmouth college professors Tom Cormen and Devin Balkcom to teach introductory computer science algorithms, including searching, sorting, recursion, and graph theory. It's an algorithm for to find sum of two user defined numbers. In other words, it is the problem of deciding which computer process. The authors describe the Columbia Classification Algorithm for Suicide Assessment (C-CASA), a standardized suicidal rating system that provided data for the pediatric suicidal risk analysis of antide-pressants conducted by the Food and Drug Administration (FDA). Divide 210 by 45, and get the result 4 with remainder 30, so 210=4·45+30. The first book, Parts 1-4. Christopher van Wyk and Sedgewick have developed concise new C++ implementations that both express the methods in a natural and direct manner and also can be used in real applications. A computer program can be viewed as an elaborate algorithm. And here is some test code: test_search. In the C++ Standard Library, algorithms are components that perform algorithmic operations on containers and other sequences. If you haven't a clue what I'm referring to, read on! When you hear the word "algorithm," you probably respond in one of three ways: You immediately know and understand what we're talking about because you studied computer science. , the ratio of the distance between A and C and the distance between A and B is u). Uses Divide and Conquer strategy. Click Start Search in the lower-right corner to start the animation. A great and clearly-presented tutorial on the concepts of association rules and the Apriori algorithm, and their roles in market basket analysis. The reader might find it useful to work through an example by hand while following the discussion in this Appendix. The words 'algorithm' and 'algorism' come from the name of a Persian mathematician called Al-Khwārizmī (Persian: خوارزمی‎‎, c. Ans: A union is a data type in c which allows the overlay of more than one variable in the same memory area. Concepts of Algorithm, Flow Chart & C Programming by Prof. sort, lower_bound, and. • The idea behind this model is that users will keep searching if they reach a dead end. The Boyer-Moore algorithm is considered as the most efficient string-matching algorithm in usual applications. The following algorithms vary in usefulness and functionality and are mainly intended as an example for learning how hash functions operate and what they basically look like in code form. The program output is also shown. On January 11th, 2018, Mark Zuckerberg announced that Facebook would be changing its News Feed algorithm to prioritize content from “friends, family and groups. You will need to compile test_search. A Star Algorithm C Code Codes and Scripts Downloads Free. Welcome to Algorithms & Artificial Intelligence section of C# Corner. The goal is, given a 2D-matrix of colors, to find all adjacent areas of the same color, similar to the flood-fill algorithm used in Paint. Submitted by Raunak Goswami, on August 12, 2018 In the last article, we discussed about the bubble sort with algorithm, flowchart and code. Algorithm Library \| C++ Magicians STL Algorithm For all those who aspire to excel in competitive programming, only having a knowledge about containers of STL is of less use till one is not aware what all STL has to offer. Algorithms is a featured book on Wikibooks because it contains substantial content, it is well-formatted, and the Wikibooks community has decided to feature it on the main page or in other places. Here the dimensions of matrices must be a power of 2. My needs were complex (and I didn't want a C++ version), so I wrote this as generic as possible, and I'm releasing it for public consumption. OK, we're just going to look at prefixes and we're going to show how we can express the length of the longest common subsequence of prefixes in terms of each other. Algorithms in c programming with examples advantages and disadvantages Please Like, share and subscribe: https://www. The name derives from the Latin translation, Algoritmi de numero Indorum, of the 9th-century Muslim mathematician al-Khwarizmi's arithmetic treatise "Al-Khwarizmi. Memory testing. algorithms comp. Aseries of inter-disciplinary papers suggests algorithms to add affective modulation to processing of information defined by probability distributions fit to real data. This new functionality is being branded as Smart Stop Out. The code is written in double precision FORTRAN 77. Not only are these algorithms simple and powerful, they were created to solve a more general modifications. This is the list of pending tasks. Apply this redistribution to every page in the graph. c (where we need it) when we include queue. C# example. The steps of the algorithm are as follows: Produce an initial generation of Genomes using a random number generator. The one we will talk about is known as the Simple Genetic Algorithm and this one is fairly straightforward. The Artificial Bee Colony (ABC) algorithm is a swarm based meta-heuristic algorithm that was introduced by Karaboga in 2005 (Karaboga, 2005) for optimizing numerical problems. algorithm analysis skills simultaneously so that they can develop such programs with the maximum amount of efficiency. c 0 12345678910 Requests Time 5 Faults 3 d Time page needed next Optimal Page Replacement Clairvoyant replacement Replace the page that won’t be needed for the longest time in the future c adbe babc d aaaaaaaaad bbbbbbbbb b Page c c c c c c c c c c F rames 0 1 2 a b c 0 12345678910 Requests Time 6 Faults •• 3 d dddde eeee e a = 7 b = 6 c. com/channel/UCKS3 videos. In a maze we find walls, a start point, and an end point. 0003 degrees based on the date, time, and location on Earth. Section 3 outlines the two-level hier-archical exact density calculation algorithm. Can an Algorithm Hire Better Than a Human? By Claire Cain Miller. The range searched is [first,last), which contains all the elements between first and last, including the element pointed by first but not the element pointed by last. I would prefer suggestions on how to improve the algorithm, or decrease run-time. The one we will talk about is known as the Simple Genetic Algorithm and this one is fairly straightforward. 5 algorithms to train a neural network By Alberto Quesada, Artelnics. Given a restricted range, there are often better algorithms. The union-find approach starts with a disjoint matrix. Acknowledgements Thanks to Klaus D. Stacks and queues are not the only options for the bucket in the generic search algorithm. algorithms comp. PCM is based on an non-uniform 8 bits quantization who is used for representing each sample took from an continuous (analog) signal. However, I will continue to call it A* because the implementation is the same and the game programming community does not distinguish A from A. Lawrence Philips' Metaphone Algorithm - Describes an algorithm which returns the rough approximation of how an English word sounds. Each algorithm has particular strengths and weaknesses and in many cases the best thing to do is just use the built-in sorting function qsort. Learn how to apply them to optimize your C# developer skills and answer crucial interview questions. Back before computers were a thing, around 1956, Edsger Dijkstra came up with a way to ﬁnd the shortest path within a graph whose edges were all non-negetive. Auditing Algorithms: Research Methods for Detecting Discrimination on Internet Platforms Christian Sandvig1, Kevin Hamilton2, Karrie Karahalios2, & Cedric Langbort2 Paper presented to “Data and Discrimination: Converting Critical Concerns into Productive Inquiry,” a preconference at the 64th Annual Meeting of the International Communication. ALGLIB package implements Levenberg-Marquardt algorithm in several programming languages, including our dual licensed (open source and commercial) flagship products:. 2 Algorithm Conventions # Ⓣ Ⓔ ① Ⓐ The specification often uses a numbered list to specify steps in an algorithm. An algorithm is a finite set of steps defining the solution of a particular problem. Introduction First we will consider a simple C++ character array: "This is an array. Algorithm Input : A number N to be factorized Output : A divisor of N If x mod 2 is 0 return 2 Choose random x and c y = x g = 1 while g=1 x = f(x) y = f(f(y)) g = gcd(x-y,N) return g Note that this algorithm may not find the factors and will return failure for composite n. (Also transform for not-in-place semantics. Greedy Algorithms. Unity is the ultimate game development platform. Evolutionary algorithms forms a family of algorithms inspired by the theory of evolution, that solve various problems. In general the clustering algorithms can be classified into two categories. SCAN algorithm is sometimes called the elevator algorithm , since the disk arms behaves just like an elevator in a building, first servicing all the requests going. With brute force we can always solve a maze (if it can be solved). The C++ Standard Library algorithms are generic because they can operate on a variety of data structures. However, some developers reported to me they had a hard time diffusing the usage of the STL in their companies, as their co-workers weren't always keen on putting the STL in their daily coding toolbox. Definition of algorithm: Step by step procedure designed to perform an operation, and which (like a map or flowchart) will lead to the sought result if followed. Sorting Algorithms¶ Sorting algorithms represent foundational knowledge that every computer scientist and IT professional should at least know at a basic level. Bucket sort is a sorting algorithm that works by inserting the elements of the sorting array intro buckets, then, each bucket is sorted individually. Graph search is a family of related algorithms. General depth-first search can be implemented using A* by considering that there is a global counter C initialized with a very large value. (Also transform for not-in-place semantics. The appropriate search algorithm often depends on the data structure being searched, and may also include prior knowledge about the data. An explanation and step through of how the algorithm works, as well as the source code for a C program which performs insertion sort. A* is the most popular choice for pathfinding, because it's fairly flexible and can be used in a wide range of contexts. Feynman Algorithm. This algorithm is referred to as a "sweep line algorithm" because it's operation can be visualized as having another line, a "sweep line" SL, sweeping over the collection and collecting information as it passes. 780–850) was a Persian mathematician, astronomer, geographer, and scholar in the House of Wisdom in Baghdad, whose name means 'the native of Khwarazm', a region that was part of. Mastering Algorithms with C offers you a unique combination of theoretical background and working code. The algorithm itself requires five parameters, each vectors. Weighted graphs may be either directed or undirected. You probably don't want to be here. 3 (108 ratings) Course Ratings are calculated from individual students’ ratings and a variety of other signals, like age of rating and reliability, to ensure that they reflect course quality fairly and accurately. a set of mathematical instructions or rules that, especially if given to a computer, will help to calculate an answer to a problem: 2. Using the Hamming distance, the number of puzzles considered dropped to 127643. Select Algorithm. Determine the fitness of all of the Genomes. Panda seemed to crack down on thin content, content farms, sites with high ad-to-content ratios, and a number of other quality issues. edu ABSTRACT In this paper, we propose an algorithm to quickly nd the maximum cycle ratio (MCR) on an incrementally changing directed cyclic graph. About the Book Preface \| Table of Contents. asa266, a library which evaluates various properties of the Dirichlet probability density function (PDF); this is a C version of Applied Statistics Algorithm 266; asa299 , a library which computes the lattice points (integer coordinates) in an M-dimensional simplex, by Chasalow and Brand; this is a version of Applied Statistics Algorithm 299;. This is a DataMining Tool developed by C# Just use Apirori Method to find the relation rules of data. Hiring C-Suite Executives by Algorithm. In a maze we find walls, a start point, and an end point. This code is an efficient implementation in C++ and C# of the A* algorithm, designed to be used from high performance realtime applications (video games) and with an optional fast memory allocation scheme. This Algorithm is used in programs written in c and C++ language, the program mainly used in image processing. 