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https://www.quantumstudy.com/a-strength-smooth-tunnel-is-dug-through-a-spherical-planet-and-is-perpendicular-to-the-planets-axis-of-rotation-which-is-fixed-in-space-the-planet-rotates/	[ "# A strength smooth tunnel is dug through a spherical planet and is perpendicular to the planet’s axis of rotation, which is fixed in space. The planet rotates with the angular velocity ω so that objects in the tunnel have no acceleration relative to the tunnel. Find the value of ω.\n\nQ: A strength smooth tunnel is dug through a spherical planet and is perpendicular to the planet’s axis of rotation, which is fixed in space. The planet rotates with the angular velocity ω so that objects in the tunnel have no acceleration relative to the tunnel. Find the value of ω.\n\n(a) $\\sqrt{\\frac{4}{3}} \\pi G \\rho_0$\n\n(b) $\\sqrt{\\frac{2}{3}} \\pi G \\rho_0$\n\n(c) $\\sqrt{\\frac{1}{3}} \\pi G \\rho_0$\n\n(d) $2 \\sqrt{\\frac{4}{3}} \\pi G \\rho_0$\n\nAns: (a)" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7484738,"math_prob":0.9942331,"size":451,"snap":"2022-27-2022-33","text_gpt3_token_len":142,"char_repetition_ratio":0.16331096,"word_repetition_ratio":0.0,"special_character_ratio":0.35254988,"punctuation_ratio":0.06451613,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9980784,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-07-04T06:31:24Z\",\"WARC-Record-ID\":\"<urn:uuid:9bbdb39b-5b3a-4d52-b1d1-2e199046f6b1>\",\"Content-Length\":\"52750\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:08f1fec9-54b3-430d-bb6b-d986833a50b5>\",\"WARC-Concurrent-To\":\"<urn:uuid:de908aea-710b-43a1-874f-f1a607a996d8>\",\"WARC-IP-Address\":\"103.133.214.171\",\"WARC-Target-URI\":\"https://www.quantumstudy.com/a-strength-smooth-tunnel-is-dug-through-a-spherical-planet-and-is-perpendicular-to-the-planets-axis-of-rotation-which-is-fixed-in-space-the-planet-rotates/\",\"WARC-Payload-Digest\":\"sha1:TH2HPVCXGSLDHHIVQPEHPWKL2K5D3VZA\",\"WARC-Block-Digest\":\"sha1:A4EWU2MC3YHLBSOK7Z6IFNDU7HF3KXDH\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656104354651.73_warc_CC-MAIN-20220704050055-20220704080055-00526.warc.gz\"}"}
https://docs.arnoldrenderer.com/exportword?pageId=70746280	[ "Date: Wed, 27 Oct 2021 10:50:50 +0000 (UTC) Message-ID: <654809145.34819.1635331850577@[52.200.175.11]> Subject: Exported From Confluence MIME-Version: 1.0 Content-Type: multipart/related; boundary=\"----=_Part_34818_473868297.1635331850575\" ------=_Part_34818_473868297.1635331850575 Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable Content-Location: file:///C:/exported.html Advanced (Toon)", null, "=20\n\n#### Indirect Diffuse\n\nThe amount of diffuse light received = from indirect sources only.\n\n=20\n=20\n=20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20", null, "", null, "", null, "=20 0 =20 0.5 =20 1 (default)\n=20\n=20\n=20\n\nIndirect_diffuse: 0 (default) creates a more two-dimensional lo= oking toon shading effect.\n\n=20\n=20\n=20 =20 =20 =20 =20 =20 =20 =20 =20 =20", null, "", null, "=20 0 (default) =20 1\n=20\n=20\n=20\n\n#### Indirect Specular\n\nThe amount of specularity received fr= om indirect sources only. Values other than 1.0 will cause the = materials to not preserve energy, and global illumination may not converge.=\n\n=20\n=20\n=20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20", null, "", null, "", null, "=20 0 =20 0.5 =20 1 (default)\n=20\n=20\n\n#### Energy Conserving\n\nThe toon shader is energy= conserving by default. If thi= s is disabled, the Toon shader simply adds base, spec= ular, and transmission. Care should be taken when disabling this option as it will affect i= ndirect illumination with the toon shader.\n\n#### AOV Prefix\n\nAn optional aov_prefix that will be prepended to the toon AOVs'= names. For instance, if aov_prefix is `\"toon_\"`, the toon diff= use AOV will be written out to `\"toon_diffuse\"`. This can be used when you need to access bo= th the toon AOVs and the core's LPE AOVs." ]	[ null, "https://docs.arnoldrenderer.com/3D\"636f0466494b77a8e15f5d20af6ab89f\"", null, "https://docs.arnoldrenderer.com/=3D\"1bd861d89f1d9886297c1b7bc6c41fca\"", null, "https://docs.arnoldrenderer.com/=3D\"bbb613714af9bdad0619ab9b5dd25cb1\"", null, "https://docs.arnoldrenderer.com/=3D\"18a1ad94961346f22d87aeecb1712e74\"", null, "https://docs.arnoldrenderer.com/=3D\"23daac9f1b392432e07dbfd71cc778d7\"", null, "https://docs.arnoldrenderer.com/=3D\"dd9e39ece665e680b242cfc2bcbf29e0\"", null, "https://docs.arnoldrenderer.com/=3D\"d53b7af35a39912d821dd6753a8ff259\"", null, "https://docs.arnoldrenderer.com/=3D\"319fed87c7519b83ba8867a7dc7b1cb3\"", null, "https://docs.arnoldrenderer.com/=3D\"368f62eec2e11c8a68b7300e70c1d553\"", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7185731,"math_prob":0.7637049,"size":1384,"snap":"2021-43-2021-49","text_gpt3_token_len":423,"char_repetition_ratio":0.14637682,"word_repetition_ratio":0.03208556,"special_character_ratio":0.37210983,"punctuation_ratio":0.15789473,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99969804,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],"im_url_duplicate_count":[null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-27T10:50:50Z\",\"WARC-Record-ID\":\"<urn:uuid:6f1fc22b-9d7f-4a99-9267-79f536677b66>\",\"Content-Length\":\"497570\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:11653b80-bfc9-4430-86fa-7e2aca587757>\",\"WARC-Concurrent-To\":\"<urn:uuid:64d01887-dbab-4468-a9df-a8faae728419>\",\"WARC-IP-Address\":\"52.200.175.11\",\"WARC-Target-URI\":\"https://docs.arnoldrenderer.com/exportword?pageId=70746280\",\"WARC-Payload-Digest\":\"sha1:LCNP2N6SED6CE3IRWB7XICBR555LOHSF\",\"WARC-Block-Digest\":\"sha1:GFMONUHZW4KZ3OQJ73EO7L3COUM7UGTH\",\"WARC-Identified-Payload-Type\":\"message/rfc822\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323588113.25_warc_CC-MAIN-20211027084718-20211027114718-00216.warc.gz\"}"}
https://www.arunma.com/2023/03/19/build-your-own-counting-bloom-filter-in-rust/	[ "March 19, 2023\n\n# Build your own Counting Bloom Filter in Rust\n\nA Counting Bloom Filter is a probabilistic data structure that helps us quickly check if an element is present in a set or not. You might argue \"Hey, can't a simple Set do it?\".\n\nYes, indeed and to top it, Counting Bloom Filter is not even 100% accurate and expects us to provide the expected \"false positive rate\". Where Bloom Filters shine is that for large volumes of data it would use much lesser memory than a regular HashSet.\n\nThis post attempts an implementation of the datastructure. At the end of the post, you'll be able to build your own Counting Bloom Filter. The final code is available here.\n\nBefore we get into the implementation, let's get some properties of CBF out of the way:\n\n1. As with our traditional BloomFilter, Quotient Filter or CuckooFilter, Counting Bloom Filter is a probabilistic data structure that is used to test whether an element is a member of a set. Essentially, these filters aim to solve a class of problems called \"Membership\". A key feature that CBF offers over traditional Bloom Filter is that CBF allows deletion of inserted items.\n2. As with any probabilistic data structure, there is a chance that we might get some false positives - essentially, the response to the question \"Is an element present in the set?\" is not 100% accurate. There are times when the bloom filter would say that the element is present but it actually was not. This \"false positive\" rate is configurable through a parameter.\n3. Unlike HashSet, you cannot iterate through the values stored in the set - we can only check whether the value is present in the CBF.\n\n## Bloom Filter vs Counting Bloom Filter\n\nThe underlying storage of a BloomFilter is just a fixed-size array of bits.\n\n💡\nThis is a fantastic write-up detailing how to implement your own Bloom Filter.\n\nA Counting Bloom Filter uses a fixed-size array of counters.\n\nTo put it in crude terms, the underlying storage of Bloom Filter is a Vec<bool> (or a BitSet). With CBF, it is a Vec<u8> or Vec<u16> or Vec<u32> or a list of unsigned integers.\n\n💡\nThe counter size does not need to be 8, 16, 32 bits etc - rather, they don't need to be aligned to the datatypes. Using arbitrary bit sizes is common and is achieved by partitioning a usize integer into arbitrary bit sizes.\n\nFor illustration, our implementation of a Counting Bloom Filter will use a u8 bit counter.\n\n### Setting the stage\n\n#### 1. Declaring our struct and instantiation\n\n#[derive(Debug)]\npub struct CountingBloomFilter<T: ?Sized + Hash> {\ncounter: Vec<u8>,\nm: usize,\nk: usize,\nhasher: SipHasher24,\nexpected_num_items: usize,\nfalse_positive_rate: f64,\nlen: usize,\n_p: PhantomData<T>,\n}\n\nAs expected, our struct encapsulates a list of counters. There are two parameters that we would be accepting from the user, which are expected_num_items and false_positive_rate.\n\n1. The expected_num_items is an approximate number of items that the user would like to store in the CBF.\n2. The false_positive_rate is an estimate of False Positives/inverse accuracy that the user can live with.\n\nThe len just stores the total number of items inserted into the data structure so far. We'll discuss the m and k fields in the next section.\n\n#### 2. Number of buckets and number of hashes\n\n##### Bloom Filter\n\nIn a BloomFilter, m is simply the size of the bit vector (or underlying bits representing the membership of the various inserted values). k is the number of hash functions that each inserted value has to be run against.\n\nSo, given a value x, it will be hashed by k number of hash functions and the corresponding bit in the m buckets will be set to True.\n\nLet's consider the illustration above. Each item ( x, y, z) are hashed by 2 hash functions (represented by arrows). The resulting output are expected to be the bucket indices. Considering the total number of buckets is just 8 in the above illustration, the output of the hash function is modulo-ed to reduce the range between 0 and 7.\n\n##### Counting bloom filter.\n\nThe number of hash functions ( k ) is 2 as above and the m is 8. However, the underlying store is not a list of bits but a list of counters.\n\nThe formulas for calculating the size of m and k given the expected_num_of_items and false_positive_rate for Counting Bloom Filter are:\n\nLet's implement the optimal_m, optimal_k and the new functions\n\n\nfn optimal_m(num_items: usize, false_positive_rate: f64) -> usize {\n-(num_items as f64 * false_positive_rate.ln() / (2.0f64.ln().powi(2))).ceil() as usize\n}\n\nfn optimal_k(n: usize, m: usize) -> usize {\nlet k = (m as f64 / n as f64 * 2.0f64.ln()).round() as usize;\nif k < 1 {\n1\n} else {\nk\n}\n}\n\nimpl<T: ?Sized + Hash + Debug> CountingBloomFilter<T> {\npub fn new(expected_num_items: usize, false_positive_rate: f64) -> Result<Self> {\nvalidate(expected_num_items, false_positive_rate)?;\nlet m = optimal_m(expected_num_items, false_positive_rate);\nlet counter = vec![0; m];\nlet k = optimal_k(expected_num_items, m);\nlet random_key = generate_random_key();\nlet hasher = create_hasher_with_key(random_key);\nOk(Self {\ncounter,\nm,\nk,\nhasher,\nexpected_num_items,\nfalse_positive_rate,\nlen: 0,\n_p: PhantomData,\n})\n}\n\nAs you can see, only the \"estimated number of items to be inserted\" and the \"false positive rate\" is needed to construct our Counting Bloom Filter.