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https://chemistry.stackexchange.com/questions/61902/is-it-possible-to-create-a-solution-with-superoxide	[ "Is it possible to create a solution with superoxide?\n\nI am not a chemist, and somehow I cannot find any material on this topic, so I decided to ask here.\n\nI am looking to create a solution with a specified amount/concentration of superoxide ($\\ce{O2^.-}$). Can you suggest some compounds that will enable me to \"isolate\" superoxide in a solution?\n\n• OK, so what you're suggesting is to mix $KO_2$ and $C_{12}H_{24}O_6$ in $(CH_3)_2SO$, then the whole thing is added to an acqueous solution, which should have $O_2^{\\bar{\\cdot}}$. Correct? – A.I. Oct 31 '16 at 16:27" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9053094,"math_prob":0.9070704,"size":291,"snap":"2019-43-2019-47","text_gpt3_token_len":71,"char_repetition_ratio":0.11149826,"word_repetition_ratio":0.0,"special_character_ratio":0.23367697,"punctuation_ratio":0.10344828,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98381037,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-23T10:07:52Z\",\"WARC-Record-ID\":\"<urn:uuid:9fe02ddc-2466-4858-b7c4-10e88932292f>\",\"Content-Length\":\"145163\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:618621be-d2ec-4621-8163-c07764603023>\",\"WARC-Concurrent-To\":\"<urn:uuid:8557976f-f479-4a41-90a1-fe7c8b318b79>\",\"WARC-IP-Address\":\"151.101.1.69\",\"WARC-Target-URI\":\"https://chemistry.stackexchange.com/questions/61902/is-it-possible-to-create-a-solution-with-superoxide\",\"WARC-Payload-Digest\":\"sha1:N5AGGQODC6MPXLTQGPAZ2YRF764J6QXP\",\"WARC-Block-Digest\":\"sha1:QA7VDU74GEOT3TOX4OUTNJVW6DGTGJQP\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570987833089.90_warc_CC-MAIN-20191023094558-20191023122058-00453.warc.gz\"}"}
https://kashi.re/blog/books.html	[ "This is a list of books I plan on get/read at some point.\n\nMathematics\n\n• Discrete Mathematics with Application (S. Epp)\n• Discrete Mathematical Structures (B. Kolman, R. Busby, S. Ross))\n• Mathematical Proofs: A Transition to Advanced Mathematics (G. Chartrand, A. Polimeni, et al)\n• An Introduction to Abstract Mathematics (R. Bond, W. Keane)\n• Fearon's Pre Algebra (L. Cardine, M. Curth, et al.)\n• College Algebra (J. Kaufmann)\n• College Algebra Essentials (R. Blitzer)\n• A Graphical Approach to Algebra & Trigonometry (J. Hornsby, M. Lial, et al.)\n• Calculus 5e (J. Steward)\n• Calculus Made Easy (S. Thompson)\n• Calculus (M. Spivak)\n• A First Course in Differential Equations with Modeling Applications (D. Zill)\n• Ordinary Differential Equations With Applications (L. Andrews)\n• Elementary Linear Algebra (H. Anton)\n• Linear Algebra (S. Friedberg, A. Insel, et al.)\n• Mathematical Statistics with Applications (D. Wackeryly, W. Mendenhall, et al.)\n• A First Course in Probability (S. Ross)\n• Fundamentals of Complex Analysis (E. Saff, A. Snider)\n• Complex Variables and Applications (J. Brown, R. Churchill)\n• Analysis I (T. Tao)\n• Analysis II (T. Tao)\n• Principles of Mathematical Analysis (W. Rudin)\n• Advanced Calculus A Course in Mathematical Analysis (P. Fitzpatrick)\n• Elementary Analysis: The Theory of Calculus (K. Ross)\n• Abstract Algebra (D. Saracino)\n• Contemporary Abstract Algebra (J. Gallian)\n• Introduction to Topology (T. Gamelin, R. Greene)\n• Applied Combinatorics (A. Tucker)\n• Naive Set Theory (P. Halmos)\n• Introduction to Functional Analysis with Applications (W. Kreyszig)\n• Graph Theory (R. Gould)\n• Real Analysis (H. Royden)\n• Real and Complex Analysis (W. Rudin)\n• Foundations and Fundamental Concepts of Mathematics (H. Eves)\n\nCryptology\n\n• A First Course in Coding Theory (R. Hill)\n• Introduction to Cryptography with Coding Theory (W. Trappe, L. Washington)\n\nNotes\n\nI'm trying to find books that help with Group Theory, Ring Theory, and Field Theory. There is also a need for books about:\n\n• Functional Analysis\n• Public Key Cryptosystems\n• Cryptanalysis\n• Cryptology\n• Cryptography" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.67752457,"math_prob":0.6983379,"size":2108,"snap":"2021-31-2021-39","text_gpt3_token_len":573,"char_repetition_ratio":0.14020912,"word_repetition_ratio":0.006024096,"special_character_ratio":0.25142315,"punctuation_ratio":0.20485175,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99715644,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-07-26T04:22:58Z\",\"WARC-Record-ID\":\"<urn:uuid:98313c3e-2325-4ea0-a018-36e3bcde009c>\",\"Content-Length\":\"3151\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4d927de6-b00a-4e93-b68e-5e9d58120bdd>\",\"WARC-Concurrent-To\":\"<urn:uuid:95027875-ba5d-4d9f-8669-5f01cf262bbe>\",\"WARC-IP-Address\":\"50.116.47.154\",\"WARC-Target-URI\":\"https://kashi.re/blog/books.html\",\"WARC-Payload-Digest\":\"sha1:ETOK7ETXGDNZL2PAMY6W7X7DKRZBOV4E\",\"WARC-Block-Digest\":\"sha1:VLXOBIDEUTMGBINVKCUYZ5EBENW2VICI\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-31/CC-MAIN-2021-31_segments_1627046152000.25_warc_CC-MAIN-20210726031942-20210726061942-00496.warc.gz\"}"}
https://encyclopedia2.thefreedictionary.com/Fermat+number	[ "# Fermat numbers\n\n(redirected from Fermat number)\nAlso found in: Wikipedia.\n\n## Fermat numbers\n\n[′fer·mä ‚nəm·bərz]\n(mathematics)\nThe numbers of the form Fn = (2(2 n )) + 1 for n = 0, 1, 2, ….\nMcGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.\nReferences in periodicals archive ?\nThe main purpose of this paper is using the elementary method to study the estimate problem of S (Fn), and give a sharper lower bound estimate for it, where Fn = [2.sup.2n] + 1 is called the Fermat number.\n152), you state, \"The first Fermat number is [2.sup.2]+1, or 5,\" and later, \"the first four Fermat numbers are prime, but [among] the rest, up to and now including the 24th, none are prime.\"\nIn a similar effort, Lenstra and Manasse also organized a massive factor-by-mail project using the Number Field Sieve in order to factor the 9th Fermat number, [2.sup.512] + 1, which as the number whose factorization was 'Most Wanted' by mathematicians.\nPapadopoulos of the University of Maryland at College Park performed a massive calculation to prove that the 24th Fermat number, which is more than 5 million digits long, is not a prime.\nIn 1990, Lenstra and a colleague used the method to factor a 155-digit Fermat number, [2.sup.m] + 1, where m = [2.sup.9] (SN: 6/23/90, p.389).\n* Two computer scientists factored a record-breaking, 155-digit Fermat number (137: 389; 138: 90).\nSystems Research Center in Palo Alto, Calif., finished factoring the tenth Fermat number, proving that this 155-digit behemoth is the product of three prime numbers.\nThey discuss topics like what prime numbers are, division and multiplication, congruences, Euler's theorem, testing for primality and factorization, Fermat numbers, perfect numbers, the Newton binomial formula, money and primes, cryptography, new numbers and functions, primes in arithmetic progression, and sequences, with examples, some proofs, and biographical notes about key mathematicians.\nHis exploration of elementary and advanced topics in classical number theory covers a range of numbers, including Fermat numbers, Mersenne primes, powerful numbers, sublime numbers, Wieferich primes, insolite numbers, Sastry numbers, and voracious numbers.\nMore recently analysis of these so-called Fermat numbers have found no other primes above [F.sub.4].\nAlong the way, the lay reader learns to appreciate how mathematicians derive such principles as Fermat numbers, the Fibonacci Series, and Godel's incompleteness theorem.\nThis capacity to exploit Fermat numbers allows the general-purpose supercomputer to surmount technical problems inherent in other machines.\nSite: Follow: Share:\nOpen / Close" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8568295,"math_prob":0.9691893,"size":2309,"snap":"2020-45-2020-50","text_gpt3_token_len":561,"char_repetition_ratio":0.1626898,"word_repetition_ratio":0.0,"special_character_ratio":0.23083586,"punctuation_ratio":0.17633928,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9912406,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-31T13:38:38Z\",\"WARC-Record-ID\":\"<urn:uuid:bb1621ce-87c9-43fa-bf3b-b474af18ded3>\",\"Content-Length\":\"42761\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:aaccacc6-ddf9-4dcc-8e07-757e88bd721e>\",\"WARC-Concurrent-To\":\"<urn:uuid:ed6d3786-7ce8-4d41-af7b-35f774539c74>\",\"WARC-IP-Address\":\"85.195.124.227\",\"WARC-Target-URI\":\"https://encyclopedia2.thefreedictionary.com/Fermat+number\",\"WARC-Payload-Digest\":\"sha1:H4BZEF7C47FMLK4X7RVDMKIMWHBO2DRL\",\"WARC-Block-Digest\":\"sha1:U2CTG5N2MJEDLWZ3JGB6KJ56Y2XU2KCL\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-45/CC-MAIN-2020-45_segments_1603107918164.98_warc_CC-MAIN-20201031121940-20201031151940-00418.warc.gz\"}"}
https://www.data-stats.com/maximum-value/	[ "Question: Given a list of non-negative integers representing the value of each cell, determine the maximum amount of value you can get keeping in mind that you can’t use adjacent cells together.\n\nNow in this question, at first you might think of a greedy solution to take the largest element one by one if none of its neighbours is taken yet. But the problem with this approach is that it does not ensures a global optimal solution.\n\nFor instance, take this array\n\nInput: [1,4,6,5]\n\nIf we try to apply greedy approach here\nWe will take 6 from the array and after that our only option left is 1 so the final answer we get is 1+6=7, which we can clearly see is not the best answer because we can get 4+5=9.\n\nLet’s try to think if we can solve this problem using Dynamic Programming.\n\nFor this we need two things, first a DP state and second is base cases.\n\nLet’s take\n\n1. DP[i] = maximum amount of value we can get till position i.\n2. A[i] = value of ith cell.\n\nBase case:\n\n1. DP=0 because we haven’t used any cell yet.\n2. DP=value of first cell.\n\nTo calculate DP[i], we have two choices:\n\n1. Choose current index: In this case we can’t use the adjacent index i.e., (i-1)th index so the answer will be (DP[i-2]+A[i])\n2. Leave the current index: In this case our answer will be same as DP[i-1].\nTo get optimal solution, we will\ntake maximum of the two choices i.e., DP[i]=max(DP[i-1],DP[i-2]+A[i]).\n#include<bits/stdc++.h>\nusing namespace std;\nint main()\n{\nint n;\ncin>>n;\nvector<int> a(n+1),dp(n+1,0);\nfor(int i=1;i<=n;i++)\n{\ncin>>a[i];\n}\ndp=0;\ndp=a;\nfor(int i=2;i<=n;i++)\n{\ndp[i]=max(dp[i-1],dp[i-2]+a[i]);\n}\ncout<<dp[n]<<endl;\nreturn 0;\n}\n\nInput:\n\n5\n\n[2 ,8, 9, 8, 1]\n\nOutput:\n\n16\n\nThat’s all I have and thanks a lot for reading 😉 . Please let me know if any corrections/suggestions. Please do share and comments if you like the post. Thanks in advance…\n\nThanks Shiva Gupta for helping us to grow day by day. Shiva is expert in Data Structure and loves to solve competitive problems.\n\n$${}$$" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.81763184,"math_prob":0.9836664,"size":1982,"snap":"2022-05-2022-21","text_gpt3_token_len":555,"char_repetition_ratio":0.08746208,"word_repetition_ratio":0.0057971017,"special_character_ratio":0.29667002,"punctuation_ratio":0.13333334,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9981752,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-01-17T19:27:30Z\",\"WARC-Record-ID\":\"<urn:uuid:d930510a-d790-40ef-a988-1ff4e5d30d8b>\",\"Content-Length\":\"77874\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d8c4cc31-61ea-412d-97d6-4e0cb6e4a4ab>\",\"WARC-Concurrent-To\":\"<urn:uuid:5ef91bb9-3412-444e-bd77-ee7142ad7db0>\",\"WARC-IP-Address\":\"148.66.138.158\",\"WARC-Target-URI\":\"https://www.data-stats.com/maximum-value/\",\"WARC-Payload-Digest\":\"sha1:V2BS33G3FG6TXCGIQBPHOMGPEVWHFUAS\",\"WARC-Block-Digest\":\"sha1:NNWDJ7NBV4GQGRUOM6I5MOBUVOFVL6LR\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-05/CC-MAIN-2022-05_segments_1642320300616.11_warc_CC-MAIN-20220117182124-20220117212124-00068.warc.gz\"}"}
https://www.exceltactics.com/calculate-net-work-hours-using-networkdays/	[ "# How to Calculate Net Work Hours Between Two Dates", null, "Counting the number of hours between dates and times in Excel is normally a straightforward process. Since Excel stores dates as decimal numbers, you can just subtract the two to get your result. But when you are working with business hours, like for time sheets or hours worked, you need to take weekends and holidays into account. Excel has a function called NETWORKDAYS, but this only works with complete days. To calculate the Net Work Hours between two dates, we need to build our own formula. Here’s how…\n\n## Details of the Net Work Hours Formula\n\nA lot of assumptions have to be made about any formula that works with dates and times. Definitions of terms like work day, weekend, and holiday can vary, so this is a summary of the assumptions of this net work hours formula:\n\n• The eligible work hours in a given work day start and end at the same time each work day.\n• Weekend days do not count as work days. The weekend is defined as Saturday and Sunday of each week (look ahead to Non-Standard Weekends with NETWORKDAYS.INTL to change this).\n• Holidays, as listed, do not count as work days.\n• Partial work days at the start and end of a date period are included for their fraction of hours.\n• If the start time of the work day is later than the end time of the work day, the formula nets zero.\n• If the start date and time is later than the end date and time, the formula nets zero.\n\nThere are modifications available to the above rules later in this tutorial:\n\nIf your definition of weekend is different than above and you have Excel 2007 or newer, you can look ahead to Non-Standard Weekends with NETWORKDAYS.INTL to change this.\n\nIf you want to include all 24 hours of each work day rather than a specified range of work hours, you can look ahead to 24 Hour Net Workday Hours Variation to tally the total number of hours each work day.\n\n## Setting Up the Variables for Net Work Hours\n\nIn order to calculate the net work hours between two dates, we need to have some information to start with. Here are the variables necessary to make the net work hours formula work.\n\n\\$Start_Time – This is the time that the work day starts, as an Excel-formatted time. (e.g. 9:00AM) No date should be attached. The variable is marked with a dollar sign because you will probably want to anchor this cell reference in the formula.\n\n\\$End_Time – This is the time that the work day ends, as an Excel-formatted time. (e.g. 5:00PM) No date should be attached. The variable is marked with a dollar sign because you will probably want to anchor this cell reference in the formula.\n\nStart_Date – This is the starting date and time for your period, as an Excel-formatted date and time (e.g. 12/30/2013 11:00AM)\n\nEnd_Date – This is the ending date and time for your period, as an Excel-formatted date and time (e.g. 1/14/2014 4:00PM)\n\nHolidayList – This is a named range containing a list of days that are to be considered holidays and not counted as work days, even if they are not on the weekend. (e.g. New Year’s Day – 1/01/2013) It is usually located elsewhere on the worksheet.\n\n### Setting Up the HolidayList Named Range\n\nThe function NETWORKDAYS takes an array input called [holidays] that lists all the non-weekend days that should be excluded from the calculation. This means that we could give the function a series of dates in brackets, but they would need to be formatted in Excel date format. This means that they need to be converted using a DATE function. (See the Definitive Guide to Using Dates and Times in Excel for an explanation.) An easier approach is to build a named range.\n\nIn a separate worksheet tab, or in an un-used portion of your current spreadsheet, make a column list of the dates you want to include as holidays in Excel date-format. You can either use a DATE function to specify the year, month, and day for each, or let Excel auto-convert as you type.\n\nWhen you finish your list, select the list (and maybe a few extra cells, if you plan to add to it later). The name of the first cell in the selection appears in the name box above the spreadsheet cells…", null, "With your list selected, select the cell name in the name box, type in “HolidayList“, and press ENTER.", null, "You have now named the selected cells as an array. It can be referred to in the net work hours formula as HolidayList instead of manually entering the range, and the NETWORKDAYS function can automatically exclude those dates from calculations.\n\n## The Net Work Hours Formula\n\nHere is the formula, in all it’s glory. Replace the variable names like \\$Start_Time and HolidayList with your specific references. You can look to the sample file at the end of this post for an example of the formula in action.\n\n```=IF(OR(\\$End_Time<\\$Start_Time,End_Date<Start_Date),0,\n(NETWORKDAYS(Start_Date,End_Date,HolidayList)\n-(NETWORKDAYS(Start_Date,Start_Date,HolidayList)\nIF(MOD(Start_Date,1)>\\$End_Time,1,\n(MAX(\\$Start_Time,MOD(Start_Date,1))-\\$Start_Time)\n/(\\$End_Time-\\$Start_Time)))\n-(NETWORKDAYS(End_Date,End_Date,HolidayList)\nIF(MOD(End_Date,1)<\\$Start_Time,1,\n(\\$End_Time-MIN(\\$End_Time,MOD(End_Date,1)))\n/(\\$End_Time-\\$Start_Time))))\n(\\$End_Time-\\$Start_Time)24)```\n\nWe’ll walk through how it works in a moment, but first, let’s look at a couple of variations that can solve different versions of the net work hours problem.\n\n## Non-Standard Weekends with NETWORKDAYS.INTL\n\nIf you have Excel 2007 or newer, you can use a modified version of the NETWORKDAYS function called NETWORKDAYS.INTL to re-define what days are the “weekend” of each week. NETWORKDAYS.INTL takes an additional input called [weekend] that defines from a list which days are included in the weekend. The syntax for NETWORKDAYS and NETWORKDAYS.INTL are as follows:\n\n`=NETWORKDAYS(start_date, end_date, [holidays])`\n`=NETWORKDAYS.INTL(start_date, end_date, [weekend], [holidays])`\n\nNote the additional input [weekend]. The [weekend] input has the following input options:", null, "If you just enter 1, NETWORKDAYS.INTL behaves identically to NETWORKDAYS. Other options give you additional flexibility on your definition of “weekend”.\n\nFor example, if you want to define the weekend as Monday and Tuesday, the net work hours formula becomes the following:\n\n```=IF(OR(\\$End_Time<\\$Start_Time,End_Date<Start_Date),0,\n(NETWORKDAYS.INTL(Start_Date,End_Date,3,HolidayList)\n-(NETWORKDAYS.INTL(Start_Date,Start_Date,3,HolidayList)\nIF(MOD(Start_Date,1)>\\$End_Time,1,\n(MAX(\\$Start_Time,MOD(Start_Date,1))-\\$Start_Time)\n/(\\$End_Time-\\$Start_Time)))\n-(NETWORKDAYS.INTL(End_Date,End_Date,3,HolidayList)\nIF(MOD(End_Date,1)<\\$Start_Time,1,\n(\\$End_Time-MIN(\\$End_Time,MOD(End_Date,1)))\n/(\\$End_Time-\\$Start_Time))))\n(\\$End_Time-\\$Start_Time)24)```\n\nNote the change to NETWORKDAYS.INTL and the use of 3 as the input for the new [weekend] term in the function.\n\n## 24 Hour Net Workday Hours Variation\n\nThe above two versions operate on the assumption that there are defined start times and end times to each work day. Sometimes, though, we need to count all 24 hours of each eligible day. This is actually a simpler problem to solve, since we don’t need to look for ineligible hours in eligible work days. Therefore, the formula becomes:\n\n```=IF(Start_Date>End_Date,0,(NETWORKDAYS(Start_Date,End_Date,HolidayList)\n-NETWORKDAYS(Start_Date,Start_Date,HolidayList)MOD(Start_Date,1)\n-NETWORKDAYS(End_Date,End_Date,HolidayList)(1-MOD(End_Date,1)))*24)```\n\nIf you need to use a non-standard weekend definition, you can replace the NETWORKDAYS functions in the formula with the NETWORKDAYS.INTL as shown in the Non-Standard Weekend section above.\n\n## How the Net Work Hours Formula Works\n\nThere are a lot of moving parts to this formula, so it is useful to step through it procedurally. We’ll use an example case to walk through the logic of the formula step-by-step.\n\nFor this example, we’ll assume the following:\n\nStart_Date is December 30, 2013 at 11:00 am.\nEnd_Date is January 9, 2014 at 12:00 pm.\n\\$Start_Time is 9:00 AM.\n\\$End_Time is 5:00 PM.\nHolidayList includes only January 1, 2014 (New Years Day).\n\n### Step 1", null, "Check if the dates or times are impossible (i.e. the period ends before it starts). If so, return zero, otherwise, continue to step 2.\n\n### Step 2", null, "Add the total number of days in the date series, minus weekends and holidays.\n\n### Step 3", null, "Remove the greater of either the \\$Start_Time or the Start_Date time on the Start_Date day. Remove the greater of the time after the \\$End_Time or the time after the End_Date time on the End_Date day. (Not shown: If the Start_Date time starts after the \\$End_Time, subtract an entire day. If the End_Date time comes before the \\$Start_Time, subtract an entire day.)\n\n### Step 4", null, "For every work day between Start_Date and End_Date, count only the hours between \\$Start_Time and \\$End_Time.", null, "This is the outcome of the subtractions in the formula.\n\n### Step 5", null, "Add up the partial days in all the periods and multiply by 24 to calculate the total net work hours.", null, "Andrew Roberts has been solving business problems with Microsoft Excel for over a decade. Excel Tactics is dedicated to helping you master it. 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https://open.library.ubc.ca/cIRcle/collections/ubctheses/24/items/1.0064956	[ "# Open Collections\n\n## UBC Theses and Dissertations", null, "## UBC Theses and Dissertations\n\n### Design of SiGe/Si quantum-well optical modulators Tasmin, Tania 2010\n\nNotice for Google Chrome users:\nIf you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.\n\n#### Download\n\nMedia\n24-ubc_2010_fall_tasmin_tania.pdf [ 1003.66kB ]\nMetadata\nJSON: 24-1.0064956.json\nJSON-LD: 24-1.0064956-ld.json\nRDF/XML (Pretty): 24-1.0064956-rdf.xml\nRDF/JSON: 24-1.0064956-rdf.json\nTurtle: 24-1.0064956-turtle.txt\nN-Triples: 24-1.0064956-rdf-ntriples.txt\nOriginal Record: 24-1.0064956-source.json\nFull Text\n24-1.0064956-fulltext.txt\nCitation\n24-1.0064956.ris\n\n#### Full Text\n\n```Design of SiGe/Si Quantum-Well Optical Modulators by Tania Tasmin B.Sc., Bangladesh University of Engineering and Technology, 2004 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in The Faculty of Graduate Studies (Electrical and Computer Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) August 2010 c© Tania Tasmin 2010 Abstract An electro-optic modulator containing a single SiGe/Si quantum-well has been designed for operation at λ0 = 1.55 µm. This single quantum-well modulator has a lower VpiLpi than the 3 quantum-well modulator recently designed and optimized by Maine et al. for operation at λ0 = 1.31 µm, for which the VpiLpi product was 1.8 V·cm . Both modulators are de- rived from the detailed design given for their modulator in . This sin- gle quantum-well modulator contains a Si0.8Ge0.2 quantum-well with Non- Intentionally Doped (NID) and P+ highly doped layers on either side. With no field applied, holes from the P+ layers are captured by and confined in the quantum-well and when a reverse bias is applied holes are released from the quantum well and drift to the P+ contact layer. Variations of the hole distri- bution lead to changes in the free-carrier absorption and the refractive index of each layer and subsequently to phase modulation of guided TE modes. The VpiLpi product of the single quantum-well modulator is estimated 1.09 V·cm for low voltage linear modulation and 1.208 V·cm for 0 to 1.6 V digital modulation, whereas the 3 quantum-well modulator gives a VpiLpi of 2.039 V·cm for 0 to 6 V digital modulation for operation at λ0 = 1.55 µm. Also, the optical loss in the single quantum-well (5.36 dB/cm at V = 0 V) is lower than that of the 3 quantum-well structure (5.75 dB/cm at V = 0 V). This ii Abstract single quantum-well modulator should also offer higher frequency operation than the 3 quantum-well modulator. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv Statement of Co-Authorship . . . . . . . . . . . . . . . . . . . . . xvi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Optical Modulation in Si and Si-based Materials . . . . . . . 3 1.2.1 Franz-Keldysh Effect (FKE) . . . . . . . . . . . . . . 5 1.2.2 Quantum-Confined Stark Effect (QCSE) . . . . . . . 9 1.2.3 Free Carrier Absorption Effect . . . . . . . . . . . . . 10 1.3 Silicon Modulators Based on FKE, QCSE, and Free Carrier Absorption Effect . . . . . . . . . . . . . . . . . . . . . . . . 14 iv Table of Contents 1.3.1 Mach-Zehnder Interferometer . . . . . . . . . . . . . . 19 1.3.2 Fabry-Perot Interferometer . . . . . . . . . . . . . . . 20 1.3.3 Ring Resonator . . . . . . . . . . . . . . . . . . . . . 21 1.4 Organization of the Thesis . . . . . . . . . . . . . . . . . . . 23 2 Material Choice, Device Structure, and Electrical and Op- tical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.1 SiGe/Si Heterostructure Properties . . . . . . . . . . . . . . 28 2.1.1 Bandgaps and Band Alignments for SiGe/Si Heterostruc- ture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.1.2 Refractive Index and Absorption Coefficient . . . . . 31 2.2 Software for Electrical Simulation . . . . . . . . . . . . . . . 32 2.2.1 Basic Steps of Electrical Analysis in ATLAS . . . . . 33 2.2.2 Models Incorporated for Device Simulation . . . . . . 34 2.3 Optical Simulations . . . . . . . . . . . . . . . . . . . . . . . 38 2.3.1 Silicon-On-Insulator (SOI) Waveguides . . . . . . . . 38 2.3.2 Mode Solver for Optical Simulation . . . . . . . . . . 41 2.4 Design of the SiGe/Si Optical Modulator . . . . . . . . . . . 42 2.4.1 Device Structure . . . . . . . . . . . . . . . . . . . . 44 2.4.2 Electrical and Optical Analysis . . . . . . . . . . . . . 45 2.5 DC analysis in 3 Quantum-Well SiGe/Si Modulator . . . . . 47 2.5.1 Electrical Analysis . . . . . . . . . . . . . . . . . . . . 49 2.5.2 Calculation of Absorption Coefficient and Refractive Index . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.5.3 Effective Index and Optical Loss Calculation . . . . . 57 v Table of Contents 2.6 Transient Analysis . . . . . . . . . . . . . . . . . . . . . . . . 59 2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3 Single Quantum-Well SiGe/Si Optical Modulator . . . . . 65 3.1 DC analysis in Single Quantum-Well SiGe/Si Modulator . . 66 3.1.1 Refractive Index Change in Single SiGe/Si Quantum- Well Optical Modulator . . . . . . . . . . . . . . . . . 67 3.1.2 Effective Index and Optical Loss Calculation . . . . . 68 3.2 Transient Analysis . . . . . . . . . . . . . . . . . . . . . . . . 71 3.3 Mach-Zehnder Interferometer Performance . . . . . . . . . . 75 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4 Summary, Conclusion, and Suggestions for Future Work . 80 4.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.2 Suggestions for Future Work . . . . . . . . . . . . . . . . . . 83 4.2.1 All-Silicon Optical Modulators . . . . . . . . . . . . . 83 4.2.2 Traveling-Wave Electrodes . . . . . . . . . . . . . . . 83 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Appendices A Mach-Zehnder Interferometer . . . . . . . . . . . . . . . . . . 94 B Mode Solver Program . . . . . . . . . . . . . . . . . . . . . . . 98 vi List of Tables 1.1 Si electro-absorption modulators . . . . . . . . . . . . . . . . 15 1.2 Free carrier absorption based silicon modulators . . . . . . . 17 vii List of Figures 1.1 Kerr effect in c-Si, re-digitized from . . . . . . . . . . . . . 5 1.2 (a) Energy band diagram, (b) absorption spectrum in bulk semiconductor with and without the electric field. . . . . . . . 6 1.3 Field dependence of refractive index change at two wave- lengths, re-digitized from . . . . . . . . . . . . . . . . . . . 8 1.4 Quantum-well wavefunctions with and without electric field. . 9 1.5 Optical absorption of c-Si showing the influence of various concentrations of (a) free electrons (b) free holes, re-digitized from . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.6 Refractive index perturbation in c-Si produced as a function of free carrier concentration at (a) λ = 1.3 µm, (b) λ = 1.55 µm, re-digitized from . . . . . . . . . . . . . . . . . . . . . 13 1.7 Schematic view of the Mach-Zehnder interferometer. Two optical Y-branch couplers are used to split and recombine the incoming light. . . . . . . . . . . . . . . . . . . . . . . . . 20 1.8 Schematic view of the Fabry-Perot cavity. . . . . . . . . . . . 21 1.9 Schematic view of the ring resonator. . . . . . . . . . . . . . . 22 viii List of Figures 2.1 Critical layer thickness of SiGe layer grown on Si as a function of the Ge mole fraction, re-digitized from . . . . . . . . . . 30 2.2 Band alignments for Si1−xGex/Si heterostructures on Si sub- strate, where χ is the electron affinity. . . . . . . . . . . . . . 32 2.3 Absorption coefficient of Si and strained SiGe, temperature is 300K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.4 The two major recombination processes in silicon are (a) SRH recombination (b) Auger recombination . . . . . . . . . . . . 35 2.5 Schematic band diagram of an abrupt heterojunction. Ef1 and Ef2 represents the electron quasi-fermi level in each semi- conductor region. JTE and JTunnel are the thermionic emis- sion and tunneling current, respectively. . . . . . . . . . . . . 37 2.6 SOI planar waveguide, the refractive indices (n) of the layers are shown for the wavelength of 1.55 µm. The z direction is taken as the direction of light propagation. . . . . . . . . . . 39 2.7 Different configurations of three dimensional waveguides (a) strip, (b) embedded strip, (c) rib (or ridge), and (d) strip- loaded waveguides. . . . . . . . . . . . . . . . . . . . . . . . . 40 2.8 Cross section of the SOI rib waveguide discussed in . . . . 41 2.9 2D Mode profile for the fundamental TE mode of the SOI rib waveguide discussed in . Each line represents an identi- cal field value (−3 dB step between lines, −45 dB minimum value). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.10 Schematic view of the 3 quantum-well SiGe/Si optical mod- ulator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ix List of Figures 2.11 Hole distribution in various layers at various reverse bias volt- ages. With the increase of reverse biasing, hole deplete from the quantum-wells. Holes starts to deplete from the NID layer on the P+ side after all the quantum-wells are fully depleted i.e. after 7.5 V NID layer starts depleting. . . . . . . . . . . . 50 2.12 (a) Valence and conduction band energy profiles for the 3 quantum-well SiGe/Si structure, black lines are for V = 0 V and red lines are for V = 6 V, the dotted line shows the quasi-fermi energy level for holes (b) electric field in the 3 quantum-well SiGe/Si structure. . . . . . . . . . . . . . . . . 51 2.13 (a) Effective index variation in a single quantum-well modu- lator with and without averaging the hole concentration (b) optical loss with and without averaging the hole concentration. 53 2.14 Electron distribution in various layers at various reverse bias voltages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.15 Refractive index changes (∆nv = nv − n0) at λ0 = 1.55 µm and λ0 = 1.31 µm (a) in the quantum-wells with dash-dot line, dotted line, and solid line for quantum- well 1, 2, 3 re- spectively. (b) in the NID layer on the P+ side . . . . . . . . 55 2.16 Refractive index changes at λ0 = 1.55 µm in the (a) NID layers (b) δ-doped-P+ layers . . . . . . . . . . . . . . . . . . . 56 2.17 (a) Effective index variation (∆neff-v = neff-v − neff-0) of the 3 quantum-well modulator; the blue point shows the effective index variation at 6 V obtained by Marris et al. in Ref. (b) optical loss at various voltages. . . . . . . . . . . . . . . . 58 x List of Figures 2.18 Hole density distribution at various times in the 3 quantum- well modulator, only the quantum-wells and the P+ layers are shown in the figure. . . . . . . . . . . . . . . . . . . . . . . . 60 2.19 Hole density distribution with time in the 3 quantum-well modulator for (a) t = 0 ps to t = 100 ps (b) t = 100 ps to t = 200 ps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.20 Effective index variation with time in the 3 quantum-well modulator for 0 to 6 V variation (a) with field dependent mobility model (b) without field dependent mobility model. . 62 3.1 Schematic view of the SiGe/Si optical modulator. . . . . . . . 66 3.2 Comparison of the refractive index change in the single quantum- well structure and in the first quantum-well of the 3 quantum- well structure at λ0 = 1.55 µm. The refractive index change in the NID layers on the P+ side are shown for both the single quantum-well and the 3 quantum-well modulator. . . . . . . . 68 3.3 Comparison of the (a) effective index variation (b) slope of the effective index variation for single quantum-well and 3 quantum-well structure. . . . . . . . . . . . . . . . . . . . . . 69 3.4 Comparison of the optical loss in the single quantum-well modulator and the 3 quantum-well modulator. . . . . . . . . 70 3.5 Hole density distribution at various times in the single quantum- well modulator, only the quantum-well and the P+ layers are shown in the figure. . . . . . . . . . . . . . . . . . . . . . . . 72 xi List of Figures 3.6 Hole density distribution with time in the single quantum- well modulator for (a) t = 0 ps to t = 100 ps (b) t = 100 ps to t = 200 ps. . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.7 Effective index variation with time in the single quantum-well modulator for 0 to 1.6 V variation. . . . . . . . . . . . . . . . 74 3.8 Ratio of output to input intensity in a MZI, dotted line shows the slope of this curve. . . . . . . . . . . . . . . . . . . . . . 77 4.1 (a) All-Si optical modulator (vertical diode), (b) all-Si optical modulator (lateral diode). . . . . . . . . . . . . . . . . . . . . 84 A.1 Schematic view of the Mach-Zehnder interferometer. Two optical Y-branch couplers are used to split and recombine the incoming light. . . . . . . . . . . . . . . . . . . . . . . . . 95 B.1 A typical finite difference mesh for an integrated waveguide. The rib waveguide is shown by the shaded region. P, N, S, E, W, NE, NW, SE and SW are used to label, respectively, the grid point under consideration, and its nearest neighbours to the north, south, east, west, north-east, north-west, south- east, and south-west. . . . . . . . . . . . . . . . . . . . . . . . 100 xii Acknowledgements I would like to express my profound gratitude to my thesis supervisor: Dr. Nicolas A. F. Jaeger for his friendly supervision and guidance through- out the course of this research work. He was always ready to offer timely suggestions and I have learned a lot from him. Sincere and special thanks go to Dr. Lukas Chrostowski (my co-supervisor) and Dr. Nicolas Rouger for the endless discussions on problems I encountered during the course of this work. Without their support and kind guidance, I would not have made it as far as I did. I am also grateful to Dr. Guangrui Xia for her valuable discussions and assistance. I would also like to thank everybody who has contributed in one way or another to my experience at UBC: Dr. Jaeger’s graduate students, Dr. Chrostowski’s graduate students, Dr. Alina Kulpa. I would like to thank my parents Md. Shafiqul Islam and Shahjat Merina for their prayers and encouragement and also my husband Sharif Fakhruz Zaman and my sweet daughter Sharif Maymunah Zaman for their support, love, and patience. I would also like to thank my grandmother Begum Fazilatunnesa for her love and prayers at every stage my life. Above all, I give thanks to Allah, for the gift of life, with a bit of knowl- xiii Acknowledgements edge and understanding and mercy. xiv Dedication To my loving parents. xv Statement of Co-Authorship Chapter two and chapter three of this thesis will be published in SPIE Photonics North Proceedings, 2010. The authors are Tania Tasmin, Nicolas Rouger, Guangrui Xia, Lukas Chrostowski, and Nicolas A. F. Jaeger. The literature review and the modeling for this research work were done by Tania Tasmin under the supervision of Dr. Nicolas Jaeger, Dr. Lukas Chrostowski, and Dr. Guangrui Xia. Tania Tasmin did the final preparation for the manuscript of the paper after careful revision and approval by Nicolas Rouger, Guangrui Xia, Lukas Chrostowski, and Nicolas A. F. Jaeger. xvi Chapter 1 Introduction Today optical interconnects dominate long-distance communications due to their low transmission loss, high bandwidth, and immunity to interference. But, in the case of short distance communications, such as chip-to-chip and on-chip interconnects, the electrical interconnects (metal wires) still domi- nate. However, due to the dramatic increase in the number of transistors per chip and the shrinkage of the transistors, electrical interconnects encounter many problems such as RC propagation delays, signal distortion, and high power consumption. Thus with the transistors getting smaller, electrical in- terconnects have become the primary bottleneck for the improvement of the chip performance . Optical interconnects offer a promising way to allevi- ate this bottleneck. The basic advantages of the optical interconnects over the electrical interconnects are reduced delay, lower power consumption, and higher bandwidth. 1.1 Motivation The improvement of the modern technology leads a dramatic increase in the number of transistors per chip as well as the shrinkage of transistors on a microprocessor . This shrinking process results in a spectacular im- 1 1.1. Motivation provement of the device speed. On the microprocessor, metallic (electrical) interconnects are used for signalling and clocking. Local interconnects, con- sisting of very thin conductors, are used to connect various parts of multiple transistors within a functional block on the chip. Global interconnects pro- vide clock and signal distribution between different functional blocks on the chip and deliver power/ground to all of the functional blocks . With the miniaturization of transistors on the chip, the length of the local intercon- nects scale down and propagation delay for the local interconnects is not an issue. But the cross-sectional area of the global interconnects reduce without changing the length which leads to the increase of the resistance of the global interconnect and hence the increase of RC propagation delays . Other problems with electrical interconnects are the power consump- tion and the signal distortion. Thus with the transistor shrinking, electrical interconnects have become the primary bottleneck for improvement of the device performance . Optical interconnects offer a promising way to alle- viate this bottleneck. The basic advantages of the optical interconnects over the electrical interconnects are reduced RC delay, lower power consumption, and higher bandwidth, particularly for the longer distance interconnects (e.g. across the chip, between processor cores). The basic building blocks of an optical interconnect system are light sources, modulators, waveguides, and detectors. Light sources can be either directly modulated (light source is turned on/off according to an electrical signal) or it can be used with an external modulator (light coming from the light source will be coupled with an external modulator and the light intensity at the modulator output is controlled by an applied electrical signal 2 1.2. Optical Modulation in Si and Si-based Materials to the modulator). Waveguides provide a means of carrying light from the light sources to the modulators and from the modulator to the detector or other parts of the chip. Detectors are used to convert the modulated light back into electrical signal. Silicon-based materials are the best platform for the various compo- nents of on-chip optical interconnects. Due to their CMOS (Complementary Metal-Oxide-Semiconductor) compatibility, both the electronic circuit and photonic circuit can be grown monolithically on the same substrate, reduc- ing the cost . Moreover, stronger optical confinement can be obtained with SOI (Silicon-On-Insulator) waveguides due to the higher refractive in- dex contrast between Si and SiO2 as compared to those obtained with III-V based optical interconnects grown on Si substrates . Significant progress has been made to develop Si-based, on-chip, optical components and, in par- ticular, the light sources, the waveguides, and the photodetectors . The main obstacle for implementing the optical interconnect system is developing a Si-based optical modulator with low drive voltage, small size, low optical loss, and high speed. In this thesis we have designed SiGe/Si quantum-well modulators having all these properties. 1.2 Optical Modulation in Si and Si-based Materials Optical modulation is defined as the process of varying a property of light e.g., phase, frequency, polarization, or intensity according to an applied electrical signal (i.e., the process of impressing information on a light car- 3 1.2. Optical Modulation in Si and Si-based Materials rier). The optical modulators can be grouped into two major types: electro- absorption modulators and electro-optic modulators. In an electro-absorption modulator, a variation of the light intensity at the modulator output, according to the applied electric field, is obtained through the variation of the optical absorption coefficient of the waveguide material. ON/OFF states correspond to the situation of low absorption coefficient and high absorption coefficient, respectively. In an electro-optic modulator, the phase of the carrier light is varied according to the applied electric field through the variation of the refractive index of the waveguide material. An integrated interferometer is typically used to convert the re- fractive index induced phase modulation into the intensity modulation. The electro-absorption effect include the Franz-Keldysh effect and the quantum-confined Stark effect. The electro-optic effect include Pockels ef- fect, Kerr effect, and free carrier absorption effect. The Pockels effect, in which the refractive index change of a material is linearly proportional to the applied electric field, is absent in Si. The Kerr effect, in which the refractive index change is proportional to the square of the applied electric field, is very small in Si as shown in Fig. 1.1 . Hence, the Franz-Keldysh effect, the quantum-confined stark effect, and the free carrier absorption effect, which are the relevant optical modulation mechanisms in Si and Si-based materials, are described in section 1.2.1, 1.2.2, and 1.2.3, respectively. 4 1.2. Optical Modulation in Si and Si-based Materials 104 105 106 10−8 10−7 10−6 10−5 10−4 Applied field (V/cm) − ∆n Figure 1.1: Kerr effect in c-Si, re-digitized from . 1.2.1 Franz-Keldysh Effect (FKE) When a strong electric field is applied to a bulk semiconductor, the ab- sorption coefficient of the semiconductor changes according to the applied electric field. Also, with the application of the electric field, there is a tilt in the valence band and in the conduction band. At some locations, the energy difference between the conduction band and the valence band is re- duced (by dE) below the bandgap energy Eg. Photons with energy higher than Eg− dE are absorbed at these locations by exciting the electrons from the valance band to the conduction band as shown in Fig. 1.2(a). So there is an absorption tail below the bandgap energy as shown in Fig. 1.2(b). This effect is known as Franz-Keldysh effect . 5 1.2. Optical Modulation in Si and Si-based Materials E<Eg E>Eg (a) With Electric Field Without Electric Field A b s o r p t i o n C o e f f i c i e n t ( α ) Eg A b s o r p t i o n C o e f f i c i e n t ( Energy (E) Eg-dE (b) Figure 1.2: (a) Energy band diagram, (b) absorption spectrum in bulk semiconductor with and without the electric field. 