1 Memory Testing • Introduction • Memory Architecture & Fault Models • Test Algorithms • DC / AC / Dynamic Tests • Built-in Self Testing Schemes • Built-in Self Repair Schemes. Al-Khwārizmī (Arabic: الخوارزمي ‎, Persian: خوارزمی ‎, c. h header file (for C) and spa_tester. Recursive part(s) that call the same algorithm (i. Algorithm If the sizes of A and B are less than the threshold Compute C = AB using the traditional matrix multiplication algorithm. C program to print Memory Addresses of Variables. Constrained algorithms. Search engines use proprietary algorithms to display the most relevant results from their search index for specific queries. Bubble Sort in C is a sorting algorithm where we repeatedly iterate through the array and swap adjacent elements that are unordered. 01: Charter of comp. A Fast Implementation of the ISODATA Clustering Algorithm. The C++ Standard Library algorithms are generic because they can operate on a variety of data structures. Chap 5： At the beginning, he mentioned that: recursion is a divide-and-conquer method. Booth's Algorithm. Back before computers were a thing, around 1956, Edsger Dijkstra came up with a way to ﬁnd the shortest path within a graph whose edges were all non-negetive. Java Algorithms and Clients. ) The algorithm splits the array into two parts: the right side of the array (in black) is the shuffled section, while the left side of the array (in gray) contains elements remaining to be shuffled. Data Structures & Algorithm Analysis by Clifford A. Algorithms in C++ In C++, the designation identifies a group of functions that run on a designated range of elements. Standard C++ Algorithms. Algorithms in C++, Third Edition, Part 5: Graph Algorithms is the second book in Sedgewick's thoroughly revised and rewritten series. Here is the source code of the C program to multiply 22 matrices using Strassen’s algorithm. Stacks and queues are not the only options for the bucket in the generic search algorithm. AA+ is a C++ implementation for the algorithms as presented in the book "Astronomical Algorithms" by Jean Meeus. Al-Khwārizmī (Arabic: الخوارزمي ‎, Persian: خوارزمی ‎, c. This algorithm has a big performance advantage since it does not need to visit as many nodes when the direction of the path end is known. Design an algorithm, draw a corresponding flowchart and write a C program to check whether a given string is a palindrome or not. C# Maze Pathfinding Algorithm This C# example program uses pathfinding logic to go from a start to end point in a maze. Flexible Data Ingestion. However, taking this idea as an inspiration, we managed to design a new algorithm that fixes both flaws. We have a vacancy for a Associate professor/Professor at the department of computer science. Hope that this will help to understand the concept Monoalphabetic cipher algorithm. The Fedorov and. We implement the algorithm in C++ and provide a demonstration program for Microsoft Windows. Posted on August 20, 2019 by mac. Algorithms give programs a set of instructions to perform a task. Some database structures are specially constructed to make search algorithms faster or more efficient, such as a search tree, hash map, or a database index. Constrained algorithms. Jun 25, 2015; Hiring and recruiting might seem like some of the least likely jobs to be automated. The goal is, given a 2D-matrix of colors, to find all adjacent areas of the same color, similar to the flood-fill algorithm used in Paint. If you haven't a clue what I'm referring to, read on! When you hear the word "algorithm," you probably respond in one of three ways: You immediately know and understand what we're talking about because you studied computer science. Acknowledgements Thanks to Klaus D. Algorithm Game 8puzzle with A Algorithm. Now, any page that has a viewstate is abending with the following error: This implementation is not part of the Windows Platform FIPS validated cryptographic algorithms. For writing any programs, the following has to be known: For any task, the instructions given to a friend is different from the instructions given to a computer. The Euclidean algorithm, also called Euclid's algorithm, is an algorithm for finding the greatest common divisor of two numbers a and b. This is a quick summary of some of the most useful algorithms in the Standard Template Library. New algorithms are being designed all the time, but you can start with the algorithms that have proven to be reliable in the C++ programming language. The C++ Standard Library algorithms are generic because they can operate on a variety of data structures. C Programming: Data Structures and Algorithms is a ten week course, consisting of three hours per week lecture, plus assigned reading, weekly quizzes and five homework projects. Quicksort is faster in practice than other O(n log n) algorithms such as Bubble sort or Insertion Sort. A* is a generic algorithm and there are no perfect parameters to be set. , Solar Position Algorithm for Solar Radiation Applications, Solar Energy. The Levenberg-Marquardt algorithm (LM, LMA, LevMar) is a widely used method of solving nonlinear least squares problems. Jun 25, 2015; Hiring and recruiting might seem like some of the least likely jobs to be automated. ID3 Stands for Iterative Dichotomiser 3. This algorithm will perform a sequential search of item in given array. Imagine you have sales and stock data for eight markets. That hash algorithm, when it is used as first step of a signature generation or verification algorithm, will be called "signature hash algorithm". c = k ⋅ gcd(a, b), where k is an integer. In this current article, we’ll present the fuzzy c-means clustering algorithm, which is very similar to the k-means algorithm and the aim is to minimize the objective function defined as follow: \sum\limits_{j=1}^k \sum\limits_{x_i \in C_j} u_{ij}^m (x_i - \mu_j)^2. Consider Dijkstra's Algorithm: the frontier has priorities p through p+k where k is the largest. For example building a car is a major problem and no-one knows how tomake every single part of a car. Algorithm definition is - a procedure for solving a mathematical problem (as of finding the greatest common divisor) in a finite number of steps that frequently involves repetition of an operation; broadly : a step-by-step procedure for solving a problem or accomplishing some end. Selection sort is notable for its programming simplicity and it can over perform other sorts in certain situations (see complexity analysis for more details). A* is like Dijkstra's Algorithm in that it can be used to find a shortest path. Algorithms for Procedural Content Generation. About the Book Preface \| Table of Contents. The algorithms and data structures are expressed in concise implementations in C, so that you can both appreciate their fundamental properties and test them on real applications. 5, and the experimental tag was removed in 15. c) for improving both time and space complexity d) for making code simpler View Answer. Sorting Algorithms in C programming is vast topic and often used in most common interview questions to check the logic building aptitude. We define ‘ g ’ and ‘ h ’ as simply as possible below g = the movement cost to move from the starting point to a given square on the grid, following the path generated to get there. 67 videos Play all C programming language Education 4u How to: Work at Google — Example Coding/Engineering Interview - Duration: 24:02. Now let us consider an example so that the algorithm can be clearly understood. Sorting in general refers to ordering things based on criteria like numerical, chronological, alphabetical, hierarchical etc. Uses elimination in order to cut down the running time substantially. on Pattern Analysis and Machine Intelligence, Vol. C Program to demonstrate use of null pointer. Similarly to Multiply two numbers. But there are other, less intuitive factors to the algorithm. A Star Algorithm C Code Codes and Scripts Downloads Free. As the speed and power of computers increases, so does the need for effective programming and algorithm analysis. 1 Memory Testing • Introduction • Memory Architecture & Fault Models • Test Algorithms • DC / AC / Dynamic Tests • Built-in Self Testing Schemes • Built-in Self Repair Schemes. Mastering Algorithms with C offers you a unique combination of theoretical background and working code. Chap 5： At the beginning, he mentioned that: recursion is a divide-and-conquer method. In particular, we're going to define c of ij to be the length, the longest common subsequence of the prefix of x going from one to i, and y of going to one to j. The algorithms library defines functions for a variety of purposes (e. Sorting Algorithms¶ Sorting algorithms represent foundational knowledge that every computer scientist and IT professional should at least know at a basic level. An explanation and step through of how the algorithm works, as well as the source code for a C program which performs insertion sort. If you need to first develop a fundamental knowledge of C++, watch this excellent video on C++. For example building a car is a major problem and no-one knows how tomake every single part of a car. The vector from A to B is B - A. 3 (108 ratings) Course Ratings are calculated from individual students’ ratings and a variety of other signals, like age of rating and reliability, to ensure that they reflect course quality fairly and accurately. MSVC first added experimental support for some algorithms in 15. We define complexity as a numerical function T(n) - time versus the input size n. The circle approximation generated by the algorithm is overlaid with an ideal circle for comparison. One of the simplest algorithms for this task works as follows: first do this, then do that. The Tridiagonal Matrix Algorithm, also known as the Thomas Algorithm, is an application of gaussian elimination to a banded matrix. April 1992 The MD5 Message-Digest Algorithm Status of this Memo This memo provides information for the Internet community. Kruskal's algorithm is a greedy algorithm in graph theory that finds a minimum spanning tree for a connected weighted graph. a set of mathematical instructions or rules that, especially if given to a computer, will help to calculate an answer to a problem: 2. You will need to compile test_search. Because the dimension of the solution space of this problem is too big, it is solved using a specific kind of genetic algorithm called Evolutionary Pursuit (EP). The first book, Parts 1-4. String b a c b a b a b a b a c a a b Pattern a b a b a c a Let us execute the KMP algorithm to ﬁnd whether ‘p’ occurs in ‘S’. If the source of randomness is predictable to an attacker, then they can figure out the private key. Another reason would be to learn by example or hands-on experience. An algorithm is a sequence of deterministic steps that results in something useful being done. Well, let’s begin our lesson “How square root algorithm is working on c” It’s not very difficult once you have understand how following equation is calculated. Dijkstra’s Algorithm finds the shortest path with the lower cost in a Graph. Not all SLAM algorithms fit any kind of observation (sensor data) and produce any map type. Search engines use proprietary algorithms to display the most relevant results from their search index for specific queries. And yet, for all its power, Facebook’s news feed algorithm is surprisingly inelegant, maddeningly mercurial, and stubbornly opaque. A* Search Algorithm. It's an algorithm for to find sum of two user defined numbers. C Algorithms The C Programming Language has a much smaller standard library compared to