\n\n#### 3. Hashers\n\nOnce we calculate the optimal number of hash functions (k), we need to create k independent hash functions. One approach is to have a repository of hash functions and initiate them with a random key. However, there is a simpler approach that is proposed by Adam Kirsch and Michael Mitzenmacher, where we could simulate additional hash functions with the help of two base hash functions.\n\nQuoting the paper:\n\nThe idea is the following: two hash functions h1(x) and h2(x) can simulate more than two hash functions of the form gi(x) = h1(x) + ih2(x). In our context i will range from 0 up to some number k − 1 to give k hash functions, and the hash values are taken modulo the size of the relevant hash table.\n\nSpecifically, only two hash functions are necessary to effectively implement a Bloom filter without any increase in the asymptotic false positive probability. This leads to less computation and potentially less need for randomness in practice.\n\nEssentially, we just need 2 independent hash functions to simulate k hash functions. Typically, Bloom Filters use 64-bit hash functions since they create enough bits for us to populate the buckets uniformly. In our implementation, we will use a 128-bit hash function and use the first and the second 64 bits to simulate two hash function outputs. We will then use these two 64-bit hashes to create more hashes.\n\nFirstly, in order to provide a random seed to our 128-bit hasher, we'll use the generate_random_key function.\n\nuse siphasher::sip128::SipHasher24;\n\n/// Generates a random 128-bit key used for hashing\nfn generate_random_key() -> [u8; 16] {\nlet mut seed = [0u8; 32];\ngetrandom::getrandom(&mut seed).unwrap();\nseed[0..16].try_into().unwrap()\n}\n\n/// Creates a SipHasher24 hasher with a given 128-bit key\nfn create_hasher_with_key(key: [u8; 16]) -> SipHasher24 {\nSipHasher24::new_with_key(&key)\n}\n\n### Building the APIs\n\nWith all the foundational functions out of the way, let us get into implementing the API - the three functions insert, contains and delete\n\n#### 1. Insert\n\nWhen an element is inserted into the filter, the counters corresponding to the element's hash values are incremented by one. This is in contrast to BloomFilter where we just set the corresponding bit. While we do this, we will also use this opportunity to keep track of the number of items inserted into the bloom filter by incrementing the len field.\n\n pub fn insert(&mut self, item: &T) {\nself.get_set_bits(item, self.k, self.m, self.hasher)\n.iter()\nself.len += 1;\n}\n\nThe get_set_bits is a helper function that takes the item to be inserted and returns the indices of all buckets for whom the counters need to be incremented. Essentially, the get_set_bits function\n\n1. Takes an item as an input\n2. Creates a 128-bit hash (done in get_hash_pair)\n3. Splits the 128-bit hash into 2 parts, thereby creating the 2 base hashes (done in get_hash_pair)\n4. Creates k hashes out of the 2 base hashes using the Kirsch and Mitzenmacher method\n5. Since we just have m buckets and the hash function output could be bigger, we do a modulus of m on each hash output. This returns k bucket indices on m . These are the indices of the counters in the bucket that needs to be incremented.\n fn get_set_bits(&self, item: &T, k: usize, m: usize, hasher: SipHasher24) -> Vec<usize> {\nlet (hash1, hash2) = self.get_hash_pair(item, hasher);\nlet mut set_bits = Vec::with_capacity(k);\nif k == 1 {\nlet bit = hash1 % m as u64;\nset_bits.push(bit as usize);\nreturn set_bits;\n}\nfor ki in 0..k as u64 {\nlet bit = hash % m as u64;\nset_bits.push(bit as usize);\n}\nassert!(set_bits.len() == k);\nset_bits\n}\n\n/// Computes the pair of 64-bit hashes for an item using the internal hasher\nfn get_hash_pair(&self, item: &T, mut hasher: SipHasher24) -> (u64, u64) {\nitem.hash(&mut hasher);\nlet hash128 = hasher.finish128().as_u128();\nlet hash1 = (hash128 & 0xffff_ffff_ffff_ffff) as u64;\nlet hash2 = (hash128 >> 64) as u64;\n(hash1, hash2)\n}\n💡\nThe wrapping_add, wrapping_mul and other wrapping functions allow for overflow of numbers without triggering a panic.\n\neg.\n\nlet a: u16 = 65534;\nlet b = a.wrapping_add(1); // 65535\nlet c = a.wrapping_add(2); // 0\n\n#### 2. Contains\n\nAs you might have guessed already, testing whether an item is present in the filter is just a matter of computing the hashes for the item and checking whether the counters in the bucket corresponding to the hashes have a non-zero value.\n\n\npub fn contains(&self, item: &T) -> bool {\nself.get_set_bits(item, self.k, self.m, self.hasher)\n.iter()\n.all(\|&i\| self.counter[i] > 0)\n}\n\n#### 3. Delete\n\nThe deletion operation is just an inverse of the insertion operation with two additional nuances.\n\n1. We can only delete an item that is already present in the filter. So, we check for it by calling the contains.\n2. Our counter can never be less than 0. So, we use the saturating_sub function to ensure that.\n\tpub fn delete(&mut self, item: &T) {\nlet is_present = self.contains(item);\nif is_present {\nself.get_set_bits(item, self.k, self.m, self.hasher)\n.iter()\n.for_each(\|&i\| {\nself.counter[i] = self.counter[i].saturating_sub(1);\n});\nself.len -= 1;\n}\n}\n\n#### 4. Estimated count\n\nAlthough Counting Bloom Filter is not the best data structure to get an estimated count of an item, it is not uncommon to see instances where the function is implemented. Estimated count is arrived at by looking at all the counter values and picking the minimum of those values. The reasoning behind choosing minimum and not maximum or median is that a counter could be incremented by several items due to hash collisions. Minimum is chosen to avoid the skew created by high-frequency elements and to be fair to the low-frequency elements.\n\n pub fn estimated_count(&self, item: &T) -> u8 {\nlet mut retu = u8::MAX;\nfor each in self.get_set_bits(item, self.k, self.m, self.hasher) {\nif self.counter[each] == 0 {\nreturn 0;\n}\nif self.counter[each] < retu {\nretu = self.counter[each];\n}\n}\nretu\n}\n💡\nWe will revisit this discussion when we build our own CountMinSketch later.\n\n### Overflow\n\nAlthough not included in our implementation, I felt that a discussion on this topic is important.\n\nIn our example, we decided that our counter is represented by an unsigned 8-bit integer with a maximum value of 255. What if subsequent item insertions attempt to increment the value of that counter? This would result in an overflow and in Rust, a panic. To avoid overflowing, the counter could be capped/saturated at the maximum value. This essentially means that subsequent insertions into that bucket will have no effect on the filter's state for that counter.\n\nThere are two common workarounds to solve overflows:\n\n1. Freezing\n2. Dynamic resizing\n\n#### 1. Freezing\n\nFreezing is a process where\n\n1. the buckets are scanned for counters that have already reached the maximum value.\n2. reset the counter to a threshold value (around 80% of the maximum value). For u8 that's approximately 200.\n\n#### 2. Dynamic resizing\n\nThe other simpler yet computationally expensive alternative is to resize the counter to accept larger values. Say, promoting the datatype from u8 to u16. This involves creating a new counter list, copying the existing counters and replacing the old list with the new one.\n\n### Code\n\nComplete code is here." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7936797,"math_prob":0.94544166,"size":12160,"snap":"2023-40-2023-50","text_gpt3_token_len":3002,"char_repetition_ratio":0.13384336,"word_repetition_ratio":0.01711984,"special_character_ratio":0.2524671,"punctuation_ratio":0.1468085,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98214024,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-22T06:13:25Z\",\"WARC-Record-ID\":\"<urn:uuid:29c2a16f-88d8-4e01-9087-253dbf404ee6>\",\"Content-Length\":\"38915\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d5f8a62c-b203-4aea-9977-c7e402f48cbf>\",\"WARC-Concurrent-To\":\"<urn:uuid:8aa19b81-2466-41e2-aa57-db8e2c39aeef>\",\"WARC-IP-Address\":\"146.75.39.7\",\"WARC-Target-URI\":\"https://www.arunma.com/2023/03/19/build-your-own-counting-bloom-filter-in-rust/\",\"WARC-Payload-Digest\":\"sha1:YO4RHPOWW4A3NVZ2ESUKOYNHMVFGAD2B\",\"WARC-Block-Digest\":\"sha1:I5BFSOTPRX6EN72QKJKMLTDW4CIZSTWZ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233506329.15_warc_CC-MAIN-20230922034112-20230922064112-00410.warc.gz\"}"}
https://books.google.com.jm/books?id=Meu1MaOdmO8C&lr=	[ "# Mathematical Dictionary and Cyclopedia of Mathematical Science: Comprising Definitions of All the Terms Employed in Mathematics - an Analysis of Each Branch, and of the Whole, as Forming a Single Science\n\nA.S. Barnes & Company, 1855 - Mathematics - 592 pages\n\n### Popular passages\n\nPage 215 - A Circle is a plane figure bounded by a curved line every point of which is equally distant from a point within called the center.\nPage 217 - If two triangles have the three sides of the one equal to the three sides of the other, each to each, the triangles are congruent.\nPage 140 - The circumference of every circle is supposed to be divided into 360 equal parts, called degrees ; and each degree into 60 equal parts, called minutes ; and each minute into 60 equal parts, called seconds ; and these into thirds, etc.\nPage 180 - The part of the equation which is on the left of the sign of equality is called the first member ; the part on the right of the sign of equality, the second member.\nPage 80 - Then multiply the second and third terms together, and divide the product by the first term: the quotient will be the fourth term, or answer.\nPage 400 - Multiply the divisor, thus augmented, by the last figure of the root, and subtract the product from the dividend, and to the remainder bring down the next period for a new dividend.\nPage 217 - If two triangles have two sides, and the included angle of the one equal to two sides and the included angle of the other, they are equal in all their parts.\nPage 124 - Subtract the cube of this number from the first period, and to the remainder bring down the first figure of the next period for a dividend.\nPage 360 - Three lines are in harmonical proportion, when the first is to the third, as the difference between the first and second, is to the difference between the second and third ; and the second is called a harmonic mean between the first and third. The expression 'harmonical proportion..." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9165518,"math_prob":0.97657585,"size":2906,"snap":"2023-40-2023-50","text_gpt3_token_len":557,"char_repetition_ratio":0.12818746,"word_repetition_ratio":0.050314464,"special_character_ratio":0.19993117,"punctuation_ratio":0.084812626,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.997434,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-05T18:02:03Z\",\"WARC-Record-ID\":\"<urn:uuid:0c4fe611-93e8-4213-8401-9d881e76e488>\",\"Content-Length\":\"63858\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f84e49c7-8996-4fd9-8a1f-97285cff58a0>\",\"WARC-Concurrent-To\":\"<urn:uuid:05779227-3ddf-4be1-aae8-b9e5cd1f6140>\",\"WARC-IP-Address\":\"172.253.115.138\",\"WARC-Target-URI\":\"https://books.google.com.jm/books?id=Meu1MaOdmO8C&lr=\",\"WARC-Payload-Digest\":\"sha1:5W3TFHCJO6BF3IMBZ4OPTOGUTXV5RKMV\",\"WARC-Block-Digest\":\"sha1:VBQC32PPN4ZUBGRYWU5INCUKEOPOJ6AQ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100555.27_warc_CC-MAIN-20231205172745-20231205202745-00182.warc.gz\"}"}