6 1.2. Optical Modulation in Si and Si-based Materials The complex refractive index of a material can be written as n + ik, where n is the refractive index and k is the optical extinction coefficient. The absorption coefficient, α is related to k by k = αλ/4pi. The change in the real part of the refractive index (∆n) and in the imaginary part of the refractive index (∆α) are related by the following Kramers-Kronig relations ∆n(E) = c pi P ∫ ∞ 0 ∆α(Ē) Ē2 − E2 dĒ (1.1) ∆α(E) = − c pi P ∫ ∞ 0 ∆n(Ē) Ē2 − E2 dĒ (1.2) The absorption change with the applied electric field is obtained by ∆α(E, ξ) = α(E, ξ)− α(E, 0) (1.3) where E is the energy of the light and ξ is the applied electric field. Hence, in the Franz-Keldysh effect, the electric field involves a change in both the absorption coefficient and in the refractive index. Soref and Benett quan- tified the change in the refractive index in silicon using the electro-absorption spectrum measured by Wendland and Chester . They plotted the change in refractive index as a function of the applied electric field (Franz-Keldysh effect) at λ0 = 1.07 µm and at λ0 = 1.09 µm which are shown in Fig. 1.3. The Franz-Keldysh effect falls very significantly at the telecommunication wavelength, so this effect is not a favourable choice for the silicon electro- optic modulators. 7 1.2. Optical Modulation in Si and Si-based Materials 40 140 240 34010 −6 10−5 10−4 E (kV/cm) ∆n λ =1.07 µ m λ =1.09 µ m Figure 1.3: Field dependence of refractive index change at two wavelengths, re-digitized from . 8 1.2. Optical Modulation in Si and Si-based Materials 1.2.2 Quantum-Confined Stark Effect (QCSE) In a quantum-well, when no electric field is applied, the electron and the hole wavefunctions are symmetrical inside the well and electron-hole are coupled together by coulombic forces to form excitons. These excitons give sharp resonance peaks at the band-edge of the absorption spectrum in absence of an electric field. With the application of the electric field, the quantum-well energy decreases with respect to the center of the well and electron and hole wavefunctions shift towards the opposite sides of the quantum-well. Due to the electron and hole separation in the quantum-well, the exciton peak is lowered at the band-edge and the band-edge is shifted towards the long wavelength (red shifting) . Ec Ev Without Electric Field With Electric Field Figure 1.4: Quantum-well wavefunctions with and without electric field. Recently the discovery of strong QCSE is reported in compressively strained Ge quantum-wells with SiGe barriers . We will discuss about 9 1.2. Optical Modulation in Si and Si-based Materials this electro-absorption modulator in section 1.3. 1.2.3 Free Carrier Absorption Effect If the energy of the incident light is so small that the photons cannot transfer electrons from the valence band to the conduction band, electrons (or holes) undergo transitions within different states in the same band by absorbing the incoming light. This is known as the free carrier absorption . The free carrier absorption depends on the free carrier concentration of the material. If carriers are injected to an undoped sample, the absorption coefficient of that material increases and the refractive index decreases . If the carriers are removed from a doped sample, the opposite effect takes place. As the refractive index change depends on both the frequency of light and the plasma of free carriers, this absorption is also called as plasma dispersion effect. In classical physics, free carriers (electrons and holes) are treated as particles, and they are forced into motion by the incoming light. The dis- placement vector of the carriers, which is related to the electric field of the light, can be found by solving the equation of motion of the carriers. By putting this value of the displacement vector into the equation for dielectric permittivity, the formula for free-carrier-induced absorption change and the refractive index change are found as [12, 13] ∆α = λ2e3 4pi2c3n\u000f0 ( ∆Ne µem∗2e + ∆Nh µhm ∗2 h ) (1.4) 10 1.2. Optical Modulation in Si and Si-based Materials ∆n = λ2e2 8pi2c2n\u000f0 ( ∆Ne m∗e + ∆Nh m∗h ) (1.5) here e is the charge of an electron, \u000f0 is the free space permittivity, n is the unperturbed refractive index, ∆Ne and ∆Nh are the change of concentra- tion of free electrons and holes, respectively; m∗e and m∗e are the effective masses of electron and hole respectively and µe and µh are the mobilities of electrons and holes, respectively. However, these equations (defined as the Drude model) for free carrier absorption ignore all of the scattering process involving phonons or other impurities for conservation of momentum. Later Soref and Benett collected some experimental results [14–16] on the absorp- tion spectrum of doped silicon (which are shown in Fig. 1.5), which include all of the scattering effects on the change of free carrier absorption. From these curves they calculated the change in free carrier absorption and from the change in absorption they calculated the change in refractive index using the Kramers-Kronig relations. Then they used these results to determine the carrier-concentration dependence of the refractive index change at two specific wavelengths: λ = 1.3 and 1.55 µm, which are shown in Fig. 1.6(a) and Fig. 1.6(b), respectively. From these figures, they concluded that the free holes are more effective in perturbing the refractive index than the free electrons. They also pro- duced the following expressions, which are now used almost universally to evaluate changes due to injection or depletion of free carriers in silicon: 11 1.2. Optical Modulation in Si and Si-based Materials 0 0.4 0.8 1.2 1.6 2 2.4 2.810 0 101 102 103 104 105 Photon Energy (eV) α (cm − 1 ) undoped0.32 6 10 24 32 100× 1018 cm−3 (a) 0 0.4 0.8 1.2 1.6 2 2.4 2.810 0 101 102 103 104 105 Photon Energy (eV) α (cm − 1 ) undoped 0.5 2.8 6 12 70 100× 1018 cm−3 (b) Figure 1.5: Optical absorption of c-Si showing the influence of various concentrations of (a) free electrons (b) free holes, re-digitized from . 12 1.2. Optical Modulation in Si and Si-based Materials 1017 1018 1019 1020 10−4 10−3 10−2 10−1 ∆N (cm−3) − ∆n λ =1.3 µm Free Electrons Free Holes (a) 1017 1018 1019 1020 10−4 10−3 10−2 10−1 ∆N (cm−3) − ∆n λ =1.55 µm Free Electrons Free Holes (b) Figure 1.6: Refractive index perturbation in c-Si produced as a function of free carrier concentration at (a) λ = 1.3 µm, (b) λ = 1.55 µm, re-digitized from . 13 1.3. Silicon Modulators Based on FKE, QCSE, and Free Carrier Absorption Effect At λ0 = 1.3 µm ∆α = 5.9× 10−18∆N + 4.0× 10−18∆P (1.6) ∆n = −6.2× 10−22∆N − 6.0× 10−18∆P 0.8 (1.7) At λ0 = 1.55 µm ∆α = 8.5× 10−18∆N + 6.0× 10−18∆P (1.8) ∆n = −8.8× 10−22∆N − 8.5× 10−18∆P 0.8 (1.9) where ∆N and ∆P are, respectively, the electron and the hole concentration variations cm−3 with respect to the intrinsic carrier concentration. 1.3 Silicon Modulators Based on FKE, QCSE, and Free Carrier Absorption Effect Table 1.1 summarized the recently reported SiGe electro-absorption mod- ulators based on FKE and QCSE. Typically electro-absorption modulators suffer from chirp which can be reduced by using electro-optic modulators as these modulators can be implemented in a Mach-Zehnder push-pull config- uration. To date, the free carrier absorption effect has been the most effective 14 1.3. Silicon Modulators Based on FKE, QCSE, and Free Carrier Absorption Effect Group Kimerling David Miller Publication Year 2008 2007 Modulation Mechanism Franz-Keldysh effect Quantum Confined Stark effect Operating Wavelength 1539− 1553 nm 1441− 1461 nm Structure Reverse biased P-I-N diode grown on a SOI substrate. The intrinsic region contains a 50 µm long, 600 nm wide and 400 nm high Si0.08Ge0.92 epitaxial layer. This layer is coupled with Si waveguides at the input and output. Reverse biased P-I- N diode grown on a Si0.1Ge0.9 buffer on a Si substrate. The intrinsic region contains 40 pairs of compres- sively strained Ge/SiGe quantum-wells (15.5 nm Ge well/33 nm Si0.16Ge0.84 barrier). Interferometric Structure - Fabry-Perot cavity Insertion Loss 3.7 dB at 1550 nm - Extinction Ratio 8 dB at 1550 nm with 7 V Peak contrast ratio 7.3 dB at 1457 nm for 0 to 10 V swing 3 dB Bandwidth 1.2 GHz - Table 1.1: Si electro-absorption modulators mechanism for varying the refractive index in silicon, which is polarization independent . Carrier injection, carrier depletion, and carrier accumula- tion are the most commonly used mechanisms that modify the free carrier concentration in Si modulators based on free carrier absorption . Three different device configurations, namely, P-Intrinsic-N (P-I-N) diodes, PN diodes or metal-oxide-semiconductors field-effect transistors (MOSFET) are used to exploit these mechanisms. In carrier-injection based modulators, free carriers are injected into the intrinsic region of a P-I-N diode by forward biasing the diode. The change 15 1.3. Silicon Modulators Based on FKE, QCSE, and Free Carrier Absorption Effect in free carriers leads to an increase in the absorption coefficient as well as a decrease in the refractive index of the device. These devices require a large forward bias current density to ensure significant carrier concentration change. Moreover, these devices have moderate response time (>> 1 ns) due to the electron-hole recombination process. In carrier-depletion based modulators, carriers are stored in the active region of a PN/P-I-N diode without an applied field, carriers are swept out of the active region by reverse biasing the diode. The change in free carrier leads to a decrease in the absorption coefficient as well as an increase in the refractive index of the device. In these modulators, only one kind of carrier (either electron or hole) is involved. Hence, no recombination process takes place and a high frequency can be expected from these devices. Moreover, current density is reduced in these devices which leads to the reduced power consumption as compared to the carrier-injection based devices. In case of the carrier- accumulation based devices, charge carriers are accumulated near the gate dielectric in a MOS capacitor when a voltage is applied to the device. Table 1.2 summarized the recently reported Si modulators based on the free carrier absorption effect. 16 1.3. Silicon Modulators Based on FKE, QCSE, and Free Carrier Absorption Effect Table 1.2: Free carrier absorption based silicon modulators Group Manipatruni Liao Liao Publication Year 2007 2005 2007 Result Type Modeling Experimental Experimental Operating Wavelength - - 1550 nm Carrier Den- sity Variation by- Injection Accumulation Depletion Modulator Structure PIN diode inte- grated with Si rib waveguide MOS capacitor integrated with Si rib waveguide PN diode inte- grated with Si rib waveguide Interferometer Ring resonator MZI with 13 mm phase shifter MZI with 5 mm phase shifter Vpi · Lpi - 3.3 V·cm 4.0 V·cm Phase shifter loss - 10 dB 7 dB 3 dB Band- width 40 Gbps RC cut-off fre- quency 10.2 GHz 20 GHz Table continued on next page 17 1.3. Silicon Modulators Based on FKE, QCSE, and Free Carrier Absorption Effect Table continued from previous page Group Marris Marris Marris Publication Year 2006 2008 2008 Result Type Experimental Experimental Modeling Operating Wavelength 1550 nm 1550 nm 1310 nm Carrier Den- sity Variation by- Depletion Depletion Depletion Modulator Structure Vertical PIN diode grown on SOI substrate, a highly doped P+ layer embedded in the intrinsic region Lateral PIN diode grown on SOI substrate, a highly doped P+ slit embedded in the intrinsic region Vertical PIN diode grown on SOI substrate, The intrinsic region con- sists of three Si0.8Ge0.2/Si quantum-well surrounded by P+ highly doped layers. Interferometer Fabry-Perot cav- ity MZI with 4 mm phase shifter phase shifter length 3 mm Vpi · Lpi 3.1 V·cm (1 V·cm by simulation) 5.0 V·cm (2 V·cm by simulation) 1.8 V·cm Phase shifter loss - 5 dB 9 dB 3 dB Band- width 90 GHz by simu- lation 10 GHz RC cut-off fre- quency 16 GHz 18 1.3. Silicon Modulators Based on FKE, QCSE, and Free Carrier Absorption Effect The change in free carrier concentration in these devices leads to the change in free carrier absorption which, in turn, changes the refractive index of the device. The phase of the light passing through the device will be changed due to the change in refractive index. This phase modulator can be converted into intensity modulator using a Mach-Zehnder interferometer structure, Fabry-Perot cavity structure, or ring resonator structure. The operating principles of these intensity modulators are described in section 1.3.1, section 1.3.2, and section 1.3.3, respectively. 1.3.1 Mach-Zehnder Interferometer An integrated Mach-Zehnder interferometer consists of an input waveguide, a splitter, two phase shifters, an output combiner, and an output waveg- uide, as illustrated in Fig. 1.7. The optical beam coming through the input waveguide is split into two optical beams by the splitter. The two optical beams travel through the two phase shifters inserted into the arms of the Mach-Zehnder interferometer, and then recombine at the output combiner. The ON state is achieved when the two optical beams arrive at the combiner in phase and interface constructively to produce a high intensity. When an electric field is applied to the phase shifters to create a relative path dif- ference between the two optical beams, the intensity is reduced. The OFF state is achieved when there is a differential phase shift of pi radians and a minimum intensity is produced at the output. The splitter (or the combiner) can be either an optical Y-branch or a 3-dB coupler. 19 1.3. Silicon Modulators Based on FKE, QCSE, and Free Carrier Absorption Effect Phase-Control Waveguide Sections Optical Y-Branch to split signal Optical Y-Branch to recombine signal O p t i c a l O u t p u t P o w e r O p t i c a l I n p u t P o w e r O p t i c a l O u t p u t P o w e r O p t i c a l I n p u t P o w e r Figure 1.7: Schematic view of the Mach-Zehnder interferometer. Two op- tical Y-branch couplers are used to split and recombine the incoming light. 1.3.2 Fabry-Perot Interferometer A Fabry-Perot cavity consists of two partially reflecting mirrors enclosing a resonator cavity as shown in Fig. 1.8. The transmission of the cavity will be maximum when the optical length of the cavity matches the resonance condition: nL = pλ/2, where n and L are, respectively, the refractive index and the length of the cavity, and p is an integer. The phase shifter is inserted into the cavity. When no bias is applied to the phase shifter, the transmission presents sharp maxima at some wavelengths. When a bias is applied to the phase shifter, the resonance condition modifies due to the change of the refractive index n, the resonance (resonant wavelengths are shifted) is shifted, and the cavity transmission at a given wavelength varies. 20 1.3. Silicon Modulators Based on FKE, QCSE, and Free Carrier Absorption Effect L Partially reflecting mirrors Reflected Light (Er) Transmitted Phase Shifter Input Light (Ei) Light (Et) Figure 1.8: Schematic view of the Fabry-Perot cavity. 1.3.3 Ring Resonator A ring resonator is based on a ring waveguide near a straight waveguide. To realize an optical modulator, the phase shifter is inserted in the ring. Light coming from the straight waveguide is coupled into the ring. The coupling of light from the straight waveguide to the ring depends on the gap between the waveguides. After propagating in the ring, light is coupled back to the straight waveguide. The transmission from the ring will be minimum 21 1.3. Silicon Modulators Based on FKE, QCSE, and Free Carrier Absorption Effect when the ring circumference matches the resonance condition: nd = pλ/2, where n and d are, respectively, the refractive index and the circumference of the ring, and p is an integer. When no bias is applied to the phase shifter integrated in the ring, the transmission presents sharp minima at some wavelengths. When a bias is applied to the phase shifter in the ring, the phase shift that is encountered by the optical mode propagating in the ring varies, and the resonance (resonant wavelengths are shifted) is shifted, and the transmission from the ring at a given wavelength varies. p+ doped Waveguide Output V n+ doped Input Ring Figure 1.9: Schematic view of the ring resonator. 22 1.4. Organization of the Thesis 1.4 Organization of the Thesis In this chapter, in order to provide some background, we have discussed various relevant optical modulation techniques in Si and the optical modu- lators based on them. At the beginning of this chapter, the Franz-Keldysh effect, the Quantum Confined Stark effect, and the free carrier absorption effect were discussed. Then, various techniques for obtaining optical mod- ulation from Si and Si-based materials using free carrier absorption effect (i.e., carrier injection, carrier depletion, and carrier accumulation) as well as a literature review of Si-based optical modulators using these techniques were presented. At the end of this chapter, the various interferometer struc- tures used to obtain intensity modulation from Si-based phase modulators were briefly discussed. In Chapter 2, Material Choice, Device Structure, and Electrical and Optical Simulations, we will describe the key components for design- ing SiGe/Si quantum-well modulators based on the free carrier depletion effect. The electrical and optical properties of Si and SiGe, which are im- portant for designing these modulators, will be discussed at the beginning of Chapter 2. The software used for the electrical simulation of these modula- tors and the models used in this simulation will be described next. Then, we discussed the mode solver program used for the optical simulation of SiGe/Si waveguide modulators. The general structure of a Si0.8Ge0.2/Si quantum- well phase modulator followed by the coupled electrical-optical analysis used to design these modulators is presented next. Using this analysis, we simu- lated a 3 quantum-well Si0.8Ge0.2/Si modulator (designed by Marris et al.) 23 1.4. Organization of the Thesis providing the most important results of the simulations. From the results of the DC analysis and the transient analysis performed on this 3 quantum-well modulator, we came to the conclusion that, a single quantum-well modu- lator may have a lower drive voltage, lower optical loss, and be capable of higher intrinsic speed than the 3 quantum-well modulator. Hence we re- moved 2 quantum-wells from the 3 quantum-well modulator and designed a single quantum-well modulator which will be described in Chapter 3. In Chapter 3, Single Quantum-Well SiGe/Si Optical Modulator, the results of the simulation on a single quantum-well Si0.8Ge0.2/Si mod- ulator, which is derived from the 3 quantum-well modulator described in Chapter 2, will be discussed. Then we will compare its performance with that of the 3 quantum-well modulator and we will find that, this single quantum-well modulator is better than the 3 quantum-well modulator in terms of the drive voltage, the optical loss, and the intrinsic speed. The performance of a Mach-Zehnder interferometer using single quantum-well phase modulators inserted into the two branches of the Mach-Zehnder in- terferometer will be described next. This chapter will conclude with the idea that, if we want to operate in a lower bias region, we can remove the quantum-wells and only the highly doped P+ layers and the NID layers are sufficient to obtain an effective index variation close to that obtained with the 3 quantum-well or single quantum-well modulator in a lower bias re- gion (V < 2 V). Removing the quantum-wells from the modulator may also increase the intrinsic speed. Chapter 4, Summary, Conclusion, and Suggestions for Future Work, will summarize our present work and will give some suggestions 24 1.4. Organization of the Thesis about the future work in this area. 25 Chapter 2 Material Choice, Device Structure, and Electrical and Optical Simulations In this chapter, we describe the key components for designing a free carrier depletion based phase modulator. The material parameters choice which ensure various critical characteristics of a free carrier depletion based phase modulator are described in section 2.1. The software used for the electrical simulation for designing these modulators is described in section 2.2. A sim- ple single mode SOI waveguide followed by the description of the mode solver used to solve for the eigen modes in these SOI waveguides are described in section 2.3. In section 2.4 the general structure of the free carrier depletion based phase modulators that we studied for operation at λ0 = 1.55 µm is described. The coupled electrical-optical analysis used to design these mod- ulators are also described in section 2.4. In the next section we describe a 3 quantum-well Si0.8Ge0.2/Si optical modulator. The results obtained using the DC analysis for the 3 quantum-well modulator is presented in section 2.5. This chapter concludes with section 2.6, which describes the transient 26 Chapter 2. Material Choice, Device Structure, and Electrical and Optical Simulations analysis performed on this 3 quantum-well Si0.8Ge0.2/Si electro-optic mod- ulator. To date, the free-carrier depletion effect has become the most commonly used approach to obtain the electro-optic effect in Si-based materials. The performance of different Si electro-optic modulators is compared in terms of the VpiLpi product (Vpi and Lpi are, respectively, the applied voltage and the corresponding length required to obtain a change in the phase at the output of the modulator of pi), the optical loss of the device, and the 3 dB bandwidth of the device. The VpiLpi product is commonly used as the figure of merit for these devices [21–25]. An efficient modulator should possess a low VpiLpi (for low drive voltage and/or small size) with low absorption losses and a high 3 dB bandwidth. Lpi can be calculated from the formula Lpi = λ0/2∆neff where, ∆neff is the effective index variation at the applied bias of Vpi. If these electro-optic modulators are inserted into two arms of a Mach-Zehnder interferometer as described in section 1.3.1, two additional performance parameters - insertion loss and extinction ratio, should be con- sidered. The insertion loss and the extinction ratio are given by the following relations, IIN , ION , IOut−max are defined as the the light input intensity, light output intensity and the maximum value of the light intensity that can be obtained at the output. IL = 10× Log(IOut−max IIN ) (2.1) ER = 10× Log( ION IOFF ) (2.2) 27 2.1. SiGe/Si Heterostructure Properties In this chapter we describe the steps for designing a free carrier depletion based phase modulator and design an efficient 3 quantum-well Si0.8Ge0.2/Si electro-optic modulator which ensures low VpiLpi product, low absorption losses, and high 3 dB bandwidth. 2.1 SiGe/Si Heterostructure Properties In section 2.1.1 the reasons for choosing the Ge mole fraction, x, of 0.2 in a Si1−xGex layer grown on a Si substrate are presented. To begin with, the bandgap requirements of the materials that will be used in a free carrier depletion based modulators are described. Then, why the thickness of the layers grown from these materials should be below a particular thickness, critical thickness, is discussed. How a Ge mole fraction of 0.2 meets both the bandgap and thickness requirements is presented next. This section con- cludes with the bandgap structure of a Si0.8Ge0.2/Si quantum-well. Section 2.1.2 describes two important properties (refractive index and absorption coefficient) of Si and SiGe. 2.1.1 Bandgaps and Band Alignments for SiGe/Si Heterostructure Si and SiGe are both indirect bandgap materials. As absorption of photons by the free carriers within the same band is the desired mechanism to change the refractive index in silicon, band-to-band absorption of the light propa- gating through the device should be avoided. This is done by keeping the indirect bandgap of both Si and SiGe above the incident photon energy (0.95 28 2.1. SiGe/Si Heterostructure Properties eV at λ0 = 1.31 µm and 0.801 eV at λ0 = 1.55 µm) . This condition is fulfilled by silicon (indirect bandgap of 1.12 eV). For a SiGe layer grown on a Si layer, the mole fraction of Ge in the SiGe layer should be chosen such that the bandgap of SiGe will be higher than the incident photon energy. When a SiGe layer is grown on a Si layer, the thickness of the SiGe layer should be below a particular thickness to avoid misfit dislocations in SiGe layers. This thickness is known as critical thickness. If a SiGe layer with a thickness below the critical thickness is grown on a Si layer to create a heterojunction device, due to the lattice mismatch between SiGe and pure Si, the SiGe films are biaxially strained . This strain has two components; the hydrostatic component and the uniaxial component. The hydrostatic component lowers the conduction band minimum and lifts the valence band maximum, as a result band gap energy will be decreased in the strained SiGe layers . The uniaxial component splits the valence band into a heavy hole band and a light hole band where light hole band becomes the topmost (lowest energy) valence band . Once the critical thickness is exceeded, these films go back to their intrinsic cubic lattice constant and misfit dislocations appear in the films. These misfit dislocations degrade the optical and electrical quality of these films. In order to avoid these misfit dislocations occuring, the thickness of the SiGe layer should be below the critical thickness. This critical thickness depends on the Ge mole fraction as shown in Fig. 2.1 . In a Si/SiGe/Si multilayer structure, in case of p pe- riods of Si (thickness dSi) and Si1−xGex layer (thickness dSiGe), the critical thickness can be evaluated by considering an equivalent structure of a single layer Si1−x′ Gex′ , whose Ge content is given by x ′ = x(dSiGe)/(dSiGe + dSi) 29 2.1. SiGe/Si Heterostructure Properties and thickness is given by p(dSiGe + dSi) . For example, if a Si/SiGe/Si multilayer structure is grown with Si thickness of dSi = 25 nm, SiGe thick- ness of dSiGe = 10 nm, and the Ge mole fraction of 0.2, the equivalent layer has a Ge fraction x ′ = 0.06. This leads to a critical thickness for such a Si/SiGe/Si multilayer structure about 125 nm as shown in Fig. 2.1. The total thickness of the Si/SiGe/Si multilayer structure (p(dSiGe + dSi)) must be below 125 nm to avoid misfit dislocations in the SiGe layers. 0 0.2 0.4 0.6 0.8 110 0 101 102 103 Ge mole fraction (x) Cr itic al T hi ck ne ss (n m) Figure 2.1: Critical layer thickness of SiGe layer grown on Si as a function of the Ge mole fraction, re-digitized from . The bandgap relation of strained Si1−xGex is given by Eg = 1.12− 0.74x. For the Ge mole fraction of 0.2 in Si1−xGex, the bandgap of SiGe will be 0.972 eV, which is above the incident light energy (0.95 eV at λ0 = 1.31 30 2.1. SiGe/Si Heterostructure Properties µm and 0.801 eV at λ0 = 1.55 µm). Thus the bandgap requirement is fulfilled. Now, for a multiple periods of SiGe/Si layers, the critical thickness depends on both the Ge mole fraction and the the thicknesses of Si and SiGe layers. We have seen that with dSi = 25 nm and dSiGe = 10 nm, if the Ge mole fraction of 0.2 is selected, the critical thickness is about 125 nm. In section 2.4, we will see that the thickness of the Si/SiGe/Si multilayer structure will be much lower than this thickness. At the heterojunction, the conduction band discontinuity is given by ∆Ec < 20 meV and the valence band discontinuity is given by 0.74x in . The band alignment of a Si0.8Ge0.2/Si heterostructure grown on a Si substrate, that we used in our simulations, is shown in Fig. 2.2. The valence band and conduction band discontinuities that we used in our simulation are quite similar to those used in . For the high valence band discontinuity, a Si0.8Ge0.2 layer sandwiched between Si layers acts as the well for holes. When no voltage is applied to this structure, holes will be confined inside the SiGe wells and, when a reverse bias voltage is applied, they will escape from the SiGe wells. Both the confinement of holes inside the quantum- wells at V = 0 V and the depletion of holes at a particular reverse voltage are important for getting larger changes of the refractive index in the SiGe layers, which will lead to higher effective index variations. 2.1.2 Refractive Index and Absorption Coefficient The refractive index of strained Si0.8Ge0.2 is given in by nSiGe(x, λ0) = nSi(λ0) + (1.16− 0.26 · λ0)× x2 (2.3) 31 2.2. Software for Electrical Simulation Eg(Si)=1.12eV Strained Eg(Si0.8Ge0.2)=0.972eV ∆Ec=χ(Si)- χ(Si0.8Ge0.2) =0.01eV ∆Ev =0.158eV Figure 2.2: Band alignments for Si1−xGex/Si heterostructures on Si sub- strate, where χ is the electron affinity. where x is the Ge mole fraction in Si1−xGex. This expression is applicable for wavelengths from 0.9 to 1.7 µm and for mole fractions less than 0.33. In our simulation, the refractive index of intrinsic Si is taken to be 3.503 and 3.475 at the wavelengths 1.31 and 1.55 µm, respectively. The absorption coefficient for the undoped Si and SiGe which can be extracted from Fig. 2.3 can be considered negligible for wavelengths of above 1.24 µm . 2.2 Software for Electrical Simulation We used the device simulator SILVACO ATLASTM for solving the pois- son and the drift-diffusion equations to predict various internal characteris- tics in our modulator under various biased conditions for DC and transient analysis. These internal characteristics include electron and hole distribu- tion, conduction band and valence band energies, electric field distribution, 32 2.2. Software for Electrical Simulation 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.410 0 101 102 103 104 Ge mole fraction (x) Ab so rp tio n Co ef fic ie nt (c m− 1 ) 1.55 µm 1.31 µm Figure 2.3: Absorption coefficient of Si and strained SiGe, temperature is 300K etc. In section 2.2.1, we describe key components we need to define in the software before simulating a structure. In section 2.2.2, the models we used in the simulations of SiGe/Si quantum-well optical modulators are described. 2.2.1 Basic Steps of Electrical Analysis in ATLAS Simulation of a device in SILVACO ATLASTM for the prediction of the in- ternal behavior follows the following procedure. At first, the user has to define the structure by defining the thickness, the material composition and the doping profile of every layer of the structure. A mesh is defined for solv- ing the various equations. The mesh is basically used to determine the nodes 33 2.2. Software for Electrical Simulation at which various equations will be solved. A good practice is to define a fine mesh in regions where the device characteristics are of particular interest (quantum-wells in our modulator). The user then defines the various mod- els for material parameters that are used in the simulation. The user then specifies the bias conditions for performing the electrical simulation. After the completion of the simulation, the device characteristics can be viewed using TONYPLOT (Graphical user interface) or these can be extracted as numerical data. 2.2.2 Models Incorporated for Device Simulation In this section, we briefly discuss the models that we used in the modeling of our SiGe/Si modulator. Details of these models are found in the ATLAS manual . 2.2.2.1 Recombination Models Shockley-Read-Hall (SRH) Recombination - Concentration depen- dent SRH model (CONSRH) is one of the recommended recombination models for Si and Si-based materials. Under the SRH recombination, re- combination of excess carriers in a heavily doped semiconductor occur in the presence of a trap, which is an energy level within the forbidden gap of the semiconductor corresponding to a crystal impurity or vacancy, which alternately captures electrons and holes, as shown in Fig. 2.4. The rate of recombination depends on the lifetime of the excess carriers. In the case of the concentration dependent SRH model, the constant carrier lifetimes of electrons and holes will be functions of the impurity concentrations. 34 2.2. Software for Electrical Simulation e- e- h+ h+ (a) (b) Figure 2.4: The two major recombination processes in silicon are (a) SRH recombination (b) Auger recombination Auger Recombination - Under Auger recombination, electrons and holes recombine by a band-to-band transition and due to the recombination, a third carrier (electron or hole) is emitted. As the Auger recombination is a 3 particle process, it is unlikely under low-level injection and therefore the SRH recombination dominates the re- combination processes . However, Auger dominates as the carrier density increases. 2.2.2.2 Mobility Models To model the mobilities of holes and electrons in Si and SiGe layers, we used concentration dependent mobility (CONMOB) and electric field dependent mobility (FLDMOB) models. The mobility of the carriers decrease with the increase of the doping con- 35 2.2. Software for Electrical Simulation centration, as the ionized impurities introduced by the dopants will enhance the scattering process. That is why we used the CONMOB model. The drift velocity of the carriers is the product of the mobility and the electric field in the direction of the current flow. The carrier velocity increases with the increase of the electric field. At high electric field, carriers that gain energy take part in a wider range of scattering process and there will be a reduction in the mobility of the carriers. Hence the velocity cannot increase much with the increase of electric field. Eventually, the velocity saturates at a constant velocity. To consider the effects of this saturation velocity, we used the FLDMOB model. We will see later in section 2.5 and 2.6 that this model does not have any effect on the internal characteristics of a device in the case of DC analysis, but it significantly affects the transient analysis. 2.2.2.3 Bandgap Narrowing Model In the presence of heavy doping (greater than 1018 cm−3), the bandgap of a semiconductor becomes doping dependent. As the doping level increases, a decrease in the bandgap separation occurs, where the conduction band is lowered by approximately the same amount as the valence band is raised. Details of this model can be found in the ATLAS manual. 2.2.2.4 Thermionic Emission and Tunneling The thermionic field emission model takes into account the thermionic emis- sion process at the abrupt heterojunction interface as well as the tunneling across the heterojunction. The band diagram of an abrupt heterojunction is 36 2.2. Software for Electrical Simulation shown in Fig. 2.5. Electron transport across the conduction band spike can be described by tunneling through the spike and thermionic emission over it [33, 34]. Ec x=0 xE JT-E JTunnel Ef2 ∆Ec Ex Ef1 Figure 2.5: Schematic band diagram of an abrupt heterojunction. Ef1 and Ef2 represents the electron quasi-fermi level in each semiconductor re- gion. JTE and JTunnel are the thermionic emission and tunneling current, respectively. In a quantum-well, suppose the barriers to the right and to the left of the well are given by Er El, respectively. A carrier of charge e and effective mass m∗ must be higher in energy than one of the barriers in order to be able to leave the well. By applying an electric field across the well, the confined carrier concentration in the quantum-well will decrease as δn(t) δt = −n0 τe (2.4) where n0 is the initial confined hole concentration and τe is escape char- acteristic time given by the thermionic and tunneling escape times 1/τe = 37 2.3. Optical Simulations 1/τth+1/τtun . The carrier population n of the well decays by thermionic emission with the time constant τth which is given by 1 τh = 1 w ( kBT 2pim∗ ) 1 2 · exp ( −Er,l − Ei − qF w 2 kBT ) (2.5) where w is the barrier width, Ei is the energy of the carrier, and F is the applied electric field to the quantum-well. If no field is applied to the quantum-well, the thermionic emission and tunneling escape time will be very large and the carriers will not decay at zero electric field. When elec- tric field is applied to the quantum-well, the thermionic emission time will decrease and the carriers can escape the quantum-well. 2.3 Optical Simulations Section 2.3.1 describes the operating principle of a simple SOI planar waveg- uide (2-dimensional waveguide). Then four possible configurations of 3- dimensional waveguides are shown. A single mode SOI rib waveguide fabri- cated by Lardenois et al. is discussed next. In section 2.3.2 we simulated the SOI rib waveguide described in with a MATLAB mode solver and found that the SOI rib waveguide is single mode with a very low leakage loss. 2.3.1 Silicon-On-Insulator (SOI) Waveguides A dielectric waveguide is formed when a high refractive index region is sur- rounded by regions of lower refractive index. Light propagating through the waveguide will be primarily confined to the high refractive index region 38 2.3. Optical Simulations due to the total internal reflection at the boundaries between the higher and lower refractive index regions . A simple SOI waveguide, shown in Fig. 2.6, consists of a silion layer (the higher refractive index material or core) sandwiched between a silica (SiO2) layer and air (the lower refractive index materials). The SiO2 layer (the lower cladding layer) should be thick enough to prevent light leaking into the silicon substrate. Buried SiO2 cladding (n=1.45) Si substrate (n=3.475) Si guiding layer (n=3.475) air (n=1.0) x y z Figure 2.6: SOI planar waveguide, the refractive indices (n) of the layers are shown for the wavelength of 1.55 µm. The z direction is taken as the direction of light propagation. The planar two-dimensional waveguide is useful for describing the the- ory of waveguides and the solution of Maxwells equations. But it confines the light only in one dimension (here the y direction in Fig. 2.6). For many 39 2.3. Optical Simulations applications, two-dimensional confinement is required. There are several configurations used for three-dimensional waveguides, schematically repre- sented in Fig. 2.7: strip (a), embedded strip (b), rib (or ridge) (c) and strip-loaded (d) waveguides. In SOI-based devices, the rib waveguide is most commonly used . n2 n1 n2 n1 n2 n1 n2 n3 n1 (a) (b) (c) (d) Figure 2.7: Different configurations of three dimensional waveguides (a) strip, (b) embedded strip, (c) rib (or ridge), and (d) strip-loaded waveguides. Lardenois et al. (in the same group of Marris-Morini) fabricated a SOI rib waveguide . The cross section of this waveguide is shown in Fig. 2.8. The buried SiO2 thickness of the SOI waveguide was 700 nm, which is large enough to prevent lightwave from leaking towards the Si substrate. The Si film thickness was 400 nm which was reduced to 380 nm during the etching process. The etched depth of this waveguide was 70 nm. The rib 40 2.3. Optical Simulations n2 n1 n0air Si SiO2 1 µm 380 nm 70 nm 700 nm Si n1 Figure 2.8: Cross section of the SOI rib waveguide discussed in . width was 1 µm, which is close to the maximum value to ensure single mode conditions . The measured propagation losses in this Si-SiO2 rib waveguide was found to be 0.4 dB/cm. We simulated this SOI rib waveguide in the MATLAB mode solver program available online [38, 39]. In the next section, The simulation results (effective index and optical loss) obtained for the SOI rib waveguide is presented. 2.3.2 Mode Solver for Optical Simulation The mode solver program that we used for the simulation of the SiGe/Si waveguide modulator is described very briefly in Appendix B. Before sim- 41 2.4. Design of the SiGe/Si Optical Modulator ulating the SiGe/Si waveguide modulator with this program, we solved the eigenmodes of the SOI rib waveguide described in section 2.3.1. We defined the refractive index and the thickness of various layers of the SOI waveg- uide; the grid spacings both in x and y directions and an initial guess for the effective index of the TE0 mode in the program. Now, as the refrac- tive index profile is symmetric about the y axis, only half of the waveguide needs to be included in the computational domain. So we defined half of the rib width in the program. Then we defined the boundary conditions which are considered for the points on the edge of the computation window as described in Appendix B. The absorbing boundary condition is used for north (upper), south (lower) and east (right) boundary and the symmetric boundary condition is used for the west (left) boundary. Then we simulated the SOI waveguide described in section 2.3.1 for operation at 1.55 µm. We found that this waveguide is single mode and the modal effective index of the fundamental mode for TE polarization was neff = 3.1193+i1.792 ·10−8. The mode profile is shown in Fig. 2.9. 2.4 Design of the SiGe/Si Optical Modulator Marris et al. designed a 3 quantum-well modulator for operation at λ0 = 1.31 µm using coupled electrical-optical simulations in Ref. . In this paper, they provided all the device parameters as well as a figure on the depletion process of the holes with the application of reverse bias voltage showing the position of all of the layers. We extracted the thicknesses of the P+ and N+ layers and the NID layers next to these P+ and N+ layers from this figure. 42 2.4. Design of the SiGe/Si Optical Modulator x y Ex (TE0 mode) Si SiO2 Si Air 0 0.5 1 1.5 2 2.50.5 1 1.5 2 2.5 −45 −40 −35 −30 −25 −20 −15 −10 −5 0 Figure 2.9: 2D Mode profile for the fundamental TE mode of the SOI rib waveguide discussed in . Each line represents an identical field value (−3 dB step between lines, −45 dB minimum value). In section 2.4.1, we describe the general structure of these modulators. The coupled electrical-optical analysis used to design these modulators is described in section 2.4.2. We performed the electrical analysis in two dif- ferent modes: DC analysis and transient analysis. We performed the DC analysis in order to investigate the electrical and optical behaviors of the de- vice with the application of various reverse bias voltages. For this analysis we varied the applied reverse voltage from 0 V to 10 V. Then we performed the transient analysis in order to investigate the frequency response of the device. For this analysis, at first a reverse bias voltage step from 0 to x V 43 2.4. Design of the SiGe/Si Optical Modulator (where x = 6 for the 3 quantum-well modulator) with a rise time of 1 fs was applied to the device and the electrical and optical behaviors of the device were observed at several times ranging from 0 to 100 ps. Then, a reverse bias voltage step from x to 0 V with a fall time of 1 fs was applied to the device and the electrical and optical behaviors of the device were observed at several times for the next 100 ps. 2.4.1 Device Structure The modulator consists of a PIN (P-Intrinsic-N) diode; the active region consists of ’k’ periodic stacks of layers (where k = 3 for the 3 quantum- well modulator). Each period consists of a 10 nm Si0.8Ge0.2 quantum-well surrounded by 10 nm Si-NID (Non-Intentionally Doped-1016 cm−3) layers and 5 nm P+ highly doped (2×1018 cm−3) layers (P+-δ-doped layers). The thickness of both of the P and N part of the PIN diode is 30 nm. The PIN diode (thickness t) is grown on a 30 nm Si-NID layer at the bottom of the PIN diode . This structure is simulated using a 700 nm SiO2 layer, which is thick enough to prevent the light leaking towards the Si substrate . For the lateral confinement of light, a rib structure is defined, simulating the partial etching of the upper layers. The dimensions of the rib structure are defined such that this modulator could be integrated with a single mode SOI rib waveguide . We discussed about this single mode SOI rib waveguide fabricated by Marris et al. in section 2.3.2. The mole fraction of Ge was chosen to be 0.2 to keep the bandgap of SiGe higher than the energy of the incident light to avoid band-to-band absorption and to maintain the total thickness below the critical thickness, as we described in section 2.1.1. 