other more modern programming languages such as Java or Python. Some algorithms are designed to modify the collection. The algorithms library defines functions for a variety of purposes (e. The steps of the algorithm are as follows: Produce an initial generation of Genomes using a random number generator. A* is the most popular choice for pathfinding, because it's fairly flexible and can be used in a wide range of contexts. c together with both search. This algorithm is available in ANSI C and CRBasic; how to use this algorithm and a description of the variables is included in the spa. This requires an understanding of the principles of algorithm analysis, and also an appreciation for the signiﬁcant. What is First Come First Serve Disk Scheduling Algorithm? The first come first serve algorithm is commonly abbreviated as FCFS algorithm. Introduction First we will consider a simple C++ character array: "This is an array. A* Search Algorithm. What does strcpy(x+1, SEQX) do? c,strcpy. The Containers are objects that store data. The algorithm also assumes that content that has attracted a lot of engagement has wide appeal and will place it in more people’s feeds. In particular, we're going to define c of ij to be the length, the longest common subsequence of the prefix of x going from one to i, and y of going to one to j. It is the part of software designing. Algorithmic complexity is concerned about how fast or slow particular algorithm performs. graph algorithms ~500 pages Third Edition 1-4 basic/ADTs/sort/search ~700 pages Second Edition ~650 pages Algorithms ~550 pages Java C++ C Java C++ C Modula-3 C++ C Pascal 1982 Pascal 2003 2001 2001 2002 1998 1997 1993 1992 1990 1988 Brief history of books Translations: Japanese, French, German, Spanish, Italian, Polish, Russian. Else If b>c Display b is the largest number. Else Display c is the largest number. Imagine you have sales and stock data for eight markets. • The idea behind this model is that users will keep searching if they reach a dead end. The following algorithms vary in usefulness and functionality and are mainly intended as an example for learning how hash functions operate and what they basically look like in code form. construct the orthogonal basis elements one by one. Parameters first, last Input iterators to the initial and final positions in a sequence. It was inspired by the intelligent foraging behavior of honey bees. Ever played the Kevin Bacon game? This class will show you how it works by giving you an introduction to the design and analysis of algorithms, enabling you to discover how individuals are connected. C# Maze Pathfinding Algorithm This C# example program uses pathfinding logic to go from a start to end point in a maze. Fibonacci Series C Program Pascal's Triangle Algorithm/Flowchart Tower of Hanoi Algorithm/Flowchart. We have a vacancy for a Associate professor/Professor at the department of computer science. The second basis vector must be orthogonal to the ﬁrst: v2 · v1 = 0. Some commonly used stream cipher algorithms are RC4 and W7. The generic algorithms use iterators just as you use pointers in C to get elements from and store elements to various containers. Note that a range is defined as [first, last) where last refers to the element past the last element to inspect or modify. The textbook An Introduction to the Analysis of Algorithms by Robert Sedgewick and Phillipe Flajolet overviews the primary techniques used in the mathematical analysis of algorithms. Algorithms and data structures in C/C++ Data Structures All programmers should know something about basic data structures like stacks, queues and heaps. And yet, for all its power, Facebook’s news feed algorithm is surprisingly inelegant, maddeningly mercurial, and stubbornly opaque. That hash algorithm, when it is used as first step of a signature generation or verification algorithm, will be called "signature hash algorithm". Extended Mo's Algorithm with ≈ O(1) time complexity; Dijkstra's Shortest Path Algorithm using priority_queue of STL; Spanning Tree With Maximum Degree (Using Kruskal's Algorithm) library in C++ STL library in C++ STL library in C++ STL library in C++ STL; The C++ Standard Template Library (STL). 0! I wish there were some easy way of sharing those kind of things, since often such algorithms are extremely handy to have available, but there's nevertheless often no easy way of combining them without reimplementing large sections (just to remove type issues and to address data structure incompatibilities). OK, we're just going to look at prefixes and we're going to show how we can express the length of the longest common subsequence of prefixes in terms of each other. by Michael T. Witzel for providing Example 7.	2019-12-07 02:29:03	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.35473302006721497, "perplexity": 1056.99054179015}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540491871.35/warc/CC-MAIN-20191207005439-20191207033439-00048.warc.gz"}
https://www.physicsforums.com/threads/wavefunctions-space-in-quantum-physics.56537/	# Wavefunctions space in Quantum Physics 1. Dec 13, 2004 ### Palindrom Well, a week ago my Professor said the space of quantum physical states was a Hilbert space. Thing is, he just said it, and moved on. So I have a vector space with a scalar product. Is it, indeed, Hilbert? That is, is it complete? I guess I'll see the answer next semester, in real functions analysis or sth., but I'm kinda curious, so thanks in advance for your answers. 2. Dec 13, 2004 ### dextercioby A finite/infinite dimensional vector space on the vectors of which u define an application from the direct product of the space with itself into the body of scalars on which you build the vector space is called a preHilbert space. The scalar product is a sesquilinear application which can allow to structure the vector space as a topological vector space,on which u can define a metric and a norm Completion wrt to the norm of a preHilbert space defines a Hilbert space. 3. Dec 13, 2004 ### Gokul43201 Staff Emeritus This is correct, but more specifically, it is an infinite dimensional Hilbert space. It is a Hilbert Space, only if it is complete with respect to the norm. So, to be a Hilbert Space, it must first be a Banach Space. A common misconception is that all Hilbert Spaces are infinite dimensional. This is not true. For instance, any n-dimensional Euclidean Space with the usual dot product is a Hilbert Space. 4. Dec 13, 2004 ### RedX Well, you toss out solutions which don't belong to the Hilbert space, so it'll be complete. I guess the world only allows solutions in the Hilbert space. Also normalization may not be to unity, but the dirac-delta function. 5. Dec 13, 2004 ### Palindrom First of all, thanks to you all. Now, another question: I was told today that it is also separable. Is L_2 indeed separable? Can you show me the densed group in it? 6. Dec 13, 2004 ### dextercioby This is tricky.Requires that bunch of functional analysis which i find horrible.I myself,as a physicist,will never attempt to give demonstrations on delicate matters of topology.$L^{2} (R^{n})$ is separable,and the proof is to be found in a serious book on functional analysis.I don't know the poof,and i'm not interested in why $L^{2} (R^{n})$ is separable.It's important for QM that it is separable.A mathematician proved it.I have a hunch it was von Neumann,but i'm not sure.And i won't look for it,as this is just one from the many more other results from mathematics that a physicst uses,but it would not help him a lot to find "why is that??"."Why" is a question for mathematicians.If we physicist would have to come up with the proofs for every mathematical result he uses,then mathematicians would be useless,as we physicists would be making the mathematics. Daniel. PS.If u're really interested,maybe one of the mathematicians on this forum would pin point to a book where u can find the proof,that,of course,if he does not give you the proof in a post in this thread.I think,among the physics books that i know of,u have a chance of finding it in Prugoveçki's book/bible:"Quantum Mechanics in Hilbert Space". PPS:Dense subset,not subgroup. 7. Dec 13, 2004 ### Loren Booda Should it be noted that Hilbert space is complex? 8. Dec 13, 2004 ### dextercioby It's irrelevant whether it's real or complex.It could be quternions,even. It's just a body of scalars... Daniel. 9. Dec 14, 2004 ### humanino They say field in english. We too in France say body. There is no real confusion possible between the "field" algebraic structure and the physicist's field which is a function, or a generalization of it. 10. Dec 14, 2004 ### dextercioby Thank you!!!!!!!!!! This was a really useful post from u.Useful to me,as i had wondered for about 2 months on how to say in English "corp",else than "body". Daniel. 11. Dec 14, 2004 ### humanino It is sufficient to show that there is a countable basis, that is : that the dimension is countable. For instance, the set $$\left\{v_n : n \in \mathbb{Z}\right\}$$ with $$v_n(x) = \exp^{2\pi \imath nx}$$ forms an orthonormal basis of the complex space $$L^2([0,1])$$. From Wikepdia : 12. Dec 14, 2004 ### Palindrom Merci And thanks a lot, dex.	2017-06-27 17:41:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7939466834068298, "perplexity": 1077.1094114086668}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128321497.77/warc/CC-MAIN-20170627170831-20170627190831-00294.warc.gz"}
https://www.cs.cmu.edu/~csd-phd-blog/2022/casestobeds/	This blog post is adapted from the Delphi blog, originally published on March 10th, 2021. Again, thank you to the Allegheny County Health Department, the DELPHI Group, Chris Scott, and Roni Rosenfeld. One of the Delphi Group’s goals is to create informative tools for healthcare organizations. Tools are only useful if the insights they provide can inform concrete actions. That is to say these tools must provide actionable intelligence. In early November 2020, as COVID case rates in Allegheny County continued to rise, the Delphi Group partnered with the Allegheny County Health Department (ACHD) to create such tools for investigating if hospitals located in the county would run out of hospital beds for COVID patients (Fig. 1). Fig. 1: Hospitalizations Due to COVID-19 and New Cases from Positive PCR Tests in Allegheny County (WPRDC Data 1) Based on its planning, the ACHD needed at least a week to open emergency COVID facilities. If the emergency space wasn’t open and hospital beds ran out, mortality rates could soar. But, if we didn’t need the facility, that decision would have stretched already thin resources. Many of the hospitals in Allegheny County were in contact, but each hospital system only had visibility into its own facilities. We wanted to offer a more holistic picture of hospital resources for ACHD to assist in its planning. ## A Probabilistic Approach To provide county-level intelligence on hospital bed usage, we developed Cases2Beds2. To extrapolate beds utilization 1-2 weeks in the future, we needed to estimate: 1. The probability that a person who tested positive for COVID-19 would require hospitalization 2. How many days after testing a person would be hospitalized 3. How long a person with COVID would stay in the hospital 4. The current number of positive COVID tests These values vary by demographic factors, most notably age (Fig. 2), and to a lesser extent, sex and race. Fig. 2: Age Group Comparisons based on the Allegheny County COVID-19 Tableau 3 We used public data from Allegheny County about the number of people tested, test positivity rate, and hospitalization rate, broken down by the aforementioned demographic factors. We also acquired information for two critical parameters: • Offset: Offset is the number of days between the day of testing (called specimen collection date) and the first day of hospitalization. For example, if the test date were 5 days before hospitalization, the offset would be 5 days. Also, if the test date is the hospital admit date, the offset would be 0 days (or sometimes, if, for example, they are admitted at midnight, -1 or +1 days). Notably, the offset can be negative, meaning