https://askncertquestions.com/searchQuestion.php?q=520	[ "Wellcome!\n\nQuestion 1. Asked on :05 February 2019:05:11:52 PM\n\n# The perimeter of a parallelogram is 60 cm and the ratio of its adjacent sides is 3:2 .If the altitude corresponding is 5 cm. Find the area of the parallelogram and the alyitude corresponding to the smaller side\n\n-Added by Himanshi Verma Mathematics » Perimeter And Area\n\nHimanshi Verma\n\nGiven\n\nRatio of side =3:2\n\nPerimeter =60cm\n\nAltitude =5cm\n\nSo let take one side as 3x and other is 2x\n\nThen perimeter of parallelogram\n\n=3x+2x+3x+2x\n\n=10x\n\nAccording to question perimeter =60cm\n\nThen\n\n10x=60cm\n\nX=60/10=6cm\n\nThen one side is =6 x 3=18\n\nAnd other is =2x6=12\n\nSo base=18cm\n\nArea of parallelogram =bxh\n\n=18x5=90cm²\n\nAltitude corresponding to smaller side\n\n90=12xh\n\nH=7.5cm\n\n-Answered by Himanshi Verma On 05 February 2019:05:26:39 PM(1295Average Rating Based on rating)\n\nYou can see here all the solutions of this question by various user for NCERT Solutions. We hope this try will help you in your study and performance.\n\nThis Solution may be usefull for your practice and CBSE Exams or All label exams of secondory examination. These solutions or answers are user based solution which may be or not may be by expert but you have to use this at your own understanding of your syllabus.\n\n#### What do you have in your Mind....\n\n* Now You can earn points on every asked question and Answer by you. This points make you a valuable user on this forum. This facility is only available for registered user and educators.\n\n## Search your Question Or Keywords\n\n#### Do you have a question to ask?\n\nUser Earned Point: Select\n\n## All Tags by Subjects:\n\nScience (2231)\nHistory (278)\nGeography (310)\nEconomics (257)\nPolitical Science (96)\nMathematics (200)\nGeneral Knowledge (5686)\nBiology (94)\nPhysical Education (20)\nChemistry (118)\nCivics (114)\nHome Science (12)\nSociology (9)\nHindi (45)\nEnglish (258)\nPhysics (1435)\nOther (97)\nAccountancy (378)" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.91508055,"math_prob":0.57907337,"size":1312,"snap":"2021-43-2021-49","text_gpt3_token_len":374,"char_repetition_ratio":0.10091743,"word_repetition_ratio":0.0,"special_character_ratio":0.2660061,"punctuation_ratio":0.10037175,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9711347,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-20T00:50:07Z\",\"WARC-Record-ID\":\"<urn:uuid:d7374fb8-0925-426f-8dff-5363d667ec8d>\",\"Content-Length\":\"22216\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4bf1f061-8670-4ec2-85ff-8668f4b6cb99>\",\"WARC-Concurrent-To\":\"<urn:uuid:61dd8a54-b3f2-4960-868c-b284c559e8ff>\",\"WARC-IP-Address\":\"148.66.137.24\",\"WARC-Target-URI\":\"https://askncertquestions.com/searchQuestion.php?q=520\",\"WARC-Payload-Digest\":\"sha1:SRID4LGXSXVR47SNXU5IOZLSSZ2LBQWN\",\"WARC-Block-Digest\":\"sha1:AJ4JBQAVUQJQISB2YNZN4TZE6VNS7HTE\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585290.83_warc_CC-MAIN-20211019233130-20211020023130-00456.warc.gz\"}"}
https://stats.stackexchange.com/questions/310545/arima-forecasting-using-exogenous-variables-with-their-own-forecast-intervals	[ "ARIMA forecasting using exogenous variables with their own forecast intervals\n\nSuppose\n\nmodel <- Arima(y , xreg=cbind(x1, x2), order=(p,d,q))\n\n\nIf I am forecasting $x_1$ and $x_2$, then for forecasting $y$:\n\n1) If I use expected forecasts for $x_1$ and $x_2$ (single numbers), I simply do:\n\nforecast(model, xreg=cbind(E(future x1) , E(future x2))\n\n\n2) How about if I want to use forecast intervals for $x_1$ and $x_2$?\n\nThis post suggests that: you can draw (a lot of) random numbers from each predictive density, plug them into the model and get a predictive distribution for $y$. Then I guess taking the average prediction interval to come up with the one forecast interval for $y$. Does this make sense?" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.79057324,"math_prob":0.9985358,"size":619,"snap":"2019-26-2019-30","text_gpt3_token_len":179,"char_repetition_ratio":0.13658537,"word_repetition_ratio":0.0,"special_character_ratio":0.2907916,"punctuation_ratio":0.13178295,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9999268,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-06-26T16:05:31Z\",\"WARC-Record-ID\":\"<urn:uuid:f3068193-573f-4301-98ff-0bc4e7eaff99>\",\"Content-Length\":\"138752\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:75925656-d87e-48d1-a8af-8c1b5a4fa9c8>\",\"WARC-Concurrent-To\":\"<urn:uuid:1f82dc6a-e7d5-4a1e-b2bb-4e01e91a0bce>\",\"WARC-IP-Address\":\"151.101.193.69\",\"WARC-Target-URI\":\"https://stats.stackexchange.com/questions/310545/arima-forecasting-using-exogenous-variables-with-their-own-forecast-intervals\",\"WARC-Payload-Digest\":\"sha1:F6WQVPNKU4BSHCZYPMPX7XPN72DBS4CO\",\"WARC-Block-Digest\":\"sha1:XAAG266C4GQJGDTGTCR25NMHJNG42Z6I\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-26/CC-MAIN-2019-26_segments_1560628000367.74_warc_CC-MAIN-20190626154459-20190626180459-00522.warc.gz\"}"}
https://epjquantumtechnology.springeropen.com/articles/10.1140/epjqt/s40507-015-0018-0	[ "# Phase diagram of a QED-cavity array coupled via a N-type level scheme\n\n## Article metrics\n\n• 1946 Accesses\n\n• 1 Citations\n\n## Abstract\n\nWe study the zero-temperature phase diagram of a one-dimensional array of QED cavities where, besides the single-photon hopping, an additional coupling between neighboring cavities is mediated by an N-type four-level system. By varying the relative strength of the various couplings, the array is shown to exhibit a variety of quantum phases including a polaritonic Mott insulator, a density-wave and a superfluid phase. Our results have been obtained by means of numerical density-matrix renormalization group calculations. The phase diagram was obtained by analyzing the energy gaps for the polaritons, as well as through a study of two-point correlation functions.\n\n## Introduction\n\nThe recent impressive advances in the field of quantum simulators allowed to probe the many-body physics of strongly correlated systems at the level of the single quantum object. At present cold atoms trapped in optical lattices can be considered among the most promising examples of quantum simulators. By means of ultracold atomic and molecular gases, it is nowadays possible to reach a degree of control and accuracy in engineering the dynamics of many-body systems that were unimaginable in previously. As a consequence, the coherent quantum dynamics emerging from carefully tailored microscopic Hamiltonians can now be tested experimentally . It has been possible, just to recall one example, to implement the Bose-Hubbard (BH) model [2, 3] and to detect its zero-temperature superfluid (SF) to Mott insulator (MI) quantum phase transition . Other models involving spinor gases, Fermi systems, Bose-Fermi mixtures, or dipolar gases have been also devised and realized, providing an even richer phase diagram (see for example the review ). We mention the stabilization of density-wave (DW) phases for bosons, as well as more peculiar topological or supersolid orderings, which can arise in the presence of finite-range interactions .\n\nMore recently a novel kind of many-body quantum simulator has been introduced, based on the idea to use single photons as quantum objects. Since photons hardly interact in open space, the most natural way to significantly increase their interactions is to trap them into an optical QED cavity, and couple the field with atoms/molecules inside it in order to create an optical nonlinearity. If the nonlinearity is sufficiently large, the so called photon blockade sets in [7, 8], namely, the presence of a single photon inside a cavity prevents a second one to enter it. In the rotating-wave approximation, the simplest light-matter interaction scheme of this type can be accurately described by the Jaynes-Cummings (JC) model. By arranging an array of cavities coupled through the photon hopping, such to generate a competition between the hopping and the on-site nonlinearities, one can devise a setup that is well described by the so called Jaynes-Cummings-Hubbard (JCH) model .\n\nIn many respects, if one ignores dissipation, the physics emerging from the JCH Hamiltonian resembles, at low-energies, that of an effective BH model. Probably the main difference between the two systems is that, instead of having neutral bosons as building blocks of the model, in the JCH Hamiltonian one has to think in terms of polaritons, i.e., combined photonic/atomic excitations. Many different works already addressed the JCH equilibrium phase diagram with analytical, as well as numerical methods, leading to a fairly complete theoretical understanding of the nature and the location of the emerging phases and phase transitions in terms of the parameters governing the system (the field has been recently reviewed in, e.g., Refs. ).\n\nAdditional interest in cavity arrays comes from the fact that these systems can be naturally considered as open-system quantum simulators. Some related features have been recently explored . In the following we will not touch on this and consider only the ‘equilibrium’ phase diagram.\n\nThis intense activity has been very recently boosted by the first experiments on QED cavity arrays . As of today, the most concrete possibility to realize controllable and scalable quantum simulators with cavity arrays involves circuit-QED cavities .\n\nSo far the coupling between cavities has been mostly considered through photon hopping. Only few works started addressing more general schemes, where the cavity coupling can be induced also by means of non-linear elements [23, 24, 31, 32]. Such configurations include cross-Kerr interactions and/or correlated hopping terms, which lead to generalizations of the JCH model in a way similar to the extended BH (EBH) Hamiltonian for atoms with large dipole momentum loaded in optical lattices . The underlying physical model is believed to possess a much richer structure, with the emergence of exotic phases of correlated polaritons. It is particularly interesting to address these schemes in one-dimensional (1D) systems, where interactions become crucial to stabilize exotic phases of matter . These notably include a series of nontrivial density-wave (DW) states, which can arise in the strong coupling regime , as well as supersolidity and phase-separation effects [39, 40]. Extension to consider also counter-rotating terms in the ultrastrong coupling regime, thus leading to the so called Rabi-Hubbard model , have been investigated . However we are not aware of numerical investigations of coupled cavity models beyond the JCH and Rabi-Hubbard model.