44 2.4. Design of the SiGe/Si Optical Modulator 2.4.2 Electrical and Optical Analysis With no field applied holes from the P+ layers are captured by and confined in the quantum-wells and when a reverse bias is applied holes are released from the quantum wells and drift to the 30 nm P+ contact layer. Varia- tion of the hole distribution leads to a free-carrier absorption change and a refractive index change in each layer and, subsequently, the phase modula- tion of the guided optical wave. We calculated the hole distribution in the various layers of the structure using Silvaco ATLASTM . The solution of Poisson-Fermi-Schroedinger equations is needed for the calculation of the hole density in the quantum-wells of the SiGe/Si modulator . However, it was shown by Marris et al. (in Ref. ) that, if the quantum-well is equal to or thicker than 10 nm, the hole density profile in the structure at V = 0 V obtained by solving the Poisson-Fermi-Schroedinger equations is similar to that obtained by solving the Poisson-Fermi equations. That is why we used the Poisson-Fermi solver in Silvaco ATLASTM for the cal- culation of the hole density distribution at various applied voltages. The carrier transport in bulk material and at heterojunctions is calculated, re- spectively, using the drift-diffusion expressions and thermionic emission ex- pressions [33, 34]. The models used by Marris et al. are the concentration de- pendent SRH (Shockley-Read-Hall) recombination model, the concentration dependent mobility model, the Auger model, and the Fermi-Dirac statistics model . We used these models as well as two additional models: the field dependent mobility model and the band gap narrowing model. These additional models do not affect the DC analysis, but the field dependent 45 2.4. Design of the SiGe/Si Optical Modulator mobility model affects the transient analysis as described in section 2.6. Experimental results showed that the higher reverse bias voltages lead to higher reverse leakage currents, and, therefore, higher leakage power that might create a self heating of the device which, in turn, affects the effec- tive index variation . This thermo-optical index variation becomes more significant at higher reverse bias voltages. However, as well as Marris et al., we ignored these effects in our calculations. The material parameters used for this simulation are the default parameters for Si and Si0.8Ge0.2 used in Silvaco ATLASTM . We tuned the bandgaps of these materials to be EG−Si = 1.12 eV and EG−Si0.8Ge0.2 = 0.972 eV as defined in Ref. and also the electron affinities to be χSi = 4.05 eV and χSi0.8Ge0.2 = 4.04 eV . The hole concentration obtained using the electrical simulation in Silvaco ATLASTM is averaged in each layer . As the layers are very thin, this approximation gives a refractive index profile similar to the refractive index profile obtained using the actual hole concentration profile. The absorption coefficient variation and refractive index variation in the doped Si layers with respect to undoped Si at λ0 = 1.55 µm, is calculated from the hole distribution using the following formulae ∆α = 8.5× 10−18∆N + 6.0× 10−18∆P (2.6) ∆n = −8.8× 10−22∆N − 8.5× 10−18∆P 0.8 (2.7) where ∆N and ∆P are, respectively, the electron and the hole concentration variations cm−3 with respect to the intrinsic carrier concentration of every 46 2.5. DC analysis in 3 Quantum-Well SiGe/Si Modulator layer. To calculate the absorption coefficient change and the refractive index change in Si0.8Ge0.2 we used equation 2.6 and 2.7 due to the unavailability of separate equations for free carrier absorption for Si0.8Ge0.2 . The intrinsic hole concentration of Si and Si0.8Ge0.2 are, respectively, taken as 0.668× 1010 cm−3 and 10× 1010 cm−3, as defined in Silvaco ATLASTM at T = 300oK. The refractive index of undoped Si and Si0.8Ge0.2 are taken as 3.475 and 3.505, respectively, for λ0 = 1.55 µm. The absorption coefficient for undoped Si and Si0.8Ge0.2 are considered to be negligible at λ0 = 1.55 µm. The free carrier absorption as well as the refractive index of each layer are used as the input to a mode solver , which provides the effective index as well as the optical loss for TE polarized light at various voltages using a 2D semi-vectorial finite-difference approach. The optical loss can be converted into dB/cm using the formula α = 10 · log10e × 4piλ0 ×Ni dB/cm, where Ni is the imaginary part of the effective index calculated in the mode solver. 2.5 DC analysis in 3 Quantum-Well SiGe/Si Modulator A schematic diagram of the 3 quantum-well SiGe/Si modulator is shown in Fig. 2.10. We re-analyzed this structure for operation at λ0 = 1.31 µm. For a V = 0 V to V = 6 V variation Marris et al. obtained the effective index 47 2.5. DC analysis in 3 Quantum-Well SiGe/Si Modulator 1 µm 70 nm 3 5 0 n m X=0 QW1 QW2 QW3 Si Substrate 700 nm SiO2 30 nm Si NID 30 nm Si P+ 1018 cm-3 120 nm Si NID 10 nm Si NID 5 nm Si P+ 2×1018 cm-3 10 nm Si0.8Ge0.2 QW 60 nm Si NID 30 nm Si N+ 1018 cm-3 Figure 2.10: Schematic view of the 3 quantum-well SiGe/Si optical modu- lator. variation of 1.7 · 10−4. For a V = 0 V to V = 6 V variation, we obtained an effective index variation of 1.65·10−4. Our result (effective index variation at V = 6 V) was consistent with their result. We, then, repeated the analysis for this structure for the wavelength of λ0 = 1.55 µm, which gives a much lower VpiLpi compared to the VpiLpi obtained for the wavelength of λ0 = 1.31 µm, the reasons will be discussed in section 2.5.2. In section 2.5.1, various device internal characteristics such as the elec- tron and hole distribution, conduction band and valence band energies, elec- tric field distribution, etc., of the 3 quantum-well SiGe/Si optical modulator under various reverse bias voltages are discussed. Then, in section 2.5.2 we describe the absorption coefficient and the refractive index calculations using the hole distribution in various layers of the 3 quantum-well structure. The 48 2.5. DC analysis in 3 Quantum-Well SiGe/Si Modulator absorption coefficient as well as the refractive index of each layer are used as an input in the mode solver we discussed in section 2.3.2, which provides the effective index as well as the optical loss. The effective index and optical loss calculation for the fundamental TE mode for operation at 1.31 and 1.55 µm using the mode solver program are presented in section 2.5.3. This section concludes with a comparison of the important performance characteristics (effective index variation and optical loss) for 1.31 and 1.55 µm. 2.5.1 Electrical Analysis For the DC analysis, we applied reverse bias voltages ranging from 0 V to 10 V to this modulator. The hole distribution in the structure is calculated using Silvaco ATLASTM at each applied reverse bias voltage. The hole concentrations for the 3 quantum-well structure at several specific voltages are shown in Fig. 2.11. With no field applied, holes from the P+ layers are captured by and confined in the quantum-wells and when a reverse bias is applied, holes are released from the quantum wells and drift to the 30 nm P+ contact layer. The band profiles and the electric field profiles in the structure at V = 0 V and at V = 6 V are shown respectively in Fig. 2.12(a) and Fig. 2.12(b). It can be seen that at V = 6 V, the valence band in the first two quantum- wells are fully bent and only partially bent in the 3rd quantum-well. In other words, the first two quantum-wells are fully depleted and the 3rd quantum- well is almost depleted for V = 6 V. Fig. 2.12(b) shows that inside the quantum-wells, the slope of the electric field is positive at V = 0 V due to the confinement of holes. With the application of a reverse bias, holes are 49 2.5. DC analysis in 3 Quantum-Well SiGe/Si Modulator 0 0.05 0.1 0.15 0.2 0.25 0.3 0.3510 0 105 1010 1015 1020 x (µm) H ol e Co nc en tra tio n (cm − 3 ) P+ NID Q W 3 QW 2 QW 1 NID N+ V= 0V V= 2V V= 4V V= 6V V= 10V Figure 2.11: Hole distribution in various layers at various reverse bias volt- ages. With the increase of reverse biasing, hole deplete from the quantum- wells. Holes starts to deplete from the NID layer on the P+ side after all the quantum-wells are fully depleted i.e. after 7.5 V NID layer starts depleting. 50 2.5. DC analysis in 3 Quantum-Well SiGe/Si Modulator 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35−8 −6 −4 −2 0 2 x(µm) Ba nd P ro file (e V) P+ NID Q W 3 QW 2 QW 1 NID N+ (a) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35−6 −5 −4 −3 −2 −1 0 1 2 x 10 5 x(µm) El ec tri c Fi el d (V /cm ) P+ NID Q W 3 QW 2 QW 1 NID N+ V=0V V=6V (b) Figure 2.12: (a) Valence and conduction band energy profiles for the 3 quantum-well SiGe/Si structure, black lines are for V = 0 V and red lines are for V = 6 V, the dotted line shows the quasi-fermi energy level for holes (b) electric field in the 3 quantum-well SiGe/Si structure. 51 2.5. DC analysis in 3 Quantum-Well SiGe/Si Modulator removed from the quantum-wells and the electric field is almost flat inside the quantum-wells at V = 6 V. 2.5.2 Calculation of Absorption Coefficient and Refractive Index The hole concentration is averaged in each layer at each applied voltage . As the layers are very thin, this approximation gives an effective index variation with applied reverse bias similar to the effective index variation obtained using the actual hole concentration profile as shown in Fig. 2.13. The absorption coefficient variation and refractive index variation in the doped Si layers with respect to undoped Si at λ0 = 1.55 µm is calculated from the hole distribution using equations 2.6 and 2.7. In our calculations, we considered only the hole concentration change in the various layers, as the electron concentration is negligible with respect to the hole concentration except in the N+ layer, where the electron concentra- tion is relatively high as shown in Fig 2.14. For this N+ layer, the refractive index and the absorption coefficient at V = 0 V is calculated from both ∆N and ∆P (∆N is considered as the doping density of this layer). Fig. 2.15(a) shows that the refractive index changes in the quantum- wells are much higher at λ0 = 1.55 µm than at λ0 = 1.31 µm which implies a higher effective index variation at λ0 = 1.55 µm than at λ0 = 1.31 µm. Fig. 2.15(a) shows that the refractive index change saturates with a satura- tion value of 1.6×10−3, 1.95×10−3, and 2.1×10−3 at 2.1 V, 4.3 V, and 7.5 V, respectively, in quantum-wells 1, 2, and 3, which indicates the quantum wells are depleted. Also, Fig. 2.15 (a) shows that the onset of the change 52 2.5. DC analysis in 3 Quantum-Well SiGe/Si Modulator 0 1 2 3 4 5 6 7 8 9 100 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 x 10 −4 Applied voltage (V) Ef fe ct ive in de x va ria tio n (∆n e ff) With approximation Without approximation (a) 0 1 2 3 4 5 6 7 8 9 100 1 2 3 4 5 6 Applied voltage (V) O pt ica l lo ss α (dB /cm ) With approximation Without approximation (b) Figure 2.13: (a) Effective index variation in a single quantum-well modula- tor with and without averaging the hole concentration (b) optical loss with and without averaging the hole concentration. 53 2.5. DC analysis in 3 Quantum-Well SiGe/Si Modulator 0 0.05 0.1 0.15 0.2 0.25 0.3 0.3510 0 105 1010 1015 1020 x (µm) El ec tro n Co nc en tra tio n (cm − 3 ) P+ NID Q W 3 QW 2 QW 1 NID N+ V= 0V V= 2V V= 4V V= 6V Figure 2.14: Electron distribution in various layers at various reverse bias voltages 54 2.5. DC analysis in 3 Quantum-Well SiGe/Si Modulator 0 1 2 3 4 5 6 7 8 9 100 0.5 1 1.5 2 2.5 x 10 −3 Applied voltage (V) R ef ra ct ive in de x ch an ge (∆ n ) in th e qu an tu m −w el ls QW3 QW2 QW1 1.55µm 1.31µm (a) 0 1 2 3 4 5 6 7 8 9 100 0.5 1 1.5 2 2.5 x 10 −4 Applied voltage (V) R ef ra ct ive in de x ch an ge (∆ n ) in th e N ID la ye r 1.55µm 1.31µm (b) Figure 2.15: Refractive index changes (∆nv = nv − n0) at λ0 = 1.55 µm and λ0 = 1.31 µm (a) in the quantum-wells with dash-dot line, dotted line, and solid line for quantum- well 1, 2, 3 respectively. (b) in the NID layer on the P+ side 55 2.5. DC analysis in 3 Quantum-Well SiGe/Si Modulator 0 1 2 3 4 5 6 7 8 9 100 0.5 1 x 10−4 Applied voltage (V) R ef ra ct ive in de x ch an ge in th e N ID la ye rs in b et we en th e qu an tu m −w el ls NID−1 NID−3 NID−5 NID−2 NID−4 NID−6 (a) 0 1 2 3 4 5 6 7 8 9 100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 x 10−3 Applied voltage (V) R ef ra ct ive in de x ch an ge in th e δ− do pe d− P+ la ye rs Delta−1 Delta−2 Delta−3 Delta−4 (b) Figure 2.16: Refractive index changes at λ0 = 1.55 µm in the (a) NID layers (b) δ-doped-P+ layers 56 2.5. DC analysis in 3 Quantum-Well SiGe/Si Modulator in refractive index in each of the quantum-wells occurs near the saturation value of the preceding quantum-well. After the depletion of all 3 quantum-wells, the NID layer on the P+ side starts depleting as well. Fig. 2.15(b) shows the refractive index changes in the NID layer on the P+ side. The refractive index changes in the other layers (NID layers in between the quantum-wells and the P+-δ-doped layers) are very small as shown in Fig. 2.16. 2.5.3 Effective Index and Optical Loss Calculation The free carrier absorption as well as the refractive index of each layer are used as the input to a mode solver , which provides the effective index as well as the optical loss for TE polarized light at various voltages. The effective index variation curve for the 3 quantum-well structure is shown in Fig. 2.17(a). When the bias is lower than 7.5 V the effective index variation in the modulator is mainly caused by the refractive index change in the quantum-wells. Above 7.5 V, the refractive index changes in all of the quantum-wells become constant. However, above V=7.5 V the contribution from the NID layer on the P+ side to the effective index variation becomes significant. In other words, after the depletion of all of the quantum-wells, the quantum-well modulator turns into a NID-layer modulator. The two most important performance parameters obtained for this mod- ulator are the effective index variation and optical loss which are found to be around 2.28 × 10−4 and 3.13 dB/cm at V = 6 V for the wavelength of λ0 = 1.55 µm as shown in Fig. 2.17(a) and Fig. 2.17(b), respectively. To evaluate the modulation efficiency of the phase modulator, a figure of merit 57 2.5. DC analysis in 3 Quantum-Well SiGe/Si Modulator 0 1 2 3 4 5 6 7 8 9 100 0.5 1 1.5 2 2.5 3 3.5 4 x 10 −4 Applied voltage (V) Ef fe ct ive in de x va ria tio n (∆n e ff) 1.55µm 1.31µm (a) 0 1 2 3 4 5 6 7 8 9 100 1 2 3 4 5 6 Applied voltage (V) O pt ica l lo ss α (dB /cm ) 1.55µm 1.31µm (b) Figure 2.17: (a) Effective index variation (∆neff-v = neff-v − neff-0) of the 3 quantum-well modulator; the blue point shows the effective index variation at 6 V obtained by Marris et al. in Ref. (b) optical loss at various voltages. 58 2.6. Transient Analysis can be defined as the VpiLpi product, where Vpi and Lpi are, respectively, the applied voltage and the length required to obtain a pi phase shift of the guided wave. Lpi can be calculated from the formula Lpi = λ0/2∆neff where, ∆neff is the effective index variation at the applied bias of Vpi. The effective index variation of 2.28× 10−4 at V = 6 V leads to a VpiLpi of 2.039 V·cm (Lpi = 0.349 cm) for the 3 quantum-well structure defined in for λ0 = 1.55 µm, which is much lower than the VpiLpi of 2.37 V·cm obtained for λ0 = 1.31 µm predicted by our analysis, at V = 6 V. The optical loss is slightly higher at λ0 = 1.55 µm as compared to that at λ0 = 1.31 µm, which is consistent with the experimental results . 2.6 Transient Analysis Transient analysis is performed to evaluate the response time of the 3 quantum- well modulator λ0 = 1.55 µm using thermionic emission, tunneling, and field dependent mobility model. Marris et al. did not use the field dependent mo- bility model while doing the transient simulations . But we found that, in case of transient analysis, the field dependent mobility significantly af- fects the transient response. The drift velocity of the carriers is the product of the mobility and the electric field in the direction of the current flow. The carrier velocity will increase with the increase of the electric field but at high electric fields, it will begin to saturate due to the reduction of the effective mobility. Hence, without using the field dependent mobility model, the response would be much faster than when using this model and the modulation speed will be overestimated. 59 2.6. Transient Analysis 0.12 0.16 0.2 0.24 0.280 1 2 3 4 5 6 7 8 x 10 18 x (µm) H ol e Co nc en tra tio n (cm − 3 ) t=0 ps QW 3 QW 2 QW 1 0.12 0.16 0.2 0.24 0.280 1 2 3 4 5 6 7 8 x 10 18 x (µm) H ol e Co nc en tra tio n (cm − 3 ) t=2 ps QW 3 QW 2 QW 1 0.12 0.16 0.2 0.24 0.280 1 2 3 4 5 6 7 8 x 10 18 x (µm) H ol e Co nc en tra tio n (cm − 3 ) t=20 ps QW 3 QW 2 QW 1 0.12 0.16 0.2 0.24 0.280 1 2 3 4 5 6 7 8 x 10 18 x (µm) H ol e Co nc en tra tio n (cm − 3 ) t=30 ps QW 3 QW 2 QW 1 0.12 0.16 0.2 0.24 0.280 1 2 3 4 5 6 7 8 x 10 18 x (µm) H ol e Co nc en tra tio n (cm − 3 ) t=40 ps QW 3 QW 2 QW 1 0.12 0.16 0.2 0.24 0.280 1 2 3 4 5 6 7 8 x 10 18 x (µm) H ol e Co nc en tra tio n (cm − 3 ) t=60 ps QW 3 QW 2 QW 1 0.12 0.16 0.2 0.24 0.280 1 2 3 4 5 6 7 8 x 10 18 x (µm) H ol e Co nc en tra tio n (cm − 3 ) t=80 ps QW 3 QW 2 QW 1 0.12 0.16 0.2 0.24 0.280 1 2 3 4 5 6 7 8 x 10 18 x (µm) H ol e Co nc en tra tio n (cm − 3 ) t=100 ps QW 3 QW 2 QW 1 0.12 0.16 0.2 0.24 0.280 1 2 3 4 5 6 7 8 x 10 18 x (µm) H ol e Co nc en tra tio n (cm − 3 ) t=102 ps QW 3 QW 2 QW 1 0.12 0.16 0.2 0.24 0.280 1 2 3 4 5 6 7 8 x 10 18 x (µm) H ol e Co nc en tra tio n (cm − 3 ) t=110 ps QW 3 QW 2 QW 1 0.12 0.16 0.2 0.24 0.280 1 2 3 4 5 6 7 8 x 10 18 x (µm) H ol e Co nc en tra tio n (cm − 3 ) t=120 ps QW 3 QW 2 QW 1 0.12 0.16 0.2 0.24 0.280 1 2 3 4 5 6 7 8 x 10 18 x (µm) H ol e Co nc en tra tio n (cm − 3 ) t=130 ps QW 3 QW 2 QW 1 0.12 0.16 0.2 0.24 0.280 1 2 3 4 5 6 7 8 x 10 18 x (µm) H ol e Co nc en tra tio n (cm − 3 ) t=140 ps QW 3 QW 2 QW 1 0.12 0.16 0.2 0.24 0.280 1 2 3 4 5 6 7 8 x 10 18 x (µm) H ol e Co nc en tra tio n (cm − 3 ) t=160 ps QW 3 QW 2 QW 1 0.12 0.16 0.2 0.24 0.280 1 2 3 4 5 6 7 8 x 10 18 x (µm) H ol e Co nc en tra tio n (cm − 3 ) t=200 ps QW 3 QW 2 QW 1 Figure 2.18: Hole density distribution at various times in the 3 quantum- well modulator, only the quantum-wells and the P+ layers are shown in the figure. 60 2.6. Transient Analysis 0 10 20 30 40 50 60 70 80 90 1000 2 4 6 8 10 12 x 10 17 Time (ps) Av er ag e Ho le C on ce nt ra tio n in th e qu an tu m −w el ls QW1 QW2 QW3 (a) 100 120 140 160 180 2000 0.5 1 1.5 2 x 10 18 Time (ps) Av er ag e Ho le C on ce nt ra tio n in th e qu an tu m −w el ls QW3 QW2 QW1 (b) Figure 2.19: Hole density distribution with time in the 3 quantum-well modulator for (a) t = 0 ps to t = 100 ps (b) t = 100 ps to t = 200 ps. 61 2.6. Transient Analysis 0 20 40 60 80 100 120 140 160 180 2000 0.5 1 1.5 2 2.5 x 10 −4 Time(ps) Ef fe ct ive in de x va ria tio n (∆n e ff) (a) 0 20 40 60 80 100 120 140 160 180 2000 0.5 1 1.5 2 2.5 x 10 −4 Time(ps) Ef fe ct ive in de x va ria tio n (∆n e ff) (b) Figure 2.20: Effective index variation with time in the 3 quantum-well modulator for 0 to 6 V variation (a) with field dependent mobility model (b) without field dependent mobility model. 62 2.6. Transient Analysis At first a reverse bias voltage step from 0 to 6 V, with a rise time of 1 fs, is applied to the device. The hole density distributions in the case of 3 quantum-well modulator obtained at several specific times during depleting the quantum-wells, are plotted in Fig. 2.18. The holes are initially confined in the three quantum-wells at t = 0. As time increases, initially the holes do not escape from the quantum-wells, they simply distribute themselves along the left side of each quantum-well. By t = 36 ps and t = 80 ps the 1st and the 2nd quantum-wells are depleted, respectively. By t = 100 ps, only a few holes are left in the 3rd quantum-well. Fig. 2.19(a) shows that the quantum-wells are depleting sequentially in time as described in Ref. . After 100 ps, the applied voltage returned to zero with a fall time 1 fs. The hole density distributions obtained at several specific times, during which the holes are returning to the quantum-wells, are plotted in Fig. 2.18. Fig. 2.19(b) shows that at t = 150 ps, the average hole distribution became constant in the 3rd and 2nd quantum-well. The hole density distribution in the 1st quantum-well becomes constant after t = 180 ps. In case of transient analysis, the field dependent mobility has significant effects on the effective index variation. The effective index variation curves with time with and without the FLDMOB are shown in Fig. 2.20(a) and Fig. 2.20(b) respectively. The initial sharp increase of effective index varia- tion is due to the hole density change of the NID layer as the quantum-wells are not depleted initially. 63 2.7. Conclusion 2.7 Conclusion In this chapter we found that, in the 3 quantum-well SiGe/Si structure, the quantum-wells deplete sequentially both with time and voltage. During the DC analysis, from Fig. 2.15(a), we found that the quantum-well 1, 2, and 3 are depleted at 2.1 V, 4.3 V, and 7.5 V, respectively. This figure also showed that below 1.5 V, the effective index variation ∆neff is mainly caused by the refractive index change in the first quantum-well. The onset of the change in the refractive index in the second quantum-well occurs after the refractive index change in the first quantum-well approaches the saturation value. If we want to operate in a low bias region (below 1.5 V), only one quantum-well may be sufficient to have the same effective index variation as we obtained in the 3 quantum-well structure. Again, during the transient analysis, from Fig. 2.19(a), we found that by t = 36 ps and t = 80 ps the 1st and the 2nd quantum-wells are depleted, respectively. By t = 100 ps, only a few holes are left in the 3rd quantum- well. If we can get rid of the last 2 quantum-wells, the hole depletion process from the quantum-well could be done by t = 36 ps. Hence we removed 2 quantum-wells in the 3 quantum-well modulator and designed a single quantum-well modulator which will be discussed in the next chapter. 64 Chapter 3 Single Quantum-Well SiGe/Si Optical Modulator In the 3 quantum-well structure, we found that when the voltage is lower than V = 1.5 V, the effective index variation ∆neff is mainly caused by the refractive index change in the first quantum-well. The onset of the change in the refractive index in the second quantum-well occurs after the refractive index change in the first quantum-well approaches the saturation value. This implies that only one quantum-well may be needed to obtain the same effective index variation as is obtained in the 3 quantum-well structure for voltages below about V = 1.5 V. This single quantum-well structure can give the additional benefit of higher intrinsic speed over the 3 quantum-well modulator as we concluded from the results of the transient analysis of the 3 quantum-well modulator in chapter 2. Therefore we designed a single quantum-well modulator keeping the thicknesses and doping levels constant for all of the layers (i.e., 2 QW layer stacks are removed). The results obtained using the DC analysis for the single quantum-well modulator are described in section 3.1. The transient response for this mod- ulator is described in section 3.2. The single quantum-well phase modulator 65 3.1. DC analysis in Single Quantum-Well SiGe/Si Modulator can be converted into an intensity modulator by inserting it into each of the arms of a Mach-Zehnder interferometer and by applying a bias voltage to each of the arms as described in section 1.3.1. The performance of such a Mach-Zehnder Interferometer is described in section 3.3. 3.1 DC analysis in Single Quantum-Well SiGe/Si Modulator A schematic diagram of the single quantum-well SiGe/Si modulator is shown in Fig. 3.1. 1 µm 70 nm 2 8 0 n m X=0 QW Si Substrate 700 nm SiO2 30 nm Si NID 30 nm Si P+ 1018 cm-3 120 nm Si NID 10 nm Si NID 5 nm Si P+ 2×1018 cm-3 10 nm Si0.8Ge0.2 QW 60 nm Si NID 30 nm Si N+ 1018 cm-3 Figure 3.1: Schematic view of the SiGe/Si optical modulator. For the DC analysis, we applied reverse bias voltages ranging from 0 V to 10 V to this modulator. The hole distribution in the structure is calculated using Silvaco ATLASTM at each applied reverse bias voltage. In section 66 3.1. DC analysis in Single Quantum-Well SiGe/Si Modulator 3.1.1 we present the results of the refractive index calculations using the hole distribution in various layers of the single quantum-well modulator. The effective index and optical loss for the fundamental TE mode for operation at 1.55 µm are presented in section 3.1.2. Section 3.1.2 concludes with a comparison of the important performance characteristics (effective index variation, optical loss, and VpiLpi) for 3 quantum-well and single quantum- well modulator. 3.1.1 Refractive Index Change in Single SiGe/Si Quantum-Well Optical Modulator The refractive index change in the single quantum-well saturates at about 2 V as shown in Fig. 3.2. The single quantum-well structure changes from a quantum-well modulator into a NID-layer modulator at this voltage due to the significant contribution of the refractive index change of the NID layer on the P+ side as shown in Fig. 3.2. 67 3.1. DC analysis in Single Quantum-Well SiGe/Si Modulator 0 1 2 3 4 5 6 7 8 9 100 0.4 0.8 1.2 1.6 2 x 10 −3 Applied voltage (V) R ef ra ct ive in de x ch an ge (∆ n ) 3QW modulator (∆n in QW1) 1QW modulator (∆n in QW) 3QW modulator (∆n in NID) 1QW modulator (∆n in NID) Figure 3.2: Comparison of the refractive index change in the single quantum-well structure and in the first quantum-well of the 3 quantum- well structure at λ0 = 1.55 µm. The refractive index change in the NID layers on the P+ side are shown for both the single quantum-well and the 3 quantum-well modulator. 3.1.2 Effective Index and Optical Loss Calculation The effective index variation curves and their slopes are shown in Fig. 3.3. Fig. 3.3(a) illustrates that below 2 V the effective index variation in the single quantum-well structure is slightly higher than that in the 3 quantum- well structure. Above 2 V, and considering only the refractive index changes in the quantum-wells, the effective index variation in the single quantum- well structure becomes smaller than that in the 3 quantum-well structure as shown by the dotted line in Fig. 3.3(a). However, due to the onset of the refractive index change in the NID layer on the P+ side, the single quantum- 68 3.1. DC analysis in Single Quantum-Well SiGe/Si Modulator 0 1 2 3 4 5 6 7 8 9 100 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 x 10 −4 Applied voltage (V) Ef fe ct ive in de x va ria tio n (∆n e ff) 3QW modulator 1QW modulator 3QW modulator (QW contribution) 1QW modulator (QW contribution) (a) 0 1 2 3 4 5 6 7 8 9 100 1 2 3 4 5 6 7 8 x 10 −5 Applied voltage (V) Sl op e of Ef fe ct ive in de x va ria tio n (V− 1 ) 3QW modulator 1QW modulator (b) Figure 3.3: Comparison of the (a) effective index variation (b) slope of the effective index variation for single quantum-well and 3 quantum-well structure. 69 3.1. DC analysis in Single Quantum-Well SiGe/Si Modulator well structure shows significantly higher effective index variation than the 3 quantum-well structure until the cross-over point at about 6.5 V. Above 6.5 V, the 3 quantum-well modulator turns into a NID-layer modulator and the effective index variation in the 3 quantum-well structure becomes greater than that in the single quantum-well structure due to the combined effects of the refractive index changes in the quantum-wells and the NID layer. The optical loss in the single quantum-well structure is lower than that of the 3 quantum-well structure due to the removal of the two quantum-wells as well as the other doped layers. The optical losses for both of the structures are shown in Fig. 3.4. 0 1 2 3 4 5 6 7 8 9 100 1 2 3 4 5 6 Applied voltage (V) O pt ica l lo ss α (dB /cm ) 3QW modulator 1QW modulator Figure 3.4: Comparison of the optical loss in the single quantum-well mod- ulator and the 3 quantum-well modulator. The highest slope of the effective index variation (0.71 × 10−4 V−1) for the single quantum-well modulator occurs at about 1.6 V, as shown 70 3.2. Transient Analysis in Fig. 3.3(b). At this voltage, the effective index variation with respect to 0 V is 1.02 × 10−4 . If we desire a Vpi of 1.6 V for digital signal modula- tion, the VpiLpi of this modulator will be 1.208 V·cm (Lpi = 0.759 cm) which is much lower as compared to the VpiLpi of the 3 quantum-well structure (2.039 V·cm) previously defined. For low voltage modulation, we can apply a voltage between 1.4 V and 1.8 V (i.e., ±0.2 V about 1.6 V). Hence, if we desire a Vpi of 0.4 V, for low voltage modulation, the VpiLpi of this modulator will be 1.09 V·cm, which is also lower than the VpiLpi of the 3 quantum-well modulator. The 3 quantum-well modulator that we studied in chapter 2 was optimized by Marris et al. to get the best performance in terms of the effective index variation and the optical loss by studying the influence of the thicknesses and doping levels of various layers in the structure. But we have not yet optimized the single quantum-well modulator. If we do the optimization of the single quantum-well modulator by doing such studies, it may give even lower VpiLpi than that of the 3 quantum-well modulator. 3.2 Transient Analysis In the case of the single quantum-well modulator, at first, a reverse bias voltage step from 0 to 1.6 V, with a rise time of 1 fs, is applied to the device. The hole density distributions obtained at several specific times, are plotted in Fig. 3.5. Most of the holes are initially confined in the quantum-well at t = 0. Most of the holes are removed from the quantum-well by t = 36 ps. By t = 100 ps, only a few holes are left in the quantum-well as shown in Fig. 3.6(a). 71 3.2. Transient Analysis 0.12 0.16 0.20 1 2 3 4 5 6 7 8 x 10 18 x (µm) H ol e Co nc en tra tio n (cm − 3 ) t=0 ps QW 1 0.12 0.16 0.20 1 2 3 4 5 6 7 8 x 10 18 x (µm) H ol e Co nc en tra tio n (cm − 3 ) t=2 ps QW 1 0.12 0.16 0.20 1 2 3 4 5 6 7 8 x 10 18 x (µm) H ol e Co nc en tra tio n (cm − 3 ) t=20 ps QW 1 0.12 0.16 0.20 1 2 3 4 5 6 7 8 x 10 18 x (µm) H ol e Co nc en tra tio n (cm − 3 ) t=30 ps QW 1 0.12 0.16 0.20 1 2 3 4 5 6 7 8 x 10 18 x (µm) H ol e Co nc en tra tio n (cm − 3 ) t=40 ps QW 1 0.12 0.16 0.20 1 2 3 4 5 6 7 8 x 10 18 x (µm) H ol e Co nc en tra tio n (cm − 3 ) t=60 ps QW 1 0.12 0.16 0.20 1 2 3 4 5 6 7 8 x 10 18 x (µm) H ol e Co nc en tra tio n (cm − 3 ) t=80 ps QW 1 0.12 0.16 0.20 1 2 3 4 5 6 7 8 x 10 18 x (µm) H ol e Co nc en tra tio n (cm − 3 ) t=100 ps QW 1 0.12 0.16 0.20 1 2 3 4 5 6 7 8 x 10 18 x (µm) H ol e Co nc en tra tio n (cm − 3 ) t=102 ps QW 1 0.12 0.16 0.20 1 2 3 4 5 6 7 8 x 10 18 x (µm) H ol e Co nc en tra tio n (cm − 3 ) t=110 ps QW 1 0.12 0.16 0.20 1 2 3 4 5 6 7 8 x 10 18 x (µm) H ol e Co nc en tra tio n (cm − 3 ) t=120 ps QW 1 0.12 0.16 0.20 1 2 3 4 5 6 7 8 x 10 18 x (µm) H ol e Co nc en tra tio n (cm − 3 ) t=130 ps QW 1 0.12 0.16 0.20 1 2 3 4 5 6 7 8 x 10 18 x (µm) H ol e Co nc en tra tio n (cm − 3 ) t=140 ps QW 1 0.12 0.16 0.20 1 2 3 4 5 6 7 8 x 10 18 x (µm) H ol e Co nc en tra tio n (cm − 3 ) t=160 ps QW 1 0.12 0.16 0.20 1 2 3 4 5 6 7 8 x 10 18 x (µm) H ol e Co nc en tra tio n (cm − 3 ) t=200 ps QW 1 Figure 3.5: Hole density distribution at various times in the single quantum- well modulator, only the quantum-well and the P+ layers are shown in the figure. 72 3.2. Transient Analysis 0 10 20 30 40 50 60 70 80 90 1000 2 4 6 8 10 x 10 17 Time (ps) Av er ag e Ho le C on ce nt ra tio n in th e qu an tu m −w el l (a) 100 120 140 160 180 2000 2 4 6 8 10 x 10 17 Time (ps) Av er ag e Ho le C on ce nt ra tio n in th e qu an tu m −w el l (b) Figure 3.6: Hole density distribution with time in the single quantum-well modulator for (a) t = 0 ps to t = 100 ps (b) t = 100 ps to t = 200 ps. 73 3.2. Transient Analysis 0 20 40 60 80 100 120 140 160 180 2000 0.5 1 1.5 2 x 10 −4 Time(ps) Ef fe ct ive in de x va ria tio n (∆n e ff) Figure 3.7: Effective index variation with time in the single quantum-well modulator for 0 to 1.6 V variation. 74 3.3. Mach-Zehnder Interferometer Performance Hence, we can say that the hole depletion process is much faster in the case of the single quantum-well modulator than in the case of the 3 quantum-well modulator. After 100 ps, the applied voltage returns to zero with a fall time 1 fs. Fig. 3.6(b) shows that at t = 140 ps, the average hole distribution becomes constant in the quantum-well. The effective index variation curve for the single quantum-well modulator is shown in Fig. 3.7. From this figure we can conclude that the transient response of the electro-optic effect in the single quantum-well modulator will be faster than that in the 3 quantum-well modulator. For the 3 quantum-well modulator the rise time (the time required for the effective index variation to increase from 10% to 90% of the maximum value) and the fall time (the time required for the effective index variation to decrease from 90% to 10% of the maximum value) are found to be around 26.6 ps and 21 ps, respectively. In case of the single quantum-well modulator these are found to be around 3.02 ps and 2.25 ps, respectively. Due to decrease of the time constants, the single quantum-well modulator will give higher intrinsic speed than the 3 quantum-well modulator. 3.3 Mach-Zehnder Interferometer Performance The single quantum-well phase modulator can be converted into an inten- sity modulator by inserting it into each of the arms of the Mach-Zehnder interferometer and by applying a bias voltage to each of the arms as we described in section 1.3.1. Both the effective index variation (∆neff related to the modification of the propagation constants ∆β) and the absorption 75 3.3. Mach-Zehnder Interferometer Performance variation (∆α) of the light beams, which are functions of the applied volt- age to the phase shifters, will have effects on the output intensity of the Mach-Zehnder interferometer. Assuming identical waveguides, and assum- ing an ideal splitter and combiner, the Mach-Zehnder intensity transmission is given by (see appendix A): Iout Iin (V ) = 1 4 [ e−α1(V )L1 + e−α2(V )L2 + 2 · e−α1L1/2+α2L2/2 · cos (β2(V )L2 − β1(V )L1) ] (3.1) where L1 and L2 are the lengths of phase shifter 1 and phase shifter 2, respectively; β1 and β2 are the propagation constants of the light beams propagating through the phase shifters, and α1 and α2 are the propagation losses of the phase shifters. Assuming reverse bias are applied to both of the arms, β1 = β0 + ∆β1 (3.2) β2 = β0 + ∆β2 (3.3) α1 = α0 −∆α1 (3.4) α2 = α0 −∆α2 (3.5) The length of the two phase shifters are taken to be L1 = 0.5 cm and L2 = 0.7 cm respectively. The voltage applied to the 0.5 cm branch is varied (modulating arm/arm 1) and the voltage at the 0.7 cm branch is fixed at 8.2 V (reference arm/arm 2) to get the minimum intensity at the OFF state. 76 3.3. Mach-Zehnder Interferometer Performance Both the loss and phase components are taken into account to calculate the output transmission. 0 1 2 3 4 5 6 7 8 9 100 0.2 0.4 0.6 0.8 1 Bias Voltage (V) I ou t / I in −0.6 −0.3 0 0.3 0.6 0.9 d(I o u t / I in ) / dV Figure 3.8: Ratio of output to input intensity in a MZI, dotted line shows the slope of this curve. The ratio of output to input intensity as well as its slope as a function of the voltage on arm 1 are given in Fig. 3.8. At Varm1 = 1.5V the slope of the Iout/Iin curve is very high (−0.43 V−1). We can bias the modulating arm at this voltage and then apply a small-signal to this arm so that the output intensity will be varied in a small range. By doing this, the modulator will operate only in the quantum-well-modulation region rather than using the NID-layer-modulation. 77 3.4. Conclusion 3.4 Conclusion In this chapter the performance of a single quantum-well modulator is dis- cussed. During the DC analysis, from Fig. 3.3(a), we found that below 2 V the effective index variation in the single quantum-well structure is slightly higher than that in the 3 quantum-well structure. Fig. 3.3(b) shows that The highest slope of the effective index variation for the single quantum-well modulator occurs at about 1.6 V. We found that, if we desire a Vpi below this voltage, it gives lower VpiLpi as compared to the VpiLpi of the 3 quantum-well modulator for both digital signal modulation and low voltage modulation. Again, during the transient analysis, from Fig. 3.5, we found that most of the holes are removed from the quantum-well by t = 36 ps. We concluded from this figure that the hole depletion process is much faster in the case of the single quantum-well modulator than in the case of the 3 quantum- well modulator. The rise time and the fall time of the single quantum-well modulator, which we calculated from Fig. 3.7, were found to be around 3.02 ps and 2.25 ps, respectively, which are lower than that of the 3 quantum- well modulator. The performance of a Mach-Zehnder interferometer using single quantum-well phase modulators inserted into the two branches of the Mach-Zehnder interferometer were described next. We can conclude with the idea that, if we want to operate in a lower bias region, we can remove the quantum-wells and only the highly doped P+ layers and the NID layers may be sufficient to obtain an effective index variation close to that obtained with the 3 quantum-well or single quantum- well modulator in a lower bias region (V < 2 V). Removing the quantum- 78 3.4. Conclusion wells from the modulator may also increase the intrinsic speed. 79 Chapter 4 Summary, Conclusion, and Suggestions for Future Work 4.1 Summary In this thesis we described the electrical and optical analysis required for designing SiGe/Si quantum-well optical modulators. Then we studied the performance of a 3 quantum-well modulator based on this analysis. Then we designed a single quantum-well modulator which has lower VpiLpi product, lower absorption losses, and higher intrinsic speed than the 3 quantum-well modulator. In Chapter 2, Material Choice, Device Structure, and Electrical and Optical Simulations, we described the key components for designing SiGe/Si quantum-well modulators based on the free carrier depletion effect. The electrical and optical properties of Si and SiGe, which are important for designing these modulators were discussed at the beginning of Chapter 2. The software used for the electrical simulation of these modulators and the models used in this simulation was described next. Then, we discussed the mode solver program used for the optical simulation of SiGe/Si waveguide 80 4.1. Summary modulators. The general structure of a Si0.8Ge0.2/Si quantum-well phase modulator followed by the coupled electrical-optical analysis used to design these modulators was presented next. Using this analysis, we simulated a 3 quantum-well Si0.8Ge0.2/Si modulator (designed by Marris et al.) providing the most important results of the simulations. The two most important performance parameters obtained for this modulator are the VpiLpi product and the optical loss which are found to be around 2.039 V·cm for 0 to 6 V digital modulation and 5.75 dB/cm at V = 0 V for the wavelength of λ0 = 1.55 µm. During the DC analysis, we found that below 1.5 V, the effective index variation ∆neff is mainly caused by the refractive index change in the first quantum-well. The onset of the change in the refractive index in the second quantum-well occurs after the refractive index change in the first quantum-well approaches the saturation value. If we want to operate in a low bias region (below 1.5 V), only one quantum-well may be sufficient to have the same effective index variation as we obtained in the 3 quantum-well structure. Again, during the transient analysis, we found that by t = 36 ps and t = 80 ps the 1st and the 2nd quantum-wells are depleted, respectively. By t = 100 ps, only a few holes are left in the 3rd quantum- well. If we can get rid of the last 2 quantum-wells, the hole depletion process from the quantum-well could be done by t = 36 ps. From the results of the DC analysis and the transient analysis performed on this 3 quantum-well modulator, we came to the conclusion that, a single quantum-well modulator may have a lower drive voltage, lower optical loss, and be capable of higher intrinsic speed than the 3 quantum-well modulator. Hence we removed 2 quantum-wells from the 3 quantum-well modulator and designed a single 81 4.1. Summary quantum-well modulator which was described in Chapter 3. In Chapter 3, Single Quantum-Well SiGe/Si Optical Modulator, the results of the simulation on a single quantum-well Si0.8Ge0.2/Si mod- ulator, which is derived from the 3 quantum-well modulator described in Chapter 2, was discussed. Then we compared its performance with that of the 3 quantum-well modulator and we found that, this single quantum-well modulator is better than the 3 quantum-well modulator in terms of the drive voltage, the optical loss, and the intrinsic speed. The VpiLpi product of the single quantum-well modulator is estimated 1.09 V·cm for low voltage linear modulation and 1.208 V·cm for 0 to 1.6 V digital modulation, whereas the 3 quantum-well modulator gives a VpiLpi of 2.039 V·cm for 0 to 6 V digital modulation for operation at λ0 = 1.55 µm. Also, the optical loss in the single quantum-well (5.36 dB/cm at V = 0 V) is lower than that of the 3 quantum-well structure (5.75 dB/cm at V = 0 V). From the transient analysis of the electro-optic effect, we calculated the rise time and the fall time of both of the modulators. The rise time and the fall time of the single quantum-well modulator are found to be around 3.02 ps and 2.25 ps, re- spectively, whereas for the 3 quantum-well modulator these are found to be around 26.6 ps and 21 ps, respectively. The performance of a Mach-Zehnder interferometer using single quantum-well phase modulators inserted into the two branches of the Mach-Zehnder interferometer was described next. This chapter concluded with the idea that, if we want to operate in a lower bias region, we can remove the quantum-wells and only the highly doped P+ layers and the NID layers are sufficient to obtain an effective index varia- tion close to that obtained with the 3 quantum-well or single quantum-well 82 4.2. Suggestions for Future Work modulator in a lower bias region (V < 2 V). Removing the quantum-wells from the modulator may also increase the intrinsic speed. 4.2 Suggestions for Future Work 4.2.1 All-Silicon Optical Modulators We want to simulate all-Si optical modulators and optimize those struc- tures for getting lower VpiLpi, lower optical loss, and higher intrinsic speed. Currently we have two structures which we want to simulate in future. The 1st device will consist of a P-I-N (P-Intrinsic-N) diode; the active region will consist of a 30 nm Si-NID layer surrounded by two 5 nm P+ highly doped (2 × 1018 cm−3) layers. The PIN diode will be placed on a 30 nm Si-NID layer at the bottom of the PIN diode. This structure will be simulated using a 2 µm SiO2 layer. The 2nd device will consist of a P-I-P-I-N diode laterally grown on a 2 µm SiO2 layer. A 200 nm wide P layer, with a doping concentration of 1017 cm−3, will be inserted into the intrinsic region of the P-I-N diode, giving the P-I-P-I-N structure. This P layer will be surrounded by two NID layers. We expect that these devices can provide lower VpiLpi, lower optical loss, and higher intrinsic speed than those of the SiGe/Si quantum-well modula- tors. 4.2.2 Traveling-Wave Electrodes In the case of the lumped electrodes, the speed of operation is limited by the RC time constants of the electrodes. Hence, we intend to investigate pos- 83 4.2. Suggestions for Future Work 1 µm 70 nm 2 8 0 n m 30 nm Si P+ 1018 cm-3 120 nm Si NID 30 nm Si NID 5 nm Si P+ 2×1018 cm-3 60 nm Si NID 30 nm Si N+ 1018 cm-3 X=0 Si Substrate 2 µm SiO2 30 nm Si NID (a) 400 nm 4 0 0 n m \u0001\u0002\u0003\u0004\u0005 \u0001\u0002\u0003\u0006\u0007\b \u0001\u0002\u0003\u0006\u0005 \u0001\u0002\u0003\u0004\u0005\u0003\n\u000e\u000f \u0001\u0002\u0003\u0006\u0007\b 200 nm200 nm Si Substrate 2 µm SiO2 (b) Figure 4.1: (a) All-Si optical modulator (vertical diode), (b) all-Si optical modulator (lateral diode). 