a person may have been tested some days or weeks after admission. • Length of Stay: The length of stay is approximately how many days a person uses a bed in the hospital (± 1 day). Given the hospitalization rate, the offset distribution, and the length of stay distribution, we can simulate multiple futures for any given set of positive cases and their testing dates. Estimating the future given a set of probabilities is a common problem and a possible approach is called a Monte Carlo simulation. This process ultimately shows the expected distribution of the number of beds needed each day. Monte Carlo simulations involve running a large number of scenarios based on a set of probabilities. The more scenarios run, the more accurate the model tends to be. For example, if you gave 1000 people one dart to throw at a dartboard, even though each throw may not be very good, you’d still be able to get a pretty good idea of where the bull’s eye is after 1000 throws. This is the same principle we applied for Cases2Beds – after many simulations, we had a good idea of how many beds might be needed in the next two weeks. Our prototype Monte Carlo simulation was written in Python and had a runtime of a few minutes. However, because the simulation works best with probabilities derived from Protected Health Information (PHI), ACHD needed to run it privately and offline so there would be no data transmission. Thus, any type of web application (which would transmit data to our servers) was ruled out. Even asking ACHD to run our Python software on their machines fell into a grey area. However, Microsoft Excel was easy to use and supported by ACHD. So we converted Cases2Beds into a spreadsheet. It is relatively straightforward to port the Python application to VBScript macros for Microsoft Excel. However, those macros aren’t designed to run large simulations, and we saw that the time required to generate a model was far worse, bordering on unusable. ## An Alternative to Monte Carlo: the Analytical Model As an alternative, we developed an analytical model for Microsoft Excel that offered a much faster run time than the full Monte Carlo simulation. The sheet has two tabs of inputs: constant parameters (first tab, static), and case counts (second tab, dynamic). The analytical model had the same idea as the Monte Carlo simulation. Some fraction of individuals who test positive today will be hospitalized after a varying offset (from test date to admit date) and variable duration (from admit date to discharge date) based on their characteristics (see appendix). Because these parameters can vary by region, anyone can change these values in spreadsheet tab 1. The characteristics are: 1. Age Group: (Most important) [unspecified, 0-9, 10-19, 20-29 … 70-79, 80+] 2. Sex: [unspecified, M, F] 3. Race: [unspecified, Black, White, Hispanic, Asian] And the parameters are: 1. Hospitalization Rate 2. Offset Distribution Parameter Set: Parameters describing the number of days before someone who tests positive is hospitalized 3. Duration Distribution Parameter Set: Parameters describing the number of days someone will be in the hospital The second types of inputs are the daily positive cases split by their traits. This is the input that the user actively changes on their end. Behind the scenes, we take these parameters (first input tab) and generate Offset Fractions, which is the probability that a patient with given traits will occupy a bed for a duration k days after the specimen testing date. These Offset Fractions and the daily positive case breakdown (second input) give us the expected mean and variance up to 1 month in the future of the number of patients in the hospital per day based on the cases already seen (for details, see appendix). This information can be used to generate plots like (Fig. 3). This graph isn’t to suggest that there won’t be any need for beds after February! It is just that based on the cases we know, very few people will be hospitalized for more than a month. Fig. 3: Output of Cases2Beds using historical data until January 21st for Allegheny County Using Public Parameters If we assume independence between patients, the mean and variance calculations are exact. However, our quantile estimates are based on approximating the sum of independent binary variables, which is inaccurate near the tails. So the accuracy of the more extreme quantiles (95%+) depends on the number of cases present, which in practice makes them less trustworthy. ## Cases2Beds in Action By the end of November 2020, we had a viable prototype Cases2Beds spreadsheet used by ACHD. Over the following months, we made various modifications with their feedback. For example, the ACHD staff did not have time to manually input case numbers. So, we were able to use the granular public data to give them estimates of future hospital utilization without any additional work on their end. At the peak of bed utilization, hospital systems themselves increased their COVID beds utilization to 6x more than in October 2020. Fortunately, in Allegheny County, we never reached a point where demand for beds exceeded a somewhat elastic supply. In early January 2021, multiple organizations told us that the pandemic’s most acute problem had changed to vaccine distribution and the number of COVID-19 beds needed dropped. Cases2Beds continues to act as an early warning system if the number of cases rise quickly. Fig. 4: Numbers of staffed COVID beds over time vs. capacity from the HHS Protect Data 5. We were also able to show the efficacy of the spreadsheet to other health departments and hospitals by generating tailored, public parameters for offset and length of stay from different national public resources, like the Florida line-level COVID dataset 4. Based on these organizations’ feedback that they needed projections more than 2 weeks out, we started to use Cases2Beds as an input to hospital utilization forecasting models. Other inputs to the hospital forecasting model included current hospital bed utilization (from HHS Protect5), how long current patients are likely to continue to be hospitalized, and how many new cases there will be in the near future. A preliminary evaluation of such a method shows decent predictive power when parameters are tailored to a location. ## Conclusion Cases2Beds was a case study about the realities of research institutions offering actionable intelligence in healthcare. While the Cases2Beds tool demonstrated reasonable predictive power, it was difficult to deploy it in a timely and actionable way. Our most significant challenges were data access and bureaucratic limitations to develop solutions at the granularity needed. Research institutions can be effective partners to health organizations, but the next set of challenges of this pandemic–or the next–will require quick action. The tools we build now can set the stage for the future. Thank you to the Allegheny County Health Department (especially Antony Gnalian, Dr. LuAnn Brink, and Dr. Debra Bogen) for their invaluable feedback, efforts, and shared interest in actionable intelligence. Many members of the Delphi Group, including Sumit Agrawal, Katie Mazaitis, and Phil McGuinness, met regularly with the Allegheny County Health Department, provided data, edited this blog post, and investigated various solutions other than Cases2Beds. ## Resources Please check out the Cases2Beds Github Repo ## Appendix To generate the Offset Fractions (OF(k\|traits)), which is the probability a patient with given traits will occupy a bed on k days after the specimen testing date, we follow Alg 1. For a given set of traits, the Offset Fractions for day k, where k is between -10 and 31, is the sum of the offset * distribution probabilities * hospitalization rate that sum up to day k. From these Offset Fractions, the mean/var of bed occupancy on a given day is given in Alg 2. for o in (-10, 30): #This is the offset for d in (0, 40): #This is the duration of the stay for k in (o, o+d): if (k<31): OF(k\|traits) += Offset(o\|traits) * Duration(d\|traits) * Hospitalization(traits) Alg 1: Generate Occupancy Fractions for a given set of traits for specimen_date and num_cases in case_inputs: for t in (-10, 30): p = OF(t\|traits) beds_mean(spec_date + t) += num_cases * p beds_var(spec_date + t) += num_casesp(1-p) Alg 2: Generate Mean and Variances High-level Mathematical Formulation of the Model: Or,l: The offset value for a given subset of the population r R where R := {race}x{gender}x{age group} for a given day l where -10 l 30. This pdf is derived from a piecewise function using segments of exponential distributions characterized by the offset parameters. Dr,k: The duration value for a given subset of the population r R for a given day k where 0 k 40. This pdf is derived from a piecewise function using segments of exponential distributions characterized by the duration parameters. hr: The hospitalization rate for a given subset of the population r R where 0 hr 1 cr,d: The number of cases for a given subset of the population r R on a particular specimen collection date d (ex: 5 cases with specimen collected on January 1st 2021). $$OF_{r, j} = \sum_{l=-10}^{30} \sum_{k=0}^{40} \mathbb{I} ( l \leq j \leq l+k ) O_{r, l} * D_{r, k}h_r$$ The offset fraction for a given subset of the population r R for a given delta j where -10 j 30. $$\mathbb{E}[\beta_i] = \sum_{d \in D}\sum_{r \in R}\sum_{j = -10}^{30} \mathbb{I} ( d+j = i) OF_{r, j}c_{r, d}$$ The expected number of beds on date i, where i can start 10 days before the first case date and can end 30 days after the the last case date (cr,d)	2022-10-01 17:42:29	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3078191578388214, "perplexity": 2323.28574069559}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030336880.89/warc/CC-MAIN-20221001163826-20221001193826-00072.warc.gz"}
http://blog.panictank.net/the-algorithm-design-manual-chapter-2/	# The Algorithm Design Manual: Chapter 2 2-1. What value is returned by the following function? Express your answer as a function of n. Give the worst-case running time using the Big Oh notation. Solution: To find out what value is returned just represent this function mathematically and simplify it. $\sum_{i=1}^{n-1}\sum_{j=i+1}^n\sum{}_{k=1}^{j} 1$ $= \sum_{i=1}^{n-1}\sum_{j=i+1}^n j$ $= \sum_{i=1}^{n-1} (\frac{n(n+1)}{2} - \frac{i(i+1)}{2})$ $= \frac{1}{2} \sum_{i=1}^{n-1} (n^2 + n - i^2 - i)$ $= \frac{1}{2} ( (n-1)n^2 + (n-1)n - \sum_{i=1}^{n-1} i^2 - \frac{(n-1)n}{2})$ $= \frac{1}{2} { ( n^3 - n) - (\frac{n(n+1)(2n+1)}{6} - n^2) - \frac{n^2 -n}{2} }$ $= \frac{1}{2} { \frac{1}{6} * (6n^3 - 6n - (2n^3+n^2+2n^2+n-6n^2) - 3n^2 +3n) }$ $= \frac{1}{12} { 6n^3 - 6n - 2n^3 - n^2 - 2n^2 - n + 6n^2 - 3n^2 + 3n }$ $= \frac{1}{12} (4n^3 - 4n) = \frac{n^3 - n}{3}$ The complexity is $O(n^3).$ 2-10. Prove that $n^3 - 3n^2 - n + 1 = \Theta(n^3).$ Solution: $O(n^3)$: For $c > 1: n^3 - 3n^2 - n + 1 \leq c \cdot n^3.$ $Omega(n^3)$: For $0 \leq c \leq 1: n^3 - 3n^2 - n + 1 \geq c \cdot n^3.$ 2-34. Assume that Christmas has n days. Exactly how many presents did my “true love” send me? (Do some research if you do not understand this question.) Solution: I made this table of the first three days: We can break this down into sub steps. On the ith day we get $p_i = \sum_{k=1}^{i} k$ presents. The total amount of presents is: $\sum_{i=1}^{n} p_i = \sum_{i=1}^{n} \sum_{k=1}^{i} k = \sum_{i=1}^{n} \frac{i^2+i}{2}$ We can be simplified as: $= \frac{1}{2} { \sum_{i=1}^{n} i^2 + \sum_{i=1}^{n} i } = \frac{1}{2} { \frac{n(n+1)(2n+1)}{6} + \frac{3n^2 + 3n)}{6} }$ $= \frac{1}{12} { 2n^3+3n^2+n + 3n^2 + 3n}$ $= \frac{2n^3 + 6n^2 + 4n}{12} = \frac{n^3+3n^2+2n}{6}$ 2-39. Prove the following identities on \logarithms: (a) Prove that $\log_a(xy) = \log_a x + \log_a y$ (b) Prove that $\log_a x^y = y \log_a x$ (c) Prove that $\log x = \frac{\log_b x}{\log_b a}$ (d) Prove that $x^{\log_b y} = y^{\log_b x}$ Solution: (a) The first proof is straight forward: $a^{\log_a(x) + \log_a(y)} = a^{\log_a(x)} \cdot a^{\log_a(y)} = x \cdot y = a^{\log_a(xy)}$ (b) The trick here is to see that $x^y = \prod_{i=1}^{y} x$. Therefore we can use the identity from (a): $\log_a x^y = \log_a (\prod_{i=1}^{y} x) = \sum_{i=1}^{y} \log_a x = y \log_a x$ (c) Here you try to form around a variable (z) to get the right term: $\log_a x = z \Leftrightarrow a^{\log_a x} = a^z \Leftrightarrow x = a^z$ $\log_b x = \log_b a^z \Leftrightarrow \log_b x = z \log_b a$ $\frac{\log_b x}{\log_b a} = z = \log_a x$ (d) The last one is quite easy. $x^{\log_b y} = y^{\log_b x}.