\n\nIn all such situations, non-perturbative, either numerical or analytical calculations are necessary. Here the density-matrix renormalization group (DMRG) algorithm [43, 44] has been employed to work out the quantitative zero-temperature phase diagram of the JCH model . This is a particularly efficient method for the statics of 1D many-body problems. Its key strategy consists in constructing a portion of the system (called block) and then recursively enlarge it. At each step, the basis of the corresponding Hamiltonian is truncated to a given value m, so that one can manage the Hamiltonian in an effective Hilbert space of fixed dimensions, as the physical system grows. This truncation is performed by retaining the eigenstates corresponding to the m highest eigenvalues of the reduced density matrix of the block.\n\nThe aim of this paper is to quantitatively study a generalization of the JCH Hamiltonian, aimed at taking into account an effective nearest-neighbor nonlinearity between cavities mediated by an N-type four-level system as discussed for two cavities in Ref. . The presence of this coupling leads to an effective cross-Kerr nonlinearity. An analysis at the mean-field level of a dissipative open EBH as an effective model for nonlinearly coupled cavities has been performed, unveiling the emergence of novel photon crystal and supersolid phases [23, 24]. Here we do not resort to the effective EBH model and analyze the full model as introduced in . Using the DMRG algorithm, we work out the 1D ground-state phase diagram. We show that a physics similar to the EBH model appears, with a rich phase diagram including gapless SF, as well as MI and DW phases of polaritons. We postpone the analysis of the interplay of driving and dissipation to a future work.\n\nThe paper is organized as follows. In the next two sections we introduce the model of coupled cavities of our interest (Section 2) and the quantities we are going to address, namely the energy gaps, and the staggered number-number correlations (Section 3). In Section 4 we discuss the zero-temperature equilibrium phase diagram, focusing on the MI/SF boundary and on the boundary separating the DW from the other phases. Finally, in Section 5 we draw our conclusions.\n\n## The model\n\nLet us consider a 1D array of QED cavities, where photons can hop between neighboring cavities. Moreover two adjacent resonators are also nonlinearly coupled to each other via a N-type four-level system, as shown in Figure 1(a). For the sake of clarity in our description, we shall divide the 1D array into coupled effective sites composed of a cavity and an atom. The four levels are denoted by $$\\{ \\vert i \\rangle \\}_{i = 1 ,\\ldots,4}$$, and are depicted in Figure 1(b). An external laser with frequency Ω resonantly drives the transition $$\\vert 3 \\rangle \\leftrightarrow \\vert 2 \\rangle$$. The transition $$\\vert 1 \\rangle \\leftrightarrow \\vert 3 \\rangle$$ is resonantly coupled to the cavity mode of the same site with strength $$g_{1}$$, while the transition $$\\vert 2 \\rangle \\leftrightarrow \\vert 4 \\rangle$$ couples to the cavity mode of its right nearest-neighbor site with strength $$g_{2}$$, and a detuning Δ.\n\nThe use of such N-type atom for generating large Kerr nonlinearity has been extensively studied in the literature [7, 8, 49], however the vast majority of the scenarios only focused on a single-mode cavity. Our work is inspired by the idea of Ref. , where the cross-Kerr nonlinearity is generated between two different and neighboring cavities, in circuit-QED systems. In practice, we use the unbalanced couplings of atomic transition $$\|1\\rangle\\leftrightarrow\|3\\rangle$$ with left cavity mode, and $$\|2\\rangle\\leftrightarrow\|4\\rangle$$ with right cavity mode respectively, in order to generate the local ($$g_{1}$$) and nonlocal ($$g_{2}$$) nonlinearities of our many-body system. This kind of four-level artificial molecule can be realized using two Josephson transmon qubits coupled by a superconducting quantum interference device.\n\nUsing the interaction picture and in the rotating-wave approximation, the system Hamiltonian reads\n\n$$\\mathcal{H}= \\sum_{i} \\bigl[ \\Delta\\sigma^{44}_{i} + \\bigl( \\Omega\\sigma^{23}_{i} + g_{1} \\sigma_{i}^{13} a_{i}^{\\dagger}+ \\mathrm{H.c.} \\bigr) + \\bigl(-t a_{i} a_{i+1}^{\\dagger}+ g_{2} \\sigma_{i}^{24} a_{i+1}^{\\dagger}+ \\mathrm{H.c.} \\bigr) \\bigr] ,$$\n(1)\n\nwhere $$\\sigma^{mn} = \\vert m \\rangle \\langle n\\vert$$ ($$m,n = 1,2,3,4$$), and a ($$a^{\\dagger}$$) is the annihilation (creation) operator of the cavity mode. The subscripts denote the site position along the 1D chain. The first three terms in the r.h.s. of Eq. (1) describe the local terms and the nonlinearities on each site. Inside the latter brackets, the first term is the photon hopping, while the second term describes the coupling of the atom to its right neighboring cavity, which generates an effective nonlocal cross-Kerr nonlinearity between the two cavities.\n\nHereafter we concentrate on the 1D model in Eq. (1) at zero temperature, specifically addressing the case without dissipation with DMRG. Let us also fix the Hamiltonian quantities in units of Ω, set $$\\hbar= 1$$, and work with open boundary conditions. We recall that, in the presence of dissipation, the problem becomes much more difficult to be handled numerically.Footnote 1\n\nFor the system we are considering here, in the strong coupling regime atoms and photons cannot be considered as two separate entities. It is thus natural to investigate the phase diagram in terms of combined atomic/photonic modes, named polaritons. The polaritonic number operator on each site i, representing the number of local excitations, is defined as $$n^{\\mathrm{pol}}_{i}=2\\sigma_{i}^{44}+\\sigma^{33}_{i}+\\sigma ^{22}_{i}+a^{\\dagger}_{i}a_{i}$$. For the closed system described by the Hamiltonian (1), the total number $$N^{\\mathrm{pol}} = \\sum_{i} n^{\\mathrm{pol}}_{i}$$ of such polaritons is a conserved quantity. In the following we work in the canonical ensemble for polaritons, and focus on the integer filling situation.\n\n## Energy gaps and correlation functions\n\nThe different nature of the various phases is sensitive to a number of properties which we are going to focus on. Here we are going to study quantities that resemble those characterizing the various phases of the EBH model .\n\nFirst of all, the ground-state energy gap is an important indicator which characterizes the presence or absence of criticality in the model. In particular, in the critical SF phase, the charge gap vanishes in the thermodynamic limit. On the other side, in the insulating MI and DW phases, such gap remains finite. In order to make connection with a similar notation in the EBH model, below we introduce the so called charge and neutral gaps referring respectively to the gaps corresponding to adding one extra particle (‘charge’ sector) or remaining with the same number of particles (‘neutral’ sector). We stress however that in the present model the excitation carry no real charge. This has to be understood only as a convention.\n\nThe charge gap is defined as\n\n$$\\Delta E_{c} = \\Delta E^{+} - \\Delta E^{-} ,$$\n(2)\n\nwhere, in the canonical ensemble, $$\\Delta E^{+}$$ ($$\\Delta E^{-}$$) denotes the extra energy needed to add (remove) one particle, i.e. one polariton, in the system. In the specific, focusing on the unit filling, $$\\Delta E^{+} = E_{L+1}-E_{L}$$ and $$\\Delta E^{-} = E_{L}-E_{L-1}$$, where $$E_{L}$$ is the ground-state energy per site of an L-sites cavity-array with exactly L excitations, and $$E_{L+1}$$ ($$E_{L-1}$$) is the corresponding energy per site with one excitation more (less). It is therefore possible to extrapolate $$\\Delta E_{c}$$ by running three different DMRG simulations with fixed number of polaritons $$N^{\\mathrm{pol}} = L-1, L, L+1$$ [52, 53].\n\nWhile the charge gap is able to detect particle-hole excitations, in some circumstances it is possible that the dominant low-energy excitations are of a different type. Their presence can be detected only by the so called neutral gap at a fixed number of particles,\n\n$$\\Delta E_{n} = E_{L}^{1} - E_{L} ,$$\n(3)\n\nwhere, again working in the canonical ensemble, $$E_{L}^{1}$$ denotes the first excited energy per site of an L-site system with L excitations.\n\nIn the following, we also focus on the analysis of the staggered diagonal order for the polaritons, in order to distinguish the DW from the other phases. We do this by investigating the two-point correlation function\n\n$$C_{\\mathrm{DW}}(r)=(-1)^{r} \\bigl\\langle \\delta n_{i}^{\\mathrm{pol}} \\delta n_{i+r}^{\\mathrm{pol}} \\bigr\\rangle ,$$\n(4)\n\nwhere $$\\delta n_{i}^{\\mathrm{pol}} = n^{\\mathrm{pol}}_{i}-\\bar{n}$$ denotes the polariton fluctuation from the average filling $$\\bar{n}$$. The order parameter identifying the DW phase is thus given by: $$\\mathcal{O}_{\\mathrm{DW}}\\equiv\\lim_{r\\rightarrow\\infty }{C_{\\mathrm{DW}}(r)}$$. A finite value of $$\\mathcal{O}_{\\mathrm{DW}}$$ indicates a tendency to establish, in the thermodynamic limit, a staggered occupation of the polaritons. On the other side, in the MI as well as the SF phases, $$C_{\\mathrm{DW}}(r)$$ vanishes exponentially with increasing distance r.\n\n## Phase diagram\n\nThe zero-temperature phase diagram of model (1), at unit polariton filling $$\\bar{n} = 1$$ and in the $$g_{2}-t$$ plane, is summarized in Figure 2. We observe that three different phases can be stabilized. Their boundaries have been obtained by means of a finite-size scaling of the numerical data, for systems up to $$L=300$$ sites. In our simulations we imposed a cutoff photon number in each cavity, such that $$n^{\\mathrm{phot}}_{i} \\leq3$$. We also truncated the effective Hilbert space dimension to a value $$m=80$$ in all the simulations, except for those shown in Figure 6 for the neutral energy gap (see the discussion in Section 4.3). We checked that, by increasing m and the local fock-space truncation over the photon number, the results concerning the charge gap and the DW order parameter do not change on the scales shown here.\n\nFor small photon hopping ($$t / \\Omega\\lessapprox0.2$$), by increasing the nonlocal nonlinearity $$g_{2}$$ the system exhibits a direct transition from the MI to the DW phase. On the other hand, for $$t / \\Omega\\gtrapprox0.2$$, the MI-to-DW transition is mediated by an extended region appearing at intermediate $$g_{2}$$ values, where the system stabilizes into a gapless SF. In the following we are going to elucidate our finite-size scaling procedure and how we were able to distinguish between the different phases.\n\n### Boundary between MI and SF phases\n\nIn the limit of small $$g_{2}$$ and t values, the dominant presence of the on-site interactions stabilize the system into a MI phase with exactly one polariton per cavity ($$\\bar{n} = 1$$), and where the charge energy gap has a finite value. As long as the hopping strength t is progressively increased (and for fixed $$g_{1}$$, $$g_{2}$$), the system eventually enters a SF phase, with a vanishing gap. The filled squares of Figure 2 denoting the MI/SF boundaries have been obtained by means of a finite-size scaling of the charge gap. We performed simulations up to $$L=100$$ sites and analyzed whether the gap closes or remains finite in the thermodynamic limit $$L \\to\\infty$$.