84 4.2. Suggestions for Future Work sible improvement using slow-wave electrodes which remove this RC time constant dependency. The RF signal will be launched onto coplanar slow wave electrodes and co-propagate with the optical signal. The slow-wave electrodes will be designed using the EM simulation software SONNET with the goal of impedance matching, reducing the microwave-light velocity mis- match, and reducing the microwave loss. 85 Bibliography G. 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Available: http://www.virginiasemi.com/ pdf/generalpropertiesSi62002.pdf A. Cordat, S. Lardenois, V. L. Thanh, and A. Koster, “SiGe/Si multiquantum well structure for light modulation,” Materials Science and Engineering B, vol. 89, no. 1-3, pp. 66–69, Feb. 2002. [Online]. Available: http://www.sciencedirect.com/science/article/ B6TXF-44YVPGS-G/2/267924d9f1f6ca50a49a33269efe277b T. E. Murphy, “Design, fabrication and measurement of integrated bragg grating optical filters,” Ph.D. dissertation, Massachusetts Insti- tute of Technology, 2001. 93 Appendix A Mach-Zehnder Interferometer An integrated Mach-Zehnder interferometer consists of an input waveguide, a splitter, two phase shifters, an output combiner, and an output waveguide, as illustrated in Fig. A.1. The optical beam coming through the input waveguide is split into two optical beams by the splitter. The two optical beams travel through the two phase shifters inserted into the arms of the Mach-Zehnder interferometer, and then recombine at the output combiner. Assuming the waveguide Y-branch splitter at the input of the interfer- ometer divides the wave evenly, the intensities in arm 1 and arm 2 of the interferometer will be the same. Suppose, the electric fields in arm 1 and arm 2 of the interferometer as E1 and E2 respectively. Eout = E1 + E2√ 2 (A.1) Suppose k1 and k2, are the complex wavenumbers in arm 1 and arm 2, respectively, which can be expressed by 94 Appendix A. Mach-Zehnder Interferometer Phase-Control Waveguide Sections Optical Y-Branch to split signal Optical Y-Branch to recombine signal O p t i c a l O u t p u t P o w e r O p t i c a l I n p u t P o w e r O p t i c a l O u t p u t P o w e r O p t i c a l I n p u t P o w e r Figure A.1: Schematic view of the Mach-Zehnder interferometer. Two optical Y-branch couplers are used to split and recombine the incoming light. k1 = β1 + iα1 = β0 + ∆β1 + i ( α0 −∆α1 2 ) (A.2) k2 = β2 + iα2 = β0 + ∆β2 + i ( α0 −∆α2 2 ) (A.3) where β0 and α0 is the propagation constant and the optical power ab- sorption coefficient of light in both of the arms without the application of voltage. ∆β1 and ∆α1 are the changes of propagation constant and absorp- tion coefficient, respectively, in arm 1. ∆β2 and ∆α2 are those changes in 95 Appendix A. Mach-Zehnder Interferometer arm 2. Eout = E√ 2 · ei(k1z1−ωt) + E√ 2 · ei(k2z2−ωt) (A.4) Eout = E√ 2 · ei(β1z1−ωt) · e−α1z1/2 + E√ 2 · ei(β2z2−ωt) · e−α2z2/2 (A.5) Pout = Eout · E∗out (A.6) Pout = E2 2 · [ ei(β1z1−ωt) · e−α1z1/2 + ei(β2z2−ωt) · e−α2z2/2 ] · [ e−i(β1z1−ωt) · e−α1z1/2 + e−i(β2z2−ωt) · e−α2z2/2 ] (A.7) Pin = Pin−1 + Pin−2 = E2 + E2 = 2 · E2 (A.8) So, Pout = Pin 4 · [ ei(β1z1−ωt) · e−α1z1/2 + ei(β2z2−ωt) · e−α2z2/2 ] · [ e−i(β1z1−ωt) · e−α1z1/2 + e−i(β2z2−ωt) · e−α2z2/2 ] (A.9) Pout = Pin 4 · [ ei(β1z1−ωt) · e−α1z1/2 + ei(β2z2−ωt) · e−α2z2/2 ] · [ e−i(β1z1−ωt) · e−α1z1/2 + e−i(β2z2−ωt) · e−α2z2/2 ] (A.10) 96 Appendix A. Mach-Zehnder Interferometer Pout = Pin 4 · [ e−α1z1 + e−α2z2 + ei(β2z2−β1z1) · e−α1z1/2+α2z2/2 + ei(β1z1−β2z2) · e−α1z1/2+α2z2/2 ] (A.11) Pout Pin = 1 4 [ e−α1z1 + e−α2z2 + 2 · e−α1z1/2+α2z2/2 · cos (β2z2 − β1z1) ] (A.12) Iout Iin (V ) = 1 4 [ e−α1(V )z1 + e−α2(V )z2 + 2 · e−α1z1/2+α2z2/2 · cos (β2(V )z2 − β1(V )z1) ] (A.13) 97 Appendix B Mode Solver Program The description of the mode solver used to solve for the eigen modes in the SOI waveguides is described in detail in [38, 45]. In this appendix, we will describe the mode solver very briefly. The full-vector eigen value equation which describes the modes of prop- agation for an integrated waveguide is given by- ∥∥∥∥∥∥∥ Pxx Pxy Pyx Pyy ∥∥∥∥∥∥∥ ∥∥∥∥∥∥∥ ex ey ∥∥∥∥∥∥∥ = β2 ∥∥∥∥∥∥∥ ex ey ∥∥∥∥∥∥∥ (B.1) where Pxx...Pyy are differential operators defined as- Pxxex = δ δx [ 1 n2 δ(n2ex) δx ] + δ2ex δy2 + n2k2ex (B.2) Pyyey = δ δy [ 1 n2 δ(n2ey) δy ] + δ2ey δx2 + n2k2ey (B.3) Pxyey = δ δx [ 1 n2 δn2ey δy ] − δ 2ey δxδy (B.4) Pyxex = δ δy [ 1 n2 δn2ex δx ] − δ 2ex δyδx (B.5) where n and β, respectively, denote the refractive index of the layers in 98 Appendix B. Mode Solver Program the waveguide and the propagation constant of the eigen mode. The two transverse electric field components, ex and ey, are the eigenfunctions, and the corresponding eigenvalue is β2. The four remaining field components ez, hx, hy, and hz are derived from these two transverse components by applying Maxwell’s equations. The two transverse field components ex and ey are coupled, i.e. these cannot be solved separately by two separate eigen value equations. Because of this coupling, the eigenmodes of an optical waveguide are usually not purely TE or TM modes, and they are often referred to as hybrid modes . However, in our simulations, we assumed that one of the two transverse field components is larger than the other, and we used the semivectorial finite difference method which neglects the smaller field component and solves the eigenvalue equations for the remaining field component by neglecting Pxy, Pyx and either Pxx or Pyy in the full vector finite difference method which is described in . For example, if we want to solve for ex, The eigen value equation is reduced to the semivectorial eigen value equation which is Pxxex = β2ex (B.6)( δ2 δx2 + δ2 δy2 + n2(x, y)k2 ) ex = β2ex (B.7) The ridge waveguide is broken up into small rectangular cells or pixels of size ∆x×∆y. In each cell, refractive index is constant and the discontinuities in the refractive index may occur at the boundaries of the pixel. This is illustrated in Fig. B.1. For every grid point located at the center of each 99 Appendix B. Mode Solver Program cell, the partial differential equation is translated into a finite difference equation. Suppose for the point P, Equation B.7 will be turned into the difference equation : y x ∆x ∆y PW E N SSW NW NE SE Figure B.1: A typical finite difference mesh for an integrated waveguide. The rib waveguide is shown by the shaded region. P, N, S, E, W, NE, NW, SE and SW are used to label, respectively, the grid point under considera- tion, and its nearest neighbours to the north, south, east, west, north-east, north-west, south-east, and south-west. φW (∆x)2 + φE (∆x)2 + φN (∆y)2 + φS (∆y)2 + ( n2Pk 2 − 2 (∆x)2 − 2 (∆y)2 ) φP = β2φP (B.8) where φP is the field sample at the grid point P and φN , φS , φE , φW are the field samples at the grid points immediately north, south, east, and west of the point under consideration, P. Thus the eigen function ex in equation B.7 is replaced by samples of field located at discrete grid points and the 100 Appendix B. Mode Solver Program operator Pxx can be represented by the following diagram which illustrates the coefficients we have to multiply with each of the sample points adjacent to a particular grid point. 0 1 (∆y)2 0 1 (∆x)2 n2Pk 2 − 2 (∆x)2 − 2 (∆y)2 1 (∆x)2 0 1 (∆y)2 0 If we apply equation B.8 for each grid point in the computational win- dow, we obtain an M number of eigenvalue equations, where M is the total number of the grid points. After constructing the sets of eigen value equa- tions, boundary conditions will be considered for points which lie on the edge of the computation window. There can be three boundary conditions: absorbing, symmetric, and antisymmetric. By ”absorbing”, we mean that the field is assumed to be zero at grid points immediately outside a certain boundary of the computation window. By ”symmetric” (or ”antisymmet- ric”), we mean that the field is assumed to be symmetric (or ”antisymmet- ric”) at grid points immediately outside a certain boundary of the compu- tation window. The modifications of the Pxx operator at the boundaries according to different boundary conditions are discussed in detail in . 101```\n\n#### Cite\n\nCitation Scheme:\n\nCitations by CSL (citeproc-js)\n\n#### Embed\n\nCustomize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.\n``` ```\n<div id=\"ubcOpenCollectionsWidgetDisplay\">\n<script id=\"ubcOpenCollectionsWidget\"\nsrc=\"{[{embed.src}]}\"\ndata-item=\"{[{embed.item}]}\"\ndata-collection=\"{[{embed.collection}]}\"\ndata-metadata=\"{[{embed.showMetadata}]}\"\ndata-width=\"{[{embed.width}]}\"\ndata-media=\"{[{embed.selectedMedia}]}\"\nasync >\n</script>\n</div>\n```\n```", null, "Our image viewer uses the IIIF 2.0 standard. 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https://pro.arcgis.com/en/pro-app/help/analysis/geoprocessing/modelbuilder/inline-variable-substitution.htm	[ "# Inline variable substitution\n\nIn ModelBuilder, the value or dataset path of a variable can be substituted for another variable by enclosing the substituting variable name in percent signs (%VariableName%). Substituting variables in this manner is called inline variable substitution.\n\nFor example, if you have a variable Name with a value of Wilson, you can construct a dataset path as C:\\Data\\Output.gdb\\Clipped_%Name% and the path will be resolved as C:\\Data\\Output.gdb\\Clipped_Wilson.\n\nYou can perform inline variable substitution using any model variables or system variables.\n\n• Model variables—Any variable in a model.\n• System variable—ModelBuilder provides a built-in system variable, n, that can be used in iteration workflows. %n% is the iteration number in the model.\n\n## Model variable substitution\n\nIn the model illustrated below, a workspace variable, Data Workspace, has a value of C:\\Data.gdb. This workspace location is substituted as an inline variable in the Project tool parameters by enclosing the variable name in percent symbols. At run time, the actual variable value, C:\\Data.gdb, is substituted for %Data Workspace%.\n\n### With iterators\n\nThe use of model variable substitution is especially important when working with ModelBuilder iterators. When the iterator Iterate Feature Classes runs, it creates an output variable for both the path and name of each feature class in a workspace. The value in the Name variable can be used to construct the path to the output Projected Feature Class. When the tool executes, %Name% is replaced with the name of the feature class.\n\n### User input to a model tool\n\nModel variable substitution can help you pass values entered by a user directly into a tool inside your model. For example, in the model illustrated below, Parcel ID is a model parameter that is specified when the model tool is run from the Geoprocessing pane. This variable is used in the Expression parameter of the Select Layer By Attribute tool as \"Parcel\" = '%Parcel ID%'. When the tool runs, %Parcel ID% in the expression is replaced with the parcel ID (9 in the case below), and only those parcels with an ID of 9 are selected.\n\n##### Note:\nThe Parcel ID variable in this example is a string. inline variables that are strings need to be enclosed within quotation marks ('%string variable%') in an expression. inline variables that are numbers do not require quotation marks\n\n### Used with Calculate Value\n\nCalculate Value is a powerful ModelBuilder utility that allows you to calculate a value based on any Python expression and use that value in your model. You can use model variable substitution to pass values from model variables into the Calculate Value Python expression, or use the output variable name that stores the calculated value in another tool in your model.\n\nThe model below contains two numeric variables: Number of Residents and Waste Per Person Per Year. These variables are used in the Calculate Value tool expression by enclosing them in percent symbols. When the Calculate Value tool runs, the variable names will be substituted with their values and multiplied together to calculate Total Waste Per Year.\n\n## System variable substitution\n\nModelBuilder provides a built-in system variable that can be used in iteration workflows. This variable, %n%, refers to the current model iteration (the first iteration is zero) when a model contains an iterator.\n\nFor example, the For iterator is used to iterate a model four times. The output of the Buffer tool is used as feedback into the tool as input. The model iterates and creates a new output at each iteration. %n% is used in the output name of Buffer to give the output of each iteration a new name." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.80780977,"math_prob":0.9209546,"size":3637,"snap":"2019-13-2019-22","text_gpt3_token_len":726,"char_repetition_ratio":0.17946601,"word_repetition_ratio":0.010238908,"special_character_ratio":0.19384108,"punctuation_ratio":0.09388971,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9781642,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-03-22T09:20:28Z\",\"WARC-Record-ID\":\"<urn:uuid:3a69d3ba-884e-45ca-9520-087fec417091>\",\"Content-Length\":\"28306\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:3dd10b8d-0b3c-4912-bb91-e2f4ee46f430>\",\"WARC-Concurrent-To\":\"<urn:uuid:71060f89-431b-47ec-8b82-83c1863e0516>\",\"WARC-IP-Address\":\"198.102.61.235\",\"WARC-Target-URI\":\"https://pro.arcgis.com/en/pro-app/help/analysis/geoprocessing/modelbuilder/inline-variable-substitution.htm\",\"WARC-Payload-Digest\":\"sha1:QTCQMIIGFDZO6MQPQP3YFDQJNOXEYZG3\",\"WARC-Block-Digest\":\"sha1:TOKYZ5LW2HTQHYBYBZ73S5RAWHN5QVSK\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-13/CC-MAIN-2019-13_segments_1552912202640.37_warc_CC-MAIN-20190322074800-20190322100800-00034.warc.gz\"}"}
https://intellipaat.com/community/32658/how-to-convert-a-column-or-row-matrix-to-a-diagonal-matrix-in-python	[ "2 views\nin Python\n\nI have a row vector A, A = [a1 a2 a3 ..... an] and I would like to create a diagonal matrix, B = diag(a1, a2, a3, ....., an) with the elements of this row vector. How can this be done in Python?\n\nUPDATE\n\nThis is the code to illustrate the problem:\n\nimport numpy as np\n\na = np.matrix([1,2,3,4])\n\nd = np.diag(a)\n\nprint (d)\n\nthe output of this code is , but my desired output is:\n\n[[1 0 0 0]\n\n[0 2 0 0]\n\n[0 0 3 0]\n\n[0 0 0 4]]\n\nby (40.4k points)\n\nYou can try using diag method like this:\n\nimport numpy as np\n\na = np.array([1,2,3,4])\n\nd = np.diag(a)\n\n# or simpler: d = np.diag([1,2,3,4])\n\nprint(d)\n\nResults in:\n\n[[1 0 0 0]\n\n[0 2 0 0]\n\n[0 0 3 0]\n\n[0 0 0 4]]\n\nIf you have a row vector, you can do this:\n\na = np.array([[1, 2, 3, 4]])\n\nd = np.diag(a)\n\nResults in:\n\n[[1 0 0 0]\n\n[0 2 0 0]\n\n[0 0 3 0]\n\n[0 0 0 4]]\n\nFor the given matrix in the question:\n\nimport numpy as np\n\na = np.matrix([1,2,3,4])\n\nd = np.diag(a.A1)\n\nprint (d)\n\nThe result is again:\n\n[[1 0 0 0]\n\n[0 2 0 0]\n\n[0 0 3 0]\n\n[0 0 0 4]]\n\nIf you wish to learn more about Python, visit the Python tutorial and Python course by Intellipaat.\n\nby (106k points)\n\nTo convert a column or row matrix into a diagonal you can use diagflat:\n\nimport numpy\n\na = np.matrix([1,2,3,4])\n\nd = np.diagflat(a)\n\nprint (d)" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.843749,"math_prob":0.9994572,"size":414,"snap":"2022-40-2023-06","text_gpt3_token_len":151,"char_repetition_ratio":0.107317075,"word_repetition_ratio":0.0,"special_character_ratio":0.41545895,"punctuation_ratio":0.21666667,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99754834,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-09-26T06:42:03Z\",\"WARC-Record-ID\":\"<urn:uuid:92da2cae-c475-4d86-b5c4-bf49cc4e98ad>\",\"Content-Length\":\"92581\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:46bb29e7-dd88-4e6d-b862-7516529c129d>\",\"WARC-Concurrent-To\":\"<urn:uuid:e0386cb6-2c37-4449-a215-e09b2b197dfd>\",\"WARC-IP-Address\":\"104.18.26.176\",\"WARC-Target-URI\":\"https://intellipaat.com/community/32658/how-to-convert-a-column-or-row-matrix-to-a-diagonal-matrix-in-python\",\"WARC-Payload-Digest\":\"sha1:LEXCOHMHCJECJCR3DHXMVDRKWYLSE66B\",\"WARC-Block-Digest\":\"sha1:EFULIX2LWBAHNOPYG4PQGEYPK6TK3OCD\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030334802.16_warc_CC-MAIN-20220926051040-20220926081040-00205.warc.gz\"}"}
https://answers.everydaycalculation.com/subtract-fractions/3-4-minus-5-7	[ "Solutions by everydaycalculation.com\n\n## Subtract 5/7 from 3/4\n\n3/4 - 5/7 is 1/28.\n\n#### Steps for subtracting fractions\n\n1. Find the least common denominator or LCM of the two denominators:\nLCM of 4 and 7 is 28\n\nNext, find the equivalent fraction of both fractional numbers with denominator 28\n2. For the 1st fraction, since 4 × 7 = 28,\n3/4 = 3 × 7/4 × 7 = 21/28\n3. Likewise, for the 2nd fraction, since 7 × 4 = 28,\n5/7 = 5 × 4/7 × 4 = 20/28\n4. Subtract the two like fractions:\n21/28 - 20/28 = 21 - 20/28 = 1/28\n\nMathStep (Works offline)", null, "Download our mobile app and learn to work with fractions in your own time:" ]	[ null, "https://answers.everydaycalculation.com/mathstep-app-icon.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7196393,"math_prob":0.9928272,"size":356,"snap":"2023-14-2023-23","text_gpt3_token_len":185,"char_repetition_ratio":0.22443181,"word_repetition_ratio":0.0,"special_character_ratio":0.53932583,"punctuation_ratio":0.048076924,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9990221,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-06-07T21:52:55Z\",\"WARC-Record-ID\":\"<urn:uuid:0f2c8433-8c94-48a9-b86f-fca9de83bae8>\",\"Content-Length\":\"8468\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d635869c-1a80-43e1-ba3f-4d5f7ba22a22>\",\"WARC-Concurrent-To\":\"<urn:uuid:fcf63ff1-66cf-439e-b4d2-94a8a337d3cb>\",\"WARC-IP-Address\":\"96.126.107.130\",\"WARC-Target-URI\":\"https://answers.everydaycalculation.com/subtract-fractions/3-4-minus-5-7\",\"WARC-Payload-Digest\":\"sha1:H357MPD43A42TWNVARST6MX2YEWTUEBB\",\"WARC-Block-Digest\":\"sha1:J3MF525B3ERHRG4VGSDJ4NMODQU5LJAV\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-23/CC-MAIN-2023-23_segments_1685224654016.91_warc_CC-MAIN-20230607211505-20230608001505-00661.warc.gz\"}"}
https://www.myclassbook.org/2017/03/heterodyne-wave-analyzer.html	[ "", null, "# Heterodyne Wave Analyzer\n\n## Heterodyne Wave Analyzer:\n\n### Introduction:\n\nAnalysis of the waveform means determination of the values of amplitude, frequency and sometime phase angle of the harmonic components.\n\nA wave analyser is an instrument designed to measure relative amplitude of signal frequency components in a complex waveform .basically a wave instruments acts as a frequency selective voltmeter which is tuned to the frequency of one signal while rejecting all other signal components.\n\nIt is well known that any periodic waveform can be represented as the sum of a d.c. component and a series of sinusoidal harmonics. Analysis of a waveform consists of determination of the values of amplitude, frequency, and sometime phase angle of the harmonic components. Graphical and mathematical methods may be used for the purpose but these methods are quite laborious. The analysis of a complex waveform can be done by electrical means using a band pass filter network to single out the various harmonic components. Networks of these types pass a narrow band of frequency and provide a high degree of attenuation to all other frequencies.\n\nA wave analyzer, in fact, is an instrument designed to measure relative amplitudes of single frequency components in a complex waveform. Basically, the instrument acts as a frequency selective voltmeter which is used to the frequency of one signal while rejecting all other signal components. The desired frequency is selected by a frequency calibrated dial to the point of maximum amplitude. The amplitude is indicated either by a suitable voltmeter or CRO.\n\nThis instrument is used in the MHz range. The input signal to be analysed is heterodyned to a higher IF by an internal local oscillator. Tuning the local oscillator shifts various signal frequency components into the pass band of the IF amplifier. The output of the IF amplifier is rectified and is applied to the metering circuit. The instrument using the heterodyning principle is called a heterodyning tuned voltmeter.\n\nThe block schematic of the wave analyser using the heterodyning principle is shown in fig. above. The operating frequency range of this instrument is from 10 kHz to 18 MHz in 18 overlapping bands selected by the frequency range control of the local oscillator. The bandwidth is controlled by an active filter and can be selected at 200, 1000, and 3000 Hz.\n\n### Applications of Wave Analyzers:\n\nWave analyzers have very important applications in the following fields:\n\n1) Electrical measurements\n\n2) Sound measurements and\n\n3) Vibration measurements.\n\nThe wave analyzers are applied industrially in the field of reduction of sound and vibrations generated by rotating electrical machines and apparatus. The source of noise and vibrations is first identified by wave analyzers before it can be reduced or eliminated. A fine spectrum analysis with the wave analyzer shows various discrete frequencies and resonances that can be related to the motion of machines. Once, these sources of sound and vibrations are detected with the help of wave analyzers, ways and means can be found to eliminate them." ]	[ null, "https://4.bp.blogspot.com/-b3N8Xz8TzlE/WzHHducCTtI/AAAAAAAACLQ/t0Qb1XiaYbgqGLde7fU9VoEzn38tUtseACK4BGAYYCw/s728/ad728.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.90248954,"math_prob":0.922695,"size":3348,"snap":"2023-40-2023-50","text_gpt3_token_len":640,"char_repetition_ratio":0.13875598,"word_repetition_ratio":0.08598131,"special_character_ratio":0.18189964,"punctuation_ratio":0.08163265,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9518846,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-08T16:54:47Z\",\"WARC-Record-ID\":\"<urn:uuid:5ac96970-a823-4cf9-89c1-414cd4f8e6ca>\",\"Content-Length\":\"258496\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2cdbc4a7-766c-4ba0-a2f3-7f38bdba3f9d>\",\"WARC-Concurrent-To\":\"<urn:uuid:ce45ad5e-a754-4b6d-bee1-381d30232666>\",\"WARC-IP-Address\":\"172.253.63.121\",\"WARC-Target-URI\":\"https://www.myclassbook.org/2017/03/heterodyne-wave-analyzer.html\",\"WARC-Payload-Digest\":\"sha1:DDHIZFFOTTQ7SE6KNIHRVPVUSDTBDLHP\",\"WARC-Block-Digest\":\"sha1:Z4W6DX2NORQQ2XCXUG3VDW7JXBZLATNM\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100762.64_warc_CC-MAIN-20231208144732-20231208174732-00569.warc.gz\"}"}
https://www.colorhexa.com/165e59	[ "# #165e59 Color Information\n\nIn a RGB color space, hex #165e59 is composed of 8.6% red, 36.9% green and 34.9% blue. Whereas in a CMYK color space, it is composed of 76.6% cyan, 0% magenta, 5.3% yellow and 63.1% black. It has a hue angle of 175.8 degrees, a saturation of 62.1% and a lightness of 22.7%. #165e59 color hex could be obtained by blending #2cbcb2 with #000000. Closest websafe color is: #006666.\n\n• R 9\n• G 37\n• B 35\nRGB color chart\n• C 77\n• M 0\n• Y 5\n• K 63\nCMYK color chart\n\n#165e59 color description : Very dark cyan.\n\n# #165e59 Color Conversion\n\nThe hexadecimal color #165e59 has RGB values of R:22, G:94, B:89 and CMYK values of C:0.77, M:0, Y:0.05, K:0.63. Its decimal value is 1465945.\n\nHex triplet RGB Decimal 165e59 `#165e59` 22, 94, 89 `rgb(22,94,89)` 8.6, 36.9, 34.9 `rgb(8.6%,36.9%,34.9%)` 77, 0, 5, 63 175.8°, 62.1, 22.7 `hsl(175.8,62.1%,22.7%)` 175.8°, 76.6, 36.9 006666 `#006666`\nCIE-LAB 35.785, -22.628, -3.423 6.136, 8.897, 10.844 0.237, 0.344, 8.897 35.785, 22.886, 188.603 35.785, -25.696, -1.459 29.827, -15.476, -0.677 00010110, 01011110, 01011001\n\n# Color Schemes with #165e59\n\n• #165e59\n``#165e59` `rgb(22,94,89)``\n• #5e161b\n``#5e161b` `rgb(94,22,27)``\nComplementary Color\n• #165e35\n``#165e35` `rgb(22,94,53)``\n• #165e59\n``#165e59` `rgb(22,94,89)``\n• #163f5e\n``#163f5e` `rgb(22,63,94)``\nAnalogous Color\n• #5e3516\n``#5e3516` `rgb(94,53,22)``\n• #165e59\n``#165e59` `rgb(22,94,89)``\n• #5e163f\n``#5e163f` `rgb(94,22,63)``\nSplit Complementary Color\n• #5e5916\n``#5e5916` `rgb(94,89,22)``\n• #165e59\n``#165e59` `rgb(22,94,89)``\n• #59165e\n``#59165e` `rgb(89,22,94)``\nTriadic Color\n• #1b5e16\n``#1b5e16` `rgb(27,94,22)``\n• #165e59\n``#165e59` `rgb(22,94,89)``\n• #59165e\n``#59165e` `rgb(89,22,94)``\n• #5e161b\n``#5e161b` `rgb(94,22,27)``\nTetradic Color\n• #07201e\n``#07201e` `rgb(7,32,30)``\n• #0c3532\n``#0c3532` `rgb(12,53,50)``\n• #114945\n``#114945` `rgb(17,73,69)``\n• #165e59\n``#165e59` `rgb(22,94,89)``\n• #1b736d\n``#1b736d` `rgb(27,115,109)``\n• #208780\n``#208780` `rgb(32,135,128)``\n• #259c94\n``#259c94` `rgb(37,156,148)``\nMonochromatic Color\n\n# Alternatives to #165e59\n\nBelow, you can see some colors close to #165e59. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #165e47\n``#165e47` `rgb(22,94,71)``\n• #165e4d\n``#165e4d` `rgb(22,94,77)``\n• #165e53\n``#165e53` `rgb(22,94,83)``\n• #165e59\n``#165e59` `rgb(22,94,89)``\n• #165d5e\n``#165d5e` `rgb(22,93,94)``\n• #16575e\n``#16575e` `rgb(22,87,94)``\n• #16515e\n``#16515e` `rgb(22,81,94)``\nSimilar Colors\n\n# #165e59 Preview\n\nText with hexadecimal color #165e59\n\nThis text has a font color of #165e59.\n\n``<span style=\"color:#165e59;\">Text here</span>``\n#165e59 background color\n\nThis paragraph has a background color of #165e59.\n\n``<p style=\"background-color:#165e59;\">Content here</p>``\n#165e59 border color\n\nThis element has a border color of #165e59.\n\n``<div style=\"border:1px solid #165e59;\">Content here</div>``\nCSS codes\n``.text {color:#165e59;}``\n``.background {background-color:#165e59;}``\n``.border {border:1px solid #165e59;}``\n\n# Shades and Tints of #165e59\n\nA shade is achieved by adding black to any pure hue, while a tint is created by mixing white to any pure color. In this example, #030f0e is the darkest color, while #feffff is the lightest one.\n\n• #030f0e\n``#030f0e` `rgb(3,15,14)``\n• #071e1d\n``#071e1d` `rgb(7,30,29)``\n• #0b2e2c\n``#0b2e2c` `rgb(11,46,44)``\n• #0f3e3b\n``#0f3e3b` `rgb(15,62,59)``\n• #124e4a\n``#124e4a` `rgb(18,78,74)``\n• #165e59\n``#165e59` `rgb(22,94,89)``\n• #1a6e68\n``#1a6e68` `rgb(26,110,104)``\n• #1d7e77\n``#1d7e77` `rgb(29,126,119)``\n• #218e86\n``#218e86` `rgb(33,142,134)``\n• #259e95\n``#259e95` `rgb(37,158,149)``\n• #29ada4\n``#29ada4` `rgb(41,173,164)``\n• #2cbdb3\n``#2cbdb3` `rgb(44,189,179)``\n• #30cdc2\n``#30cdc2` `rgb(48,205,194)``\nShade Color Variation\n• #3fd2c8\n``#3fd2c8` `rgb(63,210,200)``\n• #4fd6cc\n``#4fd6cc` `rgb(79,214,204)``\n• #5fd9d1\n``#5fd9d1` `rgb(95,217,209)``\n• #6fddd6\n``#6fddd6` `rgb(111,221,214)``\n• #7ee1da\n``#7ee1da` `rgb(126,225,218)``\n• #8ee5df\n``#8ee5df` `rgb(142,229,223)``\n• #9ee8e3\n``#9ee8e3` `rgb(158,232,227)``\n• #aeece8\n``#aeece8` `rgb(174,236,232)``\n• #bef0ec\n``#bef0ec` `rgb(190,240,236)``\n• #cef4f1\n``#cef4f1` `rgb(206,244,241)``\n• #def7f5\n``#def7f5` `rgb(222,247,245)``\n• #eefbfa\n``#eefbfa` `rgb(238,251,250)``\n• #feffff\n``#feffff` `rgb(254,255,255)``\nTint Color Variation\n\n# Tones of #165e59\n\nA tone is produced by adding gray to any pure hue. In this case, #3a3a3a is the less saturated color, while #047068 is the most saturated one.\n\n• #3a3a3a\n``#3a3a3a` `rgb(58,58,58)``\n• #353f3e\n``#353f3e` `rgb(53,63,62)``\n• #314342\n``#314342` `rgb(49,67,66)``\n• #2c4846\n``#2c4846` `rgb(44,72,70)``\n• #284c4a\n``#284c4a` `rgb(40,76,74)``\n• #23514d\n``#23514d` `rgb(35,81,77)``\n• #1f5551\n``#1f5551` `rgb(31,85,81)``\n• #1a5a55\n``#1a5a55` `rgb(26,90,85)``\n• #165e59\n``#165e59` `rgb(22,94,89)``\n• #12625d\n``#12625d` `rgb(18,98,93)``\n• #0d6761\n``#0d6761` `rgb(13,103,97)``\n• #096b65\n``#096b65` `rgb(9,107,101)``\n• #047068\n``#047068` `rgb(4,112,104)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #165e59 is perceived by people affected by a color vision deficiency. This can be useful if you need to ensure your color combinations are accessible to color-blind users.\n\nMonochromacy\n• Achromatopsia 0.005% of the population\n• Atypical Achromatopsia 0.001% of the population\nDichromacy\n• Protanopia 1% of men\n• Deuteranopia 1% of men\n• Tritanopia 0.001% of the population\nTrichromacy\n• Protanomaly 1% of men, 0.01% of women\n• Deuteranomaly 6% of men, 0.4% of women\n• Tritanomaly 0.01% of the population" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.560358,"math_prob":0.7014496,"size":3683,"snap":"2021-21-2021-25","text_gpt3_token_len":1664,"char_repetition_ratio":0.1255776,"word_repetition_ratio":0.011090573,"special_character_ratio":0.5544393,"punctuation_ratio":0.23783186,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99242556,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-22T08:28:14Z\",\"WARC-Record-ID\":\"<urn:uuid:aabf5695-0e00-4001-a025-842059fa6141>\",\"Content-Length\":\"36234\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d87a145a-d376-48d6-849a-28089efd1d8f>\",\"WARC-Concurrent-To\":\"<urn:uuid:6d737b9c-62cf-4ad1-ba19-fc8df7963ca7>\",\"WARC-IP-Address\":\"178.32.117.56\",\"WARC-Target-URI\":\"https://www.colorhexa.com/165e59\",\"WARC-Payload-Digest\":\"sha1:L34KMZTFETUHH2WZUHVATDPBWP5LL3C2\",\"WARC-Block-Digest\":\"sha1:H2ZHBUD6IMX6HKVRZT5TITDKSQGG2R76\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623488512243.88_warc_CC-MAIN-20210622063335-20210622093335-00051.warc.gz\"}"}
https://kr.mathworks.com/matlabcentral/cody/problems/42258-calculate-square-and-cube-of-number/solutions/1643379	[ "Cody\n\n# Problem 42258. Calculate square and cube of number\n\nSolution 1643379\n\nSubmitted on 12 Oct 2018 by amirhosein alimohammadi\nThis solution is locked. To view this solution, you need to provide a solution of the same size or smaller.\n\n### Test Suite\n\nTest Status Code Input and Output\n1 Pass\nx = 1; y = 1; assert(isequal(cube(x),y))\n\n2 Pass\nx = 2; y = 8; assert(isequal(cube(x),y))" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.53428483,"math_prob":0.98891735,"size":305,"snap":"2020-10-2020-16","text_gpt3_token_len":101,"char_repetition_ratio":0.14617941,"word_repetition_ratio":0.0,"special_character_ratio":0.35737705,"punctuation_ratio":0.109375,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98177737,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-02-20T10:35:45Z\",\"WARC-Record-ID\":\"<urn:uuid:3c9782e8-f97d-484c-bce6-c4cb782fcc01>\",\"Content-Length\":\"72697\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b855d337-f544-4bdf-bc2e-98aaba22c649>\",\"WARC-Concurrent-To\":\"<urn:uuid:6197d24a-edf8-4e50-a526-b8e0f9a014cd>\",\"WARC-IP-Address\":\"104.110.193.39\",\"WARC-Target-URI\":\"https://kr.mathworks.com/matlabcentral/cody/problems/42258-calculate-square-and-cube-of-number/solutions/1643379\",\"WARC-Payload-Digest\":\"sha1:WYZKXO3GQQXP5OECIS5LR7GCAKRXTZXS\",\"WARC-Block-Digest\":\"sha1:AIDOKQ667PK3YE4VWABQCZPSEKM7UCXY\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-10/CC-MAIN-2020-10_segments_1581875144722.77_warc_CC-MAIN-20200220100914-20200220130914-00400.warc.gz\"}"}
https://answers.yahoo.com/question/index?qid=20191112230100AAsxnBN	[ "# Which compound contains chlorine (Cl) with an oxidation number of +3 A) Cl2O B) Cl2O3 C) ClO2?\n\nRelevance\n• O has an oxidation number of -2 and the total molecule has to equal zero if it is not charged. If it is charged the molecule has to equal the charge.\n\na)\n\nCl2O:\n\n2Cl -2 = 0\n\n2Cl = 2\n\nCl = 1\n\nthat's not the one\n\nb)\n\nCl2O3:2Cl -32 = 0\n\n2Cl -6 = 0\n\n2Cl = 6\n\nCl = 3\n\nlooks right\n\nc)\n\nClO2:\n\n2Cl -22 = 0\n\n2Cl -4 = 0\n\n2Cl = 4\n\nCl = 2\n\nalso not 3, so b) is the best answer.\n\n• B) Cl2O3 ." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.84233373,"math_prob":0.99894637,"size":642,"snap":"2019-51-2020-05","text_gpt3_token_len":244,"char_repetition_ratio":0.14263323,"word_repetition_ratio":0.0,"special_character_ratio":0.36915886,"punctuation_ratio":0.07534247,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9890704,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-12-07T13:42:38Z\",\"WARC-Record-ID\":\"<urn:uuid:ef16e51d-459f-4fce-8689-65d5a6e57c6f>\",\"Content-Length\":\"78489\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:fc0f7880-7c28-4daa-baff-b705994d0066>\",\"WARC-Concurrent-To\":\"<urn:uuid:1398a6f1-2063-469c-9bfc-512433ca4134>\",\"WARC-IP-Address\":\"69.147.92.11\",\"WARC-Target-URI\":\"https://answers.yahoo.com/question/index?qid=20191112230100AAsxnBN\",\"WARC-Payload-Digest\":\"sha1:73HWEXME6FAFBRBMSC2C3ZIRAIW5RUNP\",\"WARC-Block-Digest\":\"sha1:PSGCWH24YARSJL7R5JSRMPQFFEOZLKA7\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-51/CC-MAIN-2019-51_segments_1575540499439.6_warc_CC-MAIN-20191207132817-20191207160817-00490.warc.gz\"}"}
https://tools.carboncollective.co/compound-interest/11058-at-12-percent-in-16-years/	[ "# What is the compound interest on $11058 at 12% over 16 years? If you want to invest$11,058 over 16 years, and you expect it will earn 12.00% in annual interest, your investment will have grown to become $67,789.89. If you're on this page, you probably already know what compound interest is and how a sum of money can grow at a faster rate each year, as the interest is added to the original principal amount and recalculated for each period. The actual rate that$11,058 compounds at is dependent on the frequency of the compounding periods. In this article, to keep things simple, we are using an annual compounding period of 16 years, but it could be monthly, weekly, daily, or even continuously compounding.\n\nThe formula for calculating compound interest is:\n\n$$A = P(1 + \\dfrac{r}{n})^{nt}$$\n\n• A is the amount of money after the compounding periods\n• P is the principal amount\n• r is the annual interest rate\n• n is the number of compounding periods per year\n• t is the number of years\n\nWe can now input the variables for the formula to confirm that it does work as expected and calculates the correct amount of compound interest.\n\nFor this formula, we need to convert the rate, 12.00% into a decimal, which would be 0.12.\n\n$$A = 11058(1 + \\dfrac{ 0.12 }{1})^{ 16}$$\n\nAs you can see, we are ignoring the n when calculating this to the power of 16 because our example is for annual compounding, or one period per year, so 16 × 1 = 16.\n\n## How the compound interest on $11,058 grows over time The interest from previous periods is added to the principal amount, and this grows the sum a rate that always accelerating. The table below shows how the amount increases over the 16 years it is compounding: Start Balance Interest End Balance 1$11,058.00 $1,326.96$12,384.96\n2 $12,384.96$1,486.20 $13,871.16 3$13,871.16 $1,664.54$15,535.69\n4 $15,535.69$1,864.28 $17,399.98 5$17,399.98 $2,088.00$19,487.97\n6 $19,487.97$2,338.56 $21,826.53 7$21,826.53 $2,619.18$24,445.72\n8 $24,445.72$2,933.49 $27,379.20 9$27,379.20 $3,285.50$30,664.70\n10 $30,664.70$3,679.76 $34,344.47 11$34,344.47 $4,121.34$38,465.81\n12 $38,465.81$4,615.90 $43,081.70 13$43,081.70 $5,169.80$48,251.51\n14 $48,251.51$5,790.18 $54,041.69 15$54,041.69 $6,485.00$60,526.69\n16 $60,526.69$7,263.20 $67,789.89 We can also display this data on a chart to show you how the compounding increases with each compounding period. As you can see if you view the compounding chart for$11,058 at 12.00% over a long enough period of time, the rate at which it grows increases over time as the interest is added to the balance and new interest calculated from that figure.\n\n## How long would it take to double $11,058 at 12% interest? Another commonly asked question about compounding interest would be to calculate how long it would take to double your investment of$11,058 assuming an interest rate of 12.00%.\n\nWe can calculate this very approximately using the Rule of 72.\n\nThe formula for this is very simple:\n\n$$Years = \\dfrac{72}{Interest\\: Rate}$$\n\nBy dividing 72 by the interest rate given, we can calculate the rough number of years it would take to double the money. Let's add our rate to the formula and calculate this:\n\n$$Years = \\dfrac{72}{ 12 } = 6$$\n\nUsing this, we know that any amount we invest at 12.00% would double itself in approximately 6 years. So $11,058 would be worth$22,116 in ~6 years.\n\nWe can also calculate the exact length of time it will take to double an amount at 12.00% using a slightly more complex formula:\n\n$$Years = \\dfrac{log(2)}{log(1 + 0.12)} = 6.12\\; years$$\n\nHere, we use the decimal format of the interest rate, and use the logarithm math function to calculate the exact value.\n\nAs you can see, the exact calculation is very close to the Rule of 72 calculation, which is much easier to remember.\n\nHopefully, this article has helped you to understand the compound interest you might achieve from investing \\$11,058 at 12.00% over a 16 year investment period." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.93024355,"math_prob":0.9989636,"size":3955,"snap":"2022-40-2023-06","text_gpt3_token_len":1210,"char_repetition_ratio":0.14325488,"word_repetition_ratio":0.015267176,"special_character_ratio":0.38432363,"punctuation_ratio":0.1799787,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9999093,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-10-07T05:04:15Z\",\"WARC-Record-ID\":\"<urn:uuid:170d26aa-bd39-43cd-97fe-b20ff6eef711>\",\"Content-Length\":\"27145\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e4847820-1b01-4c3b-aede-e9327b844864>\",\"WARC-Concurrent-To\":\"<urn:uuid:db6b7170-5684-4cbe-9c31-3e7f07a79edb>\",\"WARC-IP-Address\":\"138.197.3.89\",\"WARC-Target-URI\":\"https://tools.carboncollective.co/compound-interest/11058-at-12-percent-in-16-years/\",\"WARC-Payload-Digest\":\"sha1:6QHOQBWRXOO3HKBKMN6LFVYLUSGWKE7P\",\"WARC-Block-Digest\":\"sha1:WL3O4B4P4MX5BFCIQHQ6ORBOG4WR6YU5\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030337971.74_warc_CC-MAIN-20221007045521-20221007075521-00515.warc.gz\"}"}
https://www.vrcbuzz.com/tag/binomial-distribution-using-r/	[ "## Binomial distribution probabilities using R\n\nBinomial distribution probabilities using R In this tutorial, you will learn about how to use dbinom(), pbinom(), qbinom() and rbinom() functions in R programming language to compute the individual probabilities, cumulative probabilities, quantiles and how to generate random sample from Binomial distribution. Before we discuss R functions for binomial distribution, let us see what is …" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.74735487,"math_prob":0.9673844,"size":432,"snap":"2023-14-2023-23","text_gpt3_token_len":84,"char_repetition_ratio":0.16121495,"word_repetition_ratio":0.03508772,"special_character_ratio":0.17592593,"punctuation_ratio":0.104477614,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9994747,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-06-02T14:55:29Z\",\"WARC-Record-ID\":\"<urn:uuid:b9d017ab-2017-47c0-9dec-47f1d1ef5640>\",\"Content-Length\":\"80312\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:cabd7f6e-470e-4464-9242-2e3d974ac026>\",\"WARC-Concurrent-To\":\"<urn:uuid:cbeb8795-60eb-417b-8ece-6742139b5515>\",\"WARC-IP-Address\":\"50.16.223.119\",\"WARC-Target-URI\":\"https://www.vrcbuzz.com/tag/binomial-distribution-using-r/\",\"WARC-Payload-Digest\":\"sha1:ANC3F7BKKTZMIO7WI5MPG4VDAITHEDCA\",\"WARC-Block-Digest\":\"sha1:WJRAD5VOEZ7GMN4YPSFJKNZA7QXNBQAE\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-23/CC-MAIN-2023-23_segments_1685224648695.4_warc_CC-MAIN-20230602140602-20230602170602-00015.warc.gz\"}"}
https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Supplemental_Modules_and_Websites_(Inorganic_Chemistry)/Crystal_Field_Theory/Tetrahedral_vs._Square_Planar_Complexes	[ "# Tetrahedral vs. Square Planar Complexes\n\nLearning Objectives\n\n• Discuss the d-orbital degeneracy of square planar and tetrahedral metal complexes.\n\n## Tetrahedral Geometry\n\nTetrahedral geometry is a bit harder to visualize than square planar geometry. Tetrahedral geometry is analogous to a pyramid, where each of corners of the pyramid corresponds to a ligand, and the central molecule is in the middle of the pyramid. This geometry also has a coordination number of 4 because it has 4 ligands bound to it. Finally, the bond angle between the ligands is 109.5o. An example of the tetrahedral molecule $$\\ce{CH4}$$, or methane.\n\nIn a tetrahedral complex, $$Δ_t$$ is relatively small even with strong-field ligands as there are fewer ligands to bond with. It is rare for the $$Δ_t$$ of tetrahedral complexes to exceed the pairing energy. Usually, electrons will move up to the higher energy orbitals rather than pair. Because of this, most tetrahedral complexes are high spin.\n\n## Square Planar Complexes\n\nIn square planar molecular geometry, a central atom is surrounded by constituent atoms, which form the corners of a square on the same plane. The geometry is prevalent for transition metal complexes with d8 configuration. This includes Rh(I), Ir(I), Pd(II), Pt(II), and Au(III). Notable examples include the anticancer drugs cisplatin ($$\\ce{PtCl2(NH3)2}$$).\n\nA square planar complex also has a coordination number of 4. The structure of the complex differs from tetrahedral because the ligands form a simple square on the x and y axes. Because of this, the crystal field splitting is also different (Figure $$\\PageIndex{1}$$). Since there are no ligands along the z-axis in a square planar complex, the repulsion of electrons in the $$d_{xz}$$, $$d_{yz}$$, and the $$d_{z^2}$$ orbitals are considerably lower than that of the octahedral complex (the $$d_{z^2}$$ orbital is slightly higher in energy to the \"doughnut\" that lies on the x,y axis). The $$d_{x^2-y^2}$$ orbital has the most energy, followed by the $$d_{xy}$$ orbital, which is followed by the remaining orbtails (although $$d_{z^2}$$ has slightly more energy than the $$d_{xz}$$ and $$d_{yz}$$ orbital). This pattern of orbital splitting remains constant throughout all geometries. Whichever orbitals come in direct contact with the ligand fields will have higher energies than orbitals that slide past the ligand field and have more of indirect contact with the ligand fields. So when confused about which geometry leads to which splitting, think about the way the ligand fields interact with the electron orbitals of the central atom.", null, "Figure $$\\PageIndex{1}$$: In square planar complexes Δ will almost always be large, even with a weak-field ligand. Electrons tend to be paired rather than unpaired because paring energy is usually much less than Δ. Therefore, square planar complexes are usually low spin. (CC BY-SA; Ümit Kaya)\n\nIn square planar complexes $$Δ$$ will almost always be large (Figure $$\\PageIndex{1}$$), even with a weak-field ligand. Electrons tend to be paired rather than unpaired because paring energy is usually much less than $$Δ$$. Therefore, square planar complexes are usually low spin.\n\nAdvanced: $$\\ce{[PdCl4]^{2-}}$$ is square planar and $$\\ce{[NiCl4]^{2-}}$$ is tetrahedral\n\nThe molecule $$\\ce{[PdCl4]^{2−}}$$ is diamagnetic, which indicates a square planar geometry as all eight d-electrons are paired in the lower-energy orbitals. However, $$\\ce{[NiCl4]^{2−}}$$ is also d8 but has two unpaired electrons, indicating a tetrahedral geometry. Why is $$\\ce{[PdCl4]^{2−}}$$ square planar if $$\\ce{Cl^{-}}$$ is not a strong-field ligand?\n\nSolution\n\nThe geometry of the complex changes going from $$\\ce{[NiCl4]^{2−}}$$ to $$\\ce{[PdCl4]^{2−}}$$. Clearly this cannot be due to any change in the ligand since it is the same in both cases. It is the other factor, the metal, that leads to the difference.\n\nConsider the splitting of the d-orbitals in a generic d8 complex. If it were to adopt a square planar geometry, the electrons will be stabilized (with respect to a tetrahedral complex) as they are placed in orbitals of lower energy. However, this comes at a cost: two of the electrons, which were originally unpaired in the tetrahedral structure, are now paired in the square-planer structure:", null, "We can label these two factors as $$ΔE$$ (stabilization derived from occupation of lower-energy orbitals) and $$P$$ (spin pairing energy) respectively. One can see that:\n\n• If $$ΔE>P$$, then the complex will be square planar\n• If $$ΔE<P$$, then the complex will be tetrahedral.\n\nThis is analogous to deciding whether an octahedral complex adopts a high- or low-spin configuration; where the crystal field splitting parameter $$Δ_o$$ $$ΔE$$ does above. Unfortunately, unlike $$Δ_o$$ in octahedral complexes, there is no simple graphical way to represent $$ΔE$$ on the diagram above since multiple orbitals are changed in energy between the two geometries.\n\nInterpreting the origin of metal-dependent stabilization energies can be tricky. However, we know experimentally that $$\\ce{Pd^{2+}}$$ has a larger splitting of the d-orbitals and hence a larger $$\\Delta E$$ than $$\\ce{Ni^{2+}}$$ (moreover $$P$$ is also smaller).\n\nPractically all 4d and 5d d8 $$\\ce{ML4}$$ complexes adopt a square planar geometry, irrespective if the ligands are strong-field ligand or not. Other examples of such square planar complexes are $$\\ce{[PtCl4]^{2−}}$$ and $$\\ce{[AuCl4]^{-}}$$.\n\n## Summary\n\n• In tetrahedral molecular geometry, a central atom is located at the center of four substituents, which form the corners of a tetrahedron.\n• Tetrahedral geometry is common for complexes where the metal has d0 or d10electron configuration.\n• The CFT diagram for tetrahedral complexes has dx2−y2 and dzorbitals equally low in energy because they are between the ligand axis and experience little repulsion.\n• In square planar molecular geometry, a central atom is surrounded by constituent atoms, which form the corners of a square on the same plane.\n• The square planar geometry is prevalent for transition metal complexes with dconfiguration.\n• The CFT diagram for square planar complexes can be derived from octahedral complexes yet the dx2-y2 level is the most destabilized and is left unfilled." ]	[ null, "https://chem.libretexts.org/@api/deki/files/346366/sp-01.svg", null, "https://chem.libretexts.org/@api/deki/files/370959/Chem507f09sqvstet2.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.90844285,"math_prob":0.9973359,"size":6064,"snap":"2021-31-2021-39","text_gpt3_token_len":1540,"char_repetition_ratio":0.16716172,"word_repetition_ratio":0.073913045,"special_character_ratio":0.24093008,"punctuation_ratio":0.090064995,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99870294,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,5,null,3,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-09-20T02:28:47Z\",\"WARC-Record-ID\":\"<urn:uuid:f6e61d27-5f3b-45b4-a166-395c4637a527>\",\"Content-Length\":\"102814\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f211a2a9-2f97-4801-9400-9d96d3e72233>\",\"WARC-Concurrent-To\":\"<urn:uuid:3b97059a-adb6-4b88-9a74-784ad65e3581>\",\"WARC-IP-Address\":\"99.86.230.108\",\"WARC-Target-URI\":\"https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Supplemental_Modules_and_Websites_(Inorganic_Chemistry)/Crystal_Field_Theory/Tetrahedral_vs._Square_Planar_Complexes\",\"WARC-Payload-Digest\":\"sha1:CMTXRKNHPMFS5RIQLCUST3CXJ3W4THO2\",\"WARC-Block-Digest\":\"sha1:TENH6QWRIBS2CLV4XTCQMPFWTVUU4OTB\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780056974.30_warc_CC-MAIN-20210920010331-20210920040331-00541.warc.gz\"}"}