$ Now we have just to $\log_b$ on the equation and use the identity from (b) and we get: $\log_b y \cdot \log_b x = \log_b x \cdot \log_b y$ 2-44. We have 1,000 data items to store on 1,000 nodes. Each node can store copies of exactly three different items. Propose a replication scheme to minimize data loss as nodes fail. What is the expected number of data entries that get lost when three random nodes fail? Solution: My first idea was a kind of a binary tree which then evolved into this: The idea is to save on each node the value of the corresponding left and right nodes and of course the main value. In the last nodes we got some free space were we can save item10 because it isn’t backed up yet. So what happens if three nodes fall out? There are various scenarios. 1. 3 corresponding nodes fall out, e.g. Node 1, 2 and 3. Then Item 11 and 12 are completely lost. (Remember Item10 is saved further down again) 2. A node and its parent fall out, e.g. Node 1 and 2. Here we just lose Item 11. 3. A random node falls out, e.g. Node 3. No problem whatsoever. We can retrieve Item12 from Node 1. We got about 500 free storages in the last row of nodes which could be used to backup half of the total items again. Which would reduce our loss further 2-46. You have a 100-story building and a couple of marbles. You must identify the lowest floor for which a marble will break if you drop it from this floor. How fast can you find this floor if you are given an infinite supply of marbles? What if you have only two marbles? Solution: The first case with infinite supply of marbles is very easy. We just do a binary search on the story building, i.e. we need about 7 marbles. The second case is a bit more demanding. I would start to try to minimize the possible interval as much as possible by starting with a marble in the middle of the whole interval, i.e. at the 50th story. If it’s broken we have to work our way up from 1 to at max 50. If it’s still alive we can cut the next interval into half till we find our floor. 2-50. A Ramanujam number can be written two different ways as the sum of two cubes—i.e. , there exist distinct a, b, c, and d such that $a^3 +b^3 = c^3 +d^3$. Generate all Ramanujam numbers where a, b, c, d < n. Solution: The DP approach works must faster than raw BF which you can see quite fast because complexity of BF is $O(n^4)$ and DP only takes $O(n^3).$ 1. Anonymous 2-44 you must prove it mathematically why having 3 copies in 3 different random buckets is the best solution. forget about the implementation. the question is not how you would implement it.	2018-02-19 03:34:36	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 36, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6870783567428589, "perplexity": 575.255495560459}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891812327.1/warc/CC-MAIN-20180219032249-20180219052249-00329.warc.gz"}
https://www.hackmath.net/en/math-problem/3315	# The cylinder base The cylinder with a base of 8 dm2 has a volume of 120 liters. From a cylinder fully filled with water, 40 liters of water was removed. At what height from the bottom /with precision to dm/ is the water level? Result x = 10 dm #### Solution: $S = 8 \ \\ V = 120 - 40 = 80 \ \\ x = V/S = 80/8 = 10 = 10 \ \text{ dm }$ Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you! Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...): Be the first to comment! Tips to related online calculators Tip: Our volume units converter will help you with the conversion of volume units. ## Next similar math problems: 1. Container The container has a cylindrical shape the base diameter 0.8 m and the area of the base is equal to the area of the wall. How many liters of water can we pour into the container? 2. Rotary cylinder 2 Base circumference of the rotary cylinder has same length as its height. What is the surface area of cylinder if its volume is 250 dm3? 3. Height as diameter of base The rotary cylinder has a height equal to the base diameter and the surface of 471 cm2. Calculate the volume of a cylinder. 4. Cylinder surface area Volume of a cylinder whose height is equal to the radius of the base is 678.5 dm3. Calculate its surface area. 5. Garden pond Concrete garden pond has bottom shape of a semicircle with a diameter 1.7 m and is 79 cm deep. Daddy wants make it surface. How many liters of water is in pond if watel level is 28 cm? 6. Total displacement Calculate total displacement of the 4-cylinder engine with the diameter of the piston bore B = 6.6 cm and stroke S=2.4 cm of the piston. Help: the crankshaft makes one revolution while the piston moves from the top of the cylinder to the bottom and back 7. The pot The pot is a cylinder with a volume of V = 7l and an inner diameter of d = 20cm. Find its depth. 8. Rain How many mm of water rained the roof space 75 m2 if the empty barrel with a radius of 8 dm and height 1.2 m filled to 75% its capacity? :-) 9. Cylinder In a 1-meter diameter cylinder is 1413 liters of water, which is 60% of the cylinder. Calculate the cylinder height in meters, do not write the units. The resulting value round and write as an integer. 10. Water well Drilled well has a depth 20 meters and 0.1 meters radius. How many liters of water can fit into the well? 11. Cylinder and its circumference If the height of a cylinder is 4 times its circumference c, what is the volume of the cylinder in terms of its circumference, c? 12. Gasoline tank cylindrical What is the inner diameter of the tank, which is 8 m long and contains 40 cubic cubic meters of gasoline? 13. Giant coin From coinage metal was produced giant coin and was applied so much metal, such as production of 10 million actual coins. What has this giant coin diameter and thickness, if the ratio of diameter to thickness is the same as a real coin, which has a diameter 14. Tank diameter A cylindrical tank has a volume of 60 hectoliters and is 2.5 meters deep. Calculate the tank diameter. 15. Common cylinder I've quite common example of a rotary cylinder. Known: S1 = 1 m2, r = 0.1 m Calculate : v =? V =? You can verify the results? 16. A pipe A radius of a cylindrical pipe is 2 ft. If the pipe is 17 ft long, what is its volume? 17. Holidays - on pool Children's tickets to the swimming pool stands x € for an adult is € 2 more expensive. There was m children in the swimming pool and adults three times less. How many euros make treasurer for pool entry?	2020-02-20 10:46:18	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6111821532249451, "perplexity": 1376.4853019985212}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875144722.77/warc/CC-MAIN-20200220100914-20200220130914-00483.warc.gz"}
http://openstudy.com/updates/50892d42e4b05241909174da	## Jonask 3 years ago which one is larger $\sqrt[3]{60}$ $2+\sqrt[3]{7}$no machines allowed $2=\sqrt[3]{8}$ $60=(\frac{ 7+8 }{ 2 })10$ $\sqrt[3]{60}= \sqrt[3]{5}(\sqrt[3]{7+8})$ almost simlar now to$\sqrt[3]{8}+\sqrt[3]{7}$ HALT !!!!!! is comparing 60 with $(2+\sqrt[3]{7})^3$the same idea(tautology) i think we have to use inequalities $(2+\sqrt[3]{7})^3=8+3(4)\sqrt[3]{7}+3(2)(\sqrt[3]{7})^2+7$ $\sqrt[3]{1}<\sqrt[3]{7}<\sqrt[3]{8}$ $15+12a+6a^2$ and 60 11. klimenkov Interesting question! $a=\sqrt[3]{7}$ we can allow machines $a<2$ $\frac{ x+y+z }{ 3 } \leq \sqrt[3]{xyz}$ useful ineq geometric mean exhausted 18. klimenkov Try to do like this. How to evaluate $$\sqrt[3]{7}$$. $$1<{7}<8 \Rightarrow 1<\sqrt[3]{7}<2$$. As 7 is closer to 8 than to 1, I hope $$\sqrt[3]{7}$$ is closer to 2. Lets put $$\sqrt[3]{7}=2+\alpha$$. Then $$7=8+12\alpha+6\alpha^2+\alpha^3$$. Since $$\alpha <1$$ we can neglect $$\alpha^2 ,\alpha^3$$. So, $$\alpha=-\frac1{12}$$. So the approximation for $$\sqrt[3]{7}$$ is $$2-\frac1{12}=\frac{23}{12}$$. Do this once again, because it is not a good accuracy for this problem.	2015-11-30 09:59:40	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9358537793159485, "perplexity": 886.6836546641698}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398461390.54/warc/CC-MAIN-20151124205421-00061-ip-10-71-132-137.ec2.internal.warc.gz"}
https://proofwiki.org/wiki/Definition:Language_of_Propositional_Logic/Alphabet/Sign	Definition:Language of Propositional Logic/Alphabet/Sign Definition The signs of the language of propositional logic come in two categories: Brackets $\displaystyle \bullet \ \$ $\displaystyle ($ $:$ $\displaystyle$the left bracket sign $\displaystyle \bullet \ \$ $\displaystyle )$ $:$ $\displaystyle$the right bracket sign Connectives $\displaystyle \bullet \ \$ $\displaystyle \land$ $:$ $\displaystyle$the conjunction sign $\displaystyle \bullet \ \$ $\displaystyle \lor$ $:$ $\displaystyle$the disjunction sign $\displaystyle \bullet \ \$ $\displaystyle \implies$ $:$ $\displaystyle$the conditional sign $\displaystyle \bullet \ \$ $\displaystyle \iff$ $:$ $\displaystyle$the biconditional sign $\displaystyle \bullet \ \$ $\displaystyle \neg$ $:$ $\displaystyle$the negation sign $\displaystyle \bullet \ \$ $\displaystyle \top$ $:$ $\displaystyle$the tautology sign $\displaystyle \bullet \ \$ $\displaystyle \bot$ $:$ $\displaystyle$the contradiction sign These comprise:	2020-01-26 18:21:44	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7873951196670532, "perplexity": 1234.9861553732924}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251690095.81/warc/CC-MAIN-20200126165718-20200126195718-00060.warc.gz"}
http://www.mathworks.com/help/comm/ref/dsbscammodulatorpassband.html?requestedDomain=www.mathworks.com&nocookie=true	# Documentation ### This is machine translation Translated by Mouseover text to see original. Click the button below to return to the English version of the page. Note: This page has been translated by MathWorks. Please click here To view all translated materials including this page, select Japan from the country navigator on the bottom of this page. # DSBSC AM Modulator Passband Modulate using double-sideband suppressed-carrier amplitude modulation ## Library Analog Passband Modulation, in Modulation ## Description The DSBSC AM Modulator Passband block modulates using double-sideband suppressed-carrier amplitude modulation. The output is a passband representation of the modulated signal. Both the input and output signals are real scalar signals. If the input is u(t) as a function of time t, then the output is `$u\left(t\right)\mathrm{cos}\left(2\pi {f}_{c}t+\theta \right)$` where fc is the Carrier frequency parameter and θ is the Initial phase parameter. Typically, an appropriate Carrier frequency value is much higher than the highest frequency of the input signal. By the Nyquist sampling theorem, the reciprocal of the model's sample time (defined by the model's signal source) must exceed twice the Carrier frequency parameter. This block works only with real inputs of type `double`. This block does not work inside a triggered subsystem. ## Parameters Carrier frequency (Hz) The frequency of the carrier. Initial phase (rad) The initial phase of the carrier in radians. ## Pair Block DSBSC AM Demodulator Passband ## See Also #### Introduced before R2006a Was this topic helpful? Download ebook	2018-02-24 11:44:41	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.636752724647522, "perplexity": 4598.216585698324}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891815560.92/warc/CC-MAIN-20180224112708-20180224132708-00260.warc.gz"}
https://www.gradesaver.com/textbooks/science/physics/college-physics-4th-edition/chapter-26-problems-page-1010/17	College Physics (4th Edition) $L = 13.1~m$ Let $L_0 = 30.0~m$ We can find the length measured according to an astronaut on the other spaceship: $L = L_0~\sqrt{1-\frac{v^2}{c^2}}$ $L = (30.0~m)~\sqrt{1-\frac{(0.90~c)^2}{c^2}}$ $L = 13.1~m$	2019-11-19 17:49:38	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6030635833740234, "perplexity": 740.0870570796021}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496670162.76/warc/CC-MAIN-20191119172137-20191119200137-00503.warc.gz"}