\n\nIn Figure 3, left panel, we highlight the size-dependence of $$\\Delta E_{c}$$ as a function of $$1/L$$ for two points in the phase space close to the MI/SF transition (see points along the dotted line in Figure 2). We expect to see a quadratic dependence $$\\Delta E_{c} \\sim L^{-2}$$ (dashed line) at large L [52, 53], however a linear extrapolation (solid line) is already a good approximation to the scaling, and we can use it to determine $$\\Delta E_{c}$$ in the thermodynamics limit. Indeed, we observe that the difference between quadratic and linear extrapolation is tiny (10−3) and does not produce any distinguishable modification on the scale of Figure 2. In the specific case of Figure 3, we fixed $$t / \\Omega= 0.25$$ and chose two different values of $$g_{2} / \\Omega$$ corresponding to configurations in the gapped MI ($$g_{2} / \\Omega= 1.35$$, triangles) and in the gapless SF phase ($$g_{2} / \\Omega= 1.5$$, squares). The MI is signaled by an extrapolated finite value of $$\\lim_{L \\to \\infty} \\Delta E_{c} > 0$$, while in the SF this is zero.\n\nIn order to locate the critical $$g_{2}$$ for a given value of t (filled squares in Figure 2) we perform a linear extrapolation of the charge gaps in the vicinity of the critical value of $$g_{2}$$. An example of such procedure is shown in the right panel of Figure 3, where we plot $$\\Delta E_{c}$$ as a function of $$g_{2}$$, when this is close to the phase transition (in the specific, here we set $$t / \\Omega= 0.25$$). After a linear extrapolation, we get a critical $$g_{2}$$ value corresponding to $$g_{2}^{} / \\Omega\\approx1.379$$. An analogous procedure is repeated for all the filled squares shown in Figure 2, thus identifying the MI/SF boundary.\n\n### Boundary of the DW phase\n\nThe DW phase is characterized by a finite order parameter $${\\mathcal{O}}_{\\mathrm{DW}}$$. Let us therefore look at the two-point staggered correlator in Eq. (4). Since in DMRG simulations we are employing open boundary conditions, to minimize the border effects we analyze the correlations in such a way that the two points are taken symmetrically with respect to the center of the system.Footnote 2 The left panel of Figure 4 shows how differently such polariton correlations behave when the system goes from the MI to DW phase, for a fixed system size.\n\nTo be more accurate, in the right panel we performed a finite-size scaling and showed that the staggered correlation $$C_{\\mathrm{DW}}(r)$$ approaches the zero value exponentially with L, in the MI phase (a similar behavior occurs in the SF region). On the other hand, in the DW such correlator asymptotically converges to a finite value. In the specific, here we fix $$t/\\Omega=0.05$$ and show that for $$g_{2}/\\Omega= 1.3, 1.35$$ the DW order is exponentially suppressed with L, while for $$g_{2}/\\Omega= 1.4$$ it remains finite. The $${\\mathcal{O}}_{\\mathrm{DW}}$$ order parameter reached for $$L \\to \\infty$$ is displayed in the inset as a function of $$g_{2}$$.\n\nIn order to determine the DW boundary in the phase diagram of Figure 2, we adopted the following protocol. For a fixed value of $$t/\\Omega$$, we start increasing $$g_{2}$$ from zero up to a finite value, with a fixed increment $$\\delta g_{2} = 0.05 \\Omega$$, and to compute the DW order parameter for all such values of $$g_{2}$$. The boundary of DW phase in the $$g_{2}-t$$ plane (filled circles in Figure 2), for any fixed t, is located by the $$g_{2}^{}(t)$$ that gives the first non-vanishing order parameter $$\\mathcal{O}_{\\mathrm{DW}}$$.\n\nHere we stress that, because of the arrangement of our 1D array [see Figure 1(a)] and of the asymmetric coupling between the atom and its right/left cavity, the antiferromagnetic symmetry of the system is spontaneously broken. In particular, the state $$\\vert 4 \\rangle$$ of the L-th atom will be never occupied, since the transition $$\\vert 2 \\rangle \\leftrightarrow \\vert 4 \\rangle$$ does not couple to any cavity mode [see Eq. (1)]. As a consequence, in our simulations we do not need any symmetry-breaking potential. We can observe that the expectation value for the onsite number of polaritons explicitly exhibits a staggered behavior, in that the occupation of the $$(2n-1)$$-th site is always higher than that of the $$(2n)$$-th site (for any integer value of n). Finally we notice that such staggering persists at finite size, also for the set of parameters corresponding to the MI phase, although it is extremely tiny and decreases with L. This effect eventually disappears in the thermodynamic limit.\n\nThe extension of the DW phase depends on the cavity detuning Δ. In particular, the robustness of the order parameter increases with increasing the modulus of the detuning (see Figure 5). Quite remarkably, we note that a positive Δ will never stabilize an antiferromagnetic DW ordering.\n\n### Neutral gap\n\nThe analysis leading to the phase diagram in Figure 2 has been corroborated by a study of the neutral gap, which vanishes both in proximity of the phase transitions and in the entire superfluid region. Differently for the charge gap, it is able to detect the presence of excitations other than particle-hole, and thus locates the boundaries of insulating regions (as the DW) beyond the MI.\n\nThe data displayed in Figure 6 show the behavior of $$\\Delta E_{n}$$ as a function of $$g_{2}$$, for a fixed value of $$t / \\Omega$$. In particular we analyzed a vertical cut in the phase diagram of Figure 2 (see the rightmost vertical dotted line in that figure), where the system can be in three different phases according to the value of $$g_{2}$$. With increasing $$g_{2}$$, it goes from the MI phase (nonzero $$\\Delta E_{n}$$, for $$t/\\Omega\\lesssim1.45$$) to the SF phase (zero $$\\Delta E_{n}$$, for $$1.45 \\lesssim t/\\Omega \\lesssim1.8$$), and then to the DW phase (nonzero $$\\Delta E_{n}$$, for $$t/\\Omega\\gtrsim1.8$$).\n\nWhile we cannot see a clear signature of a finite gap for $$g_{2} = 1.8 \\Omega$$, the scaling with the size displayed in the left panel of Figure 6 seems to suggest the scenario depicted above. It is however important to stress that the DMRG simulations needed to compute the neutral gap have to target the two lowest-lying eigenstates in a single run. Thus they generally require a larger dimension m of the effective Hilbert space, as compared to all the other ground-state calculations discussed before. The analysis of the neutral gap requires a careful convergence test of the results with m, which we provide in the right panel of Figure 6. We observe that the non monotonic features that are visible in the region $$1.45 \\lesssim t/\\Omega\\lesssim1.8$$ have to be probably ascribed to the inaccuracy of the method at small m values. This signals the presence of the gapless SF phase there, in agreement with the results provided by the charge gap (MI/SF boundary) and for the DW order parameter (SF/DW boundary).\n\n## Summary\n\nUsing the density-matrix renormalization group with open boundary conditions, we studied the equilibrium phase diagram of a system of coupled QED cavities in one dimension. We provided results beyond the standard model of couplings through photon hopping, and also considered nearest-neighbor cross-Kerr nonlinearities. Our analysis is based on a finite-size scaling of the ground-state charge and neutral gaps, as well as of the density-wave order parameter, for systems up to 300 sites. We showed that, beyond the conventional Mott insulator and superfluid phases, the presence of a nearest-neighbor nonlinear coupling can also stabilize a density-wave ordering of polaritons.\n\n1. 1.\n\nIt is however possible to address the effect of dissipation with a DMRG approach in a 1D chain, when this is described by a master equation within the Lindblad formalism. In the language of tensor networks, one has to generalize the matrix-product-state ansatz to a matrix-product-density-operator ansatz for mixed states, as originally proposed in Refs. [50, 51]. 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Mixed-state dynamics in one-dimensional quantum lattice systems: a time-dependent superoperator renormalization algorithm. Phys Rev Lett. 2004;93:207205.\n\n52. 52.\n\nKühner TD, Monien H. Phases of the one-dimensional Bose-Hubbard model. Phys Rev B. 1998;58:R14741.\n\n53. 53.\n\nKühner TD, White SR, Monien H. One-dimensional Bose-Hubbard model with nearest-neighbor interaction. Phys Rev B. 2000;61:12474.\n\n## Acknowledgements\n\nWe would like to acknowledge our previous collaboration with M. Hartmann and M. Leib which was inspiring for the present work. This work was supported by Italian MIUR via FIRB Project RBFR12NLNA and PRIN Project 2010LLKJBX, by EU through IP-SIQS, and by National Natural Science Foundation of China under Grant No. 11175033 and No. 11305021.\n\n## Author information\n\nCorrespondence to Jiasen Jin.\n\n### Competing interests\n\nThe authors declare that they have no competing interests.\n\n### Authors’ contributions\n\nAll the authors participated in the design of the research, analysis of the results, and writing of the paper. The DMRG code used to run all the simulations of this research has been developed and written by DR and coworkers (see also www.dmrg.it). The DMRG simulations were performed by JJ.\n\n## Rights and permissions\n\nOpen Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.\n\nReprints and Permissions", null, "" ]	[ null, "https://epjquantumtechnology.springeropen.com/track/article/10.1140/epjqt/s40507-015-0018-0", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.85372156,"math_prob":0.9842639,"size":34554,"snap":"2019-43-2019-47","text_gpt3_token_len":8886,"char_repetition_ratio":0.12231548,"word_repetition_ratio":0.02154365,"special_character_ratio":0.2572206,"punctuation_ratio":0.14940676,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.994853,"pos_list":[0,1,2],"im_url_duplicate_count":[null,5,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-18T02:07:12Z\",\"WARC-Record-ID\":\"<urn:uuid:c38ed2f3-7511-418c-ac58-ccd3fb54fbcc>\",\"Content-Length\":\"263603\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:94a730d6-8a75-4fba-8937-51abd6d5749b>\",\"WARC-Concurrent-To\":\"<urn:uuid:575d8b9c-1ac4-439c-9c8b-762c0fb07388>\",\"WARC-IP-Address\":\"151.101.200.95\",\"WARC-Target-URI\":\"https://epjquantumtechnology.springeropen.com/articles/10.1140/epjqt/s40507-015-0018-0\",\"WARC-Payload-Digest\":\"sha1:GWHGT23STZKOAZD6BODRXEJ2B24NOK6Z\",\"WARC-Block-Digest\":\"sha1:M2QPXMIX36TUV3PIOQNOZGYIOH4MJCCH\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496669422.96_warc_CC-MAIN-20191118002911-20191118030911-00162.warc.gz\"}"}