https://math.stackexchange.com/questions/3546034/how-to-prove-frac-a-sqrta23b23c2-frac-b-sqrt3a2b23c2-frac	[ "# How to prove $\\frac a{\\sqrt{a^2+3b^2+3c^2}}+\\frac b{\\sqrt{3a^2+b^2+3c^2}}+\\frac{c}{\\sqrt{3a^2+3b^2+c^2}}\\le\\frac3{\\sqrt7}$ when $a,b,c>0$\n\nI want to prove that for $$a,b,c>0$$ we have\n\n$$\\sum_{cyc} \\frac a{\\sqrt{a^2+3b^2+3c^2}}= \\frac a{\\sqrt{a^2+3b^2+3c^2}}+\\frac{b}{\\sqrt{3a^2+b^2+3c^2}}+\\frac{c}{\\sqrt{3a^2+3b^2+c^2}}\\le\\frac3{\\sqrt7}.$$\n\nMy first attempt: By Cauchy-Schwarz we have $$\\left(\\sum_{cyc} \\frac a{\\sqrt{a^2+3b^2+3c^2}}\\right)^2\\le3\\sum_{cyc}\\frac{a^2}{a^2+3b^2+3c^2}$$ so we only need to prove that the right-hand side is always less than $$\\frac{9}{7}$$, but this is false. Failed\n\nSecond attempt: By Cauchy-Schwarz\n\n$$\\sum_{cyc} \\frac a{\\sqrt{a^2+3b^2+3c^2}}=\\sum_{cyc} \\frac 1{\\sqrt{1+3\\frac{b^2}{a^2}+3\\frac{c^2}{a^2}}}\\le\\sum_{cyc} \\frac{\\sqrt 7}{1+3\\frac{b}{a}+3\\frac{c}a}$$\n\nso it remains to prove that $$\\sum_{cyc} \\frac{a}{1+3b+3c}\\le\\frac37$$ but this is wrong for example for $$a=1,b=1,c=2$$. Failed\n\nThird attempt: Let $$S=3(a^2+b^2+c^2)$$. We need to prove $$\\sum_{cyc} \\frac{a}{\\sqrt{S-2a^2}}\\le \\frac37.$$ But $$x\\mapsto \\frac{x}{\\sqrt{S -2x^2}}$$ is convex so Jensen has the wrong direction...\n\n## 4 Answers\n\nJust an observation, the equality is achieved not only for $$a=b=c$$, but also for $$(a^2 \\colon b^2 \\colon c^2) = (8\\colon 1\\colon 1)$$\n\n$$\\bf{Added:}$$\n\nDefine $$x=\\frac{a^2}{a^2 + 3 b^2 + 3 c^2},\\ y=\\frac{b^2}{b^2 + 3 a^2 + 3 c^2},\\ z=\\frac{c^2}{c^2 + 3 a^2 + 3 b^2}$$\n\nOne checks (say by direct calculation) that $$28 x y z + 8(x y + x z + y z )+ x+y+z-1=0$$\n\nSo it is enough to show that the maximum of the function $$\\sqrt{x}+\\sqrt{y}+\\sqrt{z}$$ on the part of the above surface in the first octant is $$\\frac{3}{\\sqrt{7}}$$. The Lagrange multiplier system $$28 x y z + 8(x y + x z + y z )+ x+y+z-1=0\\\\ t - x( 28 y z + 8 (y+z) + 1)^2 =0\\\\ t - y( 28 x z + 8 (x+z) + 1)^2 =0\\\\ t - z( 28 x y + 8 (x+y) + 1)^2 =0$$\n\nis in fact not that hard to solve, if we use Groebner bases. First, by elimination we get the equation in $$t$$:\n\n$$72313663744 t^7 - 207058475232 t^6 - 212349914280 t^5 + 806857109604 t^4 + 125825565483 t^3 - 784526490225 t^2=0$$ which factor nicely as $$t^2 (1372 t - 2025) (343 t - 729) (56 t + 81) (2744 t^2 - 1944 t - 6561)=0$$\n\nNow one considers each of the possible positive values of $$t$$ and solves the system in $$x$$, $$y$$, $$z$$.\n\nCase 1. $$343 t - 729=0$$. We get $$x=y=z=\\frac{1}{7}$$\n\nCase 2. $$1372 t - 2025=0$$. We get the solution $$x=\\frac{4}{7}$$, $$y=z=\\frac{1}{28}$$ and the cyclic permutation of it.\n\nCase 3. $$t = \\frac{81(6 + 19\\sqrt{2})}{1372}$$\n\nWe get $$x=\\frac{-2 + 3 \\sqrt{2}}{7}$$, $$y=z = \\frac{-2 + 3 \\sqrt{2}}{28}$$ and the circular permutations.\n\nCase 4. $$t=0$$ gives negative solutions, so we discard it\n\nThe inequality now follows, in the cases 1, 2, the functions takes the maximum value $$\\frac{3}{\\sqrt{7}}$$, in case 3 the value is smaller.\n\nWe need to prove that: $$\\sum_{cyc}\\sqrt{\\frac{a}{a+3b+3c}}\\leq\\frac{3}{\\sqrt7},$$ where $$a$$, $$b$$ and $$c$$ are positive numbers.\n\nIndeed, by C-S $$\\left(\\sum_{cyc}\\sqrt{\\frac{a}{a+3b+3c}}\\right)^2\\leq\\sum_{cyc}\\frac{a}{(a+3b+3c)(17a+2b+2c)}\\sum_{cyc}(17a+2b+2c).$$ Thus, it's enough to prove that: $$\\sum_{cyc}\\frac{a}{(a+3b+3c)(17a+2b+2c)}\\leq\\frac{3}{49(a+b+c)}.$$ Now, let $$a+b+c=3u$$, $$ab+ac+bc=3v^2$$ and $$abc=w^3$$.\n\nThus, we need to prove that: $$\\sum_{cyc}\\frac{a}{(9u-2a)(6u+15a)}\\leq\\frac{1}{49u}$$ or $$49u\\sum_{cyc}a(9u-2b)(9u-2c)(2u+5b)(2u+5c)\\leq3\\prod_{cyc}((9u-2a)(2u+5a)).$$ We'll prove that the last inequality is true even for any reals $$a$$, $$b$$ and $$c$$.\n\nIndeed, since $$\\sum\\limits_{cyc}a(9u-2b)(9u-2c)(2u+5b)(2u+5c)$$ is a fifth degree polynomial,\n\nthe last inequality is equivalent to $$f(w^3)\\geq0,$$ where $$f(w^3)=-3000w^6+A(u,v^2)w^3+B(u,v^2).$$ But $$f$$ is a concave function.\n\nThus, $$f$$ gets a minimal value for an extreme value of $$w^3$$, which happens for equality case of two variables.\n\nSince the last inequality is an even degree, homogeneous and symmetrical, it's enough to assume $$b=c=1,$$ which gives $$\\frac{a}{(a+6)(17a+4)}+\\frac{2}{(3a+4)(2a+19)}\\leq\\frac{3}{49(a+2)}$$ or $$(a-1)^2(a-8)^2\\geq0$$ and we are done!\n\n• Can I ask: when you use Cauchy-Schwarz, do you look for $x,y,z$ such that: $$\\frac{7a}{(a+3b+3c)(xa+yb+zc)^2}=\\frac{7b}{(b+3a+3c)(xb+ya+zc)^2}=\\frac{7c}{(c+3a+3b)(xc+ya+zb)^2}$$ for both equality cases? – LHF Feb 23 '20 at 15:28\n• @Atticus Yes, of course. Because our vectors should be parallel. – Michael Rozenberg Feb 23 '20 at 15:33\n\nAlternative proof:\n\nLet $$x, y, z > 0$$ such that $$\\frac{7a^2}{a^2 + 3b^2 + 3c^2} = x^2, \\ \\frac{7b^2}{b^2 + 3c^2 + 3a^2} = y^2, \\ \\frac{7c^2}{c^2 + 3a^2 + 3b^2} = z^2$$. It is not hard to obtain $$F(x, y, z) = 4 x^2 y^2 z^2+8 x^2 y^2+8 x^2 z^2+8 y^2 z^2+7 x^2+7 y^2+7 z^2-49 = 0.$$\n\nIt suffices to prove that if $$x, y, z > 0$$ and $$F(x,y,z) = 0$$, then $$x+y+z \\le 3$$. It suffices to prove that if $$x, y, z > 0$$ and $$x + y + z > 3$$, then $$F(x, y, z) > 0$$. Note that $$F(\\alpha x, \\alpha y, \\alpha z) > F(x, y, z)$$ for any $$\\alpha > 1$$ and $$x, y, z > 0$$. Thus, it suffices to prove that if $$x, y, z > 0$$ and $$x+y+z = 3$$, then $$F(x, y, z) \\ge 0$$.\n\nWe use pqr method. Let $$p = x + y + z = 3$$, $$q = xy + yz + zx$$ and $$r = xyz$$. From $$p^2 \\ge 3q$$, we have $$q \\le 3$$. Let $$q = 3(1-u^2)$$ for $$0 \\le u\\le 1$$. We have \\begin{align} (x-y)^2(y-z)^2(z-x)^2 &= -4p^3r+p^2q^2+18pqr-4q^3-27r^2\\\\ & = 108u^6 - 27(3u^2+r-1)^2 \\end{align} which results in $$108u^6 - 27(3u^2+r-1)^2 \\ge 0$$ and hence $$r \\le (2u+1)(1-u)^2 \\le 3$$. Thus, we have \\begin{align} F(x, y, z) &= 7p^2-16pr+8q^2+4r^2-14q-49 \\\\ &= 4(6-r)^2+72u^4-102u^2-100\\\\ &\\ge 4(6 - (2u+1)(1-u)^2)^2 + 72u^4-102u^2-100\\\\ &= 2u^2(2u^2-4u+9)(2u-1)^2\\\\ &\\ge 0. \\end{align} We are done.\n\nWe need to prove that:\n\n$$\\sqrt{\\frac{a}{a+3b+3c}}+\\sqrt{\\frac{b}{b+3c+3a}}+\\sqrt{\\frac{c}{c+3a+3b}}\\leq\\frac{3}{\\sqrt7}$$\n\nLet's normalize with $$a+b+c=3$$. Then the inequality is equivalent with:\n\n$$\\sqrt{\\frac{a}{9-2a}}+\\sqrt{\\frac{b}{9-2b}}+\\sqrt{\\frac{c}{9-2c}}\\leq \\frac{3}{\\sqrt{2}}$$\n\nWithout loss of generality suppose that $$a\\le b\\le c$$. Then we have $$a+b\\leq 2$$ and we will prove:\n\n$$\\sqrt{\\frac{a}{9-2a}}+\\sqrt{\\frac{b}{9-2b}} \\leq \\sqrt{\\frac{2(a+b)}{9-a-b}}$$\n\nSquaring twice, this is equivalent with:\n\n$$\\frac{(a-b)^2[729+81(a+b)^2-486(a+b)-16ab(a+b)]}{(9-2a)^2(9-2b)^2(9-a-b)^2}\\geq 0$$\n\nWe have $$16ab(a+b)\\leq 16(a+b)$$ and\n\n$$729+81x^2-502x\\geq 0, \\text{ when }x \\leq 2$$\n\nIt remains to prove that:\n\n$$\\sqrt{\\frac{14(3-c)}{6+c}}+\\sqrt{\\frac{7c}{9-2c}}\\leq 3$$\n\nSquaring twice this is equivalent with:\n\n$$\\frac{81(c-1)^2 (12 - 5 c)^2}{(9 - 2 c)^2 (6 + c)^2}\\geq 0$$\n\nwhich gives the two equality cases." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5731804,"math_prob":1.0000099,"size":970,"snap":"2021-04-2021-17","text_gpt3_token_len":469,"char_repetition_ratio":0.20393375,"word_repetition_ratio":0.0,"special_character_ratio":0.44536084,"punctuation_ratio":0.06504065,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000098,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-04-15T23:41:31Z\",\"WARC-Record-ID\":\"<urn:uuid:efa03519-b4f6-4c34-b77b-a408ad77ef00>\",\"Content-Length\":\"200820\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4517a5b2-2583-469a-b113-291aafc15025>\",\"WARC-Concurrent-To\":\"<urn:uuid:cb26e3ff-4491-4e97-b60e-03070783291b>\",\"WARC-IP-Address\":\"151.101.65.69\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/3546034/how-to-prove-frac-a-sqrta23b23c2-frac-b-sqrt3a2b23c2-frac\",\"WARC-Payload-Digest\":\"sha1:JHIGES2Z6JCEW4BNB6RLK5QFD34IX3V5\",\"WARC-Block-Digest\":\"sha1:DWX3YYDWO2HYMFHP4V76AMYA7QXRPQZV\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-17/CC-MAIN-2021-17_segments_1618038088264.43_warc_CC-MAIN-20210415222106-20210416012106-00176.warc.gz\"}"}
https://www.elitenicheresearch.com/search/body-mass-index-calculator-for-adults	[ "Keyword Analysis & Research: body mass index calculator for adults\n\nKeyword Research: People who searched body mass index calculator for adults also searched\n\nHow do you figure out your body mass index?\n\nTo use the body mass index calculator to find your BMI, Enter your current weight. (kilograms or pounds) Fill in your height (feet and inches or just inches) Press ‘Calculate’. You now have your basic Body Mass Index calculation. Furthermore you will see how YOU compare to others in the general population that are the same age and sex as you.\n\nHow do you calculate mass body index?\n\nTo calculate your body mass index (BMI), start by measuring your height in meters and then squaring it. Then, divide your weight in kilograms by your height in meters squared to find your BMI. If you're using imperial measurements, start by measuring your height in inches and then squaring it.\n\nWhat is body mass index used to calculate?\n\nBody Mass Index (BMI) is a person’s weight in kilograms divided by the square of height in meters. A high BMI can be an indicator of high body fatness. BMI can be used to screen for weight categories that may lead to health problems but it is not diagnostic of the body fatness or health of an individual. Calculate Your BMI.\n\nHow to calculate body mass index properly?\n\nBody Mass Index Calculator - Moose and Doc Enter your current weight. (kilograms or pounds) Fill in your height (feet and inches or just inches) Press 'Calculate'." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7972041,"math_prob":0.9341664,"size":1769,"snap":"2022-05-2022-21","text_gpt3_token_len":475,"char_repetition_ratio":0.16147308,"word_repetition_ratio":0.124590166,"special_character_ratio":0.30977952,"punctuation_ratio":0.105405405,"nsfw_num_words":1,"has_unicode_error":false,"math_prob_llama3":0.9542505,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-01-27T19:42:01Z\",\"WARC-Record-ID\":\"<urn:uuid:2dd3ffd5-fd0a-42c8-8395-25a6a2b37914>\",\"Content-Length\":\"111424\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9717eb95-faf6-413c-9b22-60aed8369bf4>\",\"WARC-Concurrent-To\":\"<urn:uuid:4e33c8cd-91e8-404a-8d8c-5e21f9ffccfa>\",\"WARC-IP-Address\":\"172.67.180.160\",\"WARC-Target-URI\":\"https://www.elitenicheresearch.com/search/body-mass-index-calculator-for-adults\",\"WARC-Payload-Digest\":\"sha1:ND3SLBKNOH2TSGTNU3UA7EPPD3EJ4C4Q\",\"WARC-Block-Digest\":\"sha1:NRXZWTBF3YY4CUHBFUXS6CBI7UQLYORK\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-05/CC-MAIN-2022-05_segments_1642320305288.57_warc_CC-MAIN-20220127193303-20220127223303-00256.warc.gz\"}"}
https://algebra-net.com/intermediate-algebra-class-notes.html	[ "Try the Free Math Solver or Scroll down to Resources!\n\n Depdendent Variable\n\n Number of equations to solve: 23456789\n Equ. #1:\n Equ. #2:\n\n Equ. #3:\n\n Equ. #4:\n\n Equ. #5:\n\n Equ. #6:\n\n Equ. #7:\n\n Equ. #8:\n\n Equ. #9:\n\n Solve for:\n\n Dependent Variable\n\n Number of inequalities to solve: 23456789\n Ineq. #1:\n Ineq. #2:\n\n Ineq. #3:\n\n Ineq. #4:\n\n Ineq. #5:\n\n Ineq. #6:\n\n Ineq. #7:\n\n Ineq. #8:\n\n Ineq. #9:\n\n Solve for:\n\n Please use this form if you would like to have this math solver on your website, free of charge. Name: Email: Your Website: Msg:\n\n# Intermediate Algebra Class Notes\n\nWhat you need to do this week (after attending the first class):\n\nA. Look over Class Notes for 01/14/07 and 01/16/07 (online links found in the\nright-hand pane of the homepage).\n\nB. Read the following three (3) webpages (online links found in the left-hand\npane of the homepage):\n\n1. Course Syllabus\n2. Course Schedule\n3. Course Info & Policies (covered by Quiz #01* on Wed., Jan.23rd)\n\n* This first quiz consists of ten multiple-choice and True-False questions. All\nsubsequent quizzes will cover math problems selectively picked from the most\nrecent HomeWork exercises, and will be given without any prior notice.\n\nI. Sets -\n\nA. Symbols:", null, "B. Number Sets (p.8) - See Figure 1.2\n1. N = {1, 2, 3, ...}\n2. W = {0, 1, 2, 3, ...}\n3.", null, "= {..., -3, -2, -1, 0, 1, 2, 3, ...}\nN, W &", null, "are all in “roster” notation", null, "= {x \| x is a terminating or a repeating decimal}\n5. J = {x \| x is a non-terminating, non-repeating decimal}\n6.", null, "= {x \| x is a decimal}", null, "Q, J &", null, "are all in “set-builder” notation\n\nII. Examples (pp.10-12): Exercises #34,68,14,80\n\npp.10-12 / Exercises #1-97 (every other odd)\n\nI. Absolute Value (p.15):\nl x l = distance from zero (on a number line)\n\nII. Opposites:\n\nA. - (-a) = a\nB. a – (-b) = a + b\n\nIII. Division involving Zero:\n\nA. 0 ÷ a = 0\ne.g., 0 ÷ 7 = 0 (since 0 = 7 × 0)\nB. a ÷ 0 is “undefined”\ne.g., 7 ÷ 0 = ? (requires 7 = 0 × ?)\nnote: no such number exists, i.e., undefined\n\nIV. Distributive Property (pp.21-22):\n\na(b ± c) = ab ± ac\n\nV. Examples (pp.24-25): Exercises #8,46,90,100,\n122" ]	[ null, "https://algebra-net.com/articles_imgs/5880/interm7.jpg", null, "https://algebra-net.com/articles_imgs/5880/interm8.gif", null, "https://algebra-net.com/articles_imgs/5880/interm8.gif", null, "https://algebra-net.com/articles_imgs/5880/interm9.jpg", null, "https://algebra-net.com/articles_imgs/5880/interm10.gif", null, "https://algebra-net.com/articles_imgs/5880/interm11.gif", null, "https://algebra-net.com/articles_imgs/5880/interm10.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8106464,"math_prob":0.9653646,"size":1611,"snap":"2020-24-2020-29","text_gpt3_token_len":565,"char_repetition_ratio":0.077162415,"word_repetition_ratio":0.026143791,"special_character_ratio":0.40099317,"punctuation_ratio":0.24694377,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9847642,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"im_url_duplicate_count":[null,3,null,6,null,6,null,3,null,6,null,3,null,6,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-08T06:01:43Z\",\"WARC-Record-ID\":\"<urn:uuid:f346c40a-8bef-4842-9229-59fbf210df00>\",\"Content-Length\":\"81366\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f7bada0c-61b7-4b3d-86ae-6aa81d758723>\",\"WARC-Concurrent-To\":\"<urn:uuid:d01bf664-dc91-4812-8ce2-20de2af2ba11>\",\"WARC-IP-Address\":\"54.197.228.212\",\"WARC-Target-URI\":\"https://algebra-net.com/intermediate-algebra-class-notes.html\",\"WARC-Payload-Digest\":\"sha1:YXUQUBHP3ETUVMSCEIYEQLJAEACM34HV\",\"WARC-Block-Digest\":\"sha1:HUQOV3FKLP3V73W6SWUHMBJQLCKCPMT5\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593655896374.33_warc_CC-MAIN-20200708031342-20200708061342-00098.warc.gz\"}"}
https://www.betadvisor.com/en/freetipsters/baseball.html	[ "", null, "Just trust a free tipster and start to dream. Free\n\n111 Free Tipster(s) Found\n\nOrder by\n\n1st of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit\n\n2nd of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit\n\n3rd of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit\n\n4th of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit\n\n5th of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit\n\n6th of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit\n\n7th of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit\n\n8th of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit\n\n9th of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit\n\n10th of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit\n\n11th of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit\n\n12th of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit\n\n13th of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit\n\n14th of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit\n\n15th of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit\n\n16th of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit\n\n17th of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit\n\n18th of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit\n\n19th of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit\n\n20th of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit\n\n21st of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit\n\n22nd of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit\n\n23rd of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit\n\n24th of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit\n\n25th of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit\n\n26th of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit\n\n27th of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit\n\n28th of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit\n\n29th of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit\n\n30th of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit\n\n31st of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit\n\n32nd of 111\n\nCalculate how much you can earn\n\n0,00\n\npotential profit" ]	[ null, "https://www.facebook.com/tr", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8346199,"math_prob":0.9986145,"size":4221,"snap":"2021-31-2021-39","text_gpt3_token_len":1345,"char_repetition_ratio":0.19136827,"word_repetition_ratio":0.63793105,"special_character_ratio":0.36271027,"punctuation_ratio":0.10289017,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99984884,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-09-24T02:43:10Z\",\"WARC-Record-ID\":\"<urn:uuid:eafee807-6856-478b-8c1b-97d06c494c48>\",\"Content-Length\":\"592635\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:90f5a63b-441e-4983-9b31-67919c499a20>\",\"WARC-Concurrent-To\":\"<urn:uuid:6874af0f-4822-43c0-9fcc-f739a9df9a42>\",\"WARC-IP-Address\":\"54.36.250.129\",\"WARC-Target-URI\":\"https://www.betadvisor.com/en/freetipsters/baseball.html\",\"WARC-Payload-Digest\":\"sha1:L3NIQD2HSIOKBIA5M62M7DU2JMYIFROB\",\"WARC-Block-Digest\":\"sha1:2HQITNXEK4JB6HUCMXPM7Z2ZR3GBJLKB\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780057496.18_warc_CC-MAIN-20210924020020-20210924050020-00704.warc.gz\"}"}
https://hackage.haskell.org/package/data-category-0.3.1.1/docs/src/Data-Category-Discrete.html	[ "```{-# LANGUAGE TypeFamilies, TypeOperators, GADTs, RankNTypes, ScopedTypeVariables, FlexibleContexts, FlexibleInstances, UndecidableInstances #-}\n-----------------------------------------------------------------------------\n-- \|\n-- Module : Data.Category.Discrete\n-- Copyright : (c) Sjoerd Visscher 2010\n--\n-- Maintainer : sjoerd@w3future.com\n-- Stability : experimental\n-- Portability : non-portable\n--\n-- Discrete n, the category with n objects, and as the only arrows their identities.\n-----------------------------------------------------------------------------\nmodule Data.Category.Discrete (\n\n-- * Discrete Categories\nDiscrete(..)\n, Z, S\n, Void\n, Unit\n, Pair\n\n-- * Functors\n, Next(..)\n, DiscreteDiagram(..)\n\n-- * Natural Transformations\n, voidNat\n\n) where\n\nimport Prelude hiding ((.), id, Functor, product)\n\nimport Data.Category\nimport Data.Category.Functor\nimport Data.Category.NaturalTransformation\n\ndata Z\ndata S n\n\n-- \| The arrows in Discrete n, a finite set of identity arrows.\ndata Discrete :: * -> * -> * -> * where\nZ :: Discrete (S n) Z Z\nS :: Discrete n a a -> Discrete (S n) (S a) (S a)\n\nmagicZ :: Discrete Z a b -> x\nmagicZ x = x `seq` error \"we never get this far\"\n\n-- \| @Discrete Z@ is the discrete category with no objects.\ninstance Category (Discrete Z) where\n\nsrc = magicZ\ntgt = magicZ\n\na . b = magicZ (a `seq` b)\n\n-- \| @Discrete (S n)@ is the discrete category with one object more than @Discrete n@.\ninstance Category (Discrete n) => Category (Discrete (S n)) where\n\nsrc Z = Z\nsrc (S a) = S \\$ src a\n\ntgt Z = Z\ntgt (S a) = S \\$ tgt a\n\nZ . Z = Z\nS a . S b = S (a . b)\n_ . _ = error \"Other combinations should not type-check.\"\n\n-- \| @Void@ is the empty category.\ntype Void = Discrete Z\n-- \| @Unit@ is the discrete category with one object.\ntype Unit = Discrete (S Z)\n-- \| @Pair@ is the discrete category with two objects.\ntype Pair = Discrete (S (S Z))\n\ntype family PredDiscrete (c :: * -> * -> ) :: -> * -> \ntype instance PredDiscrete (Discrete (S n)) = Discrete n\n\ndata Next :: -> * where\nNext :: (Functor f, Dom f ~ Discrete (S n)) => f -> Next f\n\ntype instance Dom (Next f) = PredDiscrete (Dom f)\ntype instance Cod (Next f) = Cod f\ntype instance Next f :% a = f :% S a\n\ninstance (Functor f, Category (PredDiscrete (Dom f))) => Functor (Next f) where\nNext f % Z = f % S Z\nNext f % (S a) = f % S (S a)\n\ninfixr 7 :::\n\n-- \| The functor from @Discrete n@ to @(~>)@, a diagram of @n@ objects in @(~>)@.\ndata DiscreteDiagram :: (* -> * -> ) -> -> * -> * where\nNil :: DiscreteDiagram (~>) Z ()\n(:::) :: Obj (~>) x -> DiscreteDiagram (~>) n xs -> DiscreteDiagram (~>) (S n) (x, xs)\n\ntype instance Dom (DiscreteDiagram (~>) n xs) = Discrete n\ntype instance Cod (DiscreteDiagram (~>) n xs) = (~>)\ntype instance DiscreteDiagram (~>) (S n) (x, xs) :% Z = x\ntype instance DiscreteDiagram (~>) (S n) (x, xs) :% (S a) = DiscreteDiagram (~>) n xs :% a\n\ninstance (Category (~>))\n=> Functor (DiscreteDiagram (~>) Z ()) where\nNil % f = magicZ f\n\ninstance (Category (~>), Category (Discrete n), Functor (DiscreteDiagram (~>) n xs))\n=> Functor (DiscreteDiagram (~>) (S n) (x, xs)) where\n(x ::: _) % Z = x\n(_ ::: xs) % S n = xs % n\n\nvoidNat :: (Functor f, Functor g, Category d, Dom f ~ Void, Dom g ~ Void, Cod f ~ d, Cod g ~ d)\n=> f -> g -> Nat Void d f g\nvoidNat f g = Nat f g magicZ\n```" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5471581,"math_prob":0.96769595,"size":2197,"snap":"2021-43-2021-49","text_gpt3_token_len":738,"char_repetition_ratio":0.23073415,"word_repetition_ratio":0.056962024,"special_character_ratio":0.39736003,"punctuation_ratio":0.19907407,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99986076,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-28T06:13:55Z\",\"WARC-Record-ID\":\"<urn:uuid:a4b9e686-a3e6-440f-aa6b-29369e863c3a>\",\"Content-Length\":\"28272\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:199ee09b-a480-46d9-b5ad-e7c8f23c8eaa>\",\"WARC-Concurrent-To\":\"<urn:uuid:0e70b5ea-e876-4592-95f5-bb197b0cd93a>\",\"WARC-IP-Address\":\"199.232.64.68\",\"WARC-Target-URI\":\"https://hackage.haskell.org/package/data-category-0.3.1.1/docs/src/Data-Category-Discrete.html\",\"WARC-Payload-Digest\":\"sha1:6NI45VZ5BD4E5344VO526DW43O7JWNT3\",\"WARC-Block-Digest\":\"sha1:4ESHIF7AZREJKOGHDBRKUIAZ6HGFJUM2\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323588257.34_warc_CC-MAIN-20211028034828-20211028064828-00195.warc.gz\"}"}
https://lbs-to-kg.appspot.com/12.9-lbs-to-kg.html	[ "Pounds To Kg\n\n# 12.9 lbs to kg12.9 Pounds to Kilograms\n\nlbs\n=\nkg\n\n## How to convert 12.9 pounds to kilograms?\n\n 12.9 lbs * 0.45359237 kg = 5.851341573 kg 1 lbs\nA common question is How many pound in 12.9 kilogram? And the answer is 28.4396318218 lbs in 12.9 kg. Likewise the question how many kilogram in 12.9 pound has the answer of 5.851341573 kg in 12.9 lbs.\n\n## How much are 12.9 pounds in kilograms?\n\n12.9 pounds equal 5.851341573 kilograms (12.9lbs = 5.851341573kg). Converting 12.9 lb to kg is easy. Simply use our calculator above, or apply the formula to change the length 12.9 lbs to kg.\n\n## Convert 12.9 lbs to common mass\n\nUnitMass\nMicrogram5851341573.0 µg\nMilligram5851341.573 mg\nGram5851.341573 g\nOunce206.4 oz\nPound12.9 lbs\nKilogram5.851341573 kg\nStone0.9214285714 st\nUS ton0.00645 ton\nTonne0.0058513416 t\nImperial ton0.0057589286 Long tons\n\n## What is 12.9 pounds in kg?\n\nTo convert 12.9 lbs to kg multiply the mass in pounds by 0.45359237. The 12.9 lbs in kg formula is [kg] = 12.9 * 0.45359237. Thus, for 12.9 pounds in kilogram we get 5.851341573 kg.\n\n## 12.9 Pound Conversion Table", null, "## Alternative spelling\n\n12.9 lb to Kilogram, 12.9 lb in Kilogram, 12.9 Pounds to kg, 12.9 Pounds in kg, 12.9 Pound to Kilogram, 12.9 Pound in Kilogram, 12.9 Pounds to Kilograms, 12.9 Pounds in Kilograms, 12.9 Pounds to Kilogram, 12.9 Pounds in Kilogram, 12.9 Pound to kg, 12.9 Pound in kg, 12.9 lbs to kg, 12.9 lbs in kg, 12.9 lb to Kilograms, 12.9 lb in Kilograms, 12.9 lb to kg, 12.9 lb in kg" ]	[ null, "https://lbs-to-kg.appspot.com/image/12.9.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.83897954,"math_prob":0.93610436,"size":1126,"snap":"2019-51-2020-05","text_gpt3_token_len":412,"char_repetition_ratio":0.26024956,"word_repetition_ratio":0.009803922,"special_character_ratio":0.43161634,"punctuation_ratio":0.23125,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9654094,"pos_list":[0,1,2],"im_url_duplicate_count":[null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-12-07T11:43:55Z\",\"WARC-Record-ID\":\"<urn:uuid:aacc6c72-c34b-4ab6-8f3d-11a9e465543e>\",\"Content-Length\":\"28391\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ec36027d-c591-4583-8424-7d4b0ff920f7>\",\"WARC-Concurrent-To\":\"<urn:uuid:626e56e4-6bef-4376-bc42-8efbbd91d1c6>\",\"WARC-IP-Address\":\"172.217.13.84\",\"WARC-Target-URI\":\"https://lbs-to-kg.appspot.com/12.9-lbs-to-kg.html\",\"WARC-Payload-Digest\":\"sha1:ANKWO3TDCO3RWPRVPVJGM2WWZTZMOFC5\",\"WARC-Block-Digest\":\"sha1:TEEZLZ7SEKUYLC34BVIEK3WHMWT7I7K3\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-51/CC-MAIN-2019-51_segments_1575540499389.15_warc_CC-MAIN-20191207105754-20191207133754-00451.warc.gz\"}"}
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https://id.scribd.com/document/364957094/cpp-functions-pdf	[ "Anda di halaman 1dari 12\n\n# Q.\n\n## 1 Let f be a real valued function such that\n\n2002\nf (x) + 2 f = 3x\nx\nfor all x > 0. The value of f (2), is\n(A) 1000 (B) 2000 (C) 3000 (D) 4000\n\nQ.2 The number k is such that tan arc tan(2) arc tan(20k ) = k. The sum of all possible values of k is\n19 21 1\n(A) (B) (C) 0 (D)\n40 40 5\nx for 0 x 1\nLet f 1 x 1 for x 1 and f 2 x f 1 x for all x\nQ.3\n0 otherwise\nf3 x f 2 x for all x\nf4 x f 3 x for all x\nWhich of the following is necessarily true?\n(A) f 4 x f 1 x for all x (B) f 1 x f3 x for all x\n(C) f 2 x f 4 x for all x (D) f 1 x f3 x 0 for all x\n\n## Q.4 Domain of definition of the function f x log 10 3x 2 9 x 1\n\n1 cos 1 1 x is\n(A) [0, 1] (B) [1, 2] (C) (0, 2) (D) (0, 1)\n\n12 5\nQ.5 If Sin , cos , 0 2 . Consider the following statements.\n13 13\n1 5 1 12 12\nI. cos II. sin III. sin 1\n13 13 13\n1 12 12\nIV. tan V. tan 1\n5 5\nthen which of the folloing statements are true ?\n(A) I, II and IV only (B) III and V only (C) I and III only (D) I, III and V only\n\nQ.6 Which one of the following depicts the graph of an odd function?\n\n(A) (B)\n\n(C) (D)\n\n2\n1 3\nQ.7 The sum tan 2 is equal to\nn 1 n n 1\n3\n(A) cot 1 2 (B) cot 1 3 (C) (D) tan 1 2\n4 2 2\n\n1 1 1\nQ.8 The value of tan tan 2A tan cot A tan 1 cot 3 A for 0 A / 4 is\n2\n(A) 4 tan 1 1 (B) 2 tan 1\n2 (C) 0 (D) None\n\n8 8 4 4\nQ.9 Given f x and g x then g(x) is\n1 x 1 x f sin x f cos x\n(A) periodic with period / 2 (B) periodic with period\n(C) periodic with period 2 (D) aperiodic\n\n## Q.10 sin 1 cos sin 1 x and cos 1 sin cos 1 x , then :\n\n(A) tan cot (B) tan cot (C) tan tan (D) tan tan\n\nsin x cos x\nQ.11 The period of the function f x sin x cos x\n(A) /2 (B) /4 (C) (D) 2\n\n1\n1 1 a 1 1 a\nQ.12 The value of tan sin tan sin , where (0 < a < b), is\n4 2 b 4 2 b\nb a b2 a 2 b2 a 2\n(A) (B) (C) (D)\n2a 2b 2b 2a\n\nQ.13 The sides of a triangle are 9, 12, 15. The area of the triangle formed by its income, centroid and orthocentre, is\n(A) 2 (B) 3 (C) 3/2 (D) 5\n\n1\nQ.14 f x sin 23 x cos 22 x and g x tan 1 x , then the number of values of x in interval\n1\n2\n10 , 20 satisfying the equation f x sgn g x , is\n(A) 6 (B) 10 (C) 15 (D) 20\n\n## Q.15 Number of solutions of the equation 2 cot 1 2 cos 1 3 / 5 cos ec 1x is\n\n(A) 0 (B) 1 (C) 2 (D) more than 2\n\nx l nx\nQ.16 f x and g x . Then identify the CORRECT statement\nl nx x\n1 1\n(A) g x and f(x) are identical functions (B) and g(x) are identical functions\nf x\n1\n(C) f x . g x 1 x 0 (D) f x . g x 1 x 0\n\n3\n1 x 1 x\nQ.17 The number of solutions of the equation tan tanh tan 1 x is\n3 2\n(A) 3 (B) 2 (C) 1 (D) 0\nQ.18 Let f (x) = sin2x + cos4x + 2 and g (x) = cos (cos x) + cos (sin x). Also let period of f (x) and g (x) be\nT1 and T2 respectively then\n(A) T1 = 2T2 (B) 2T1 = T2 (C) T1 = T2 (D) T1 = 4T2\n1 2x 2 1\nQ.19 Which of the following is the solution set of the equation 2 cos1(x) = cot ?\n2x 1 x 2\n(A) (0, 1) (B) (1, 1) {0} (C) (1, 0) (D) [1, 1]\nx x\nQ.20 Let f x e e sgn x and g x e e sgn x , x R where { x }and [ ] denotes the fractional part and\nintegral part functions respectively. Also h (x) l n f(x) l n g(x) then for all real x, h (x) is\n(A) an odd function (B) an even function\n(C) neither an odd nor an even function (D) both odd as well as even function\n\n## Q.21 Find the range of the function f x cot 1 x sec 1\n\ncos es 1 x.\n3 3 5 3\n(A) , (B) , ! ,\n2 2 2 4 4 2\n3 3\n(C) , ! , (D) , ! ,\n2 2 2 2\nQ.22 Which of the following function is surjective but not injective\n(A) f : R \" R f x x 4 2 x 3 x 2 1 (B) f : R \" R f x x3 x 1\n(C) f : R \" R f x 1 x2 (D) f : R \" R f x x 3 2x 2 x 1\n\n1 1 7 2\nQ.23 cos cos sin is equal to\n2 5 5\n23 23 3 17\n(A) (B) (C) (D)\n20 20 20 20\n2\nQ.24 Let f x ;g x cos x and h x x 3 then tha range of the composite function fogoh, is\nx 1\n(A) R (B) R {0} (C) 1, (D) R {1}\n\n1 x2\nQ.25 There exists a positive real number x satisfying cos tan x 1\nx. The value of cos 2 is\n2 4\n(A) (B) (C) (D)\n10 5 5 5\nmin( x , y )\nQ.26 If f ( x , y) max ( x , y) and g ( x , y) max( x , y) min( x , y), then\n3\nf g 1, , g ( 4, 1.75) equals\n2\n(A) - 0.5 (B) 0.5 (C) 1 (D) 1.5\n1 1 1 1 1 2\nQ.27 The number of solutions of the equation tan tan tan is\n2x 1 4x 1 x2\n(A) 0 (B) 1 (C) 2 (D) 3\n\n4\nQ.28 If the solution set for f (x) < 3 is (0, ) and the solution set for f (x) > 2 is ( , 5), then the true solution\nset for f ( x ) 2 # f (x) + 6, is\n(A) ( , + ) (B) ( , 0] (C) [0, 5] (D) ( , 0] ! [5, )\n1\nQ.29 The range of the value of p for which the equation sin cos cos(tan 1 x ) p has a solution is :\n\n1 1 1\n(A) , (B) 0, 1 (C) ,1 (D) 1, 1\n2 2 2\n\n## 1, x # 1 . Then the set S\n\n2\nQ.30 Let f x x 1 x : f ( x ) f 1 ( x ) is\n\n3 i 3 3 i 3\n(A) 0, 1, 2\n,\n2 (B) 0,1, 1\n\n## (C) 0, 1 (D) empty\n\nIf x 1 2 \\$x y 3\n2 2\nQ.31 y 2 is a parabola, then \\$\n1 3 1\n(A) 1 (B) (C) (D)\n2 2 4\nQ.32 2 cot cot 1 (3) cot 1 (7) cot 1 (13) cot 1 (21) has the value equal to\n(A) 1 (B) 2 (C) 3 (D) 4\n\n## Q.33 The graph of the function y = g (x) is shown.\n\n1\nThe number of solutions of the equation g ( x ) 1 , is\n2\n(A) 4 (B) 5\n(C) 6 (D) 8\nx rx\nQ.34 Let f x and let g x . Let S be the set off all real numbers r such that\n1 x 1 x\nf g(x) g f (x) for infinitely many real number x. The number of elements in set S is\n(A) 1 (B) (C) 3 (D) 5\n1\nQ.35 Number of natural solution(s) of the equation sin sin x cos 1 cos x in 0,5 is\n(A) 2 (B) 3 (C) 4 (D) infinite\n1 1\nQ.36 Range of the function f ( x ) tan [x ] [ x ] 2 x is\nx2\nwhere [] is the greatest integer function.\n1 1 1 1\n(A) , (B) ! 2, (C) ,2 (D) ,2\n4 4 4 4\nQ.37 The set of values of ox, satisfying the equation tan 2 sin 1 x 1 is\n\n2 2\n(A) [ 1,1] (B) ,\n2 2\n\n2 2 2 2\n(C) 1,1 , (D) [ 1,1] ,\n2 2 2 2\n5\n0 if x is irrational 0 if x is irrational\nQ.38 Let f ( x ) and g ( x )\nx if x is rational x if x is rational\nThen the function (f - g) x is\n(A) odd (B) even (C) neither odd nor even (D) odd as well as even\n\n1 2 1 1 1 x2\nQ.39 The solution set of the equation sin 1 x cos x cot sin 1 x\nx\n\n## Q.40 The period of the function cos 2 x cos 2 x is :\n\n(A) (B) 2 (C) 2 (D) none of these\n1\nQ.41 The value of the angle tan tan 650 2 tan 40 0 in degrees is equal to\n(A) 20 0 (B) 20 0 (C) 250 (D) 40 0\n{x}\nQ.42 Range of the function f ( x ) where {x} denotes the fractional part function is\n1 {x}\n\n1 1 1\n(A) 0,1 (B) 0, (C) 0, (D) 0,\n2 2 2\nQ.43 Consider the function g (x) defined as\n2008 2007\ng(x ) x ( 2 1)\n1 ( x 1) ( x 2 1) ( x 4 1)....... x 2 1 1.\nthe value of g (2) equals\n(A) 1 (B) 2 2008 1 (C) 2 2008 (D) 2\nQ.44 The range of the function, f (x) (1 sec -1x) (1 cos 1x) is\n\n## (A) , (B) , 0 ! 4, (C) 1, (1 )2 (D) [0, (1 )2 ]\n\nQ.45 Which of the following is true for a real valued function y = f (x), defined on [-a, a]?\n(A) f (x) can be expressed as a sum or a difference of two even function.\n(B) f (x) can be expressed as a sum or a difference of two odd function.\n(C) f (x) can be expressed as a sum or a difference of an odd and an even function.\n(D) f (x) can never be expressed as a sum or a difference of an odd and an even function.\nQ.46 Which of the following represents an odd function?\n\n(1 e x ) 2\n(A) f ( x ) (B) g ( x ) sec 1 (sec x )\nex\n(C) h ( x ) cos (cos 1 x ) (D) k ( x ) cot 1 (cot x )\n\n1 1 x 1 x 3\nQ.47 Let f be a real valued function defined by f x sin cos . Then domain of f(x) is\n3 5\ngiven by :\n(A) 4, 4 (B) 0, 4 (C) 3, 3 (D) 5, 5\n\n6\nQ.48 Given the graphs of the two functions, y = f(x) & y = g(x). In the adjacent\nfigure from point A on the graph of the function y = f(x) corresponding\nto the given value of the independent variable (say x0), a straight line is\ndrawn parallel to the X-axis to intersect the bisector of the first and the third y=g(x)\nquadrants at point B . From the point B a straight line parallel to the Y-axis\nis drawn to intersect the graph of the function y = g(x) at C. Again a straight\ny-f(x)\nline is drawn from the point C parallel to the X-axis, to intersect the line NN%\nat D . If the straight line NN% is parallel to Y-axis, then the co-ordinates\nof the point D are\n(A) f(x0), g(f(x0)) (B) x0, g(x0) (C) x0, g(f(x0)) (D) f(x0), f(g (x0))\n1\nQ.49 If x and x 1 y 1 2 the the radian measue of cot 1`\nx cot 1 y is\n2\n3\n(A) (B) (C) (D)\n2 3 4 4\nQ.50 Given f (x) is a polynomial function of x, satisfying f(x) . f(y) = f(x) + f(y) + f(xy) - 2 and that f (2) = 5.\nthen f (3) is equal to\n(A) 10 (B) 24 (C) 15 (D) none\nQ.51 Let cos 1 x cos 1 2 x cos 1 3x . If x satisfies the cubic ax 3 bx 2 ` cs 1 0, then (a + b + c)\nhas the value equal to\n(A) 24 (B) 25 (C) 26 (D) 27\n\nxx if x 1\nQ.52 Let f ( x ) [1 x ] [1 x ] if 1 x 1\nxx if x #1\n\n## where [x] denotes the greatest integer function then F(x) is\n\n(A) even (B) odd\n(C) neither odd nor even (D) even as well as odd\narc cot x\nQ.53 The domain of the function f x , where [x] denotes the greatest integer not greater than\nx 2 [x 2 ]\nx, is :\n(A) R (B) R - {0}\n\n## (C) R & n : n 1 ! {0} (D) R {n : n 1}\n\n1 2 1 1 1\nQ.54 If tan tan , then which one of the following can not be equal to\n2 1 2\n\n1 1 1 1\n(A) 2 tan (B) cot 2 cot 3\n2\n\n1 1 1\n(C) tan 1 2 tan 1 3 (D) tan 3 sin\n5\n\n7\nQ.55 The range of the function, f ( x ) cot 1 log 0.5 x 4 2 x 2 3 is :\n\n3 3 3\n(A) 0, (B) 0, (C) , (D) ,\n4 4 2 4\n\n1 1 1 1\nQ.56 The value of cos cos 6 2 cos 6 2 equals\n4 4\n\n1 3 1\n(A) (B) (C) (D) 0\n2 2 2\nQ.57 If f x ay, x- ay axy then f x, y is equal to :\n\nx2 y2 x2 y2\n(A) (B) (C) 4 xy (D) none\n4 4\nQ.58 If f (x) px q and f f f (x) 8x 21, where p and q are real number, then p + q equals\n(A) 3 (B) 5 (C) 7 (D) 11\nQ.59 The value of x satisfying the equation sin(tan 1 x ) cos cot 1 ( x 1) is\n1 1\n(A) (B) (C) 2 1 (D) no finite value\n2 2\nQ.60 If f ( x ) 2 tan 3x 5 1 cos 6x ; g ( x ) is a function having the same period as that of f(x), then which\nof the following can be g(x).\n(A) (sec 2 3 x cosec 2 3 x) tan 2 3 x (B) 2 sin 3x 3 cos 3x\n\n## (C) 2 1 cos 2 3 x cosec 3 x (D) 3 cosec 3 x 2 tan 3 x\n\nQ.61 Let f (x) max . sin t : 0 t x\ng( x ) min . sin t : 0 t x\nand h ( x ) f (x ) g ( x )\nwhere [ ] denotes greatest integer function, then the range of h(x) is\n(A) {0, 1} (B) {1, 2} (C) {0, 1, 2} (D) {-3, -2, -1, 0, 1, 2, 3}\n\n[COMPREHENSION TYPE]\nParagraph for question nos. 62 to 64\nConsider a function y = f (x) satisfying the equation tan-1y = tan-1x + C where y = 1 when x = 0.\n\n## Q.62 The domain of the explicit form of the function is\n\n(A) ,1 (B) R { 1,1} (C) ( 1,1) (D) 0,\nQ.63 Range of the function is\n(A) R { 1} (B) 1, (C) 1, (D) ,1\nQ.64 For the function y = f (x) which one of the following does not hold good ?\n(A) f (x) is injective (B) f (x) is neither odd nor even\nx 1\n(C) f (x) is aperiodic (D) explicit form of f (x) is\nx 1\n8\nParagraph for question nos. 65 to 68\nLet f ( x ) x 2x 1 2\nx R. Let f : , a \" b, , where a is the largest real number for which\nf (x) is bijective.\nQ.65 The value of (a + b) is equal to\n(A) - 2 (B) - 1 (C) 0 (D) 1\nQ.66 Let f : R \" R , g ( x ) f (x ) 3x 1, then the least value of function y g x is\n(A) - 9/4 (B) - 5/4 (C) - 2 (D) - 1\nQ.67 Let f : a , \" b, , then f ( x ) is given by\n1\n\n## (A) 1 x 2 (B) 1 x 3 (C) 1 x 2 (D) 1 x 3\n\nQ.68 Let f : R \" R , then range of values of k for which equation f x k has 4 distinct real roots is\n(A) (- 2, - 1) (B) (- 2, 0) (C) (- 1, 0) (D) (0, 1)\n[MULTIPLE OBJECTIVE TYPE]\nQ.69 Which of the following function (s) is/are Transcendental ?\n2 sin 3x\n(A) f ( x ) 5 sin x (B) f ( x ) 2\nx 2x 1\n(C) f ( x ) x2 2x 1 (D) f ( x ) x 2 3 . 2x\nQ.70 sin 1 sin 3 sin 1 sin 4 sin 1 sin 5 when simplified reduces to\n(A) an irrational number (B) a rational number\n(C) an even prime (D) a negative integer\nQ.71 The functions which are aperiodic are :\n(A) y = [x + 1] (B) y = sin x 2 (C) y = sin2 x (D) y = sin-1 x\nwhere [x] denotes greatest integer function\nQ.72 Which of the following pairs of functions are identical ?\n1\n(A) f ( x ) e n sec\nand g ( x ) sec 1 x (B) f x tan tan 1 x and g ( x ) cot cot1 x\n(C) f ( x ) sgn ( x ) and g ( x ) sgn(sgn( x )) (D) f ( x ) cot 2 x . cos 2 x and g ( x ) cot 2 x cos 2 x\nQ.73 Which of the functions defined below are one-one function(s) ?\n(A) f ( x ) x 1 , x # 1 (B) g ( x ) x 1 / x ( x 0)\n(C) h ( x ) x2 4 x 5, ( x 0) (D) f ( x ) e x , x # 0\n1 1 14\nQ.74 The value of cos 2 cos cos is :\n5\n7 2 3\n(A) cos (B) sin (C) cos (D) cos\n5 10 5 5\n1 1 1\nQ.75 If cos x cos y cos z , then\n2 2 2\n(A) x y z 2xyz 1\n(B) 2 sin x sin 1 y sin 1 z\n1\ncos 1 x cos 1 y cos 1 z\n(C) xy + yz + zx = x + y + z - 1\n1 1 1\n(D) x y z #6\nx y z\nQ.76 Which of the following functions are homogeneous ?\n(A) x sin y y sin x (B) x e y / x y e x / y (C) x 2 xy (D) arc sin xy\n\n9\n1\nQ.77 Let tan f (x) cos 1 x then which of the following do/does not hold good ?\n(A) Domain of f(x) is [-1, 1] (B) Range of f(x) is ( , )\n(C) f (x) is bounded (D) f(x) is odd\nQ.78 Suppose f (x) = ax + b and g (x) = bx + a, where a and b are positive integers. If f g (50) g f (50) 28\nthen the product (ab) can have the value equal to\n(A) 12 (B) 48 (C) 180 (D) 210\nQ.79 Which of the following function(s) have the same domain and range ?\n1\n(A) f ( x ) 1 x2 (B) g ( x ) (C) h ( x ) x (D) l ( x ) 4 x\nx\nQ.80 2 tan tan 1 ( x ) tan 1 ( x 3 ) where x R { 1,1}is equal to\n\nl 2x\n(A) (B) tan 2 tan 1 x\n1 x2\n(C) tan cot 1 ( x ) cot 1 ( x ) (D) tan 2 cot 1 x\nQ.81 Which pair(s) of function(s) is/are equal ?\n1 x2 2x\n(A) f ( x ) cos( 2 tan 1 x ) ; g ( x ) (B) f ( x ) ; g( x ) sin 2 cot 1 x\n1 x2 1 x2\n1\n(D) f ( x ) x a , a 0; g (x ) a x , a\n1\nn (sgn cot x) n [1 { x }]\n(C) f ( x ) e ; g( x) e 0\nwhere {x} and [x] denotes the fractional part and integral part functions.\nQ.82 Which of the following function(s) would represent a non singular mapping.\nf\n(A) f : R \" R f (x) x Sgn x where Sgn denotes Signum function\n(B) g : R \" R g (x) x 3/ 5\n(C) h : R \" R h(x) x4 3 x2 1\n\n3 x2 7 x 6\n(D) k : R \" R k (x)\nx x2 2\nQ.83 Let x1 , x 2 , x 3 , x 4 be four non zero numbers satisfying the equation\n\n1 a 1 b 1 c 1 d\ntan tan tan tan\nx x x x 2\n4\n\n(A) xi a b c d\ni 1\n\n4\n1\n(B) 0\ni 1 xi\n4\n\n(C) 'x\ni 1\ni abcd\n\n(D) x1 x 2 x3 x2 x3 x4 x3 x4 x1 x 4 x1 x 2 abcd\nQ.84 If the function f(x) = ax + b has its own inverse then the ordered pair (a, b) can be\n(A) (1, 0) (B) (-1, 0) (C) (-1, 1) (D) (1, 1)\n\n10\nQ.85 Let y (sin x sin 2x sin 3x ) 2 (cos x cos 2 x cos 3x ) 2 then which of the following is correct ?\n\ndy 3 5\n(A) when x is 2 (B) value of y when x is\ndx 2 5 2\n\n1 2 3\n(C) value of y when x is (D) y simplifies to (1 2 cos x ) in [0, ]\n12 2\nQ.86 Which of the following trigonometric equation(s) has/have no solution x R?\n\n2 x 2 1\n(A) 2 cos sin x x2 (B) sin 4 x 2 sin 2 x 1 0\n2 x2\n(C) cos e x 5x 5 x\n(D) sin 2 x [1 sin 2 x ] [1 cos 2 x ]\nwhere [ ] denotes greatest integer function.\n[MATCH THE COLUMN]\nQ.87 Column-I Column-II\nx\n(A) The period of the function f ( x ) sin cos cos(sin x ) equals k (P) 2\n2\nthen k is equal to\n(B) The integral value(s) in the domain of definition of the function, (Q) 3\nf\n3x 2 7 x 8\nf ( x ) arc cos\n1 x2\nwhere [] denotes the greatest integer functin, is\n(C) Let f ( x ) sin [a ] x. If f is periodic with fundamental period , then (R) 4\nthe possible integral vlaue(s) of a is/are\n(where [ ] denotes the greatest integer function)\n2\n(D) If the values of x satisfying the equation x 5[ x ] 6 0, then integral (S) 5\nvalue of x, is/are\n(where [ ] denotes the greatest integer function)\n\n## Q.88 Column-I Column-II\n\nn n\nC2\n(A) Lim\nn\" n equals (P) 0\nn 1 2\n\n## (B) Let the roots of f(x) = 0 are 2, 3, 5, 7 and 9 (Q) 1\n\nand the roots of g(x) = 0 are -1, 3, 5, 7 and 8.\nf x\nNumber of solutions of the euqation = 0 is\ngx\n\nsin 3 x cos 3 x\n(C) Let y where 0 x , (R) 3/2\ncos x sin x 2\nthen the minimum value of y is\n(D) A circle passes through vertex D of the square ABCD, and is tangent (S) 2\nto the sides AB and BC. If AB = 1, the radius of the circle can be\nexpressed as p q 2 , then p + q has the value equal to\n11\nQ.89 Column-I Column-II\n3 3\n(A) sin x cos 3 x cos x sin 3 x, 0 x 2 , is (P) , ! , ! ,\n4 4 4 4\n3\n(B) 4 sin 2 x 8 sin x 0, 0 x 2 , is (Q) ,2 ! {0}\n2\n\n## (C) tanx 1 and x [ , ] , is (R) 0,\n\n4\n5\n(D) cos x sin x # 1 and 0 x 2 (S) ,\n6 6\n\nx\nQ.90 Let : f : R \" , , f (x) x 2 3ax b, g ( x ) sin 1\n( R ).\n4\nColumn-I Column-II\n(A) The possible integral values of a for which f(x) is many one in (P) - 2\ninterval [-3, 5] is/are\n(B) Let a = - 1 and gof(x) is defined for x [ 1,1] then possible (Q) - 1\nintegral values of b can be\n(C) Let a 2, 8 the value(s) of which f(x) is surjective is/are (R) 0\n(D) If a = 1, b = 2, then integers in the range of fog(x) is/are (S) 1\nQ.91 Column-I Column-II\n1 0\n(A) cot tan( 37 ) (P) 1430\n(B) cos cos( 2330 )\n1\n(Q) 127 0\n1 1 1 3\n(C) sin cos (R)\n2 9 4\n1 1 2\n(D) cos arc cos (S)\n2 8 3\n\n1 x 1\nQ.92 Let f (x) xand g(x) .\nx x 2\nMatch the composite function given in Column-I with their respective domains given in Column-II.\nColumn-I Column-II\n(A) fog (P) R - {-2, -5/3}\n(B) gof (Q) R - {-1, 0}\n(C) fof (R) R - {0}\n(D) gog (S) R - {-2, -1}\nQ.93 Column-I Column-II\n1 1 1\n(A) f (x) sin sin 2 x cosec (cosec 2 x) tan(tan 2 x) (P) odd function\n1\n(B) g (x) sin {x}, where {x} denotes fractional part function (Q) injective mapping\n2\n(C) h (x) cosec 1 (sgn x) (R) range contian two\ninteger only\n(D) k ( x ) cos 1 ( sin x cos x ) (S) aperiodic\n[SUJBECTIVE]\n\n## Q.94 Find the value of x satisfying the equation,\n\n1\nlog10 5 cot 1 x 1 log10 2 cot 1 x 3 log10 5 1.\n2\nQ.95 Let the straight line L : tan(cot 1 2) x y 4 be rotated through an angle cot -1 3 about the point\nM(0, - 4) in anticlockwise sense. After rotation the line become tangent to the circle which lies in\n4th quadrant and also touches coordinate axes. Find the sum of radii of all possible circles.\n\nETOOS Academy Pvt. Ltd. : F-106, Road No. 2, Indraprastha Industrial Area, End of Evergreen Motors 13\n(Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005)" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.68491596,"math_prob":0.9999316,"size":11147,"snap":"2019-35-2019-39","text_gpt3_token_len":4929,"char_repetition_ratio":0.16090819,"word_repetition_ratio":0.03893924,"special_character_ratio":0.47008163,"punctuation_ratio":0.086556755,"nsfw_num_words":2,"has_unicode_error":false,"math_prob_llama3":0.99950373,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-09-23T15:43:37Z\",\"WARC-Record-ID\":\"<urn:uuid:d1fef164-771e-4f02-acbe-9249a2a03d24>\",\"Content-Length\":\"368966\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:dd6a3e1b-4ecc-4290-b2c5-5e12479eaef6>\",\"WARC-Concurrent-To\":\"<urn:uuid:5ad42046-c6b1-447d-af5f-ef5265c508e8>\",\"WARC-IP-Address\":\"151.101.250.152\",\"WARC-Target-URI\":\"https://id.scribd.com/document/364957094/cpp-functions-pdf\",\"WARC-Payload-Digest\":\"sha1:FNJF646VQNJMAEAVZBFATMVZ4GALRRJE\",\"WARC-Block-Digest\":\"sha1:K7ACVQOT6DVJUMGSMQJ56M6R3UG2N2ZU\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-39/CC-MAIN-2019-39_segments_1568514577363.98_warc_CC-MAIN-20190923150847-20190923172847-00164.warc.gz\"}"}