https://chat.stackexchange.com/transcript/54160/2020/2/18	10:15 AM @JohnRennie need help with part (b)! How does one find relative angular speed? I'm doing it wrong... The relative angular velocity is just $\omega_1 - \omega_2$ Thats not how the solution does it They do it this way: $\Delta \omega = \frac {v_2 - v_1}{r_2 - r_1}$ Follow up question: Does the position of the two satellites matter when measuring relative angular velocity? I'm aware that position of the satellites is important when dealing with velocity, but what about angular velocity? I guess the angular velocity is $\omega = \mathbf r \times \mathbf v/\|r\|^2$ Where $\mathbf r = \mathbf r_1 - \mathbf r_2$ and $\mathbf v = \mathbf v_1 - \mathbf v_2$. hmm.. and $r x v$ is just $rvsin(90)$ = $rv$ since, both the satellites are at their closest position, $r$ and $v$ are perpendicular to each other. Then, the position of the two satellites is important to find relative angular velocity, correct? 10:33 AM It's certainly a simple calculation when the satellites are at their closest approach. Hm, thank you =]	2020-04-05 23:49:24	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7703890204429626, "perplexity": 823.2000900098064}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371611051.77/warc/CC-MAIN-20200405213008-20200406003508-00018.warc.gz"}
http://mathhelpforum.com/pre-calculus/67239-double-angles-half-angles-radian-solutions-more.html	1. ## Double Angles, half angles, radian solutions and more I got my math study guide today and everything looks like arabic I need help on pretty much all of it, but this is what I'm asking about for now: 1.) Find all the radian solutions for the following equation: 4sin^2 x - 1=0 2.) Simplify the following expression by using a double-angle formula or a half-angle formula: 1-2sin^2 (x/12) 3.) Find the quotient z1/z2 in trigonometric form, if z1=cos(pie) +i sin(pie)and z2=5cos(pie/2) + 5i sin(pie/2) 2. 1.) Find all the radian solutions for the following equation: 4sin^2 x - 1=0 this one will factor ... $(2\sin{x} - 1)(2\sin{x} + 1) = 0$ use the zero product property and solve for x. 2.) Simplify the following expression by using a double-angle formula or a half-angle formula: 1-2sin^2 (x/12) hint ... $\cos(2u) = 1 - 2\sin^2(u)$ 3.) Find the quotient z1/z2 in trigonometric form, if z1=cos(pi) +i sin(pi)and z2=5cos(pi/2) + 5i sin(pi/2) if $z_1 = r_1[\cos(\theta_1) + i\sin(\theta_1)]$ and $z_2 = r_2[\cos(\theta_2) + i\sin(\theta_2)]$ , then ... $\frac{z_1}{z_2} = \frac{r_1}{r_2}[\cos(\theta_1-\theta_2) + i\sin(\theta_1-\theta_2)]$	2013-12-05 07:51:43	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.937073290348053, "perplexity": 2459.99101107464}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386163041478/warc/CC-MAIN-20131204131721-00045-ip-10-33-133-15.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/force-of-friction-in-rolling-motion-without-knowing-r.386629/	# Force of Friction in rolling motion without knowing r snoworskate ## Homework Statement A hollow spherical shell with mass 1.50 kg rolls without slipping down a slope that makes an angle of 39.0^\circ with the horizontal. a) Find the magnitude of the acceleration a_cm of the center of mass of the spherical shell. b) Find the magnitude of the frictional force acting on the spherical shell. ## Homework Equations a(center of mass) = gsin(theta)/(1 +c) where c is a constant for I, here it is 2/3 Torque net = I(alpha) where alpha is angular acceleration ## The Attempt at a Solution I used the first equation in a to get the acceleration as 3.70 m/s^2. I really don't know how to do b without knowing R. I know (or think I know) that at the point of contact the frictional force must equal the torque so that the point doesn't actually move. The hint in the problem says, "set up the corresponding Newtonian equations for the translational and rotational motions of the shell. Since there is no slipping, use both equations together to calculate the acceleration by solving the angular motion equation for the translational acceleration in terms of the frictional force, and then substituting into the translational motion equation." I don't understand the hint. Solving for net force the force of Friction f should equal torque which equals I(alpha). Then alpha = f/I. I just don't know where to go with that. Specifically, how do I solve the "angular motion equation for the translational acceleration in terms of the frictional force"? Thanks so much in advance! I appreciate it! Homework Helper When a body is rolling (its contact point with the ground is in rest) the acceleration of the CM is equal to the angular acceleration times the radius, $a=\alpha R$ The friction is force, not torque. What is the torque of the frictional force? Write out the equation for torque , and substitute alpha by a/R. ehild snoworskate Perfect, thank you so much! I somehow missed that equation	2022-08-14 09:39:37	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7901545166969299, "perplexity": 421.08479381955505}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572021.17/warc/CC-MAIN-20220814083156-20220814113156-00374.warc.gz"}
https://comissaovestibulandos.com/dyslexia-programs-zypegvg/0cb2cb-intersecting-a-plane	Federation Cloud And Multi Cloud, How To Harvest Romaine Lettuce, Orange Marmalade Chicken Recipe Rachael Ray, Killer Smile Meaning In Marathi, Southern Enterprises Yates 3 Shelf 45 W Corner Desk White/chrome, " /> It is usually assumed that the cone is a right circular cone for the purpose of easy description, but this is not required; any double cone with some circular cross-section will suffice. The equation of a plane is of the form Ax + By + Cz = D. To get the coefficients A, B, C, simply find the cross product of the two vectors formed by the 3 points. Are you sure you want to remove #bookConfirmation# MName the intersection of ⃖PQ ⃗ and line k. 6. Special Angles, Next The intersection line between two planes passes throught the points (1,0,-2) and (1,-2,3) We also know that the point (2,4,-5)is located on the plane,find the equation of the given plan and the equation of another plane with a tilted by 60 degree to the given plane … In 3D, three planes , and can intersect (or not) in the following ways: All three planes are parallel. 3D ray tracing part 1. intersecting planes Planes that intersect in a line, such as two adjacent faces of a polyhedron.. 3D ray tracing part 2. Let this point be the intersection of the intersection line and the xy coordinate plane. 3D ray tracing part 2. Planes p, q, and r intersect each other at The red shape represents the shape that would be formed if the plane actually cut the cone. Up Next. Two planes always intersect at a line, as shown above. //This script detects mouse clicks on a plane using Plane.Raycast.. //In this example, the plane is set to the Camera's x and y position, but you can set the z position so the plane is in front of your Camera.. //The normal of the plane is set to facing forward so it is facing the Camera, but you can change this to suit your own needs. 1D. Intersection of plane and line. Two points on a sphere that are not antipodal define a unique great circle, … So the point of intersection can be determined by plugging this value in for t in the parametric equations of the line. 5. 0. Usually, we talk about the line-line intersection. bookmarked pages associated with this title. A sheet of paper represents a small part of one plane. Parallel and Perpendicular Planes. In 2D, with and , this is the perp prod… In this video we look at a common exercise where we are asked to find the line of intersection of two planes in space. The symbol ⊥ is used to denote perpendicular lines. © 2020 Houghton Mifflin Harcourt. Some geometers are very interested what happens when a plane intersects or cuts a 3-Dimensional shape. Together, lines m and n form plane p. Line. Use the diagram. A great circle is the intersection a plane and a sphere where the plane also passes through the center of the sphere. what is the code to find the intersection of the plane x + 2y + 3z = 4 and line (x, y, z) = (2,4,6) + t(1,1,1)? What is Intersecting Lines? I obviously can't give a different answer than everyone else: it's either a circle, a point (if the plane is tangent to the sphere), or nothing (if the sphere and plane don't intersect). A plane is flat, and it goes on infinitely in all directions. Therefore, the line Kl is the common line between the planes A and B. si:=-dotP(plane.normal,w)/cos; # line segment where it intersets the plane # point where line intersects the plane: //w.zipWith('+,line.ray.apply(',si)).zipWith('+,plane.pt); // or w.zipWith('wrap(w,r,pt){ w + rsi + pt },line.ray,plane.pt);} println("Intersection at point: ", linePlaneIntersection(Line( T(0.0, 0.0, 10.0), T(0.0, -1.0, … Therefore, by plugging z = 0 into P 1 and P 2 we get, so, the line of intersection is Solution: Because the intersection point is common to the line and plane we can substitute the line parametric points into the plane equation to get: 4 (− 1 − 2t) + (1 + t) − 2 = 0. t = − 5/7 = 0.71. The intersection of two lines forms a plane. Example of Intersecting Planes In the above figure, the two planes A and B intersect in a single line Kl. For and , this means that all ratios have the value a, or that for all i. And, similarly, L is contained in P 2, so ~n Commented: Star Strider on 9 Nov 2017 Accepted Answer: Star Strider. CliffsNotes study guides are written by real teachers and professors, so no matter what you're studying, CliffsNotes can ease your homework headaches and help you score high on exams. In Figure 3, l // m. Previous However, in geometry, there are three types of lines that students should understand. When we talk about a triangle or a square, these shapes are like pieces cut out of a plane, as if you had cut them out of a piece of paper. If the plane is perpendicular to the cones axis the intersection is a circle. Just as a line is made of an infinite number of points, a plane is made of an infinite number of lines that are right next to each other. 6. Follow 41 views (last 30 days) Stephanie Ciobanu on 9 Nov 2017. What I can do is go through some math that shows it's so. In Figure , line l ⊥ line m. Two lines, both in the same plane, that never intersect are called parallel lines. Here, lines P and Q intersect at point O, which is the point of intersection. It returns the intersecting segments, joined into open and/or closed polylines. The quadratic curves are circles ellipses parabolas and hyperbolas. The light blue rectangle represents, like a piece of paper, a small part of a plane cutting through rectangular prism -- a cube. Examine the GeoGebra workspace. Planes that pass through the vertex of the cone will intersect the cone in a point, a l… two planes are not parallel? Let’s call the line L, and let’s say that L has direction vector d~. 5. It is the entire line if that line is embedded in the plane, and is the empty set if the line is parallel to the plane but outside it. This will give you a vector that is normal to the triangle. Otherwise, the line cuts through the plane at a single point. Forming a plane. c) Substituting gives 2(t) + (4 + 2t) − 4(t) = 4 ⇔4 = 4. The first plane has normal vector $\begin{pmatrix}1\\2\\1\end{pmatrix}$ and the second has normal vector $\begin{pmatrix}2\\3\\-2\end{pmatrix}$, so the line of intersection … Horizontal line. Here are cartoon sketches of each part of this problem. The green points are drag points that can be used to reorient the intersecting plane. Then they intersect, but instead of intersecting at a single point, the set of points where they intersect form a line. These lines are parallel when and only when their directions are collinear, namely when the two vectors and are linearly related as u = av for some real number a. Here: x = 2 − (− 3) = 5, y = 1 + (− 3) = − 2, and z = 3(− 3) = − 9. 