http://drorbn.net/dbnvp/LD15_Kuno-1.php	[ "© \| Dror Bar-Natan: Talks: LesDiablerets-1508: < >\n\n# Kuno's Talk 1\n\n width: 400 720 1280 download ogg/LD15_Kuno-1_400.ogg For now, this video can only be viewed with web browsers that support\n\nNotes on LD15_Kuno-1: [edit, refresh]\n\nrefresh\npanel\nManaged by dbnvp: The small-font numbers on the top left of the videos indicate the available native resolutions. Click to test.", null, "0:05:35 The mapping class group and $\\pi_1$.", null, "0:09:25 The Johnson filtration.", null, "0:12:50 The Johnson filtration, 2.", null, "0:14:50 The Johnson homomorphism. In the Artin case, this is the action of the Drinfel'd-Kohno Lie algebra on the free Lie algebra.", null, "0:19:35 A geometric construction.", null, "0:22:55 This seems to be the theorem on page 7 of Kawazumi's notes. The completion $\\widehat{{\\mathbb Q}\\pi}$ is relative to powers $(I\\pi)^p$ of the augmentation ideal. The completion $\\widehat{{\\mathbb Q}\\hat\\pi}$ is relative to ${\\mathbb Q}{\\mathbb 1}+\|(I\\pi)^p\|$.", null, "0:23:18 According to page 8 of Kawazumi's notes, $\\tau$ is related to the Massuyeau Johnson map.", null, "0:25:34 $\\tau$ and simple closed curves.", null, "0:31:06 $\\tau(t_C) = L(C) = \\left\|\\frac12(\\log x)^2\\right\|$", null, "0:34:32 Proof of $\\tau(t_C) = L(C) = \\left\|\\frac12(\\log x)^2\\right\|$.", null, "0:38:03 Generalized Dehn twists.", null, "0:42:52 Given a symplectic expansion $\\theta$...", null, "0:48:00 $\\delta\\circ\\tau=0$.", null, "0:50:55 $\\delta\\circ\\tau=0$, part II. What's the geometry behind this?", null, "0:57:52 Is there $\\theta$ such that $\\delta^\\theta=\\delta^{alg}$?" ]	[ null, "http://drorbn.net/dbnvp/frames/LD15_Kuno-1@0-05-35.jpg", null, "http://drorbn.net/dbnvp/frames/LD15_Kuno-1@0-09-25.jpg", null, "http://drorbn.net/dbnvp/frames/LD15_Kuno-1@0-12-50.jpg", null, "http://drorbn.net/dbnvp/frames/LD15_Kuno-1@0-14-50.jpg", null, "http://drorbn.net/dbnvp/frames/LD15_Kuno-1@0-19-35.jpg", null, "http://drorbn.net/dbnvp/frames/LD15_Kuno-1@0-22-55.jpg", null, "http://drorbn.net/dbnvp/frames/LD15_Kuno-1@0-23-18.jpg", null, "http://drorbn.net/dbnvp/frames/LD15_Kuno-1@0-25-34.jpg", null, "http://drorbn.net/dbnvp/frames/LD15_Kuno-1@0-31-06.jpg", null, "http://drorbn.net/dbnvp/frames/LD15_Kuno-1@0-34-32.jpg", null, "http://drorbn.net/dbnvp/frames/LD15_Kuno-1@0-38-03.jpg", null, "http://drorbn.net/dbnvp/frames/LD15_Kuno-1@0-42-52.jpg", null, "http://drorbn.net/dbnvp/frames/LD15_Kuno-1@0-48-00.jpg", null, "http://drorbn.net/dbnvp/frames/LD15_Kuno-1@0-50-55.jpg", null, "http://drorbn.net/dbnvp/frames/LD15_Kuno-1@0-57-52.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.676739,"math_prob":0.9970307,"size":914,"snap":"2019-43-2019-47","text_gpt3_token_len":311,"char_repetition_ratio":0.12307692,"word_repetition_ratio":0.015503876,"special_character_ratio":0.35667396,"punctuation_ratio":0.15656565,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9971815,"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],"im_url_duplicate_count":[null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-22T00:51:13Z\",\"WARC-Record-ID\":\"<urn:uuid:12ee63ae-c101-4740-bf9b-6e0f2c29a867>\",\"Content-Length\":\"19129\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4dfb2711-7adb-4af1-bf2d-fc92773f27ac>\",\"WARC-Concurrent-To\":\"<urn:uuid:03f3861c-f551-4c9b-b5f5-d545923c91e8>\",\"WARC-IP-Address\":\"128.100.68.50\",\"WARC-Target-URI\":\"http://drorbn.net/dbnvp/LD15_Kuno-1.php\",\"WARC-Payload-Digest\":\"sha1:UY64XWMU7SWJOST4LLWUAMOYQMFA5MIT\",\"WARC-Block-Digest\":\"sha1:QDCDX3NEZNZUQDNHUK2MYD7VCEXJMHF5\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496671053.31_warc_CC-MAIN-20191121231600-20191122015600-00181.warc.gz\"}"}
https://web2.0calc.com/questions/in-how-many-different-ways-can-2-15-be-represented	[ "+0\n\nIn how many different ways can 2/15 be represented as 1/a + 1/b, if a and b are positive integers with a>b\n\n+1\n245\n2\n\nIn how many different ways can 2/15 be represented as 1/A + 1/B, if A and B are positive integers with A less than or equal to B\n\nJan 21, 2019\nedited by Guest Jan 21, 2019\n\n#1\n+1\n\nIs this what you mean?\n\n1/8 + 1 / 120 = 2/15\n1/9 + 1 / 45 = 2/15\n1/10 + 1 / 30 = 2/15\n1/12 + 1 / 20 = 2/15\n\n1/15 + 1/15 = 2/15\n\nJan 22, 2019\nedited by Guest Jan 22, 2019\n#2\n+8\n\nIn how many different ways can 2/15 be represented as 1/A + 1/B,\n\nif A and B are positive integers with A less than or equal to B\n\n$$\\text{For odd n>2 there is always at least one decomposition into exactly two unit fractions: \\dfrac{2}{n} = \\dfrac{1}{A} + \\dfrac{1}{B} } \\\\ \\text{Finding all possibilities.}\\\\ \\text{The prime factorization of n^2 results in all possible decompositions into two unit fractions.}$$\n\n$$n=15\\ \\text{is odd} \\\\ n^2 =225 \\\\ \\text{All divisors of n^2=225 are: 1,\\ 3,\\ 5,\\ 9,\\ 15,\\ 25,\\ 45,\\ 75,\\ 225}\\\\ \\text{Let n^2=p\\times q }$$\n\n$$\\begin{array}{\|r\|r\|r\|c\|c\|c\|c\|c\|c\|c\|c\|c\| } \\hline & p & q & n^2 & s & t & r & k & A & B & A\\le B & \\\\ & & & =p\\cdot q & = \\frac{p+q}{2} & = \\frac{p-q}{2} & =\\frac{t}{2} & = \\frac{15+\\sqrt{15^2+t^2} }{2} & =k-r & =k+r & \\\\ \\hline 1. & 225 & 1 & 225=225 \\cdot 1 & 113 & 112 & 56 & 64 & 8 & 120 & \\checkmark & \\mathbf{\\dfrac{2}{15} = \\dfrac{1}{8} + \\dfrac{1}{120}} \\\\ \\hline 2. & 75 & 3 & 225= 73 \\cdot 3 & 39 & 36 & 18 & 27 & 9 & 45 & \\checkmark & \\mathbf{\\dfrac{2}{15} = \\dfrac{1}{9} + \\dfrac{1}{45}} \\\\ \\hline 3. & 45 & 5 & 225= 45 \\cdot 5 & 25 & 20 & 10 & 20 & 10 & 30 & \\checkmark & \\mathbf{\\dfrac{2}{15} = \\dfrac{1}{10} + \\dfrac{1}{30}} \\\\ \\hline 4. & 25 & 9 & 225= 25 \\cdot 9 & 17 & 8 & 4 & 16 & 12 & 20 & \\checkmark & \\mathbf{\\dfrac{2}{15} = \\dfrac{1}{12} + \\dfrac{1}{20}} \\\\ \\hline 5. & 15 & 15 & 225= 15 \\cdot 15 & 15 & 0 & 0 & 15 & 15 & 15 & \\checkmark & \\mathbf{\\dfrac{2}{15} = \\dfrac{1}{15} + \\dfrac{1}{15}} \\\\ \\hline \\end{array}$$", null, "Jan 22, 2019\nedited by heureka Jan 22, 2019\nedited by heureka Jan 22, 2019" ]	[ null, "https://web2.0calc.com/img/emoticons/smiley-laughing.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5429228,"math_prob":0.9999355,"size":1765,"snap":"2019-43-2019-47","text_gpt3_token_len":829,"char_repetition_ratio":0.15388983,"word_repetition_ratio":0.1589041,"special_character_ratio":0.5858357,"punctuation_ratio":0.053333335,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9987674,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-14T14:20:34Z\",\"WARC-Record-ID\":\"<urn:uuid:cf3f1b24-6f85-4dea-912c-19556981f7f6>\",\"Content-Length\":\"25359\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:dad4f558-44e7-4af9-a2a7-7ada22502b40>\",\"WARC-Concurrent-To\":\"<urn:uuid:ce3e970b-4363-4f9e-bdad-cfa5e836059f>\",\"WARC-IP-Address\":\"144.76.186.3\",\"WARC-Target-URI\":\"https://web2.0calc.com/questions/in-how-many-different-ways-can-2-15-be-represented\",\"WARC-Payload-Digest\":\"sha1:INR6HDJIFAEQF7HR6DXSYF7HOUNYOBQO\",\"WARC-Block-Digest\":\"sha1:CB3O6YEQFKZSPI7Q3KO3PQZVQHFEZWCF\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570986653247.25_warc_CC-MAIN-20191014124230-20191014151730-00507.warc.gz\"}"}
https://pennstate.pure.elsevier.com/en/publications/improved-bounds-for-the-nystr%C3%B6m-method-with-application-to-kernel	[ "# Improved bounds for the nyström method with application to kernel classification\n\nRong Jin, Tianbao Yang, Mehrdad Mahdavi, Yu Feng Li, Zhi Hua Zhou\n\nResearch output: Contribution to journalArticlepeer-review\n\n25 Citations (SciVal)\n\n## Abstract\n\nWe develop two approaches for analyzing the approximation error bound for the Nyström method that approximates a positive semidefinite (PSD) matrix by sampling a small set of columns, one based on a concentration inequality for integral operators, and one based on random matrix theory. We show that the approximation error, measured in the spectral norm, can be improved from O(N/√ m) to O(N/m1-1) in the case of large eigengap, where N is the total number of data points, m is the number of sampled data points, and \\rho \\in (0, 1/2) is a positive constant that characterizes the eigengap. When the eigenvalues of the kernel matrix follow a p-power law, our analysis based on random matrix theory further improves the bound to O(N/mp-1) under an incoherence assumption. We present a kernel classification approach based on the Nyström method and derive its generalization performance using the improved bound. We show that when the eigenvalues of the kernel matrix follow a p-power law, we can reduce the number of support vectors to N2p/(p2-1) which is sublinear in N when p > 1+√2, without seriously sacrificing its generalization performance.\n\nOriginal language English (US) 6547995 6939-6949 11 IEEE Transactions on Information Theory 59 10 https://doi.org/10.1109/TIT.2013.2271378 Published - 2013\n\n## All Science Journal Classification (ASJC) codes\n\n• Information Systems\n• Computer Science Applications\n• Library and Information Sciences\n\n## Fingerprint\n\nDive into the research topics of 'Improved bounds for the nyström method with application to kernel classification'. Together they form a unique fingerprint." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8451339,"math_prob":0.56983507,"size":1511,"snap":"2021-43-2021-49","text_gpt3_token_len":350,"char_repetition_ratio":0.1128069,"word_repetition_ratio":0.051948052,"special_character_ratio":0.22700198,"punctuation_ratio":0.06130268,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95459753,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-12-01T14:00:27Z\",\"WARC-Record-ID\":\"<urn:uuid:f5627fa9-9a1e-4d0d-958f-e847d3e0983a>\",\"Content-Length\":\"46820\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:73266a66-ce39-428a-862b-043a228c2d82>\",\"WARC-Concurrent-To\":\"<urn:uuid:4ba65091-601d-42b3-8fe9-5048c8ca9bce>\",\"WARC-IP-Address\":\"3.90.122.189\",\"WARC-Target-URI\":\"https://pennstate.pure.elsevier.com/en/publications/improved-bounds-for-the-nystr%C3%B6m-method-with-application-to-kernel\",\"WARC-Payload-Digest\":\"sha1:22SAOJKGBAISIE2I624RKQ6TRLKPHB7G\",\"WARC-Block-Digest\":\"sha1:K3MD435ZZVRJ6GJPN7BQMFFZWTRRBIZR\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964360803.0_warc_CC-MAIN-20211201113241-20211201143241-00085.warc.gz\"}"}
https://arcadianfunctor.wordpress.com/2007/09/25/m-is-for-magic/	[ "## M is for Magic\n\nAs we have seen, Carl Brannen’s QFT uses circulant matrices. By resetting a mass scale, one may renormalise a 1-circulant\n\nXYZ\nZXY\nYZX\n\nby a constant $\\lambda = \\frac{1}{Y + Z – 2X}$ so that the resulting circulant obeys the condition $2X = Y + Z$. This turns the circulant into a magic square. For 2-circulants the condition is instead $2Z = X + Y$. Although perhaps not a very useful observation, it is certainly entertaining! The total number of $5 \\times 5$ normal (ie. matrices built from the first few ordinals) magic squares was only computed in 1973, and the number of $6 \\times 6$ ones is still unknown. There is only one $3 \\times 3$ normal square, up to rotation and reflection.\n\nA paper by A. Adler uses circulants to find an algorithm for generating higher order normal magic n-cubes, by playing with p-adic L functions. For $p = 3$, Adler constructs two cute normal magic cubes: a $3 \\times 3 \\times 3$ cube and a $27 \\times 27 \\times 27$ cube. I was further intrigued by this paper of Adler’s, containing the conjecture that magic n-cubes always form a free monoid. It shows first that sets of magic squares contain prime squares, out of which all others are constructed, and then that generating functions built from cardinalities for magic cubes have the remarkable property of being everywhere divergent!\n\n## 2 Responses so far »\n\n1. 1", null, "### Matti Pitkanen said,\n\nThis monoid property of magic would be nice. The divergence of generating function as real function everywhere would mean that generating function for magic squares would exist as p-adic number for some p:s (probably very many). This would give additional strong support for the idea that p-adic generating functions are more natural than those with real argument.\n\n2. 2", null, "### Kea said,\n\nHi Matti. Unfortunately, as Adler writes and Google indicates, it appears that little research is being done on magic squares." ]	[ null, "https://0.gravatar.com/avatar/", null, "https://0.gravatar.com/avatar/", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9219846,"math_prob":0.9958174,"size":1900,"snap":"2021-43-2021-49","text_gpt3_token_len":468,"char_repetition_ratio":0.12447257,"word_repetition_ratio":0.0062305294,"special_character_ratio":0.24210526,"punctuation_ratio":0.10354223,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9808083,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-24T15:26:21Z\",\"WARC-Record-ID\":\"<urn:uuid:96d518b8-b164-471c-99ec-27e32ffd13e3>\",\"Content-Length\":\"78343\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:6d555c49-277b-492c-8179-7125ce2e05dd>\",\"WARC-Concurrent-To\":\"<urn:uuid:a88a565f-7d2b-4a6a-a4b2-2760fc0f0b0e>\",\"WARC-IP-Address\":\"192.0.78.12\",\"WARC-Target-URI\":\"https://arcadianfunctor.wordpress.com/2007/09/25/m-is-for-magic/\",\"WARC-Payload-Digest\":\"sha1:DF4DXNZ5CUI6PJUOP5W6UHJKM5TYAC7Z\",\"WARC-Block-Digest\":\"sha1:KL5ITVLUICVDHDVI3NR6EEJVBXTZTE3G\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323586043.75_warc_CC-MAIN-20211024142824-20211024172824-00008.warc.gz\"}"}