http://mathandmultimedia.com/tag/log-2/	[ "## Proof that log 2 is an irrational number\n\nBefore doing the proof, let us recall two things: (1) rational numbers are numbers that can be expressed as", null, "$\\frac{a}{b}$ where", null, "$a$ and", null, "$b$ are integers, and", null, "$b$ not equal to", null, "$0$; and (2) for any positive real number", null, "$y$, its logarithm to base", null, "$10$ is defined to be a number", null, "$x$ such that", null, "$10^x = y$. In proving the statement, we use proof by contradiction.\n\nTheorem: log 2 is irrational\n\nProof:\n\nAssuming that log 2 is a rational number. Then it can be expressed as", null, "$\\frac{a}{b}$ with", null, "$a$ and", null, "$b$ are positive integers (Why?). Then, the equation is equivalent to", null, "$2 = 10^{\\frac{a}{b}}$. Raising both sides of the equation to", null, "$b$, we have", null, "$2^b = 10^a$. This implies that", null, "$2^b = 2^a5^a$. Notice that this equation cannot hold (by the Fundamental Theorem of Arithmetic) because", null, "$2^b$ is an integer that is not divisible by 5 for any", null, "$b$, while", null, "$2^a5^a$ is divisible by 5. This means that log 2 cannot be expressed as", null, "$\\frac{a}{b}$ and is therefore irrational which is what we want to show." ]	[ null, "http://s0.wp.com/latex.php", null, "http://s0.wp.com/latex.php", null, "http://s0.wp.com/latex.php", null, "http://s0.wp.com/latex.php", null, "http://s0.wp.com/latex.php", null, "http://s0.wp.com/latex.php", null, "http://s0.wp.com/latex.php", null, "http://s0.wp.com/latex.php", null, "http://s0.wp.com/latex.php", null, "http://s0.wp.com/latex.php", null, "http://s0.wp.com/latex.php", null, "http://s0.wp.com/latex.php", null, "http://s0.wp.com/latex.php", null, "http://s0.wp.com/latex.php", null, "http://s0.wp.com/latex.php", null, "http://s0.wp.com/latex.php", null, "http://s0.wp.com/latex.php", null, "http://s0.wp.com/latex.php", null, "http://s0.wp.com/latex.php", null, "http://s0.wp.com/latex.php", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.966963,"math_prob":1.0000033,"size":869,"snap":"2023-40-2023-50","text_gpt3_token_len":196,"char_repetition_ratio":0.13410404,"word_repetition_ratio":0.0,"special_character_ratio":0.2301496,"punctuation_ratio":0.118644066,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000091,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-29T06:05:06Z\",\"WARC-Record-ID\":\"<urn:uuid:055b86b0-46bf-4d5a-95a2-e40e46786d04>\",\"Content-Length\":\"50757\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7802cff8-06ab-4ff9-acda-088660d24d5f>\",\"WARC-Concurrent-To\":\"<urn:uuid:92182da4-7a4a-48a4-9e11-ebcb6688ac45>\",\"WARC-IP-Address\":\"166.62.28.131\",\"WARC-Target-URI\":\"http://mathandmultimedia.com/tag/log-2/\",\"WARC-Payload-Digest\":\"sha1:JLLWZ4EVNCX6GFQ5U6TOI7MNUESH34LW\",\"WARC-Block-Digest\":\"sha1:DSAKYJBUJALN5GPD6344EJBECIGQ5QQ5\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233510498.88_warc_CC-MAIN-20230929054611-20230929084611-00418.warc.gz\"}"}
https://www.mathclasstutor.com/2020/08/computer-graphics-in-mathematics.html	[ "# Computer graphics in mathematics 2D graphics.\n\nThe main reason graphical objects are described by a collection of straight line segments is that the standard transformation in computer graphics map line segment on to other line segments.\nComputer aided design (CAD)is an integral part of many engineering process such as the aircraft design process described .\nmost interactive computer software for business and industry makes use of computer graphics in the screen display and for other functions such as a graphical display a of data desktop publishing and slide production for commercial and educational presentations. Consequently anyone studying a computer language invariably spends time learning how to use at least two dimensional (2D) graphics.\nThe basic mathematics used to manipulate and display a graphical image such as a wire frame model of an airplane.sex and image consists of a number of coins, connecting lines for cobs, and information about how to fill in close regions bounded by the lines and COD lines of approximated by short straight linecode lines of approximated by short straight line segments and it is defined mathematically by a list of points. Among the simplest 2D graphics symbol are letters used for labels on the screen. Some letters are stored as right frame of objects other that have COD portions are stored with additional mathematical formula for the cal formula for the curves.\n\n## Composite transformation\n\nThe moment of of a figure on a computer screen off and requires two or more basic transformation.the composition of 6 transformations corresponds to matrix multiplication when homogenous coordinates are used.\n\n### 3D computer graphics\n\nthree dimensional graphics, a biologist can examine or simulated protein molecules and search for active sites that might accept of drug molecule.the biologist can rotates and translate and experimental drug and attempt to attach it to the protein.this ability to visualise Pro X shall chemical reaction is visual to modern drop and Cancer research.\nHomogenous 3D coordinates\nWe say that (x,y,z1) homogenous coordinates for the point(x,y,z) in R^3.\nIn general (X,Y,Z,H) are homogenous coordinates for(x,y,z) ifH will not equal zero\nX=X/H\ny=Y/H\nz=Z/H." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9034762,"math_prob":0.94816685,"size":2236,"snap":"2020-45-2020-50","text_gpt3_token_len":424,"char_repetition_ratio":0.114247315,"word_repetition_ratio":0.011661808,"special_character_ratio":0.17978533,"punctuation_ratio":0.07052897,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9518926,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-25T18:43:21Z\",\"WARC-Record-ID\":\"<urn:uuid:991e06e6-90a5-4229-99cc-37e0c907aae1>\",\"Content-Length\":\"260131\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:139781e8-d880-406b-9601-7f6cc9ea60ff>\",\"WARC-Concurrent-To\":\"<urn:uuid:73d699cd-0ab2-42d8-b900-9ac4482f081f>\",\"WARC-IP-Address\":\"172.217.12.243\",\"WARC-Target-URI\":\"https://www.mathclasstutor.com/2020/08/computer-graphics-in-mathematics.html\",\"WARC-Payload-Digest\":\"sha1:2PUKQDNP3AVMVXW2DI2PKFT4ZWAG64EN\",\"WARC-Block-Digest\":\"sha1:CWO4P2ENZVGKF3MCXHYVGKB4CMI4LAE7\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-45/CC-MAIN-2020-45_segments_1603107889651.52_warc_CC-MAIN-20201025183844-20201025213844-00696.warc.gz\"}"}
https://vulcanhammer.info/2018/04/24/analyzing-sheet-pile-walls-with-spw-2006-part-ii-cantilever-walls/	[ "# Analyzing Sheet Pile Walls with SPW 2006: Part II, Cantilever Walls\n\nIn our last post, we introduced the SPW 2006 sheet piling software, intended for educational purposes. The software can be downloaded here. In this installment we’ll look at its application to cantilever walls, i.e., those walls with no additional support other than the soil itself. These are used in temporary works. The file for this can be found with the software.\n\nThe problem is this one, taken from the BSC Piling Handbook, Fourth Edition (1984).", null, "This is a fairly simple problem except that it has two different soil layers and properties. We’ll use the active and passive earth pressure coefficient values given in the example, although these can easily be computed from equations given in Verruijt or DM 7.\n\nBased on this, the left side soil profile after input looks like this:", null, "And the right side:", null, "We note the following:\n\n• The difference between the two is the first layer on the left side, as we would expect.\n• We have a uniform surcharge", null, "$q$ which is carried from the top downwards. It’s possible to vary that surcharge with depth; however, the program has no method of automatically computing variations in surcharge loading due to surface loads such as line and strip loads. Also, note that the surcharge is only on the right side. You have to manually check to make sure this is so; it is rare that there will be a surcharge on the left side.\n• The water table level is shown in all layers.\n• The passive earth pressure coefficients have been reduced by a factor of 1.5. There is more than one way to include a factor of safety for earth pressure; these methods are discussed in Sheet Pile Design by Pile Buck.\n• The Kn (“neutral” or “at-rest” earth pressure coefficient) has been computed using Jaky’s Equation, discussed here.\n• The stroke is probably the “stickiest wicket” in terms of soil properties. There are several ways of computing this, depending upon the amount of information on the soil you have at hand. Probably the simplest way to do this is to use a chart such as appears in DM 7, which is reproduced below.", null, "Selecting the proper case from the table at the bottom, the stroke can be computed as follows:", null, "$D_w = H\\left( \\frac{Y}{H}^* \\right)$\n\nIt is possible to be very precise with this calculation. For example, one could estimate the penetration below the dredge line", null, "$D$ to use as a value of", null, "$H$, but this becomes very tedious during the iteration process. It’s also possible (and probably better) to use the different values of passive ratios on the left side vs. active ones on the right, since these pressures predominate on their respective sides. Neither of these was used in the example, although the latter option is probably the more realistic one.\n\nIn any case the soil profile looks like this:", null, "The correspondence of the sections with the original problem is easily seen.\n\nNow we select a sheeting length and a profile. We’ll select a length of 10 m (you will need to iterate from a short length, perhaps 6m and go upwards until you get a result that does not produce an error.) We’ll also start by assuming Profile #1 (Hoesch 95.) Running this yields the following beam diagrams:", null, "The maximum moment is around 235 kN-m/m. But is this section suitable for this level of moment? The simplest way is to compute the maximum moment the sheeting section is capable of, and this can be done using the equation", null, "$M_{max} = \\frac{\\sigma_{max}\\left[EI \\right]}{Eh}$\n\nHere", null, "$\\sigma_{max}$ is the maximum allowable bending stress,", null, "$E$ is the modulus of elasticity, and", null, "$h$ is the distance from the neutral axis to the extreme fiber of the sheet (see previous post for a discussion of this.) The sheeting database is reproduced below:", null, "Assuming that the sheeting is made of ASTM A572 Fr. 50 with an allowable stress of 220 MPa, for Hoesch 95 the maximum moment is as follows:", null, "$M_{max} = \\frac{(220)(1000)(14863.6)}{(210)(1000000)(.19)} = 82 \\frac{kN-m}{m}$\n\nObviously this is too light of a section for the moment level. This indicates that the EI of an acceptable section should be", null, "$\\frac{235}{82} = 2.9$ times the current one, or about AZ13-700. As an exercise this should be checked. This ratio method is indicative and not absolute; since the program uses soil-structure interaction, the stiffness of the sheets affects the moment distribution, as is the case in actual application.\n\nThe solution printout is here.\n\nIn the next post, we will consider the case of an anchored wall.\n\n## 2 thoughts on “Analyzing Sheet Pile Walls with SPW 2006: Part II, Cantilever Walls”\n\nThis site uses Akismet to reduce spam. 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http://newton.cam.ac.uk/event/ascw01/timetable	[ "skip to content\n\n# Timetable (ASCW01)\n\n## Challenges in optimal recovery and hyperbolic cross approximation\n\nMonday 18th February 2019 to Friday 22nd February 2019\n\n 09:00 to 09:25 Registration 09:25 to 09:35 Welcome from Christie Marr (INI Deputy Director) 09:40 to 10:15 Henryk Wozniakowski (Columbia University); (University of Warsaw)Exponential tractability of weighted tensor product problems INI 1", null, "", null, "10:20 to 11:00 Morning Coffee 11:00 to 11:35 Aicke Hinrichs (Johannes Kepler Universität)Random sections of ellipsoids and the power of random information We study the circumradius of the intersection of an $m$-dimensional ellipsoid~$\\mathcal E$ with half axes $\\sigma_1\\geq\\dots\\geq \\sigma_m$ with random subspaces of codimension $n$. We find that, under certain assumptions on $\\sigma$, this random radius $\\mathcal{R}_n=\\mathcal{R}_n(\\sigma)$ is of the same order as the minimal such radius $\\sigma_{n+1}$ with high probability. In other situations $\\mathcal{R}_n$ is close to the maximum~$\\sigma_1$. The random variable $\\mathcal{R}_n$ naturally corresponds to the worst-case error of the best algorithm based on random information for $L_2$-approximation of functions from a compactly embedded Hilbert space $H$ with unit ball $\\mathcal E$. In particular, $\\sigma_k$ is the $k$th largest singular value of the embedding $H\\hookrightarrow L_2$. In this formulation, one can also consider the case $m=\\infty$, and we prove that random information behaves very differently depending on whether $\\sigma \\in \\ell_2$ or not. For $\\sigma \\notin \\ell_2$ random information is completely useless. For $\\sigma \\in \\ell_2$ the expected radius of random information tends to zero at least at rate $o(1/\\sqrt{n})$ as $n\\to\\infty$. In the proofs we use a comparison result for Gaussian processes a la Gordon, exponential estimates for sums of chi-squared random variables, and estimates for the extreme singular values of (structured) Gaussian random matrices. This is joint work with David Krieg, Erich Novak, Joscha Prochno and Mario Ullrich. INI 1", null, "", null, "11:40 to 12:15 Yuri Malykhin (Steklov Mathematical Institute, Russian Academy of Sciences); (Moscow State University)On some lower bounds for Kolmogorov widths INI 1", null, "", null, "12:20 to 13:40 Lunch at Churchill College 13:40 to 14:15 Jan Vybiral (Charles University, Prague)Approximation of Ridge Functions and Sparse Additive Models The approximation of smooth multivariate functions is known to suffer the curse of dimension. We discuss approximation of structured multivariate functions, which take the form of a ridge, their sum, or of the so-called sparse additive models. We give also results about optimality of such algorithms. INI 1", null, "", null, "14:20 to 14:55 Alexander Litvak (University of Alberta)Order statistics and Mallat--Zeitouni problem Let $X$ be an $n$-dimensional random centered Gaussian vector with independent but not necessarily identically distributed coordinates and let $T$ be an orthogonal transformation of $\\mathbb{R}^n$. We show that the random vector $Y=T(X)$ satisfies $$\\mathbb{E} \\sum \\limits_{j=1}^k j\\mbox{-}\\min _{i\\leq n}{X_{i}}^2 \\leq C \\mathbb{E} \\sum\\limits_{j=1}^k j\\mbox{-}\\min _{i\\leq n}{Y_{i}}^2$$ for all $k\\leq n$, where $j\\mbox{-}\\min$'' denotes the $j$-th smallest component of the corresponding vector and $C>0$ is a universal constant. This resolves (up to a multiplicative constant) an old question of S.Mallat and O.Zeitouni regarding optimality of the Karhunen--Lo\\eve basis for the nonlinear reconstruction. We also show some relations for order statistics of random vectors (not only Gaussian), which are of independent interest. This is a joint work with Konstantin Tikhomirov. INI 1", null, "", null, "15:00 to 15:30 Afternoon Tea 15:30 to 16:05 Mario Ullrich (Johannes Kepler Universität)Construction of high-dimensional point sets with small dispersion Based on deep results from coding theory, we present an deterministic algorithm that contructs a point set with dispersion at most $\\eps$ in dimension $d$ of size $poly(1/\\eps)*\\log(d)$, which is optimal with respect to the dependence on $d$. The running time of the algorithms is, although super-exponential in $1/\\eps$, only polynomial in $d$. INI 1 16:10 to 16:45 Dicussion INI 1 17:00 to 18:00 Welcome Wine Reception at INI\n 09:00 to 09:35 Winfried Sickel (Friedrich-Schiller-Universität Jena)The Haar System and Smoothness Spaces built on Morrey Spaces For some Nikol'skij-Besov spaces $B^s_{p,q}$ the orthonormal Haar system can be used as an unconditional Schauder basis. Nowadays necessary and sufficient conditions with respect to $p,q$ and $s$ are known for this property. In recent years in a number of papers some modifications of Nikol'skij-Besov spaces based on Morrey spaces have been investigated. In my talk I will concentrate on a version called Besov-type spaces and denoted by $B^{s,\\tau}_{p,q}$. It will be my aim to discuss some necessary and some sufficient conditions on the parameters $p,q,s,\\tau$ such that one can characterize these classes by means of the Haar system. This is joined work with Dachun Yang and Wen Yuan (Beijing Normal University). INI 1", null, "", null, "09:40 to 10:15 Dachun Yang (Beijing Normal University)Ball Average Characterizations of Function Spaces It is well known that function spaces play an important role in the study on various problems from analysis. In this talk, we present pointwise and ball average characterizations of function spaces including Sobolev spaces, Besov spaces and Triebel-Lizorkin spaces on the Euclidean spaces. These characterizations have the advantages so that they can be used as the definitions of these function spaces on metric measure spaces. Some open questions are also presented in this talk. INI 1", null, "", null, "10:20 to 11:00 Morning Coffee 11:00 to 11:35 Wen Yuan (Beijing Normal University)Embedding and continuity envelopes of Besov-type spaces In this talk, we discuss about the sharp embedding properties between Besov-type spaces and Triebel-Lizorkin-type and present some related necessary and sufficient conditions for these embedding. The corresponding continuity envelopes are also worked out. INI 1", null, "", null, "11:40 to 12:15 Bin Han (University of Alberta); (University of Alberta)Directional Framelets with Low Redundancy and Directional Quasi-tight Framelets Edge singularities are ubiquitous and hold key information for many high-dimensional problems. Consequently, directional representation systems are required to effectively capture edge singularities for high-dimensional problems. However, the increased angular resolution often significantly increases the redundancy rates of a directional system. High redundancy rates lead to expensive computational costs and large storage requirement, which hinder the usefulness of such directional systems for problems in moderately high dimensions such as video processing. In this talk, we attack this problem by using directional tensor product complex tight framelets with mixed sampling factors. Such introduced directional system has good directionality with a very low redundancy rate $\\frac{3^d-1}{2^d-1}$, e.g., the redundancy rates are $2$, $2\\frac{2}{3}$, $3\\frac{5}{7}$, $5\\frac{1}{3}$ and $7\\frac{25}{31}$ for dimension $d=1,\\ldots,5$. Our numerical experiments on image/video denoising and inpainting show that the performance of our proposed directional system with low redundancy rate is comparable or better than several state-of-the-art methods which have much higher redundancy rates. In the second part, we shall discuss our recent developments of directional quasi-tight framelets in high dimensions. This is a joint work with Chenzhe Diao, Zhenpeng Zhao and Xiaosheng Zhuang. INI 1", null, "", null, "12:20 to 13:40 Lunch at Churchill College 13:40 to 14:15 Clayton Webster (University of Tennessee); (Oak Ridge National Laboratory)Polynomial approximation via compressed sensing of high-dimensional functions on lower sets This talk will focus on compressed sensing approaches to sparse polynomial approximation of complex functions in high dimensions. Of particular interest is the parameterized PDE setting, where the target function is smooth, characterized by a rapidly decaying orthonormal expansion, whose most important terms are captured by a lower (or downward closed) set. By exploiting this fact, we will present and analyze several procedures for exactly reconstructing a set of (jointly) sparse vectors, from incomplete measurements. These include novel weighted $\\ell_1$ minimization, improved iterative hard thresholding, mixed convex relaxations, as well as nonconvex penalties. Theoretical recovery guarantees will also be presented based on improved bounds for the restricted isometry property, as well as unified null space properties that encompass all currently proposed nonconvex minimizations. Numerical examples are provided to support the theoretical results and demonstrate the computational efficiency of the described compressed sensing methods. INI 1", null, "14:20 to 14:55 Oscar Dominguez (Universidad Complutense de Madrid)Characterizations of Besov spaces in terms of K-functionals Besov spaces occur naturally in many fields of analysis. In this talk, we discuss various characterizations of Besov spaces in terms of different K-functionals. For instance, we present descriptions via oscillations, Bianchini-type norms and approximation methods. This is a joint work with S. Tikhonov (Barcelona). INI 1 15:00 to 15:30 Afternoon Tea 15:30 to 16:05 Wenrui Ye (The University of International Business and Economics - Beijing)Local restriction theorem and maximal Bochner-Riesz operator for the Dunkl transforms In this talk, I will mainly describe joint work with Dr. Feng Dai on the critical index for the almost everywhere convergence of the Bochner-Riesz means in weighted Lp-spaces with p=1 or p>2. Our results under the case p>2 are in full analogy with the classical result of M. Christ on estimates of the maximal Bochner-Riesz means of Fourier integrals and the classical result of A. Carbery, José L. Rubio De Francia and L. Vega on a.e. convergence of Fourier integrals. Besides, I will also introduce several new results that are related to our main results, including: (i) local restriction theorem for the Dunkl transform which is significantly stronger than the global one, but more difficult to prove; (ii) the weighted Littlewood Paley inequality with Ap-weights in the Dunkl non-commutative setting; (iii) sharp local point-wise estimates of several important kernel functions. INI 1", null, "", null, "16:10 to 16:45 Discussion INI 1\n 09:00 to 09:35 Holger Rauhut (RWTH Aachen University)Recovery of functions of many variables via compressive sensing The talk will report on the use of compressive sensing for the recovery of functions of many variables from sample values. We will cover trigonometric expansions as well as expansions in tensorized orthogonal polynomial systems and provide convergence rates in terms of the number of samples which avoid the curse of dimensionality. The technique can be used for the numerical solution of parametric operator equations. INI 1", null, "09:40 to 10:15 Dũng Dinh (Vietnam National University)Dimension-dependence error estimates for sampling recovery on Smolyak grids We investigate dimension-dependence estimates of the approximation error for linear algorithms of sampling recovery on Smolyak grids parametrized by $m$, of periodic $d$-variate functions from the space with Lipschitz-H\\\"older mixed smoothness $\\alpha > 0$. For the subsets of the unit ball in this space of functions with homogeneous condition and of functions depending on $\\nu$ active variables ($1 \\le \\nu \\le d$), respectively, we prove some upper bounds and lower bounds (for $\\alpha \\le 2$) of the error of the optimal sampling recovery on Smolyak grids, explicit in $d$, $\\nu$, $m$ when $d$ and $m$ may be large. This is a joint work with Mai Xuan Thao, Hong Duc University, Thanh Hoa, Vietnam. INI 1", null, "10:20 to 11:00 Morning Coffee 11:00 to 11:35 Martin Buhmann (Justus-Liebig-Universität Gießen)Recent Results on Rational Approximation and Interpolation with Completely and Multiply Monotone Radial Basis Functions We will report on new results about approximations to continuous functions of multiple variables. We shall use either approximation with interpolation or approximation by rational functions. For these kinds of approximations, radial basis functions are particularly attractive, as they provide regular, positive definite or conditionally positive definite approximations, independent of the spatial dimension and independent the distribution of the data points we wish to work with. These interpolants have very many applications for example in solving nonlinear partial differential equations by collocation. In this talk, we classify radial basis and other functions that are useful for such scattered data interpolation or for rational approximations from vector spaces spanned by translates of those basis functions (kernels); for this we study in particular multiply and/or completely monotone functions. We collect special properties of such monotone functions, generalise them and find larger classes than the well known monotone functions for multivariate interpolation. Furthermore, we discuss efficient ways to compute rational approximations using the same type of kernels. INI 1 11:40 to 12:15 Lutz Kaemmerer (Technische Universität Chemnitz)Multiple Rank-1 Lattices as Sampling Schemes for Approximation The approximation of functions using sampling values along single rank-1 lattices leads to convergence rates of the approximation errors that are far away from optimal ones in spaces of dominating mixed smoothness. A recently published idea that uses sampling values along several rank-1 lattices in order to reconstruct multivariate trigonometric polynomials accompanied by fast methods for the construction of these sampling schemes as well as available fast Fourier transform algorithms motivates investigations on the approximation properties of the arising sampling operators applied on functions of specific smoothness, in particular functions of dominating mixed smoothness which naturally leads to hyperbolic cross approximations. INI 1", null, "12:20 to 13:40 Lunch at Churchill College 13:40 to 18:00 Free afternoon 19:30 to 22:00 Formal Dinner at Emmanuel College (Old Library) LOCATIONEmmanuel CollegeHow to get thereMENU Starters Butternut Squash Veloute with Pumpkin Seed Oil (V) Main course Supreme of Chicken, Parsley Risotto, Crispy Chorizo Roasted Artichoke, Parsley Risotto, Crispy Kale (V) Selection of vegetables Dessert Crème Brulee Coffee and MintsDRESS CODESmart casual", null, "09:00 to 09:35 Thomas Kuehn (Universität Leipzig)Preasymptotic estimates for approximation of multivariate periodic Sobolev functions Approximation of Sobolev functions is a topic with a long history and many applications in different branches of mathematics. The asymptotic order as $n\\to\\infty$ of the approximation numbers $a_n$ is well-known for embeddings of isotropic Sobolev spaces and also for Sobolev spaces of dominating mixed smoothness. However, if the dimension $d$ of the underlying domain is very high, one has to wait exponentially long until the asymptotic rate becomes visible. Hence, for computational issues this rate is useless, what really matters is the preasymptotic range, say $n\\le 2^d$. In the talk I will first give a short overview over this relatively new field. Then I will present some new preasymptotic estimates for $L_2$-approximation of periodic Sobolev functions, which improve the previously known results. I will discuss the cases of isotropic and dominating mixed smoothness, and also $C^\\infty$-functions of Gevrey type. Clearly, on all these spaces there are many equivalent norms. It is an interesting effect that - in contrast to the asymptotic rates - the preasymptotic behaviour strongly depends on the chosen norm. INI 1", null, "09:40 to 10:15 Konstantin Ryutin Best m-term approximation of the \"step-function\" and related problems The main point of the talk is the problem of approximation of the step-function by $m$-term trigonometric polynomials and some closely related problems: the approximate rank of a specific triangular matrix, the Kolmogorov width of BV functions. This problem has its origins in approximation theory (best sparse approximation and Kolmogorov widths) as well as in computer science (approximate rank of a matrix). There are different approaches and techniques: $\\gamma_2$--norm, random approximations, orthomassivity of a set.... I plan to show what can be achieved by these techniques. INI 1", null, "10:20 to 11:00 Morning Coffee 11:00 to 11:35 Michael Gnewuch (Christian-Albrechts-Universität zu Kiel)Explicit error bounds for randomized Smolyak algorithms and an application to infinite-dimensional integration Smolyak's method, also known as hyperbolic cross approximation or sparse grid method, is a powerful %black box tool to tackle multivariate tensor product problems just with the help of efficient algorithms for the corresponding univariate problem. We provide upper and lower error bounds for randomized Smolyak algorithms with fully explicit dependence on the number of variables and the number of information evaluations used. The error criteria we consider are the worst-case root mean square error (the typical error criterion for randomized algorithms, often referred to as randomized error'') and the root mean square worst-case error (often referred to as worst-case error''). Randomized Smolyak algorithms can be used as building blocks for efficient methods, such as multilevel algorithms, multivariate decomposition methods or dimension-wise quadrature methods, to tackle successfully high-dimensional or even infinite-dimensional problems. As an example, we provide a very general and sharp result on infinite-dimensional integration on weighted reproducing kernel Hilbert spaces and illustrate it for the special case of weighted Korobov spaces. We explain how this result can be extended, e.g., to spaces of functions whose smooth dependence on successive variables increases (`spaces of increasing smoothness'') and to the problem of L_2-approximation (function recovery). INI 1", null, "11:40 to 12:15 Heping Wang (Capital Normal University)Monte Carlo methods for $L_q$ approximation on periodic Sobolev spaces with mixed smoothness In this talk we consider multivariate approximation of compact embeddings of periodic Sobolev spaces of dominating mixed smoothness into the $L_q,\\ 2< q\\leq \\infty$ space by linear Monte Carlo methods that use arbitrary linear information. We construct linear Monte Carlo methods and obtain explicit-in-dimension upper estimates. These estimates catch up with the rate of convergence. INI 1", null, "12:20 to 13:40 Lunch at Churchill College 13:40 to 14:15 Robert J. Kunsch (Universität Osnabrück)Optimal Confidence for Monte Carlo Integration of Smooth Functions We study the complexity $n(\\varepsilon,\\delta)$ of approximating the integral of smooth functions at absolute precision $\\varepsilon > 0$ with confidence level $1 - \\delta \\in (0,1)$ using function evaluations as information within randomized algorithms. Methods that achieve optimal rates in terms of the root mean square error (RMSE) are not always optimal in terms of error at confidence, usually we need some non-linearity in order to suppress outliers. Besides, there are numerical problems which can be solved in terms of error at confidence but no algorithm can guarantee a finite RMSE, see . Hence, the new error criterion seems to be more general than the classical RMSE. The sharp order for multivariate functions from classical isotropic Sobolev spaces $W_p^r([0,1]^d)$ can be achieved via control variates, as long as the space is embedded in the space of continuous functions $C([0,1]^d)$. It turns out that the integrability index $p$ has an effect on the influence of the uncertainty $\\delta$ to the complexity, with the limiting case $p = 1$ where deterministic methods cannot be improved by randomization. In general, higher smoothness reduces the effort we need to take in order to increase the confidence level. Determining the complexity $n(\\varepsilon,\\delta)$ is much more challenging for mixed smoothness spaces $\\mathbf{W}_p^r([0,1]^d)$. While optimal rates are known for the classical RMSE (as long as $\\mathbf{W}_p^r([0,1]^d)$ is embedded in $L_2([0,1]^d)$), see , basic modifications of the corresponding algorithms fail to match the theoretical lower bounds for approximating the integral with prescribed confidence. Joint work with Daniel Rudolf R.J. Kunsch, E. Novak, D. Rudolf. Solvable integration problems and optimal sample size selection. To appear in Journal of Complexity. M. Ullrich. A Monte Carlo method for integration of multivariate smooth functions. SIAM Journal on Numerical Analysis, 55(3):1188-1200, 2017. INI 1", null, "14:20 to 14:55 Markus Weimar (Ruhr-Universität Bochum)Optimal recovery using wavelet trees This talk is concerned with the approximation of embeddings between Besov-type spaces defined on bounded multidimensional domains or (patchwise smooth) manifolds. We compare the quality of approximations of three different strategies based on wavelet expansions. For this purpose, sharp rates of convergence corresponding to classical uniform refinement, best $N$-term, and best $N$-term tree approximation will be presented. In particular, we will see that whenever the embedding of interest is compact, greedy tree approximation schemes are as powerful as abstract best $N$-term approximation and that (for a large range of parameters) they can outperform uniform schemes based on a priori fixed (hence non-adaptively chosen) subspaces. This observation justifies the usage of adaptive non-linear algorithms in computational practice, e.g., for the approximate solution of boundary integral equations arising from physical applications. If time permits, implications for the related concept of approximation spaces associated to the three approximation strategies will be discussed. INI 1", null, "15:00 to 15:30 Afternoon Tea 15:30 to 16:05 Discussion INI 1 16:10 to 16:45 Discussion INI 1\n 09:00 to 09:35 Jürgen Prestin (Universität zu Lübeck)Shift-invariant Spaces of Multivariate Periodic Functions One of the underlying ideas of multiresolution and wavelet analysis consists in the investigation of shift-invariant function spaces. In this talk one-dimensional shift-invariant spaces of periodic functions are generalized to multivariate shift-invariant spaces on non-tensor product patterns. These patterns are generated from a regular integer matrix. The decomposition of these spaces into shift-invariant subspaces can be discussed by the properties of these matrices. For these spaces we study different bases and their time-frequency localization. Of particular interest are multivariate orthogonal Dirichlet and de la Valle\\'e Poussin kernels and the respective wavelets. This approach also leads to an adaptive multiresolution. Finally, with these methods we construct shearlets and show how we can detect jump discontinuities of given cartoon-like functions. INI 1 09:40 to 10:15 Bastian Bohn (Universität Bonn)Least squares regression on sparse grids In this talk, we first recapitulate the framework of least squares regression on certain sparse grid and hyperbolic cross spaces. The underlying numerical problem can be solved quite efficiently with state-of-the-art algorithms. Analyzing its stability and convergence properties, we can derive the optimal coupling between the number of necessary data samples and the degrees of freedom in the ansatz space.Our analysis is based on the assumption that the least-squares solution employs some kind of Sobolev regularity of dominating mixed smoothness, which is seldomly encountered for real-world applications. Therefore, we present possible extensions of the basic sparse grid least squares algorithm by introducing suitable a-priori data transformations in the second part of the talk. These are tailored such that the resulting transformed problem suits the sparse grid structure.Co-authors: Michael Griebel (University of Bonn), Jens Oettershagen (University of Bonn), Christian Rieger (University of Bonn) INI 1", null, "10:20 to 11:00 Morning Coffee 11:00 to 11:35 Song Li (Zhejiang University); (Zhejiang University)Some Sparse Recovery Methods in Compressed Sensing In this talk, I shall investigate some sparse recovery methods in Compressed Sensing. In particular, I will focus on RIP approach and D-RIP approach. As a result, we confirmed a conjecture on RIP, which is related to Terence. Tao and Jean. Bourgain's works in this fields. Then, I will also investigate the relations between our works and statistics. 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https://math.stackexchange.com/questions/1181509/undamped-spring-mass-system	[ "# Undamped spring mass system\n\nI have this study guide for an upcoming test for DE class I'm trying to figure out.\n\nA mass of 400 grams stretches a spring by 5 centimeters.\n(a) Find the spring constant k, the angular frequency ω, as well as the period T and frequency f of free undamped motion for this spring-mass system.\n(b) Find the general solution of the DE for the free spring-mass system.\n(c) Suppose that an exterior force of F(t) = 27sin(13t) Newtons\n\nacts on the spring-mass system. Find the equation of motion of the system if the mass initially is at rest in its equilibrium position.\n\nI know K is 784 (or do I need to convert to 5 centimeters to 0.05 meters?) and w is sqrt(k/m), but I'm not sure what I need to find T and F. I can find the general solution, but then I have no clue on what to do with part c.\n\n## 2 Answers\n\nYou don't need to convert because your units of mass are grams, so you are using CGS system of units. For a: $\\omega=2 \\pi/T$ For c: You have to add the new force to Newton's law.\n\nbefore you can set the equation you need $k,$ the spring constant. we will do all this in metric system. $k = \\frac{5/100}{400 \\times 9.8/1000} = 0.0127\\,N/m, \\omega^2 = \\frac{k}{m} = (0.1785)^2sec^{-2}, T = \\frac{2\\pi}{\\omega} = 35.185 \\, sec$\n\nequation of motion is $$m\\frac{d^2x}{dt^2} + kx = 0 \\to \\frac{d^2x}{dt^2} + \\omega^2 x = 0 \\text{ where x is deviation from equilibrium.}$$ the general solution is $$x = A\\cos(\\omega t - \\phi).$$\n\nthe equation of motion for a forced system is $$m\\frac{d^2x}{dt^2} + kx = 27 \\sin 13 t \\to \\frac{d^2x}{dt^2} + \\omega^2 x = \\frac{27 \\sin 13 t}m = 67.5 \\sin 13 t.$$\n\n• Is k not = mg/s, as opposed to your s/(mg)? – user3032755 Mar 9 '15 at 1:06\n• @user3032755, the unit of $kx$ that is $unit \\,of\\, k \\times m = Newton$ – abel Mar 9 '15 at 1:07\n• In your calculations you have the units of $k$ be $\\text{m/N}$. The answer should be 1 over that number – Dylan Mar 10 '15 at 2:41" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.92416155,"math_prob":0.99954987,"size":788,"snap":"2019-35-2019-39","text_gpt3_token_len":207,"char_repetition_ratio":0.12117347,"word_repetition_ratio":0.0,"special_character_ratio":0.25888324,"punctuation_ratio":0.07954545,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9999794,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-08-18T19:20:00Z\",\"WARC-Record-ID\":\"<urn:uuid:56432a31-db08-494f-a522-f25fce41794e>\",\"Content-Length\":\"143837\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:5f97ee91-10d7-43e7-a895-84bd89e91b14>\",\"WARC-Concurrent-To\":\"<urn:uuid:57157cc4-dd0c-4fb3-8588-02735d771a9b>\",\"WARC-IP-Address\":\"151.101.1.69\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/1181509/undamped-spring-mass-system\",\"WARC-Payload-Digest\":\"sha1:GOLSBF5OTRFGUIMD7IZ3IBWHKGGDENP6\",\"WARC-Block-Digest\":\"sha1:Y7YJ7YF7SEYOAVOXV2WFQJ42ARV7INPH\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-35/CC-MAIN-2019-35_segments_1566027313996.39_warc_CC-MAIN-20190818185421-20190818211421-00042.warc.gz\"}"}