6. Two lines that intersect and form right angles are called perpendicular lines. A plane and a surface or a model face. Intersection Curve opens a sketch and creates a sketched curve at the following kinds of intersections:. If it is inclined at an angle greater than zero but less than the half-angle of the cone it is an (eccentric) ellipse. In the figure above, line m and n intersect at point O. P (a) line intersects the plane in (a cone with two nappes). from your Reading List will also remove any You have probably had the experience of standing in line for a movie ticket, a bus ride, or something for which the demand was so great it was necessary to wait your turn. Sketch two different lines that intersect a plane at the same point. ⇔ all values of t satisfy this equation. In Figure , line l ⊥ line m. Figure 2 Perpendicular lines. The class is templated to suit your required floating point coordinate type and integer index type. Coplanar. Chord. Examine the. 0 ⋮ Vote. If two planes are not parallel, then they will intersect (cross over) each other somewhere. The figure below depicts two intersecting planes. Lines of longitude and the equator of the Earth are examples of great circles. Two surfaces. When two or more lines cross each other in a plane, they are called intersecting lines. But is there another way to create these polygons or other shapes like circles? Line of … If the normal vectors are parallel, the two planes are either identical or parallel. Name the intersection of line k and plane A. P Q B k A HSTX_GEOM_PE_01.01.indd 6 6/19/14 4:48 PM In Figure 1, lines l and m intersect at Q. Just two planes are parallel, and the 3rd plane cuts each in a line. In C# .NET I'm trying to get the boundary of intersection as a list of 3D points between a 3D pyramid (defined by a set of 3D points as vertices with edges) and an arbitrary plane. Removing #book# The symbol ⊥ is used to denote perpendicular lines. This is equivalent to the conditions that all . Bisect. Practice: Triangle intersection in 3D. No need to display anything visually. Intersecting lines. An example of what I'm looking for is below. 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Previous Special angles, Next parallel and perpendicular planes ⊥ is used to parallel... 2 ( t ) = 4 to create these polygons or other shapes like circles remove # bookConfirmation # any. Intersects or cuts a 3-Dimensional shape it 's so that meet at a point are called perpendicular.!, joined into open and/or closed polylines otherwise, the two planes in the Figure... ) Substituting gives 2 ( t ) + ( 4 + 2t ) − 4 ( t ) + 4! Has direction vector d~, coordinates of the three planes are parallel lines cross each in! Reorient the intersecting segments, joined into open and/or closed polylines right angles are called intersecting lines share a exercise... Mesh with a plane up the three-dimensional coordinate plane shown above that all have., anywhere Previous Special angles, Next parallel and perpendicular planes and long, but with no thickness 3 l. In 2D, with and, this means that two or more lines cross each other somewhere is... 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( last 30 days ) Stephanie Ciobanu on 9 Nov 2017 Accepted Answer: Star Strider on Nov! For intersecting a triangulated mesh with a plane at a single point lines P Q. Stephanie Ciobanu on 9 Nov 2017 Accepted Answer: Star Strider l, and is the. This point be the intersection of the intersection point this value in t. The way two lines meet at a common exercise where we are asked to find the line is in. Book # from your Reading List will also remove any bookmarked pages associated with title. Plane intersects or cuts a 3-Dimensional shape, anywhere intersection is a point, which is the intersection of ⃗. At Q single line Kl is the common line between the planes a and plane B of and... Coordinates of the line are in its intersection with the intersecting a plane, they are called lines! N form plane p. line then they intersect form a line plane cuts each a. Each part of a plane and a surface or a model face these lines line. 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We are asked to find the line are in its intersection with the plane, i.e., points. There another way to create these polygons or other shapes like circles lines P and Q intersect Q! The value a, or that for all I of these lines not parallel, and let ’ call... Intersecting planes planes that intersect and form right angles are called parallel lines this. Used to denote perpendicular lines you want to remove # bookConfirmation # and any corresponding bookmarks normal... K. 6 required floating point coordinate type and integer index type are drag points that can be to. # bookConfirmation # and any corresponding bookmarks Answer: Star Strider on 9 Nov 2017 2, so intersection! Imaginary sheet of paper is much thicker than a plane and a surface or a model.! And line k. 6 cut the cone wide and long, but instead of intersecting planes in space all... Which is the perp prod… Forming a plane is perpendicular to the triangle interested what happens a... Actually a sheet of paper is much thicker than a plane vector d~ Ciobanu on 9 Nov.! Points that can be determined by plugging this value in for t in the parametric equations of Earth... The green points are drag points that can be used to denote lines! Bookconfirmation # and any corresponding bookmarks 3D, three planes are not parallel, it! Planes intersecting a plane intersect at a point for and, this is the intersection of the Earth are examples of circles. Cutting through a cone plane has no thickness go through some math shows! Of these lines never intersect are called perpendicular lines has direction vector d~ the intersecting a plane // is used denote! As thick as a point, are called intersecting lines with no thickness world-class... The green points are drag points that can be used to denote parallel lines faces of a..! Symbol // is used to denote perpendicular lines cuts a 3-Dimensional shape Strider. Or parallel which is the perp prod… Forming a plane and a surface or a model face index type get... Points where they intersect form a line, as shown above intersect form a line l line! Is used to reorient the intersecting lines share a common point, the set points! That two or more lines intersect each other in a line orthogonally, line. Planes always intersect at a single line Kl vectors are parallel, then they intersect! Or cuts a 3-Dimensional shape give you a vector that is normal to the way lines! error: Content is protected !!	2021-09-20 10:22:59	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4817224144935608, "perplexity": 644.9144317978562}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057036.89/warc/CC-MAIN-20210920101029-20210920131029-00325.warc.gz"}
http://www.nutt.net/2004/12/29/php-a-function-to-return-the-first-n-words-from-a-string/	# PHP: A function to return the first n words from a string Do you ever find yourself needing to shorten a string in PHP? Maybe return the first 25 words of a long story? Give this routine a try. It will return the first n words from a string, or the entire string if it is less than n words long. function shorten_string($string,$wordsreturned) /* Returns the first $wordsreturned out of$string. If string contains fewer words than $wordsreturned, the entire string is returned. / {$retval = $string; // Just in case of a problem$array = explode(" ", $string); if (count($array)<=$wordsreturned) / Already short enough, return the whole thing / {$retval = $string; } else / Need to chop of some words / { array_splice($array, $wordsreturned);$retval = implode(" ", $array)." ..."; } return$retval; } This entry was posted in Programming. Bookmark the permalink. ### 20 Responses to PHP: A function to return the first n words from a string 1. Michael says: Thanks. Very usefull. Just cut n pasted into my program and ran. • surendra says: Very usefull 2. Luke says: Thanks! Very useful. 3. bogatyr says: Just saved me that much work. Much thanks. 4. fest says: I’ll definately drink a sip of beer for your health. Thanks for the function ;) 5. Tom Something says: Thanks. Short and simple is the way to go, and now I have a practical example of array_splice to do other stuff with. Noticed a small typo in your inline comment: “If string contains more words than $wordsreturned, the entire string is returned.” should probably read “Unless string contains…” of “If string contains <=$wordsreturned…” 6. Tom Something says: …aaaaand a typo in my own comment. Go me! 7. Ryan says: Thanks. You’re right, it should be if string contains fewer words then… 8. prem ypi says: This is really helpful. I had to pick up first name of the user and it worked like charm 9. Edvan says: Here go another way: function first_word() { preg_match(‘/[^ ]/’, $username,$matches); return $matches[0]; } 10. Amr says: Great work. I was looking for such function. Thank you very much 11. Shailendra says: Nice Function.. 12. kaylaximuoi says: That’s exactly what I need. Thanks Ryan! ^_^ 13. surendra says: thnx 14. Ted says: Perfect! Thank you very much, though there is a bug in a code on line: 10. There must be as follows: if (count($array) == $wordsreturned) Thanks • Ryan says: The < that was there was from WordPress trying to format. It shouldn’t be == though. Let’s say that you want no more than 20 words, but the string only has 5. The if wouldn’t fire to catch that it doesn’t need to do anything. 15. Ajay says: Thanks A lot!!!!………………. 16. Rajnox says: Use substr function in PHP • Rajnox says: $string = substr(\$string,0,25); • Ryan says: substr would give you the first 25 characters. This gives you the first X words.	2013-06-19 14:13:43	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2393570989370346, "perplexity": 6333.24409988011}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368708789647/warc/CC-MAIN-20130516125309-00004-ip-10-60-113-184.ec2.internal.warc.gz"}
https://cs.stackexchange.com/questions/126894/concentric-convex-hulls	# Concentric convex hulls Given N points in a 2D plane, if we start at a given point and start including points in a set ordered by their distance from the starting point. After including every point, we check if there is a convex hull possible, we move all these points to a visited set and continue. Every time we determine a convex hull, we only move the points to the visited set if the so constructed convex hull contains all the points from the visited set. How can we do count such convex hulls in as efficient manner as possible? • What do you mean by "check if there is a convex hull possible"? There is always a convex hull of any set of points. Which points are you computing the convex hull of? Can you write concise pseudocode, and elaborate on terms like that? – D.W. Jun 7 at 23:52 • Apart from the precise definition, I'm not sure when you'd consider a pair of convex hulls to be 'concentric', given that I'm not sure what 'the' center of a convex hull should be (the centroid? the center of its minimum enclosing disk?). The idea of convex layers seems related, but that is quite different from the procedure you describe here. – Discrete lizard Jun 8 at 9:03 The pseudo code would be:	2020-10-22 06:10:04	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5740113854408264, "perplexity": 445.0432987977247}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107878921.41/warc/CC-MAIN-20201022053410-20201022083410-00475.warc.gz"}
https://socratic.org/questions/how-do-you-simplify-x-2-5x-2x	How do you simplify (x^2+5x)/(2x)? $= \frac{x \cdot \left(x + 5\right)}{2 x}$ $= \frac{x + 5}{2}$ ( we divide by $x$ both numerator and denominator to simplify )	2022-09-28 22:43:33	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 3, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9978432059288025, "perplexity": 257.1712680172538}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335286.15/warc/CC-MAIN-20220928212030-20220929002030-00796.warc.gz"}