https://financial-calculators.com/author/morris	[ "The Rule-of-78s : The Formula and How It Works\n\nOutside of banking circles, the Rule-of-78′s is little understood, even though it is commonly applied to many consumer and business loans. For the borrower, it tends to have a pernicious effect in the nature of a hidden prepayment penalty. The borrower’s disadvantage is heightened by the fact that the operation of the Rule-of-78′s is often referred to as a \"Rebate of the Finance Charge.\" Any consumer who heard the word \"rebate\" is always tempted to say, \"Where do I sign?\"\n\nNot so fast!\n\nHere’s how it works: The name comes from the sum of the numbers one through 12, there being 12 months in a year. Yes, that adds to 78.\n\nThe theory of the Rule-of-78′s is that at the moment a borrower signs the Note, the borrower is immediately obligated to pay back all of the principal and ALL of the interest that will accrue in the future over the agreed term of the loan.\n\nNow, if the borrower prepays, the lender \"generously\" forgives some of the interest EVEN THOUGH at the time for it to accrue has not yet elapsed and so that additional interest has not been earned. That’s the so-called \"Rebate.\" Lenders argue that the uncertainty created about an early payoff entitles them to some compensation for being at the borrower’s whim for payoff. In a time of falling interest rates, that argument may have more merit than when interest rates are rising as the lender gets to put the money back to work at a higher rate and earn more.\n\nIn any event, the Rebate is calculated by summing the number of payments elapsed in inverse order as a numerator for the fraction in which the sum of the term is the denominator. That fraction times all interest over the life of the loan is the amount earned by the lender.\n\nWatch this example:\n\nAssume a two-year loan (so we’ll assume the numbers 1 through 24) for \\$10,000 with interest at 12% per year. Using our online amortization schedule calculator, we know the monthly payment is \\$470.73. The amortization schedule’s \"Running Totals\" also tells us that over the life of the loan the total amount of aggregate interest to be paid would be \\$1,297.56 (when the \"1st payment date\" is one month after the \"loan date\").\n\nAfter the fourth month, our borrower reaps a windfall and wants to prepay the whole loan. The fraction of the total interest earned by the lender is:\n\n(Sum 24 to 21) over (Sum 1 through Sum 24)\n\n90/300 = 30%\n\nNow, let’s compare that to the interest actually paid to date to see what the penalty will be. From running the \"Loan Table\" module, we found the total interest to be \\$1,297.65 and the interest paid after four payments is \\$377.61, so the penalty is:\n\nEarned Interest Per Rule: (30%) (\\$1,297.65) = \\$389.30\n\nInterest Paid to Payoff: \\$377.61\n\nAdditional Interest Owed: \\$11.69\n\nMaybe that doesn’t look like too big a number, but it’s an additional 3.1% interest. Had this been a \\$100,000 loan, the increased penalty works out to ten times as much, \\$116.84.\n\nPaying off at different times for different maturities and different interest rates produces differing penalty sizes. Two general rules of thumb can be deduced:\n\n1. The higher the interest rate, the greater the penalty amount.\n\n2. The earlier the prepayment in relation to the term, the greater the penalty amount.\n\nSo if you’re a lender, you should love using the Rule-of-78′s. If you’re a borrower, you should try to avoid it. A caution for lenders: Some states have usury and other laws that may limit use of the Rule-of-78′s." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.94582736,"math_prob":0.92328054,"size":3671,"snap":"2019-43-2019-47","text_gpt3_token_len":871,"char_repetition_ratio":0.13744205,"word_repetition_ratio":0.0093896715,"special_character_ratio":0.24652684,"punctuation_ratio":0.109375,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95160276,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-21T20:42:36Z\",\"WARC-Record-ID\":\"<urn:uuid:1c5cf001-08ab-45bd-8414-3deec273aed5>\",\"Content-Length\":\"27677\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f9823b10-cc10-4db2-9511-330c89ab9de3>\",\"WARC-Concurrent-To\":\"<urn:uuid:14b3057b-810f-4d00-b373-d44fb4432e5a>\",\"WARC-IP-Address\":\"194.1.147.58\",\"WARC-Target-URI\":\"https://financial-calculators.com/author/morris\",\"WARC-Payload-Digest\":\"sha1:WRQMFEXZJJVKA3H3YYGV3TGFNINKD3OU\",\"WARC-Block-Digest\":\"sha1:DL6HORDJGFRSUPOGQHANV66W5BQHSIMW\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570987787444.85_warc_CC-MAIN-20191021194506-20191021222006-00337.warc.gz\"}"}
https://faculty.math.illinois.edu/Macaulay2/doc/Macaulay2-1.19/share/doc/Macaulay2/Graphs/html/_nonneighbors.html	[ "# nonneighbors -- returns the non-neighbors of a vertex in a graph\n\n## Synopsis\n\n• Usage:\nN = nonneighbors(G,v)\n• Inputs:\n• G, an instance of the type Graph,\n• v, ,\n• Outputs:\n• N, a set, the non-neighbors of vertex v in graph G\n\n## Description\n\nThe non-neighbors of a vertex v are all the vertexSet of G that are not adjacent to v. That is, if u is a non-neighbor to v, {u,v} is not an edge in G.\n\n i1 : G = graph({1,2,3,4},{{2,3},{3,4}}); i2 : nonneighbors(G,2) o2 = set {1, 4} o2 : Set" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.86037874,"math_prob":0.97917247,"size":290,"snap":"2023-14-2023-23","text_gpt3_token_len":105,"char_repetition_ratio":0.13636364,"word_repetition_ratio":0.0,"special_character_ratio":0.39310345,"punctuation_ratio":0.22619048,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98799485,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-05-30T03:44:39Z\",\"WARC-Record-ID\":\"<urn:uuid:02bb2e4c-faf3-404c-8651-c9219f68d61e>\",\"Content-Length\":\"5449\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0bf3e34d-07d9-4bd3-aa8d-ed686e1a83d0>\",\"WARC-Concurrent-To\":\"<urn:uuid:bd996e5f-9a88-481c-9cce-eb157ea86db2>\",\"WARC-IP-Address\":\"128.174.199.46\",\"WARC-Target-URI\":\"https://faculty.math.illinois.edu/Macaulay2/doc/Macaulay2-1.19/share/doc/Macaulay2/Graphs/html/_nonneighbors.html\",\"WARC-Payload-Digest\":\"sha1:S6H4OJTXYRW4PZRW7JE36HEZU2IWQOQX\",\"WARC-Block-Digest\":\"sha1:WSFSHFTAJ5CLYT3QSOBINCLAMN5L2ZLY\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-23/CC-MAIN-2023-23_segments_1685224645089.3_warc_CC-MAIN-20230530032334-20230530062334-00398.warc.gz\"}"}
https://afd-wandsbek.de/site/article.php?e62e8c=clustering-with-scikit-with-gifs	[ "considers at each step all the possible merges. The two farthest subclusters are taken and for centroids to be the mean of the points within a given region. make_blobs() uses these parameters: n_samples is the total number of samples to generate. This should be all over Facebook!!!”. building block for a Consensus Index that can be used for clustering using sklearn.neighbors.kneighbors_graph to restrict Propagation on a synthetic 2D datasets with 3 classes. This global clusterer can be set by n_clusters. But it’s not all bad news. algorithm, and can be considered a generalization of DBSCAN that relaxes the Values closer to zero indicate a better This would happen when a non-core sample In the second This algorithm can be viewed as an instance or data reduction method, Single linkage is the most brittle linkage option with regard to this issue. exhaust system memory using HDBSCAN, OPTICS will maintain n (as opposed However, it’s also currently not included in scikit (though there is an extensively documented python package on github). Well, here’s the gif. cluster analysis as follows: The computation of Davies-Bouldin is simpler than that of Silhouette scores. Affinity propagation (AP) describes an algorithm that performs clustering by passing messages between points. homogeneous but not complete: v_measure_score is symmetric: it can be used to evaluate of core samples, which are samples that are in areas of high density. transform method of a trained model of KMeans. This technique is the application of the general expectation maximisation (EM) algorithm to the task of clustering. computations. as a dendrogram. To prevent the algorithm returning sub-optimal clustering, the kmeans method includes the n_init and method parameters. The contingency table calculated is typically utilized in the calculation For instance, in the swiss-roll example below, the connectivity The key difference Journal of the American Statistical Association. ‘Cutting’ the samples. If Different distance metrics can be supplied via the metric keyword. reachability-plot dendrograms, and the hierarchy of clusters detected by the 226–231. And in the world of big data, this matters. size of the clusters themselves. We’ve spent the past week counting words, and we’re just going to keep right on doing it. plot above has been color-coded so that cluster colors in planar space match labels_pred, the adjusted Rand index is a function that measures Given enough time, K-means will always converge, however this may be to a local style cluster extraction can be performed repeatedly in linear time for any Maybe humans (and data science blogs) will still be needed for a few more years! It is especially computationally efficient if the affinity matrix is sparse enable only merging of neighboring pixels on an image, as in the Why, you ask? shorter run time than OPTICS; however, for repeated runs at varying eps Two feature extraction methods can be used in … adjusted for chance and will tend to increase as the number of different labels As a rule of thumb if I’ll still provide some GIFs, but a mathematical description might be more informative in this case (i.e. observations of pairs of clusters. requires manual assignment by human annotators (as in the supervised learning In practice this difference in quality can be quite Algorithm description: It is based on minimization of the following objective function: In this equation, The The KMeans algorithm clusters data by trying to separate samples in n Due to this rather generic view, clusters Any sample that is not a also make the algorithm faster, especially when the number of the samples between DBSCAN and OPTICS is that the OPTICS algorithm builds a reachability Dremio. If GIFs aren’t your thing (what are you doing on the internet? with a small, all-equal, diagonal