https://www.varsitytutors.com/sat_ii_math_ii-help/equations-based-on-word-problems	[ "# SAT II Math II : Equations Based on Word Problems\n\n## Example Questions\n\n### Example Question #1 : Single Variable Algebra\n\nWhich of the following phrases can be written as the algebraic expression", null, "?\n\nThe absolute value of the difference of eight and a number\n\nThe absolute value of the product of negative eight and a number\n\nThe absolute value of the difference of a number and eight\n\nEight subtracted from the absolute value of a number\n\nEight decreased by the absolute value of a number\n\nThe absolute value of the difference of eight and a number\n\nExplanation:", null, "is the absolute value of", null, ", which is the difference of eight and a number. Therefore,", null, "is \"the absolute value of the difference of eight and a number.\"\n\n### Example Question #1 : Equations Based On Word Problems\n\nWhich of the following phrases can be written as the algebraic expression", null, "?\n\nThe opposite of a number decreased by seven\n\nSeven decreased by the absolute value of a number\n\nSeven decreased by the opposite of a number\n\nThe absolute value of the difference of seven and a number\n\nThe opposite of the difference of seven and a number\n\nSeven decreased by the opposite of a number\n\nExplanation:", null, "is seven decreased by", null, ", which is the opposite of a number; therefore,", null, "is \"seven decreased by the opposite of a number.\"\n\n### Example Question #1 : Single Variable Algebra\n\nWhich of the following phrases can be represented by the algebraic expression", null, "?\n\nOne divided into the difference of nine and a number\n\nNine decreased by the multiplicative inverse of a number\n\nThe multiplicative inverse of the difference of nine and a number\n\nThe multiplicative inverse of the difference of a number and nine\n\nNine less than by the multiplicative inverse of a number\n\nThe multiplicative inverse of the difference of nine and a number\n\nExplanation:", null, "is the multiplicative inverse of", null, ", which is the difference of nine and a number. Therefore,", null, "is \"the multiplicative inverse of the difference of nine and a number\".\n\n### Example Question #1 : Equations Based On Word Problems\n\nWhich of the following phrases can be represented by the algebraic expression", null, "Ten less than three times the square root of a number\n\nThe cube root of the difference of a number and ten\n\nTen decreased by three times the square root of a number\n\nTen decreased by the cube root of a number\n\nTen less than the cube root of a number\n\nTen less than the cube root of a number\n\nExplanation:", null, "is ten less than", null, ", which is the cube root of a number; therefore,", null, "is \"ten less than the cube root of a number\".\n\n### Example Question #1 : Single Variable Algebra\n\nWhich of the following phrases can be represented by the algebraic expression", null, "Twenty less than the square root of a number\n\nNegative twenty multiplied by the square root of a number\n\nTwenty decreased by the square root of a number\n\nThe square root of the difference of twenty and a number\n\nThe square root of the difference of a number and twenty\n\nTwenty decreased by the square root of a number\n\nExplanation:", null, "is twenty decreased by", null, ", which is the square root of a number, so", null, "is \"twenty decreased by the square root of a number\".\n\n### Example Question #1 : Equations Based On Word Problems\n\nAdult tickets to the zoo sell for", null, "; child tickets sell for", null, ". On a given day, the zoo sold", null, "tickets and raised", null, "in admissions. How many adult tickets were sold?", null, "", null, "", null, "", null, "", null, "", null, "Explanation:\n\nLet", null, "be the number of adult tickets sold. Then the number of child tickets sold is", null, ".\n\nThe amount of money raised from adult tickets is", null, "; the amount of money raised from child tickets is", null, ". The sum of these money amounts is", null, ", so the amount of money raised can be defined by the following equation:", null, "To find the number of adult tickets sold, solve for", null, ":", null, "", null, "", null, "", null, "", null, "", null, "", null, "adult tickets were sold.\n\n### Example Question #1 : Single Variable Algebra\n\nSarah sells lemonade at the concession stands. She charges fifty cents per cup of lemonade, and twenty five cents for refills. What is the equation that represents the total that she will make from the lemonade stand using the variables cups", null, "and refills", null, "?", null, "", null, "", null, "", null, "", null, "", null, "Explanation:\n\nSarah charges fifty cents per cup of lemonade:", null, "Sarah charges twenty five cents for refills:", null, "Set up the equation by adding the totals.\n\nThe answer is:", null, "### All SAT II Math II Resources", null, "" ]	[ null, "https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/210995/gif.latex", null, "https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/210996/gif.latex", null, "https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/210997/gif.latex", null, "https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/210998/gif.latex", null, "https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/210976/gif.latex", null, 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https://space.stackexchange.com/questions/35231/propagating-satellites-without-tle-in-python?noredirect=1	[ "# Propagating satellites without TLE in python\n\nI know how to propagate satellites in python using ephem module (that uses TLE data). However, I am not sure how I would do the same for my own set of orbital parameters (i.e, without TLE data) in python.\n\nThanks.\n\n• Maybe you should somehow emulate real TLE data for the library? – peterh - Reinstate Monica Apr 2 '19 at 6:35\n• TLE is essentially Euler angles plus a couple non-essential parameters formatted into a rigidly defined text representation. If your orbital parameters are Euler angles, you should be able to turn them into TLE trivially. If you use a state vector (position+velocity), it gets a bit trickier. – SF. Apr 2 '19 at 6:58\n\nnote: Alas, PyEphem is deprecated, so to plot from TLE's use Skyfield, written/maintained by the same person as PyEphem. I show an example here.\n\n## Step 1: Convert your orbital parameters to a state vector\n\nA state vector is a 6 dimensional vector which is the combination of a position vector $$\\mathbf{x}$$ and a velocity vector $$\\mathbf{v}$$. In a deterministic system like a simple two-body orbit calculation, it's all you need to determine mathematically the orbit and how the body will move at any given time afterward.\n\nThis answer explains how to do this and contains a python script.\n\nand I think you can learn a lot from poliastro at http://docs.poliastro.space/en/latest/\n\nChoose the method that you are most comfortable with.\n\n## Step 2: Roll your own orbit propagator:\n\nI took the plotting from here and added a simple Python integration of an initial state vector from here or here or even here. I've only added the Earth's spherically symmetric force, if you want to add the effects due to oblateness for better accuracy, it's shown in two of those linked scripts. Look for the $$J_2$$ term.\n\nThe equation of motions are:\n\n$$\\dot{\\mathbf{x}} = \\mathbf{v}$$\n\n$$\\ddot{\\mathbf{x}} = - \\frac{\\mathbf{GM}}{x^2} \\mathbf{\\hat{x}} = -GM \\frac{\\mathbf{x}}{x^3}$$\n\nand those are scripted in the function deriv(X, t) which is independent of time. This is called by the integrator scipy.integrate.odeint. It uses variable step sizes internally, then re-interpolates to user-specified time points.\n\nOutput:\n\nperiod 6307.12290204 seconds or 105.118715034 minutes\ninclination 57.0 degrees\ninitial position [ 7378.137 0. 0. ] km\ninitial velocity [ 0. 4003.17019194 6164.34152276] m/s\ninitial speed 7350.13455625 m/s", null, "def makecubelimits(axis, centers=None, hw=None):\nlims = ax.get_xlim(), ax.get_ylim(), ax.get_zlim()\nif centers == None:\ncenters = [0.5sum(pair) for pair in lims]\n\nif hw == None:\nwidths = [pair - pair for pair in lims]\nhw = 0.5max(widths)\nax.set_xlim(centers-hw, centers+hw)\nax.set_ylim(centers-hw, centers+hw)\nax.set_zlim(centers-hw, centers+hw)\nprint(\"hw was None so set to:\", hw)\nelse:\ntry:\nhwx, hwy, hwz = hw\nprint(\"ok hw requested: \", hwx, hwy, hwz)\n\nax.set_xlim(centers-hwx, centers+hwx)\nax.set_ylim(centers-hwy, centers+hwy)\nax.set_zlim(centers-hwz, centers+hwz)\nexcept:\nprint(\"nope hw requested: \", hw)\nax.set_xlim(centers-hw, centers+hw)\nax.set_ylim(centers-hw, centers+hw)\nax.set_zlim(centers-hw, centers+hw)\n\nreturn centers, hw\n\ndef deriv(X, t):\nx, v = X.reshape(2, -1)\nacc = -GMe * x * ((x2).sum())-1.5\nreturn np.hstack((v, acc))\n\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy.integrate import odeint as ODEint\nfrom mpl_toolkits.mplot3d import Axes3D\n\nhalfpi, pi, twopi = [fnp.pi for f in (0.5, 1, 2)]\n\nkm = 0.001\nGMe = 3.986E+14 # m^3/s^2\nRe = 6378137. # meters\nalt = 1E+06 # meters\na = Re + alt\nT = twopi np.sqrt(a*3/GMe)\n\nv0 = np.sqrt(GMe/a)\nincdegs = 57.\n\ncinc, sinc = [f(inc) for f in (np.cos, np.sin)]\n\nX0 = np.array([Re+alt, 0, 0] + [0, cincv0, sincv0])\n\nprint 'period {} seconds or {} minutes'.format(T, T/60.)\nprint 'inclination {} degrees'.format(incdegs)\nprint 'initial position {} km'.format(km X0[:3])\nprint 'initial velocity {} m/s'.format(X0[3:])\nprint 'initial speed {} m/s'.format(v0)\n\ntimes = np.linspace(0, T, 201)\n\nanswer, info = ODEint(deriv, X0, times, full_output=True)\n\ntheta = np.linspace(0, twopi, 201)\ncth, sth, zth = [f(theta) for f in (np.cos, np.sin, np.zeros_like)]\nlon0 = Renp.vstack((cth, zth, sth))\nlons = []\nfor phi in radsnp.arange(0, 180, 15):\ncph, sph = [f(phi) for f in (np.cos, np.sin)]\nlon = np.vstack((lon0cph - lon0sph,\nlon0cph + lon0sph,\nlon0) )\nlons.append(lon)\n\nlat0 = Renp.vstack((cth, sth, zth))\nlats = []\nfor phi in radsnp.arange(-75, 90, 15):\ncph, sph = [f(phi) for f in (np.cos, np.sin)]\nlat = Renp.vstack((cthcph, sthcph, zth+sph))\nlats.append(lat)\n\nif True:\nfig = plt.figure(figsize=[10, 8]) # [12, 10]\n\nax = fig.add_subplot(1, 1, 1, projection='3d')\n\nax.plot(kmx, kmy, kmz)\nfor x, y, z in lons:\nax.plot(kmx, kmy, kmz, '-k')\nfor x, y, z in lats:\nax.plot(kmx, kmy, kmz, '-k')\n\ncenters, hw = makecubelimits(ax)\n\nprint(\"centers are: \", centers)\nprint(\"hw is: \", hw)\n\nplt.show()\n\n• Disclaimer: poliastro author Two minor details: matplotlib 3D capabilities are not enough to properly plot orbits (as can be seen in your image), so I recommend using Plotly instead (poliastro can do it). Also, OrbitalPy is unmaintained (last commit 2015) whereas poliastro is still active. – astrojuanlu Apr 2 '19 at 15:12\n• @astrojuanlu please feel free to add another answer, or to edit this one. Thanks! – uhoh Apr 2 '19 at 15:14" ]	[ null, "https://i.stack.imgur.com/iX2d6.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6854763,"math_prob":0.9868253,"size":4920,"snap":"2020-10-2020-16","text_gpt3_token_len":1624,"char_repetition_ratio":0.098657444,"word_repetition_ratio":0.12551723,"special_character_ratio":0.33699188,"punctuation_ratio":0.20070733,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9986274,"pos_list":[0,1,2],"im_url_duplicate_count":[null,5,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-02-25T09:57:10Z\",\"WARC-Record-ID\":\"<urn:uuid:cb0e4c3a-1e25-442a-8090-0b1ea9eedf46>\",\"Content-Length\":\"151949\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e10c3fb5-ddf2-4d87-a70c-765c5db12bb0>\",\"WARC-Concurrent-To\":\"<urn:uuid:1651fcbd-583f-4701-b931-5b70a1a13d46>\",\"WARC-IP-Address\":\"151.101.193.69\",\"WARC-Target-URI\":\"https://space.stackexchange.com/questions/35231/propagating-satellites-without-tle-in-python?noredirect=1\",\"WARC-Payload-Digest\":\"sha1:T55RHNL4YXE7S6GD67JKK6KZZJE4HOON\",\"WARC-Block-Digest\":\"sha1:ALVCRGSFV3Q2DHT7T4V5ADY2QEFRDECM\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-10/CC-MAIN-2020-10_segments_1581875146064.76_warc_CC-MAIN-20200225080028-20200225110028-00227.warc.gz\"}"}
https://blog.robofied.com/identity-function/	[ "", null, "# Identity Function\n\nAn Identity Function or a Linear Function in which the output remains the same as the input.", null, "x: an input data point", null, "Range:", null, "``````#Identity Fucntion\ndef Identity(x):\n'''\nParameters:\nx (real number) input data point\nReturns:\nx\n'''\nreturn x ``````\n\nUse cases:\n\nUsed to build simple neural architectures.\n\nPros:\n\nLess computational power.\n\nIdeal for model interpretability.\n\nCons:\n\nSince it is a linear function, the neural network model can’t find complex or non-linear patterns in the data. Even when the network is stacked up with multiple layers of neurons, the network will still be a linear model.\n\n## Share\n\n### Sigmoid Activation Function\n\nA Sigmoid Function is also known as the Logistic Function or Squashing Function. It is" ]	[ null, "data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==", null, "data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==", null, "data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==", null, "data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.68948656,"math_prob":0.9377839,"size":579,"snap":"2020-45-2020-50","text_gpt3_token_len":127,"char_repetition_ratio":0.1252174,"word_repetition_ratio":0.0,"special_character_ratio":0.21416235,"punctuation_ratio":0.13913043,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.979519,"pos_list":[0,1,2,3,4,5,6,7,8],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-11-24T11:45:18Z\",\"WARC-Record-ID\":\"<urn:uuid:f13ea892-9c9e-4a7e-884d-563e00e51f82>\",\"Content-Length\":\"65175\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:aade2d19-39e2-46eb-b79f-3b7f5f507182>\",\"WARC-Concurrent-To\":\"<urn:uuid:0173df4a-8e58-4144-b8f1-35538252a9c5>\",\"WARC-IP-Address\":\"104.28.19.125\",\"WARC-Target-URI\":\"https://blog.robofied.com/identity-function/\",\"WARC-Payload-Digest\":\"sha1:6NESZYH2W3TD5RTRF6EIRTUNBI4WS3GQ\",\"WARC-Block-Digest\":\"sha1:46DHOCIXOKR6DYZYUVPPDRWQXLRKUQJY\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-50/CC-MAIN-2020-50_segments_1606141176256.21_warc_CC-MAIN-20201124111924-20201124141924-00028.warc.gz\"}"}
https://www.askiitians.com/forums/Modern-Physics/18/27632/charges.htm	[ "", null, "×", null, "", null, "", null, "", null, "", null, "#### Thank you for registering.\n\nOne of our academic counsellors will contact you within 1 working day.\n\nClick to Chat\n\n1800-1023-196\n\n+91-120-4616500\n\nCART 0\n\n• 0\n\nMY CART (5)\n\nUse Coupon: CART20 and get 20% off on all online Study Material\n\nITEM\nDETAILS\nMRP\nDISCOUNT\nFINAL PRICE\nTotal Price: Rs.\n\nThere are no items in this cart.\nContinue Shopping\n```\nIf 3 different but eqeul charges are forming an equilateral triangle, then where inside the triangle the potential would be 0 and why????? Pls explain!!!!!!!\n\n```\n9 years ago\n\n```\t\t\t\t\t\t\thi,\npotential due to three chrges at a point is= -kq/r1-kq/r2-kq/r3.\nthus potential=-kq(1/r1+1/r2+1/r3)\nif potential has to be zero then that point should be at infinity ( r1r2r3 should be at infinity)\nplzz approve if you like the answer\n```\n9 years ago\n```\t\t\t\t\t\t\tdear shiven , first u draw the figure according to these steps\n1)draw a equilateral triangle ABC such that its base AB is on x axis , center of base is at origin...third vertice (C) is on y axis ...\n2)y axis divides the base in two halves , AO ,BO ...\n3) consider any point P(0,y) on y axis then\n4) place -q , -q at A,B & q at C...\nnow , potential at P will be\nV = k(-q)/AP + K-(q)/BP + kq/PC .......1\nAO = BO = a/2\nPC = asin60 - y = [aroot3/2 - y] ........2\nAP = BP = [y^2 + a^2/4]^1/2 ......3 ( by pythagorus theorem)\nfor potential to become 0 , eq 1 = 0 so\nq/AP + q/BP = q/PC\n2/AP = 1/PC\nAP = 2pC .........4\nputting values of AP,PC from eq 2 & 3 we get a quadratic expression\n3y^2 - (4root3a)y + 11a^2/4 = 0\n\nfrom here Y = a[4root3 +(-) root15]/6\n= a[ 4root3 - root15]/6 or [4root3 + root15]/6 ( not acceptable coz this point lies ouside)\n\ntherefore the only point is Y = a[4root3-root15]/6\n\nthis is the required point where potential will be zero ...\n```\n9 years ago\nThink You Can Provide A Better Answer ?\n\n## Other Related Questions on Modern Physics\n\nView all Questions »", null, "", null, "### Course Features\n\n• 101 Video Lectures\n• Revision Notes\n• Previous Year Papers\n• Mind Map\n• Study Planner\n• NCERT Solutions\n• Discussion Forum\n• Test paper with Video Solution", null, "", null, "### Course Features\n\n• 110 Video Lectures\n• Revision Notes\n• Test paper with Video Solution\n• Mind Map\n• Study Planner\n• NCERT Solutions\n• Discussion Forum\n• Previous Year Exam Questions" ]	[ null, "https://www.askiitians.com/resources/images/badge.png", null, "https://www.askiitians.com/Resources/images/neet-iit/user.png", null, "https://www.askiitians.com/Resources/images/neet-iit/envelope.png", null, "https://www.askiitians.com/Resources/images/neet-iit/phone-receiver.png", null, "https://www.askiitians.com/Resources/images/neet-iit/reading.png", null, "https://www.askiitians.com/Resources/images/neet-iit/reading.png", null, "https://files.askiitians.com/static/ecom/cms/bottom/physics-12.png", null, "https://www.askiitians.com/Resources/images/youtube_placeholder.jpg", null, "https://files.askiitians.com/static/ecom/cms/right/physics-11.jpg", null, "https://www.askiitians.com/Resources/images/youtube_placeholder.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7947042,"math_prob":0.9806863,"size":3089,"snap":"2020-45-2020-50","text_gpt3_token_len":1034,"char_repetition_ratio":0.11345219,"word_repetition_ratio":0.822335,"special_character_ratio":0.3629006,"punctuation_ratio":0.18364611,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96313536,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-24T18:24:19Z\",\"WARC-Record-ID\":\"<urn:uuid:f2418d0a-a92a-435f-87de-90f29907dd0b>\",\"Content-Length\":\"229593\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b5647b75-27e3-44bd-b71d-0bb7b11cb353>\",\"WARC-Concurrent-To\":\"<urn:uuid:99efc9b2-01d4-4db5-b535-00da5a13bdcb>\",\"WARC-IP-Address\":\"3.7.36.8\",\"WARC-Target-URI\":\"https://www.askiitians.com/forums/Modern-Physics/18/27632/charges.htm\",\"WARC-Payload-Digest\":\"sha1:MM2P4XDK4BSMKUFFAX77GSE4QW5AZBBR\",\"WARC-Block-Digest\":\"sha1:ACKRL7FVE5T6D3E4X5JQZ5CE3NKQN6EZ\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-45/CC-MAIN-2020-45_segments_1603107884322.44_warc_CC-MAIN-20201024164841-20201024194841-00065.warc.gz\"}"}
http://bartleylawoffice.com/recommendations/what-are-the-3-forms-of-ohms-law.html	[ "# What are the 3 forms of ohm’s law?\n\n## What are the three forms of Ohms law?\n\nThere are basically three types of Ohm’s law formulas or equations.\n\nThey are;\n\n• I = V / R.\n• V = IR.\n• R = V / R.\n\n## What is the correct form of Ohm’s law?\n\nOhm’s law states that the voltage or potential difference between two points is directly proportional to the current or electricity passing through the resistance, and directly proportional to the resistance of the circuit. The formula for Ohm’s law is V=IR.\n\n## What are the three ways to write the Ohm’s law formula?\n\nOhm’s law can be expressed as an equation three ways:\n\n• I (current) = V/R.\n• V = IR.\n• R = V/I.\n\n## What is Ohm’s law in physics?\n\nOhm’s Law is a formula used to calculate the relationship between voltage, current and resistance in an electrical circuit. To students of electronics, Ohm’s Law (E = IR) is as fundamentally important as Einstein’s Relativity equation (E = mc²) is to physicists. E = I x R.\n\n## What is Watt’s law?\n\nWatt’s Law: States the relationships between power (watts), current (amps) and voltage. Watts = Volts x Amps.\n\n## What do you mean by 1 ohm?\n\nThe ohm is defined as an electrical resistance between two points of a conductor when a constant potential difference of one volt, applied to these points, produces in the conductor a current of one ampere, the conductor not being the seat of any electromotive force.\n\n## How do I calculate resistance?\n\nThe resistance R in ohms (Ω) is equal to the voltage V in volts (V) divided by the current I in amps (A): Since the current is set by the values of the voltage and resistance, the Ohm’s law formula can show that if you increase the voltage, the current will also increase.\n\nYou might be interested: What is the law on drones\n\n## What is the formula for current?\n\nCurrent is usually denoted by the symbol I. Ohm’s law relates the current flowing through a conductor to the voltage V and resistance R; that is, V = IR. An alternative statement of Ohm’s law is I = V/R.\n\n## What is the basic principle of Ohm’s law?\n\nOhm’s Law states that the current flowing in a circuit is directly proportional to the applied potential difference and inversely proportional to the resistance in the circuit. In other words by doubling the voltage across a circuit the current will also double.\n\n## How do you explain ohms?\n\nOhm defines the unit of resistance of “1 Ohm” as the resistance between two points in a conductor where the application of 1 volt will push 1 ampere, or 6.241×10^18 electrons. This value is usually represented in schematics with the greek letter “Ω”, which is called omega, and pronounced “ohm”.\n\n## What is Ohm’s law easy definition?\n\nOhm’s law. [ ōmz ] A law relating the voltage difference between two points, the electric current flowing between them, and the resistance of the path of the current. Mathematically, the law states that V = IR, where V is the voltage difference, I is the current in amperes, and R is the resistance in ohms.\n\n## How do I calculate power?\n\nPower is a measure of the amount of work that can be done in a given amount of time. Power equals work (J) divided by time (s). The SI unit for power is the watt (W), which equals 1 joule of work per second (J/s)." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9407692,"math_prob":0.99599075,"size":3082,"snap":"2023-40-2023-50","text_gpt3_token_len":740,"char_repetition_ratio":0.16504224,"word_repetition_ratio":0.014234875,"special_character_ratio":0.23556133,"punctuation_ratio":0.10534591,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9998741,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-02T06:11:05Z\",\"WARC-Record-ID\":\"<urn:uuid:476bcf44-2738-4548-a1b5-b7dd99187650>\",\"Content-Length\":\"65691\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:380827c7-1ecb-4222-8a39-402abf9c384c>\",\"WARC-Concurrent-To\":\"<urn:uuid:fa8144a5-13ba-490a-848b-f66ce3729db8>\",\"WARC-IP-Address\":\"172.67.157.116\",\"WARC-Target-URI\":\"http://bartleylawoffice.com/recommendations/what-are-the-3-forms-of-ohms-law.html\",\"WARC-Payload-Digest\":\"sha1:AJ4OWB6B6RNAAOB3QJK32ZP4ASBTRRCX\",\"WARC-Block-Digest\":\"sha1:GNHQFLHXBKHNFFYXZGVI7KRSRNYJZWKR\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100327.70_warc_CC-MAIN-20231202042052-20231202072052-00504.warc.gz\"}"}
https://px.01ny.cn/m/h_812d/	[ "", null, "", null, "•", null, "首页\n•", null, "培训机构", null, "", null, "", null, "### 高三物理辅导优学选课平台\n\n612家高三物理辅导机构，243门高三物理辅导课程，百余名资深的教育顾问助您轻松选学校\n\n• 城市\n• 高三物理辅导\n• 小学辅导\n• 小升初\n• 初中辅导\n• 中考辅导\n• 高中辅导\n• 高考辅导\n• 艺考生文化课\n• 一对一辅导\n• 国际学校\n• 高考复读\n• 托管班\n• 暑假辅导\n• 特色辅导\n• 小班辅导\n• 自主招生辅导\n• 寒假辅导\n• 军考辅导\n• 学前教育\n• 春季高考\n• 中考体育辅导\n• 一年级辅导\n• 二年级辅导\n• 三年级辅导\n• 五年级辅导\n• 四年级辅导\n• 六年级辅导\n• 初一辅导\n• 初二辅导\n• 初三辅导\n• 高一辅导\n• 高二辅导\n• 高三辅导\n• 全托学校\n• 在线一对一\n• 高考志愿填报\n• 中考复读\n• 一年级语文辅导\n• 一年级数学辅导\n• 一年级英语辅导\n• 二年级语文辅导\n• 二年级数学辅导\n• 二年级英语辅导\n• 三年级语文辅导\n• 三年级数学辅导\n• 三年级英语辅导\n• 五年级语文辅导\n• 五年级数学辅导\n• 五年级英语辅导\n• 四年级语文辅导\n• 四年级数学辅导\n• 四年级英语辅导\n• 六年级语文辅导\n• 六年级数学辅导\n• 六年级英语辅导\n• 初一语文辅导\n• 初一数学辅导\n• 初一英语辅导\n• 初一科学辅导\n• 初一生物辅导\n• 初一地理辅导\n• 初二语文辅导\n• 初二数学辅导\n• 初二英语辅导\n• 初二物理辅导\n• 初二科学辅导\n• 初二生物辅导\n• 初二地理辅导\n• 初三语文辅导\n• 初三数学辅导\n• 初三英语辅导\n• 初三物理辅导\n• 初三化学辅导\n• 高一数学辅导\n• 高一语文辅导\n• 高一英语辅导\n• 高一物理辅导\n• 高一化学辅导\n• 高一生物辅导\n• 高一地理辅导\n• 高一政治辅导\n• 高一历史辅导\n• 高二语文辅导\n• 高二数学辅导\n• 高二英语辅导\n• 高二物理辅导\n• 高二化学辅导\n• 高二生物辅导\n• 高二政治辅导\n• 高二历史辅导\n• 高二地理辅导\n• 高三数学辅导\n• 高三语文辅导\n• 高三英语辅导\n• 高三物理辅导\n• 高三化学辅导\n• 高三生物辅导\n• 高三政治辅导\n• 高三历史辅导\n• 高三地理辅导\n• 北京\n• 天津\n• 河北\n• 陕西\n• 辽宁\n• 吉林\n• 黑龙江\n• 山东\n• 上海\n• 江苏\n• 浙江\n• 安徽\n• 福建\n• 江西\n• 河南\n• 湖北\n• 湖南\n• 广东\n• 重庆\n• 四川\n\n### 最新机构", null, "40门课程人气数：0", null, "25门课程人气数：0", null, "85门课程人气数：0", null, "45门课程人气数：0", null, "6门课程人气数：0", null, "36门课程人气数：0\n\n### 飙升榜\n\n• 机构\n• 课程\n• 动态\n• 专题", null, "1", null, "2", null, "3", null, "4", null, "5", null, "6\n\n• 热门城市\n• 小贴士" ]	[ null, "https://px.01ny.cn/statics/wap/images/nyw_h5_lcbdh.png", null, "https://px.01ny.cn/statics/wap/images/nyw_h5_cb_logo.png", null, "https://px.01ny.cn/statics/wap/images/nyw_h5_cb_sy.png", null, "https://px.01ny.cn/statics/wap/images/nyw_h5_cb_pxjg.png", null, "https://px.01ny.cn/statics/wap/images/nyw_h5_logo.png", null, "https://px.01ny.cn/statics/wap/images/nyw_h5_rsearch.png", null, "https://px.01ny.cn/statics/wap/images/nyw_h5_csjh_banner.png", null, "https://px.01ny.cn/upimages/8B/4F/44/69/6AAD203A.jpeg", null, "https://px.01ny.cn/upimages/2E/6E/D9/9E/E6DC0D6E.jpeg", null, "https://px.01ny.cn/upimages/CF/F3/29/69/1DBF2181.jpeg", null, "https://px.01ny.cn/upimages/4E/2B/B3/E9/6944C227.jpeg", null, "https://px.01ny.cn/upimages/0C/8E/61/45/A4908611.jpeg", null, "https://px.01ny.cn/upimages/1E/71/84/42/97D2A02A.jpeg", null, "https://px.01ny.cn/statics/wap/images/nyw_h5_phb_1.png", null, "https://px.01ny.cn/statics/wap/images/nyw_h5_phb_2.png", null, "https://px.01ny.cn/statics/wap/images/nyw_h5_phb_3.png", null, "https://px.01ny.cn/statics/wap/images/nyw_h5_phb_more.png", null, "https://px.01ny.cn/statics/wap/images/nyw_h5_phb_more.png", null, "https://px.01ny.cn/statics/wap/images/nyw_h5_phb_more.png", null ]	{"ft_lang_label":"__label__zh","ft_lang_prob":0.9496258,"math_prob":0.40463108,"size":190,"snap":"2020-24-2020-29","text_gpt3_token_len":196,"char_repetition_ratio":0.20967741,"word_repetition_ratio":0.0,"special_character_ratio":0.25789472,"punctuation_ratio":0.13793103,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96233654,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,10,null,8,null,null,null,null,null,null,null,10,null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-05-31T15:54:53Z\",\"WARC-Record-ID\":\"<urn:uuid:40afe6fa-c985-4974-87be-3152a03002ca>\",\"Content-Length\":\"49515\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:244cea1b-a942-43a7-b0d5-bb74e62f6380>\",\"WARC-Concurrent-To\":\"<urn:uuid:837fad7d-1dd4-41dd-bbfb-d139ae898617>\",\"WARC-IP-Address\":\"121.207.229.189\",\"WARC-Target-URI\":\"https://px.01ny.cn/m/h_812d/\",\"WARC-Payload-Digest\":\"sha1:PEBMFENWMKPSUZ2J32HRJJ7TQPD2HAXH\",\"WARC-Block-Digest\":\"sha1:ONEMEBGUHRZMF556RWIGDQRNNFTKQB6S\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-24/CC-MAIN-2020-24_segments_1590347413551.52_warc_CC-MAIN-20200531151414-20200531181414-00507.warc.gz\"}"}
http://xelerus.de/index.php?s=functions&function=16	[ "Functions Documentation", null, "", null, "Name divide Syntax (divide x y) -> z Argument List x: The dividend of the two numbers. y: The divisor of the two numbers. Returns z: The result of the two numbers divided with everything after the decimal point dropped. Category math Description Returns the result of the two numbers divided and dropping everything after the decimal point. Example `(divide 5 3)` This will return a 1. `(divide 5 2)` This will return a 2. `(divide 5 (add 1 1))` This will also return a 2. Comment Basic math function. If you hear the term integer division this is what that is referring too." ]	[ null, "http://xelerus.de/images/icons/view.png", null, "http://xelerus.de/images/icons/edit_data.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6410635,"math_prob":0.97880447,"size":614,"snap":"2020-45-2020-50","text_gpt3_token_len":155,"char_repetition_ratio":0.1557377,"word_repetition_ratio":0.03809524,"special_character_ratio":0.26058632,"punctuation_ratio":0.102564104,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9986628,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-22T05:40:37Z\",\"WARC-Record-ID\":\"<urn:uuid:0c91194d-04f7-4513-9de7-ad2ae722dde9>\",\"Content-Length\":\"9568\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:26532607-1f22-44ea-83c6-fec3b4ef069d>\",\"WARC-Concurrent-To\":\"<urn:uuid:49056844-7695-4efc-950f-3b0946abd1c4>\",\"WARC-IP-Address\":\"109.237.132.20\",\"WARC-Target-URI\":\"http://xelerus.de/index.php?s=functions&function=16\",\"WARC-Payload-Digest\":\"sha1:7U2GPX6RLOCJZISN4RNHVLM6PNZ6JIGI\",\"WARC-Block-Digest\":\"sha1:WUQGOL3MGQNBHMKJIW7GN2VRMFGGYND5\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-45/CC-MAIN-2020-45_segments_1603107878921.41_warc_CC-MAIN-20201022053410-20201022083410-00009.warc.gz\"}"}
http://ahay.org/blog/2008/08/03/extending-python-interface/	[ "Work is under way on extending the Python interface to Madagascar. With new tools, you should be able to use an interactive Python session rather than a Unix shell to run Madagascar modules. Here are some examples:\n\nimport m8r as sf\nimport numpy, pylab\n\nf = sf.spike(n1=1000,k1=300)\n\n# sf.spike is an operator\n# f is an RSF file object\n\nf.attr()\n\n# Inspect the file with sfattr\n\nb = sf.bandpass(fhi=2,phase=1)[f]\n\n# Now f is filtered through sfbandpass\n\nc = sf.spike(n1=1000,k1=300).bandpass(fhi=2,phase=1)\n\n# c is equivalent to b but created with a pipe\n\n# g is a Vplot file object\n\ng.show()\n\n# Display it on the screen\n\nd = b - c\n\n# Elementary arithmetic operations on files are defined\n\ng = g + d.wiggle(wanttitle=False)\n\n# So are operations on plots\n\ng.show()\n\n# This shows a movie\n\npylab.show(pylab.plot(b))\nc = numpy.clip(b,0,0.01)\n\n# RSF file objects can be passed to pylab or numpy\n\nc.attr()\ns = c[300:310]\n\nprint s\n\n# Taking a slice outputs a numpy array\n\nc = sf.clip(clip=0.01)[b]\nc.attr()\n\n# Alternatively, apply sfclip\n\n\nFor doing reproducible research, one can also use the new syntax inside SConstruct files, as follows:\n\nfrom rsf.proj import *\nimport m8r as sf\n\nFlow('filter',None,sf.spike(n1=1000,k1=300).bandpass(fhi=2,phase=1))", null, "" ]	[ null, "http://ahay.org/blog/wp-content/uploads/2015/08/sage1.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7056257,"math_prob":0.9442298,"size":1533,"snap":"2020-24-2020-29","text_gpt3_token_len":453,"char_repetition_ratio":0.086330935,"word_repetition_ratio":0.0,"special_character_ratio":0.2941944,"punctuation_ratio":0.14285715,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99713224,"pos_list":[0,1,2],"im_url_duplicate_count":[null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-05-29T16:47:17Z\",\"WARC-Record-ID\":\"<urn:uuid:7af87ded-3b21-493d-a694-206b13430ea6>\",\"Content-Length\":\"47059\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:473f1f95-474d-44bb-a942-350424f1a980>\",\"WARC-Concurrent-To\":\"<urn:uuid:eb335210-8f7d-4ac7-83cc-8c27bc911734>\",\"WARC-IP-Address\":\"66.33.210.126\",\"WARC-Target-URI\":\"http://ahay.org/blog/2008/08/03/extending-python-interface/\",\"WARC-Payload-Digest\":\"sha1:VPAOW2UVOSR6A3MUHJICMVVMHIW6X7C3\",\"WARC-Block-Digest\":\"sha1:MWKGHLSY5BINGYVJVFESNLHWXPJJCST2\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-24/CC-MAIN-2020-24_segments_1590347405558.19_warc_CC-MAIN-20200529152159-20200529182159-00064.warc.gz\"}"}
http://en.communpedia.org/Transformation_problem	[ "#### Transformation problem\n\nHome » Marxist theory » Transformation problem\nThe transformation problem is a problem in classical and Marxian economics. It appears in the work of Ricardo where he notes a difficulty in equating values, as given by his labour theory of value (LTV), to prices – the problem there being that given reasonable assumptions about competition, differences in the proportion of labour to non-labour costs between industries will skew the value-price relation between industries. This kink in the LTV lay unresolved by Ricardo and his immediate successors\n\nIn somewhat more detail, the problem is that, according to the LTV, the value of an industry’s output is proportional to the labour that the industry uses; but equalisation of the profit rate by competition demands that the price of its output be proportional to the labour and non-labour capital that it uses (since the rate of profit, π = [profit]/[all costs].) So an industry with a high proportion of non-labour costs (a `capital-intensive’ industry) for example, must either sell its output above value or have a subnormal profit rate. Non-labour capital costs apparently drive a wedge between value and price. Hence the LTV and profit-rate equalisation are apparently in conflict.\n\nAttention to the transformation problem greatly intensified after the publication of volume III of Marx’s Capital in 1894 because chapter nine of that book contained what was taken by some to be a solution to the Ricardian problem. Marx held that in economy-wide aggregate, price equals value, but that in particular sectors or industries output might sell at above or below value. Specifically, in `capital intensive’ industries, it would sell at above value and in labour-intensive industries it would sell at below value. Importantly, he saw this as amounting to a transfer of surplus value from the labour-intensive to the `capital intensive’ sector; the labour-intensive sector does nor `realise’ all of the surplus value that it produces, while the `capital-intensive’ sector realises not only all of the surplus value it produces, but the unrealised portion from the labour intensive sector as well.\n\nAnti-Marxists, noteably Böhm-Bawerk, quickly pounced on perceived flaws in Marx’s analysis, characterising it as a failed defence of the unworkable LTV, and contending furthermore that the entire structure of Marxian economic theory could not stand because it incorporated the LTV as a premise. The Marxian defence to this lay in two directions. One accepted the general conceptual framework of the Böhm-Bawerk-type criticisms, but provided solutions to the transformation problem that were mathematical improvements over Marx’s. This has resulted in solutions that are, arguably, adequate, although so far none meet all of the restrictive conditions that might ideally be applied. The other line of defence was to challenge Böhm-Bawerk’s conceptual framework, claiming that he had not understood what Marx was trying to do in Chapter IX of Capital III. This line holds that Marx’s purpose there was not to provide a theory of price determination at all, but to investigate the post-production distribution of surplus value throughout the economy – an investigation made important by the fact that this distribution tends to obscure, or mystify, the original creation of surplus value by labour during the production process. This investigation is seen to occupy much of Capital III; in Chapter IX, proponents of this line contend, price shifts are considered by Marx only as one means by which surplus-value transfers might be accomplished. Viewed in this light, the value-price transformation is of merely auxilliary significance to the over-all economic theory, and any sketchiness in Marx’s mathematics of the value-price transformation cannot be considered to strike at fundamentals. This line of defence is associated with a Hegelian-dialectical reading of Marx which sees Capital volume I as presenting concepts such as value and surplus value at a very abstract level that employs particular examples only as representatives of society-wide aggregate behaviour, whereas Capital volume III is at a more concrete level of analysis where some particulars are introduced – the important ones for this discussion being different industrial sectors and differing proportions between labour and non-labour capital – and the effects of these particulars in modifying, or imposing further ‘determinations’ on, the abstract principles is considered. Regarding the LTV, it is argued that although in Capital volume I Marx sometimes, à la Ricardo, equated prices with labour times, this was only done as a simplifying assumption at the most abstract level, and, as the later volumes make clear, he did not think that real prices are determined only by labour times, not even on long-term average: in short, Marx did not hold a Ricardian-type LTV.\n\nA more recent criticism of Marx, noteably by Joan Robinson and Sraffians such as Ian Steedman (1977), is that although it is possible to adequately compute prices using the Marxian transformation-problem method, that method is needlessly roundabout because it is possible to go directly from specifications of physical quantities to prices by using, eg., the mathematical techniques of Sraffa, without going through the intermediate step of considering values at all. On this view, value is an ‘unneccessary detour’, a redundant concept that would be best forgotten about. This of course would leave a large body of Marxian discourse looking rather pointless. Against this it is argued that, although the concept may not be needed for the determination of price, it remains useful for understanding other dynamics of capitalism, for linking quantitative economics to considerations of how humans expend their life energies, and for linking economics to ethics.\n\nA still different hue is cast on the debate by Goregnani (1991), who believes there is an epidemic among readers of Marx, including most of the partisans on both sides of the debate just mentioned, which consists in their thinking that the concept of value was given a prominent place by Marx because of its sociological and philosophical implcations. Goregnani says that on the contrary, Marx employed the concept of value because of its importance to a technical issue in classical economics. At that time the technique of simultaneous linear equations, to say nothing of Sraffa’s methods, had not yet arrived in economics, and thus the result established by them that the wage and the rate of profit are not independent but that, other things being equal, a rise in one will result in a fall in the other, was not well recognised. Adam Smith’s method of adding up factor costs to obtain price, for example, treated them as being independent. Ricardo, and then more firmly Marx, introduced value as a means of demonstrating their interdependent, antagonistic, relation.\n\n 1 Ricardian problem 2 Bortkiewicz-Sweezy solution 3 Makoto Itoh solution 4 Notes 5 Further reading 5.1 On paper 5.2 Free on the internet\n\n## Ricardian problem\n\nCompetitive assumptions:\n\n1. Assume that competition for investment causes the rate of profit to be the same in all industries.\n2. Assume that labour mobility causes the hourly wage to be the same in all industries.\n\nPrice of production of commodity:\n\n```p = c + W + s Where: p = price of production of commodity\nc = non-labour costs of capitalist\nw = wage costs of capitalist\ns = gross profit\n```\n\nRate of profit:\n\n``` s\nπ = ------- = same in all industries\nc + w\n```\n\nCombinig above formulae:\n\n```p = (c + w)(1 + π)\np = (c + w)(k1) where: k1 is a constant = 1 + π\n\nResult A:\n\np = k1·c + k1·w\n---------------\n```\n\nBut, by labour theory of value,\nvalue-added is proportional to time worked,\nand by assumption of equal hourly wage,\ntime worked is proportional to wage, so:\n\n```n α w where: n = value added\nn = k2·w w = wage\nk2 = a constant\n```\n\nValue of commodity:\n\n```v = c + n\n\nResult B:\n\nv = c + k2·w\n------------\n```\n\nComparing results A and B we see that price (p) and value (v) are given by different formulae that are incommensurable except in very particular cases (such as k1 = k2 = 1 or c = 0). In the realistic case of positive profit, k1 and k2 will both be greater than one, which will cause prices to be relatively high, compared to values, in industries with relatively high non-labour costs, and prices will be relatively low in labour-intensive industries.\n\nThis result is at variance with the simple assumption, used elsewhere by Ricardo, that values are related to prices by a simple proportionality constant that is the same in all industries. The discrepancy was noted by Ricardo but he did not offer a solution.\n\n## Bortkiewicz-Sweezy solution\n\nStarted by L. von Bortkiewicz (1868-1931) and revived by Paul Sweezy in his The Theory of Capitalist Development (1942). Their contribution is to convert the cost prices (ci + vi) in each department i from values, which is the form Marx left them in, to prices. They do this by introducing three unknowns, x, y, z which designate the ratio of price of production to value of output, in each of the three departments I, II, III respectively. This leads to a system of three equations:\n\nI c1x + v1y + r(c1x + v1y) = (c1 + c2 + c3)x\n\nII c2x + v2y + r(c2x + v2y) = (v1 + v2 + v3)y\n\nIII c3x + v3y + r(c3x + v3y) = (s1 + s2 + s3)z\n\nNote: c is constant capital; v, variable capital; s, surplus value; r, rate of profit.\n\nThere are three equations but four unknowns (x, y, z, r). To obtain a solution the theorists set z = 1. For convenience the following notations are introduced:\n\nσ = 1 + r\n\nf = c/v\n\ng = (c + v + s)\n\nExpressed using these notations the solution is:\n\nσ = ( -(f2g1 + g2) + [(f2g1 + g2)2 + 4(f1 – f2)g1g2]1/2 ) / 2(f1 – f2)\n\nSweezy gave the following numerical example:\n\nValue calculation\nDepartment of production Constant capital (ci) Variable capital (Vi) Surplus-value (si) Value of products (ai)\nI. Producers goods 225 90 60 375\nII. Necessary consumer goods 100 120 80 300\nIII. Luxury consumer goods 50 90 60 200\nTotal 375 300 200 875\n\nGiven the above numbers, it can be calculated that, σ = 5/4, r = 25 percent, x = 32/25, y = 16/15. From this is calculated the prices (as distinguished from the values) of each item in each department:\n\nPrice calculation\nDepartment of production Constant capital (cix) Variable capital (viy) Profit (pi) Price of product (Pi)\nI. Producers goods 288 96 96 480\nII. Necessary consumer goods 128 128 64 320\nIII. Luxury consumer goods 64 96 40 200\nTotal 480 320 200 1000\n\nSource: information is as reprised in Makoto Itoh, The Basic Theory of Capital (1988), pp 211-213.\n\n## Makoto Itoh solution\n\nMakoto Itoh has adressed some problems still remaining after the Bortkiewicz-Sweezy solution (and the debates following it) by making careful and explicit distinction between the form of value, which is exchangeability, or `the request of commodities to be exchanged’, and the substance of value, which is embodied labour time. This results in careful separation of the value and price domains.\n\nItoh gives a numerical example consisting of three tables of numbers (rather than Bortkiewicz-Sweezy’s two). His first table is the same as in the Bortkiewicz-Sweezy (B-S) example, using the same numbers, except that he is very explicit – leaves no ambiguity – that the units in the table are labour time (millions of hours), not, say, a monetary or quasi-monetary unit; and he labels the table `substance of value produced (ai)’. His second table is like the B-S second table, except Itoh chooses a value of z = 1/2 rather than 1, so that all numbers in this table are exactly half those in the B-S second table. He is explicit that the units in this table are money (millions of dollars); and he labels the table `prices of production (Pi)’.\n\nThe big difference between Itoh, on one hand, and Bortkiewicz and many other commentators on the other, is this: They see the failure of a B-S type of solution to satisfy Marx’s agregation conditions, namely, that the sum of the values equals the sum of the prices, and the sum of the surplus-values equals the sum of the profits, to indicate more or less serious flaws in Marx’s theory. Itoh, however, regards the non-satisfaction of those conditions as entirely unremarkable, unproblematic, and in fact to be expected, since value and surplus-value are in one domain, and price and profit are in another domain which is only determined in a rather elastic way by the first. What Marx was getting at with his agregation conditions, but failed to spell out clearly enough because he himself was somewhat lax domain-wise, not always maintaining clear separation, is, according to Itoh revealed in a third table which can be constructed, that shows the substance of value acquired in each of the three departments.\n\nSubstance of value acquired (a’i)\nDepartment of production ci vi s’i a’i\nI 225 90 96 411\nII 100 120 64 284\nIII 50 90 40 180\nTotals 375 300 200 875\n\nNote that the table is in the value domain; the units are millions of hours. The ci and vi numbers are the same in this table as in table 1, which they must be since they are technologically determined. The surplus values (s’i), however, are different, since they are the surplus values acquired, or realized, by the capitalists in each department, not, as in table 1, the surplus-values produced. The values-of-products acquired (realized), being the sum a’i = ci + vi + s’i, are also different than in table 1. But now look at the bottom row, the totals. They are the same as in table 1 all the way across: c, v, s’ and a’. Here, says Itoh, is the agregate invariance that Marx sensed. It is not that the sum of the values is the same as the sum of the prices, but that the sum of the values produced is the same as the sum of the values acquired; and it isn’t that total surplus-value is the same as total profit, but that the total surplus-value in production is the same as the total surplus-value acquired." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9085779,"math_prob":0.9605085,"size":17314,"snap":"2019-13-2019-22","text_gpt3_token_len":4168,"char_repetition_ratio":0.12368573,"word_repetition_ratio":0.024244554,"special_character_ratio":0.24540834,"punctuation_ratio":0.12648582,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9687207,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-03-22T18:56:31Z\",\"WARC-Record-ID\":\"<urn:uuid:ea08c71c-a82e-4ded-a38d-653268ccfac3>\",\"Content-Length\":\"48784\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7ad1d6e5-d612-40d2-9855-239925473120>\",\"WARC-Concurrent-To\":\"<urn:uuid:b67df58d-fb92-422a-ba93-761d7e88c16b>\",\"WARC-IP-Address\":\"172.96.191.122\",\"WARC-Target-URI\":\"http://en.communpedia.org/Transformation_problem\",\"WARC-Payload-Digest\":\"sha1:AAVNBQRQQKTENX4WV5ID64UFWYA4EHUV\",\"WARC-Block-Digest\":\"sha1:ZQIXAZTEJPMGKVT5JYLDVLMTO64DPMLO\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-13/CC-MAIN-2019-13_segments_1552912202688.89_warc_CC-MAIN-20190322180106-20190322202106-00468.warc.gz\"}"}