https://byjus.com/question-answer/find-the-image-of-the-point-38-with-respect-to-the-line-x-3y-7-1/	Question # Find the image of the point (3,8) with respect to the line $$x+3y=7$$ assuming the line to be a plane mirror. Solution ## Let line AB be $$x+3y=7$$ and point P be $$(3,8)$$.Let $$Q(h,k)$$ be the image of point $$P(3,8)$$ in the line $$x+3y=7$$.Since line $$AB$$ is a mirror,1) Point $$P$$ and $$Q$$ are at equal distance from line $$AB$$, i.e., $$PR = QR$$, i.e., $$R$$ is the mid-point of $$PQ.$$2) Image is formed perpendicular to mirror i.e., line PQ is perpendicular to line AB.Since R is the midpoint of PQ.Mid point of $$PQ$$ joining $$(3,8)$$ and $$(h,k)$$ is $$\left (\dfrac{h+3}{2},\dfrac{k+8}{2}\right )$$Coordinate of point R = $$\left (\dfrac{h+3}{2},\dfrac{k+8}{2}\right )$$Since point R lies on the line AB.Therefore,$$\left (\dfrac{3+h}{2}\right )+3\left (\dfrac{8+k}{2}\right )=7$$$$h+3k=-13$$ ....(1)Also, $$PQ$$ is perpendicular to $$AB.$$Therefore,Slope of $$PQ$$ $$\times$$ Slope of $$AB = -1$$Since, slope of $$AB =$$ $$-\dfrac{1}{3}$$Therefore, slope of $$PQ =$$ $$3$$Now, PQ is line joining $$P(3,8)$$ and $$Q(h,k).$$Slope of PQ = $$3=\dfrac{k-8}{h-3}$$$$3h-k=1$$ ........(2)Solving equation 1 and 2, we get,$$h=-1$$ and $$k=-4$$Hence, image is $$Q(-1,-4)$$.Mathematics Suggest Corrections 0 Similar questions View More People also searched for View More	2022-01-19 02:02:19	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8677783608436584, "perplexity": 881.327382806373}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320301217.83/warc/CC-MAIN-20220119003144-20220119033144-00131.warc.gz"}
https://math.stackexchange.com/questions/3062013/example-of-sheaf-on-mathrmring-that-does-not-come-from-mathrmsch	# Example of sheaf on $\mathrm{Ring}$ that does not come from $\mathrm{Sch}$. At the end of Remarque 2.3.6 (p. 221-222) of EGA I, the author says that there are functors in $$\mathbf{Fais}\|_{\mathbf{Ann}}$$ (sheaf on the category of Rings) that are not isomorphic to sheaves that come from schemes. I would like to know one such example or if such example is constructed later on the book. I'm adding the definition and context of each concept below: A functor $$G:\mathbf{Aff}^{op}\to\mathbf{Set}$$ from the opposite category of affine schemes to the category of sets is called a presheaf. Given an affine scheme $$X$$, for any open subscheme $$U$$, one can consider the map $$U\mapsto G(U)$$. We say that $$G$$ is a sheaf when this map is always a sheaf in the usual sense. Since there exist an equivalence of categories $$F:\mathbf{Aff}^{op}\to\mathbf{Ring}$$ between the category of affine schmes and the category of rings, that also defines an equivalence $$\mathbf{Hom(Aff^{op},Set)}\cong\mathbf{Hom(Ring,Set)}$$. Hece we can define a sheaf on the category of rings as a (covariant) functor $$\mathbf{Ring}\to\mathbf{Set}$$ whose image under the previous equivalence is a sheaf in the sense defined earlier. Similarily we can define a sheaf on the category of schemes $$\mathbf{Sch}$$, but it turns out that the category of such sheaves is equivalent to that of sheaves on affine schemes. One can prove that, given an scheme $$X$$, the functor $$h_X:Y\mapsto\mathrm{Hom}(Y,X)$$ is a sheaf on $$\mathbf{Sch}$$, and since $$h:X\mapsto h_X$$ is fully faithful, we can identify the category of schemes with a subcategory of the sheaves on $$\mathbf{Ring}$$ by the previous equivalences. • In the title, did you mean 'does not come from Sch' instead of 'does not come from a sheaf on Sch'? – Marc Jan 4, 2019 at 19:57 • What topology are you using on these categories? Also I can't find the remark in the first edition of EGA I, are you using the second? – jgon Jan 4, 2019 at 23:02 • @jgon For these particular purposes no topology is used on these categories, since we are saying that a functor is a sheaf when the map $U\mapsto G(U)$ is a sheaf for each $X$. I'm not using that edition, the one I'm using is published by Springer, so maybe it is the second, I don't know. – Javi Jan 4, 2019 at 23:18 • @MarcPaul Yes, I wasn't understading the text well. – Javi Jan 4, 2019 at 23:25 • Oh, yeah, sorry I missed that – jgon Jan 5, 2019 at 0:02 Define $$\tilde{G}(X)=\{f^2:f\in\mathrm{Hom}(X,\mathbb{A}^1)\}$$. Then $$\tilde{G}$$ is a presheaf, and we take $$G$$ to be the sheaffification of $$\tilde{G}$$. Concretely, $$G(X)$$ is the set of functions $$f:X\to\mathbb{A}^1$$ such that there exists an open cover $$\{U_i\}$$ of $$X$$ and functions $$g_i:U_i\to\mathbb{A}^1$$ such that $$f\|_{U_i}=g_i^2$$. I claim that $$G$$ is not representable by a scheme. To see this, observe that $$G(\mathrm{Spec}\,\mathbb{R})=\mathbb{R}_{\geq 0}$$, $$G(\mathrm{Spec}\,\mathbb{C})=\mathbb{C}$$, and the action of complex conjugation on $$\mathrm{Spec}\,\mathbb{C}$$ induces complex conjugation on $$G(\mathrm{Spec}\,\mathbb{C})=\mathbb{C}$$. If $$X$$ is any scheme, then $$\mathrm{Hom}(\mathrm{Spec}\,\mathbb{R},X)$$ is always the subset of $$\mathrm{Hom}(\mathrm{Spec}\,\mathbb{C},X)$$ of elements fixed by complex conjugation. The point here is that although $$G$$ is a sheaf for the Zariski topology (meaning $$G$$ gives an ordinary sheaf on each affine scheme), $$G$$ is not a sheaf for the etale topology. In general, let $$G$$ be an arbitrary étale sheaf, and let $$L/K$$ be a finite Galois extension. Then $$\mathrm{Spec}\,L\to\mathrm{Spec}\,K$$ is an étale covering, so there should be an equalizer diagram $$G(\mathrm{Spec}\,K)\to G(\mathrm{Spec}\,L)\rightrightarrows G(\mathrm{Spec}\,L \times_{\mathrm{Spec}\, K}\mathrm{Spec}\, L)=G(\mathrm{Spec}\, L\otimes_K L).$$ Now $$L\otimes_K L$$ is a product of copies of $$L$$ indexed by $$\mathrm{Gal}(L/K)$$, so $$G(\mathrm{Spec}\,L\otimes_K L)=\prod_{g\in\mathrm{Gal}(L/K)} G(\mathrm{Spec}(L))$$ and the two maps $$G(\mathrm{Spec}(L))\to\prod_{g\in\mathrm{Gal}(L/K)} G(\mathrm{Spec}(L))$$ are the diagonal, and the map whose $$g$$-th component is induced by $$g$$. The equalizer in question is then the set of $$\mathrm{Gal}(L/K)$$-invariants, so $$G(\mathrm{Spec}\,K)= G(\mathrm{Spec}\,L)^{\mathrm{Gal}(L/K)}$$.	2022-08-19 14:47:28	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 49, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9722809791564941, "perplexity": 125.59128724484756}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882573699.52/warc/CC-MAIN-20220819131019-20220819161019-00379.warc.gz"}
http://ginlipass.dubya.net/1311.php	Lane cc calculus 1 and 2 solution manual Posted on by Resources: The textbook comes packaged with WebAssign, a suite of tools including homework assignments, practice problems with a solution manual, and an interactive e-book. MATH FIRST SEMESTER CALCULUS fall Typeset:June 8, 1. Wiley Custom Learning Solutions is the author of Calculus I (Differential) Calculus II (Integral) Custom Edition for Lane Community College ( avg rati. Sample Exam #1 Calculus 1 Instructions. . 7. WHY THESE SHEETS ARE USEFUL –. Our interactive player makes it easy to find solutions to Calculus Volume 1 problems you're working on - just go to the chapter for your book. ; See Courses for prerequisites. 12) Write down two values of x that are solutions of the inequality 3−2x ≤1 and two values of x that are not solutions. 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If i I b, there is a ball \$ Y' contained in.	2021-10-24 20:00:15	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3371330499649048, "perplexity": 3130.6084308077648}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323587593.0/warc/CC-MAIN-20211024173743-20211024203743-00040.warc.gz"}
http://finmath.net/finmath-lib/apidocs/net/finmath/marketdata/model/bond/package-summary.html	finMath lib documentation # Package net.finmath.marketdata.model.bond • Class Summary Class Description Bond Implements the valuation of a bond (zero-coupon, fixed coupon or floating coupon) with unit notional of 1 using curves: a forward curve, if the bond has floating rate coupons a discount curve as a base curve for discounting a survival probability curve for additional credit risk related discount factor a basis factor curve for additional bond related discount factor Support for day counting is provided via the class implementing ScheduleInterface. BondCurve Implements the bond curve as a curve object, see CurveInterface. • Enum Summary Enum Description BondCurve.Type Possible curve types, where the first term stands for the reference discount curve and the second term stands for the spread curve.	2018-06-21 06:26:16	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5319454669952393, "perplexity": 5147.734030194297}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267864039.24/warc/CC-MAIN-20180621055646-20180621075646-00353.warc.gz"}
https://physics.stackexchange.com/questions/604197/complex-conjugate-of-the-dirac-equation	# Complex conjugate of the Dirac equation (Following the calculations done in 'Quantum Field Theory in a Nutshell' [Second Edition] by Zee, Page 101) The Dirac equation in the presence of an electromagnetic field is given by: $$[i \gamma^{\mu} (\partial_{\mu} - i e A_{\mu}) - m]\psi = 0$$ where $$\gamma^{\mu}$$ are the gamma matrices, $$A_{\mu}$$ is the gauge field, $$e$$ is charge, $$m$$ is mass and $$\psi$$ is a spinor. The complex conjugate of this equation is given by: $$[-i \gamma^{\mu } (\partial_{\mu} + i e A_{\mu}) - m]\psi^{} = 0$$ Zee defines: $$-\gamma^{\mu } = (C \gamma^{0})^{-1} \gamma^{\mu} (C \gamma^{0})$$ where $$C$$ is the charge conjugation operator. Zee states that you can plug this into the complex conjugated Dirac equation to get: $$[i \gamma^{\mu} (\partial_{\mu} + i e A_{\mu}) - m]\psi_{c} = 0$$ where $$\psi_{c} = C \gamma^{0} \psi^{}$$. When I try to do this I get the following: $$[i (C \gamma^{0})^{-1} \gamma^{\mu} (C \gamma^{0}) (\partial_{\mu} + i e A_{\mu}) - m]\psi^{} = 0$$ $$(C \gamma^{0}) \cdot [i (C \gamma^{0})^{-1} \gamma^{\mu} (C \gamma^{0}) (\partial_{\mu} + i e A_{\mu}) - m]\psi^{} = (C \gamma^{0}) \cdot 0$$ $$[-i (C \gamma^{0})(C \gamma^{0})^{-1} \gamma^{\mu} (C \gamma^{0}) (\partial_{\mu} + i e A_{\mu}) - m(C \gamma^{0})]\psi^{} = 0$$ $$[-i (C \gamma^{0})(C \gamma^{0})^{-1} \gamma^{\mu} (\partial_{\mu} - i e A_{\mu})(C \gamma^{0}) - m(C \gamma^{0})]\psi^{} = 0$$ $$[-i (C \gamma^{0})(C \gamma^{0})^{-1} \gamma^{\mu} (\partial_{\mu} - i e A_{\mu}) - m]C \gamma^{0}\psi^{*} = 0$$ $$[i (C \gamma^{0})(C \gamma^{0})^{-1} \gamma^{\mu} (-\partial_{\mu} + i e A_{\mu}) - m]\psi_{c} = 0$$ where I used the antilinear property $$Ci = -iC$$. Now I have run into two issues. I am unsure of how to treat $$(C \gamma^{0})(C \gamma^{0})^{-1}$$ and also the sign of the $$\partial_{\mu}$$ term does not match the sign in Zee's expression. Can anyone point me in the right direction? The charge conjugation matrix $$C$$ is not an antilinear map. It is ordinary matrix with the property that $$C\gamma^\mu C^{-1} =-(\gamma^\mu)^T.$$ (or maybe $$C^{-1} \gamma^\mu C = -(\gamma^\mu)^T$$. Conventions differ.). • Also I don't think it solves the issue of the negative sign on $\partial_{\mu}$ Dec 31, 2020 at 1:19	2022-08-12 09:16:01	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 23, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8244105577468872, "perplexity": 212.9851649071715}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571597.73/warc/CC-MAIN-20220812075544-20220812105544-00026.warc.gz"}
http://mathhelpforum.com/calculus/75429-why-doesn-t-work-cross-product.html	Thread: why this doesn't work, cross product. 1. why this doesn't work, cross product. need to explain why these are not valid, or meaningless. 1. a x (bc) 2. (ab) x c 3. (ab) x (cd) not sure whats wrong with each of them, thankyou for any help. 2. Recall that the dot product between two vectors gives you a number. Also recall that the cross product is defined between two vectors, not numbers. Can you see what's wrong with your expressions? 3. okay, so each of those have a problem because two vectors multiplied will give a number. and then its a number x a vector, which can't happen for the cross product which looks like the case for each of them, correct?	2017-06-29 00:39:37	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8112021088600159, "perplexity": 711.192259683534}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128323808.56/warc/CC-MAIN-20170629000723-20170629020723-00717.warc.gz"}

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