https://www.enhan.eu/how-to-in-fp-3-collectfirst/	[ "# 3/n - How do I in FP... trying several input for the same function\n\nIn the previous entry of this series, I wrote about error recovery in a functional way. But sometimes, we want to try several inputs until we retrieve (optionally) a result. This is the case if you have a function which compute a location from a string (think geo-search), and which optionally returns a result, if a place matched. We may try this function on several input strings and only care in the first result. In this article, we will see how to do this in a functional way.\n\n## Problem setup\n\nGiven a function `def computeLocation(str: String): Option[Location]` which tries to infer a location from a string and a list of input strings sorted by order of relevance (meaning that the first element may result in a more interesting location than the second and so on), let's retrieve the location (optionally). To sum it up:\n\n``````// The computeLocation function:\ndef computeLocation(s: String): Option[Location] = // the implementation\n\n// this call may generate a long List\nval inputs: List[String] = createInputsFromForm(aForm)\n``````\n\nHow implement `val result: Option[Location] = ???` ?\n\n## From scratch\n\nThe first solution is to call the `computeLocation` function for every string in the input list:\n\n``````inputs.flatMap(computeLocation).headOption\n``````\n\nThis solution is the most simple ever. Let's break it into pieces to analyze it.\nLet's start from the end. As we care only about the first meaningful result (which may not exist), we call `headOption`.\nNow, back to the beginning of the line, we are calling `flatMap` on a `List`. The proper signature of `flatMap` in Scala standard library is slightly different from the one we are used to:\n\n``````def flatMap[B](f: A => IterableOnce[B]): List[B] = //...\n``````\n\nIn the standard library, `flatMap` expects a function producing an `IterableOnce`. Fortunately `Option` implements this trait. The definition commonly used for `flatMap` is more like this one:\n\n``````// Given that F is a type with one parameter\ndef flatMap[A, B](fa: F[A])(f: A => F[B]): F[B]\n\n// For List this results in:\ndef flatMap[A, B](fa: List[A])(f: A => List[B]): List[B]\n``````\n\nIn the standard library, the trick with `IterableOnce` spares us from writing something like:\n\n``````inputs\n.flatMap{ s =>\ncomputeLocation(s).map(l => List(l)).getOrElse(Nil)\n}\n``````\n\nIn this code, we are manually transforming an `Option` into a `List`.\n\nWhat's wrong with this implementation ? If you provide a dummy implementation for both `computeLocation` and `createInputsFromForm`, you will see that `computeLocation` is called for every single item in the list. If we consider that computing the location is a costly operation, it is sad to waste resource computing, while we only care about the first meaningful answer. How to make the `flatMap` sequence stop when we hit a result or the end of the list ?\n\n## Recursion by hand\n\nIf we break our problem in terms of imperative programming, we will easily come up with a `while` statement (in pseudo code):\n\n``````while no result and list has elements left\nresult = computeLocation with next element in the list\ndone\n``````\n\nLet's encode this with recursion:\n\n``````@scala.annotation.tailrec\ndef recur(next: List[String]): Option[Location] = {\nnext match {\ncase Nil => None\nif (r.isDefined) r else recur(tail)\n}\n}\nrecur(inputs)\n``````\n\nWhat we do is just manually traverse the list recursively until `computeLocation` gives a meaningful result. This is strictly the same as the imperative `while` version, but in FP, we like to look further. We could for instance abstract over the `computeLocation` and take as an extra parameter the function returning an `Option`:\n\n``````@scala.annotation.tailrec\ndef recur(next: List[String],\nf: String => Option[Location]): Option[Location] = {\nnext match {\ncase Nil => None\nif (r.isDefined) r else recur(tail, f)\n}\n}\nrecur(inputs, computeLocation)\n``````\n\nLooking at this code, do we really have to deal with `String` and `Location` ? We do not use any of the properties of these types in `recur`. Let's make them parameters.\n\n``````@scala.annotation.tailrec\ndef recur[A, B](next: List[A], f: A => Option[B]): Option[B] = {\nnext match {\ncase Nil => None\nif (r.isDefined) r else recur(tail, f)\n}\n}\nrecur(inputs, computeLocation)\n``````\n\nFollowing this reasoning we could go even further and add some mechanism to also deal with something else than `Option` and `List`. In fact, cats has such functions to hide recursion from you. It is called `tailRecM` but as it deserves an entire post for itself, I won't talk about it here.\n\n## Back with the Scala standard library...\n\nIn the standard library, we can also find the `collectFirst` function which, well do pretty much what we need !\n\n``````inputs.collectFirst { str =>\ncomputeLocation(str) match {\ncase Some(r) => r\n}\n}\n``````\n\nWell, the careful reader would have noticed that the function we pass to `collectFirst` is not total: it does not produce an output for every input.\n\nAs `computeLocation` is total, then it is a bit sad to write a non-total version of it... The standard library also provides a function to transform a total function returning `Option` to a partial one: `def unlift[T, R](f: T => Option[R]): PartialFunction[T, R]`. So we can now write:\n\n``````inputs.collectFirst(Function.unlift(computeLocation))\n``````\n\nIf you are using cats, you will find `collectFirstSome` which will make this possible:\n\n``````inputs.collectFistSome(computeLocation)\n``````\n\n## What if `computeLocation` perform other effects ?\n\nLet's change the problem a bit now. What if the `computeLocation` function performs some network calls, or read a database to find a result ? These things are known to fail at one point or another. How to deal with that ?\n\nWithout adding too much complexity, let's consider that `computeLocation` has a more expressive output: `Either[Throwable, Option[Location]]`. This type means that `computeLocation` may fail with a `Throwable` or give a result if it was able to compute one (the absence of result being completely normal). The complete signature is now:\n\n``````def computeLocation(rawStr: String): Either[Throwable, Option[Location]]\n``````\n\nAs you can see, the `Option` result is now boxed into `Either`. This makes both `collectFirst` and `collectFirstSome` impossible to use, as the type do not match now. In FP, this kind of boxing appears frequently, so that authors of functional libraries like cats already wrote functions to deal with this. In cats, you will often see functions whose name ends with a capital `M`. In our case, it is easy to find `collectFirstSomeM` which has this signature:\n\n``````def collectFirstSomeM[G[_], B](f: A => G[Option[B]])\n(implicit F: Foldable[F], G: Monad[G]): G[Option[B]]\n``````\n\nAs you can see, this definition is abstract. Note that this function is part of a class parametrized on `F`, where `F` is itself a type with a \"hole\" : `F[_]`. In our example, this `F` will be `Either[Throwable,_]` (ie `Either` with only one type parameter applied). To make it easier, here is a version (pseudo scala) where abstract types are replaced with the one we use in our example (without the implicit part):\n\n``````def collectFirstSomeM[Either[Throwable,_], Location](f: String => IO[Option[Location]]): Either[Throwable,Option[Location]]\n``````\n\nLook how `computeLocation` has exactly the expected signature for the `f` parameter ! What about the implicits parameters now ? In this case, we can read the definition as: \"the `collectFirstSomeM` method is available if and only if for the `F` type we can implement the trait `Foldable` and for the `G` type the `Monad` trait\". In our case, with `List` and `Either` it is already done for us in cats so we can do:\n\n``````import cats.implicits._\nval inputs: List[String] = computeInputs(param)\nval result: Either[Throwable, Option[Location]] = inputs.collectFirstSomeM(computeLocation)\n``````\n\nYet, if you write a simple sample for this, like :\n\n``````\ndef computeLocation(rawStr: String): Either[Throwable, Option[Location]] = {\nprintln(s\"Computing with \\$rawStr\") // to see something\n// Fake implementation\nif (rawStr.length == 1) Left(new IllegalStateException(\"wrong state\"))\nelse if (rawStr.length % 4 == 0) Right(Some(Location(2.0, 4.0, \"Plop\")))\nelse Right(None)\n}\n\nval inputs = List(\"a\", \"impair?\", \"plop\", \"other\")\n\nimport cats.implicits._\nprintln(\ninputs.collectFirstSomeM(computeLocation)\n)\n``````\n\nYou will notice that the output will be:\n\n``````Computing with a\nLeft(java.lang.IllegalStateException: wrong state)\n``````\n\nThe execution is stopping as soon as a `Left` is returned ! This is perfectly normal as `Either` is a fail fast structure. And the requirement for `Monad` should have caught our attention: `Monad` express sequence, the next operation using the result of the previous one as input parameter.\nSo, as the caller of `computeLocation` in our situation, we have two options:\n\n• consider a `Left` as a blocker, and indeed, keep it,\n• try to recover from it by composing `computeLocation` with a function handling the situation with care.\n\nHere is an example of the second choice, being ignoring errors (you probably don't want to do that, and perform a selective recovery, composing with things we saw earlier):\n\n``````inputs.collectFirstSomeM(\ns => computeLocation(s).recoverWith(_ => Either.right(None))\n)\n``````\n\n## Conclusion\n\nIn this article, we saw how to deal with something simple: calling a function for elements in a list and keeping the first meaningful result (if any). Even if it was simple, what I would like to highlight is how this solution is composable. Every type in the design models an intent:\n\n• `Option` models that the result may be irrelevant or present\n• `Either` models that a computation may fail\n• `Foldable` express that we want to build a single value from an initial structure (in our case, we fold the list to a single result)\n• `Monad` express that we compute in sequence: indeed, there is no parallelism involved in our solution, we try the input strings one after the other.\n\nYet, the work here is not over. Next time, we will see how we can adapt this example to make it less \"sequential\"." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8393933,"math_prob":0.8939502,"size":9855,"snap":"2023-40-2023-50","text_gpt3_token_len":2278,"char_repetition_ratio":0.14780225,"word_repetition_ratio":0.04256671,"special_character_ratio":0.22688991,"punctuation_ratio":0.13833158,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9767943,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-11-29T08:32:06Z\",\"WARC-Record-ID\":\"<urn:uuid:2da44d48-29d4-4c94-a844-8914bd142b47>\",\"Content-Length\":\"33955\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9418c709-a682-4ff6-b361-c0d519d73ef1>\",\"WARC-Concurrent-To\":\"<urn:uuid:6a8766d0-7807-45b5-bfb1-362e33af7edc>\",\"WARC-IP-Address\":\"146.75.35.7\",\"WARC-Target-URI\":\"https://www.enhan.eu/how-to-in-fp-3-collectfirst/\",\"WARC-Payload-Digest\":\"sha1:PPYSKOYDRWUAQCNAVSFMT7EDENPVZNYM\",\"WARC-Block-Digest\":\"sha1:4ZRBZLPDIRNVQEM6ZZUNW5BYGDJGPLDQ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100057.69_warc_CC-MAIN-20231129073519-20231129103519-00533.warc.gz\"}"}
https://cs.stackexchange.com/questions/28264/generating-a-set-of-minimal-length-strings-that-together-invoke-every-producti	[ "# Generating a set of minimal-length strings that, together, invoke every production of a context free language\n\nProblem (tl;dr)\n\nGiven a context free grammar, $G$, find a set of strings that take $G$ through every production it has at least once.\n\nHow and how fast can it be done?\n\nBackground\n\nI'm working on a compiler whose parser is implemented with a tool similar to Yacc+Antlr. I've written up most of the parser code, and I'd like to generate some code of the object language that invokes every production of the grammar at least once so that I can feed it to the parser and make sure that nothing is wrong.\n\nIn the interest of good testing, what I'd really like is one, short test file that has a particular production \"under test\" -- so, for each production rule, I want to generate a minimal string that takes the parser from the start state, through the production being tested, to a set of terminals.\n\nPossible solutions\n\nI imagine there is an elegant solution using graph theory, but I'm not quite sure what it is. I would like to just use Dijkstra's algorithm to find shortest paths through some appropriate structure, but I think that a string is parsed by a context free grammar in a tree structure rather than a path, so I don't know how to make that work.\n\nI think there might be a clever way to pose it as a network flow problem. Something like this: take a graph that has a vertex for every symbol (terminal and nonterminal) and a vertex for every production. If a nonterminal has a production, add a directed edge from the nonterminal to the production. If a production produces a symbol, add a directed edge from the production to the symbol. Add a source with some capacity $c$ and attach it to the vertex corresponding to the start symbol. Add a sink with infinite capacity and attach it to each terminal.\n\nIf a nonterminal has an in-arc with a capacity $k$, add an arc from the nonterminal to each of its productions with capacity $k$. If a production has an in-arc with capacity $k$ and it has an out-arc to $n$ nonterminals, add an arc with capacity $\\frac k n$ from the production to each nonterminal.\n\nThen run some maximum flow algorithm on the network and let the productions \"trickle down\" from the start symbol to the terminals. You should end up with a flow $c$ coming out of your source, and you can return all of the terminals you hit with a nonzero flow as your result string. Then you end up with something like $O(n^3)$ time complexity for each run, where $n$ is the sum of the number of terminals and nonterminals in your grammar -- not too bad.\n\nHowever, I'm still not really sure what this graph looks like: I think that it needs to be infinite and I'm not sure if you can find the maximum flow of an infinite flow network. Past that, I'm not sure how to \"remove\" a production from consideration so that I'm guaranteed to get a new one for each test run.\n\nI Googled and I couldn't find anything. Is there a nice solution to this problem?\n\n• Can the input grammar be ambiguous? If so, does a string count for all productions in all derivations? Can we assume that the grammar is reduced, i.e. contains not \"dead end\" rules (every rules is reachable from the start symbol and has a right side that can derive to a word)? – Raphael Jul 8 '14 at 0:59\n• @Raphael The particular grammar that I'm working with is ambiguous in a few minor places but that can be ignored for simplicity if necessary. I'd rather an ambiguous string not count for all of its possible derivations, because my aim is to force the parser to hit every code branch, but as I say that I realize it might not be possible. The grammar is reduced. – Patrick Collins Jul 8 '14 at 1:30\n• Try it backward. Build a set of rule where the right-hand-side (RHS) is a terminal strings (that includes the empty word, which is an empty string of terminals). Then look at all remaining rules successively, to see if one has a RHS that can be derived into a terminal string with the rules in your set ... and try to finish this yourself. Two remarks though: I am very surprised you programmed a parser \"similar to Yacc+Antlr\" with such a limited knowledge of context-free grammars. Second point is that I doubt your test is enough: rules may interact in the parsing process. – babou Jul 8 '14 at 7:53\n• Definitely, code-coverage is known to be a very rough testing strategy. What's more, you have a programming language with ambiguous syntax? Oh dear. – Raphael Jul 8 '14 at 8:51\n• @Raphael Ambiguous grammar for a programming language does not bother me. I have been advocating them for many years. The USA Defence has been using one for some 30 years. But, the technology depends on how ambiguous, and on how ambiguity is to be dealt with. If the technology is to be Yacc+Antlr like, then the parser behaves as if the language was unambiguous: if forces a choice. Other technologies may keep the ambiguity. But I am still bothered by the fact it all requires a level of understanding that does not seem to be present in the question. – babou Jul 8 '14 at 9:16\n\n# In a nutshell\n\nNot knowing enough the literature, I worked out a solution which is presented in the next section, together with a proof for the hardest part. Once I knew what was needed, I could search the literature for the right ideas. Here is a quick presentation of the algorithm, based on the literature, which is essentially the same I developed.\n\nThe first thing to do is to find a size-minimal terminal string $\\sigma(U)$ for every non-terminal $U$ of the grammar. This can be done by using Knuth's extension to conc-or graphs (also known as CF grammars) and and-or graphs of Dijkstra's shortest path algorithm. Example B in Knuth's paper does what is needed, almost.\n\nActually, Knuth computes only the length of these terminal strings, but it is quite easy to modify his algorithm to actually compute one such terminal string $\\sigma(U)$ for each non-terminal $U$ (as I do it in my own version below). We also define $\\sigma(a)=a$ for every terminal $a$, and we extend $\\sigma$ as usual into a string homomorphism.\n\nThen we consider a directed graph where non-terminals are the nodes, and there is an arc $(U,V)$ iff there is a rule $U\\rightarrow \\alpha V\\beta$. If several such rules can produce the same arc $(U,V)$, we keep one such that the length $\|\\sigma(\\alpha\\beta)\|$ is minimal. The arc is labeled with that rule, and that minimal length $\|\\sigma(\\alpha\\beta)\|$ becomes the weight of the arc.\n\nFinally, using Dijkstra's shortest path algorithm, we compute the shortest path from the initial non-terminal $S$ to each non-terminal of the grammar. Given the shortest path for a non-terminal $U$, the rule labels on the arcs may be used to get a derivation $S\\overset{}{\\Longrightarrow}\\alpha U \\beta$. Then, to every rule of the form $U\\rightarrow\\gamma$ in the grammar, we associate the size-minimal terminal string $\\sigma(\\alpha\\gamma\\beta)$ which can be derived using that rule.\n\nTo achieve low complexity, both Dijkstra's algorithm and Knuth's extension are implemented with heaps, AKA priority queues. This gives for Dijkstra's algorithm a complexity of $O(n\\log n +t)$, and for Knuth's algorithm a complexity $O(m \\log n +t)$, where there are $m$ grammar rules and $n$ non-terminals, and $t$ is the total length of all rules. The whole is dominated by the complexity of Knuth's algorithm since $m\\geq n$.\n\nWhat follows is my own work, before I produced the short answer above.\n\n# Deriving the solution from the useless symbols elimination algorithm.\n\nThere are several aspects to this algorithm. For better intuition I chose to present it in three successive versions that introduce progressively more features. The first version does not answer the question, but is a standard algorithm for useless symbols elimination that suggests a solution. The second version answers the question without the minimality constraint, The third version gives an answer to the question, satisfying thye minimality constraint. This third solution is then improved by using an adaptation to and-or graphs of Dijkstra's shortest path algorithm.\n\nThe end result is a very simple algorithm, that avoids reconsidering computations already done. But it is less intuitive and does require a proof.\n\nThis answer only tries to answer the question as made precise by the OP's comment: \"for each production rule, I want to generate a minimal string that takes the parser from the start state, through the production being tested, to a set of terminals.\" Hence I only try to get a set of strings such that for each rule, there is a string in the set that is one of the size-minimal strings of the language having a derivation using the rule.\n\nIt must be however noted that the fact that a string \"invokes\" a rule, that is has a derivation using that rule, does not necessarily means that the rule will be considered by a parser that work with ambiguous grammars and resolves ambiguities arbitrarily. Handling such a situation would probably require more precise knowledge of the parser, and might well be a more complex question.\n\n## The basic algorithm\n\nTo solve this question, one can start with the classical algorithm for useless symbols removal in context-free grammars. It is in section 4.4, pp 88-89, of Hopcroft & Ullman, 1979 edition. But the presentation here may be a bit different.\n\nThe algorithm aims precisely at proving the existence of such a covering as requested by the OP, and consists in two parts:\n\n• lemma 4.1 of H&U, page 88: removal of all unproductive non-terminals. This is done by trying find for each terminal a terminal string it can derive on. A simple way to explain it is as follow: You create a set $Prod$ od productive symbols, which you initialize with all terminals. Then for each rule, not yet processed, that has all its right-hand-side (RHS) symbols in $Prod$, you add the left-hand-side (LHS) non-terminal to the set $Prod$, and you remove all rules with the same LHS non-terminal from the set of rules to be processed. You iterate the process until there is no rule left with all its RHS symbols in $Prod$. The remaining non-terminals, not in $Prod$ at the end of this process, are non-productive: they cannot be derived into a terminal string, and can thus be removed from the grammar.\n\n• lemma 4.2 of H&U, page 89: removal of all unreachable symbols. This is done by the classical node reachability in directed graphs, by considering non-terminals as nodes and having an arc $(U,V)$ iff there is a rule $U\\rightarrow \\alpha$ such that $V$ occurs in $\\alpha$. You create a set $Reach$ of reachable symbols which is initialized with only the initial symbol $S$. Then, for every non-terminal symbol $U$ in $Reach$ or later added to it, and for every rule $U\\rightarrow \\alpha$, you add to $Reach$ all the symbols in $\\alpha$. When all non-terminals in $Reach$ have been thus processed, all symbols (terminal or non-terminals) that are not included in $Reach$ cannot appear in a string derived from the initial symbol, and are therefore useless. Thus they can be removed from the grammar.\n\nThese two basic algorithms are useful to simplify the raw results of some grammar construction techniques, such as used for the intersection of a context-free language and a regular set. In particular, this is useful in cleaning up the results of general CF parsers.\n\nUseless non-terminal symbols removal is necessary in the context of solving the question asked, as the rules using them cannot be \"invoked\" (i.e. used in its derivation) by any string of the language.\n\n## Building a set of string that invoke every rule\n\n(We are not looking yet for minimal strings.)\n\nNow answering specifically the question, one must indeed remove all useless symbols, whether unreachable symbols or unproductive non-terminal symbols, a well as useless rules having such useless non-terminals as LHS. They have no chance of being ever invoked usefully while parsing a terminal string (though some may well waste the processing time of a parser when they are not removed; which ones may waste time depends on the parser technology).\n\nWe now consider, for each (useful) rule, the production of a terminal string that invokes it, i.e. that may be generated by using this rule. This is essentially what is done by these two algorithms above, though they do not keep the information, as they are satisfied with proving the existence of these strings to ensure that non-terminals are both reachable and productive.\n\nWe modify the first algorithm (lemma 4.1) by keeping with each non-terminal $U$ in the set $Prod$ a terminal string $\\sigma(U)$ it derives on: $U\\overset{}{\\Longrightarrow}\\sigma(U)$. For every terminal we define the $\\sigma$ as the identity mapping. When $U$ is added to the set $Prod$ because a rule $U\\rightarrow\\gamma$ has all its RHS symbols in $Prod$, then we define $\\sigma(U)=\\sigma(\\gamma)$, extending $\\sigma$ as a homomorphism on strings, and we remove all $U$-rules, that is all rules with $U$ as LHS.\n\nWe modify the second algorithm (lemma 4.2) by keeping with each non-terminal symbol $U$ added to $Reach$ the path used to reach it from the intial symbol $S$, which gives the successive rules to get a derivation $S\\overset{}{\\Longrightarrow}\\alpha U \\beta$.\n\nThen, for each rule $U\\rightarrow\\gamma$ in the grammar, we produce a terminal string that \"invokes\" this rule as follows. We take from the result of the second algorithm the derivation $S\\overset{}{\\Longrightarrow}\\alpha U \\beta$. Then we apply the rule to get the string $\\alpha \\gamma \\beta$. A terminal string \"invoking\" the rule $U\\rightarrow\\gamma$ is $\\sigma(\\alpha \\gamma \\beta)$\n\n## Building a set of minimal strings that \"invoke\" every rule\n\nWe ignore the issue of eliminating useless symbols, which can be a by-product of these modified algorithms.\n\nBuilding a set of minimal strings relies on first getting a minimal derived string for each non-terminal. This is done by further modifying the first algorithm (lemma 4.1). First we remove from the set of rules to be processed all recursive rules (i.e. with a LHS symbol occurring in the RHS string). It is obvious that none of these rules can derive to a shorter terminal string than the non-recursive rules with the same LHS. And there must be at least one non-recursive rule if the LHS is not a useless non-terminal (because non-productive).\n\nThen we procede as before to build the set $Prod$ of productive symbols, associating with each synbol $U$ a terminal string, which we note $\\sigma(U)$. The string $\\sigma(U)$ is produced as before by application of the rule $U\\rightarrow\\gamma$, substituting each non-terminal $V$ occurring in $\\gamma$ with $\\sigma(V)$. So far, it was necessary to apply this to only one rule with a given non-terminal $U$ as its LHS, the first that would have all its RHS non-terminals in $Prod$, and then ignore the others, because any such derived string would do. But we are now looking for a minimal derived string. Hence, for a non-terminal $U$, this has to be done for all rules with $U$ as LHS. But we keep only one terminal string $\\sigma(U)$, replacing the current one by the newly found one, whenever the new one is smaller.\n\nFurthermore, whenever the string $\\sigma(U)$ is replaced by a smaller one, all rules with an occurrence of $U$ in the RHS that had been already processed have to be put back in the set of rules to be processed, since change allows deriving their RHS on a shorter string. So doing this will call for more iterations, but will eventually end since none of these strings ever gets much shorter than the empty string.\n\nAt the end of this first algorithm, the string $\\sigma(U)$ is one of the smallest strings that can be derived from $U$. There may be others.\n\nNow we also have to modify the second algorithm to get, for every non-terminal $U$, (one of) the shortest string containing U as the only non-terminal. To do this, we keep the same directed graph with non-terminals as nodes, and having an arc $(U,V)$ iff there is a rule $U\\rightarrow \\alpha V \\beta$. But now we put weights on the arcs, to compute the minimum length of the terminal context that has to be associated with reachable non-terminals. The weight associated with the arc $(U,V)$ above is the length $\|\\sigma(\\alpha\\beta)\|$, where the mapping $\\sigma$ is extended to terminals as the identity, and then extended again as a string homomorphism. It is the length of (one of) the shortest terminal strings that can be derived from the string $\\alpha\\beta$. Note that $V$ is removed in this calculation. HOwever, when there are several occurences of $V$ in the RHS, only one must be removed. There may be several possible $(U,V)$ arcs, with different weights, if there are several rules with $U$ as LHS and $V$ in the RHS. In such a case, only (one of) the lighter such arc is kept.\n\nIn this graph, we no longer look just for reachability of nodes from $S$, but for the shortest weighted path that reaches every node from the initial symbol $S$. This can be done with Dijkstra's algorithm.\n\nGiven the shortest path for a non-terminal $U$, we read it as before as a sequence of rules, from which we get a dérivation $S\\overset{}{\\Longrightarrow}\\alpha U \\beta$. Then to every rule of the form $U\\rightarrow\\gamma$ in the grammar, we produce a minimal terminal string that \"invokes\" this rule as $\\sigma(\\alpha\\gamma\\beta)$\n\nRemark: The same minimal string may probably be used for several rules. But the fact that one of the strings uses a rule $\\rho$ in its derivation does not necessarily mean it is a minimal string for that rule $\\rho$, as it may have been found for another rule, while a shorter one can be found for $\\rho$. It may be possible to increase the likelyhood that the same minimal string will be found for several rules by using some priority policy whenever there is flexibility. But is it worth the trouble?\n\n## A faster algorithm for minimal terminal string deriving from a non-terminal\n\nBuilding the function $\\sigma$ such that $\\sigma(U)$ is a minimal terminal string deriving from $U$ is done above with a rather naive technique that requires iteratively reconsidering work already done when a new smaller derived string is found for some non-terminal. This is wasteful, even if the process will clearly terminate.\n\nWe propose here a more efficient algorithm, that is, in essence, an adaptation to the CF grammar graph of an extension of Dijkstra's shortest path algorithm to and-or graphs, with a proper definition of the path-concept for an and-or graph. This variant of the algorithm probably exists in the literature (assuming it is correct), but I have been unable to find it in the resources I can access. Hence I am describing it in more details, together with a proof.\n\nAs previously, we first remove from the set of rules to be processed all recursive rules (i.e. rules with a LHS symbol occurring in the RHS string). It is obvious that none of these recursive rules can derive to a shorter terminal string than the non-recursive rules with the same LHS. And, for a LHS $U$ there must be at least one non-recursive $U$-rule if the symbol $U$ is not a useless non-terminal (because non-productive). This is not strictly necessary, but reduces the number of rules to be considered later.\n\nThen we procede as before to build the set $Prod$ of productive symbols, associating with each synbol $X$ a terminal string, which we note $\\sigma(X)$, which is a size-minimal terminal string derivable from $X$ (in the previous algorithm, that was true only after termination).The set $Prod$ is initialized with all terminal symbols, and for each terminal symbol $a$, we define $\\sigma(a)=a$.\n\nThen we consider every rule $U\\rightarrow\\gamma$ such that all RHS symbols are in $Prod$, and we choose one such that $\\sigma(\\gamma)$ is size-minimal. Then we add $U$ to $Prod$, with $\\sigma(U)=\\sigma(\\gamma)$, and remove all $U$-rules. We iterate until all productives terminals have been entered in $Prod$. Any non-terminal $U$, once entered in $Prod$, never has to be considered again to change $\\sigma(U)$ for a smaller string.\n\nProof:\n\nThe previous algorithms were more or less intuitively obvious. This one is a bit trickier, because of the and-or character of the graph, and a proof seems more necessary. All we need is actually the following lemma, which establishes the correctness of the algorithm when applied to the last iteration.\n\nLemma: After each iteration of the algorithm, $\\sigma(X)$ is a size-minimal terminal string derivable from $X$, for all $X$ in $Prod$.\n\nThe base step is obvious, since this is true by definition for all terminals in $Prod$ when it is initialized.\n\nThen, assuming it is true after some non-terminals have been added to $Prod$, let $U\\rightarrow\\gamma$ be the rule chosen to add a new non-terminal to $Prod$. We know that this rule is chosen because $\\gamma\\in{Prod}^$ and $\\sigma(\\gamma)$ is size-minimal over all RHS of all rules with a RHS in ${Prod}^$. Then $U$ is added to $Prod$, and we have only to prove that $\\sigma(\\gamma)$ is a size-minimal terminal string derivable from $U$.\n\nThis is obviously the case for all derivations beginning with the rule $U\\rightarrow\\gamma$, since by induction hypothesis, application of mapping $\\sigma$ is such that all non-terminals in $\\sigma$ are substituted with size-minimal terminal strings deriving from them. Hence no other derivation can produce a shorter terminal string.\n\nWe thus consider only derivations starting with another $U$-rule $U\\rightarrow\\beta$, such that $\\beta\\overset{}{\\Longrightarrow}w\\in\\Sigma^$, where $\\Sigma$ is the set of terminal symbols.\n\nIf $\\beta\\in{Prod}^$, then a minimal string it can derive on is $\\sigma(\\beta)$. But, since we chose the rule $U\\rightarrow\\gamma$, it must be that $\|\\sigma(\\beta)\|\\geq\|\\sigma(\\gamma)\|$. So the rule $U\\rightarrow\\beta$ does not derive on a smaller terminal substring.\n\nThe last case to consider is when $\\beta\\notin{Prod}^$, and we then consider a derivation $\\beta\\overset{}{\\Longrightarrow}w\\in\\Sigma^$. If that derivation involves only non-trminals in $Prod$, then $\\beta\\in{Prod}^$, which is a case we have already seen. Hence we consider only derivations that have steps using a rule with its LHS not in $Prod$. Let $V\\rightarrow\\alpha$ be such a rule, such that $\\alpha\\in{Prod}^$.There must be at least one such rule since they are partially ordered by derivation order, and $w\\in{Prod}^$.\n\nThus we have $U\\Longrightarrow\\beta\\overset{}{\\Longrightarrow}\\mu V\\nu$. We know that $\\mu$ and $\\nu$ derive on a string of size at least 0, and since no $V$-rule with a RHS in ${Prod}^$ was chosen, they derive on terminal strings of length at least equal to $\|\\sigma(\\gamma)\|$. Hence, with the rule $U\\rightarrow\\beta$, $U$ derives on a terminal string of length at least equal to $\|\\sigma(\\gamma)\|$. $\\blacksquare$\n\n• looks plausible but suggest it needs to be converted at least to pseudocode. also it seems plausible the problem or near variants has been studied in the literature somewhere... – vzn Jul 16 '14 at 2:19\n• @vzn I doubt this problem has been published, because I do not see an application that does require this minimality. The useless symbols elimination algorithm is a classical technique, found probably in all textbooks on CF languages. And, as I said, it has some applications in general CF parsing. I never heard of pushing it further for producing test sentences. I chose to try explain its derivation, rather than give pseudo code. As it is the answer is already very long. Maybe in a second answer, if that is acceptable on the site ... as the text is becoming too long to handle. – babou Jul 16 '14 at 21:47\n• @vzn What could be in the literature is the extension of Dijkstra's shortest path algorithm to and-or graphs. Unfortunately, I do not have access to resources for finding out. I could write it up in a simplified form, without the CF grammar stuff. But then, I do not think it is a big deal. But it does gives the right hints to implement with low complexity. – babou Jul 16 '14 at 21:58\n• minimization & \"coverings\" are common considerations in the theoretical literature. further thought, maybe a normal form either Greibach or Chomsky could be key to understanding it better (there is some connection there also with removing unproductive terminals/nonreachable symbols). think this is worthwhile question for tcs.se, may ask it at some pt there & cite this one (afaik migrations tend to lose comments.) – vzn Jul 16 '14 at 22:37\n• @vzn I doubt Greibach or Chomsky normal form will bring any light. This is grammar dependent more than language dependent. Changing the grammar changes the problem. Actually, I first thought it was a homework dump. The minimality requirement does give it some interest, as finding a good algorithm was less obvious, especially my last version. I wrote the proof because I could not convinced myself any other way that it actually works. It is somehow less intuitive that the straight Dijkstra's algorithm from which I derived it. – babou Jul 16 '14 at 23:12\n\n## A simple solution\n\nIf you don't care too much about the number of strings, there is an easy solution. Basically, for each production, we will generate a string that covers that production.\n\nHow do we do that? It's a straightforward worklist algorithm. Suppose we want to cover the production $A ::= BCx$. Basically, we do breadth-first search starting from the start non-terminal $S$ to find a derivation that includes the non-terminal $A$ on the right hand side: initially, all non-terminals are unmarked, and the set of reachable non-terminals is $\\{S\\}$; in each iteration, we pick one unmarked reachable non-terminal, say $X$, and for each production with $X$ on the left-hand side, we add all the non-terminals on the right to the set of reachable terminals, then we mark $X$. Repeat until this process finds that $A$ is reachable. When it does, by backtracing, you obtain a derivation of the form $S \\to \\cdots A \\cdots$. You can extend this to a derivation of the form $S \\to \\cdots A \\cdots \\to \\cdots BCx \\cdots$.\n\nNext, we need to complete the derivation, by picking some way to complete it to be all non-terminals. This is also easy. For each non-terminal, we find the shortest string that is in the language generated by that non-terminals. This can be obtained by a worklist iteration. Let $\\ell(A)$ denote the length of the shortest string we've found so far in the language generated by $A$. Initially, we set $\\ell(A)=\\infty$ for all non-terminals $A$, unless there is a production $A ::= xyz$ where all the symbols on the right-hand side are terminals; in that case, we set $\\ell(A)$ accordingly (e.g., $\\ell(A)=3$ if the production tells us that $xyz$ is in $L(A)$). Now we iteratively look for a way to find a shorter string. If you have a production like $A ::= xBCy$, you know that $\\ell(A) \\le 2 + \\ell(B) + \\ell(C)$; so any time you find a new shorter string in $L(B)$ or $L(C)$ (i.e., any time you update $\\ell(B)$ or $\\ell(C)$ to be smaller), you can check whether this gives you a new shorter string in $L(A)$, and if so, reduce the value of $\\ell(A)$... which may in turn cause other cascading changes. Keep applying all cascading changes until no further changes are triggered. At that point, you have reached a fixed point, and by backtracing, for each non-terminal you know the shortest string (of terminals) that can be derived from that non-terminal.\n\nNow, given your derivation of the form $S \\to \\cdots BCx \\cdots$, find a way to complete the derivation: replace each non-terminal in $\\cdots BCx \\cdots$ with the shortest string (of terminals) that can be derived from it.\n\nThis gives you a string (of terminals) that covers the production $A ::= BCx$. Do this once for each production, until all of them are covered.\n\nThe number of strings is equal to the number of productions. The running time is quadratic: the process is linear-time for each production. (You could probably reduce it to be a linear-time algorithm in total, but I suspect this will be good enough in practice.)\n\n## Optimizations\n\nIf you want to reduce the number of strings, you might notice that each string will typically cover many productions, so a subset of these strings will probably suffice.\n\nThere are many ways you could reduce the number of strings. One simple one is to use the standard greedy approximation algorithm for set cover. Start by generating all of the strings as above, one for each production, and count how many productions each string covers. Keep the string that covers the most productions; that one you definitely want, so add it to your set of keepers. Now some of the productions are covered by this keeper, so we don't need any new strings that cover them again. Thus, you should update your set of counts for each strings: for string $s$, count the number of productions that are covered by $s$ but aren't covered by any keeper. Pick the string that has the largest number, add it to your set of keepers, and update the counts again. Repeat until all productions have been covered. This will likely give you a significantly smaller set of strings that cover all productions in your grammar." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8938298,"math_prob":0.9819963,"size":17722,"snap":"2019-35-2019-39","text_gpt3_token_len":4323,"char_repetition_ratio":0.16486059,"word_repetition_ratio":0.07585729,"special_character_ratio":0.22954519,"punctuation_ratio":0.09191713,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99735785,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-09-23T01:11:16Z\",\"WARC-Record-ID\":\"<urn:uuid:e5903e9c-0bf4-4636-9dfc-b04231752656>\",\"Content-Length\":\"182158\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:057f76b4-117a-4228-9a36-47872ad7c5a1>\",\"WARC-Concurrent-To\":\"<urn:uuid:8cc08ce4-a1e9-4c2b-bcd1-6311f94a614c>\",\"WARC-IP-Address\":\"151.101.65.69\",\"WARC-Target-URI\":\"https://cs.stackexchange.com/questions/28264/generating-a-set-of-minimal-length-strings-that-together-invoke-every-producti\",\"WARC-Payload-Digest\":\"sha1:OQLJ7HEG6MXJ2KK3XKN3VTEMKVYDE7F5\",\"WARC-Block-Digest\":\"sha1:6VYZCOZJF3EBEPL7LMXIDJCRBZYFCPUS\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-39/CC-MAIN-2019-39_segments_1568514575844.94_warc_CC-MAIN-20190923002147-20190923024147-00136.warc.gz\"}"}