covariance matrix. and a set of non-core samples that are close to a core sample (but are not The code is modeled after the clustering algorithms in scikit-learn and has the same familiar interface. Instead, through the medium of GIFs, this tutorial will describe the most common techniques. extraction with OPTICS looks at the steep slopes within the graph to find Hierarchical clustering is a general family of clustering algorithms that clusters can be merged together), through a connectivity matrix that defines AP can suffer from non-convergence, though appropriate calibration of the damping parameter can minimise this risk. Given the knowledge of the ground truth class assignments labels_true and The best GIFs are on GIPHY. case of a signed distance matrix, is common to apply a heat kernel: See the examples for such an application. or within-cluster sum-of-squares criterion: Inertia can be recognized as a measure of how internally coherent clusters are. Prerequisite: Optimal value of K in K-Means Clustering K-means is one of the most popular clustering algorithms, mainly because of its good time performance. of pair of points that belong to the same clusters in the true labels and not max_eps to a lower value will result in shorter run times, and can be And it is not always possible for us to annotate data to certain categories or classes. clusters and ground truth classes, a completely random labeling will Search, discover and share your favorite Clustering GIFs. of cluster $$q$$, $$c_E$$ the center of $$E$$, and $$n_q$$ the The possibility to use custom metrics is retained; cluster is therefore a set of core samples, each close to each other when it is used jointly with a connectivity matrix, but is computationally Rosenberg and Hirschberg further define V-measure as the harmonic “A comparative analysis of ‘sqrt’ and ‘sum’ averages are the geometric and arithmetic means; we use these data is provided in a different order. A cluster with an index less than $$n$$ corresponds to one of the $$n$$ original observations. Some heuristics for choosing this parameter have been 2. Intuitively, these samples Sort: Relevant Newest # spot # cluster # kmeans # scikit # dashee87githubio spot # cluster # kmeans # scikit # dashee87githubio # season 3 # lisa simpson # episode 18 # watching # speaking (2017). Today, the majority of the mac… We’ll do an overview of this widely used module and get a bit more exposure to statistical learning algorithms. We’ll also explore an unsupervised learning technique - K-means cluster analysis (via R and then via Python using scikit-learn). red clusters are adjacent in the reachability plot, and can be hierarchically cosine distance is interesting because it is invariant to global Wikipedia entry for Davies-Bouldin index. If this split node has a parent subcluster and there is room the centroid of that cluster – also know as cluster diameter. is updated by taking the streaming average of the sample and all previous centers is the number of centers to generate. from one to another. set_option (\"display.max_columns\", 100) % matplotlib inline Even more text analysis with scikit-learn. Of them, none is in predicted cluster 0, one is in Set n_clusters to a required value using criterion is fulfilled. the impact of the dataset size on the value of clustering measures concepts of clusters, such as density based clusters like those obtained when given the same data in the same order. Visual inspection can often be useful for understanding the structure for any value of n_clusters and n_samples (which is not the It’s easy to imagine where you should overlay 4 balls on the first dataset. can be run over this with metric='precomputed'. is an example of such an evaluation, where a which define formally what we mean when we say dense. Average linkage minimizes the average of the distances between all how to find the optimal number of clusters). the silhouette analysis is used to choose an optimal value for n_clusters. For each sample in the mini-batch, the assigned centroid It is a Schubert, E., Sander, J., Ester, M., Kriegel, H. P., & Xu, X. through DBSCAN. measure class: center, middle ### W4995 Applied Machine Learning # Clustering and Mixture Models 03/27/19 Andreas C. Müller ??? 49-60. and DBSCAN one can also input similarity matrices of shape entropy of clusters $$H(K)$$ are defined in a symmetric manner. All the tools you’ll need are in Scikit-Learn, so I’ll leave the code to a minimum. while not robust to noisy data, can be computed very efficiently and can calculated using a similar form to that of the adjusted Rand index: For normalized mutual information and adjusted mutual information, the normalizing This page is based on a Jupyter/IPython Notebook: download the original .ipynb import pandas as pd pd. Demonstration of k-means assumptions: Demonstrating when Max no. Likewise for $$V$$: With $$P'(j) = \|V_j\| / N$$. observations of pairs of clusters. distances tend to become inflated We will use the models imported from Scikit-Learn. HC typically comes in two flavours (essentially, bottom up or top down): Another important concept in HC is the linkage criterion. E. B. Fowkles and C. L. Mallows, 1983. reports the intersection cardinality for every true/predicted cluster pair. More formally, we define a core sample as being a sample in the dataset such assignments that are largely independent, while values close to one The AgglomerativeClustering object performs a hierarchical clustering The algorithm is concisely illustrated by the GIF below. true cluster is “a”. The former just reruns the algorithm with n different initialisations and returns the best output (measured by the within cluster sum of squares). makes no distinction how the points are distributed within the ball), but, in some cases, a Gaussian kernel might be more appropriate. K-means is equivalent to the expectation-maximization algorithm discussed in the literature, for example based on a knee in the nearest neighbor uneven cluster sizes. There are two types of hierarchical clustering: Agglomerative and Divisive. The means are commonly called the cluster The Fowlkes-Mallows index (sklearn.metrics.fowlkes_mallows_score) can be No assumption is made on the cluster structure: can be used In particular random labeling won’t yield zero This initializes the centroids to be As we’ll find out though, that distinction can sometimes be a little unclear, as some algorithms employ parameters that act as proxies for the number of clusters. (generally) distant from each other, leading to provably better results than than a thousand and the number of clusters is less than 10. labelings), similar clusterings have a positive ARI, 1.0 is the perfect for the given data. to other points in their area, and will thus sometimes be marked as noise matrix. k-means performs intuitively and when it does not, A demo of K-Means clustering on the handwritten digits data: Clustering handwritten digits, “k-means++: The advantages of careful seeding” a(k,k) = \\sum_{i' \\neq k} \\max(0, r(i',k)). it into a global clusterer. separated by areas of low density. should be the exemplar for sample $$i$$. For this purpose, the two important approach. The centre of the ball is iteratively nudged towards regions of higher density by shifting the centre to the mean of the points within the ball (hence the name). These can be obtained from the functions L. Hubert and P. Arabie, Journal of Classification 1985, Wikipedia entry for the adjusted Rand index. As its name suggests, it constructs a hierarchy of clusters based on proximity (e.g Euclidean distance or Manhattan distance- see GIF below). In other words, it repeats of the ground truth classes while almost never available in practice or Visualization of cluster hierarchy, 2.3.10. BIRCH: An efficient data clustering method for large databases. k-means++ initialization scheme, which has been implemented in scikit-learn DBSCAN. The Birch algorithm has two parameters, the threshold and the branching factor. convergence. reducing the log-likelihood). number of points in cluster $$q$$. MiniBatch code, General-purpose, even cluster size, flat geometry, not too many clusters, Many clusters, uneven cluster size, non-flat geometry, Graph distance (e.g. Various Agglomerative Clustering on a 2D embedding of digits: exploration of the Peter J. Rousseeuw (1987). a non-flat manifold, and the standard euclidean distance is It can thus be used as a consensus and our clustering algorithm assignments of the same samples be used (e.g., with sparse matrices). However MI-based measures can also be useful in purely unsupervised setting as a cluster $$k$$, and finally $$n_{c,k}$$ the number of samples used when the ground truth class assignments of the samples is known. It’s clear that the default settings in the sklearn implementation of AP didn’t perform very well on the two datasets (in fact, neither execution converged). Andrew Y. Ng, Michael I. Jordan, Yair Weiss, 2001, “Preconditioned Spectral Clustering for Stochastic Clustering text documents using k-means. HDFS stands for Hadoop Distributed File System. clusters. Marina Meila, Jianbo Shi, 2001, “On Spectral Clustering: Analysis and an algorithm” This has the additional benefit of decreasing runtime (less steps to reach convergence). In most of the cases, data is generally labeled by us, human beings. Small affinity matrix between samples, followed by clustering, e.g., by KMeans, The messages sent between points belong to one of two categories. A simple choice to construct $$R_ij$$ so that it is nonnegative and representative of themselves. One method to help address this issue is the “A Dendrite Method for Cluster Analysis”. First, even though the core samples which is the accumulated evidence that sample $$i$$ sklearn.neighbors.kneighbors_graph. The score ranges from 0 to 1.\n\n## clustering with scikit with gifs\n\nCarrabba's Pasta Recipes, Cloud Computing Meaning, Expedition Lands Zendikar Rising Price List, East Kilbride Map, Patanjali Corona Kit Online, Aquatic Experts Bonded Filter Pad, Large Trumpet Vines For Sale, Fall Fashion Colors 2020, Tuba Büyüküstün News, Process Essays Topics, Is It Illegal To Curse At An Employee, When You Rise In The Morning Sun Lyrics," ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6190845,"math_prob":0.931687,"size":442,"snap":"2022-05-2022-21","text_gpt3_token_len":121,"char_repetition_ratio":0.10045662,"word_repetition_ratio":0.027027028,"special_character_ratio":0.22171946,"punctuation_ratio":0.15384616,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97539824,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-05-22T17:31:40Z\",\"WARC-Record-ID\":\"<urn:uuid:68bf0483-8df0-4328-94fb-10535f4c9319>\",\"Content-Length\":\"19051\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d9a0713d-1a73-4249-ab43-f31a11d210a9>\",\"WARC-Concurrent-To\":\"<urn:uuid:144903f0-7b62-4fa8-8ed9-48d63b80ccfe>\",\"WARC-IP-Address\":\"88.99.139.237\",\"WARC-Target-URI\":\"https://afd-wandsbek.de/site/article.php?e62e8c=clustering-with-scikit-with-gifs\",\"WARC-Payload-Digest\":\"sha1:K53EWQOCRFNEPZZM6G34E6UCEJN6GJTI\",\"WARC-Block-Digest\":\"sha1:BJQN5LQSRCSHDQWWY2HDOZ22NN5RBPRD\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-21/CC-MAIN-2022-21_segments_1652662545875.39_warc_CC-MAIN-20220522160113-20220522190113-00770.warc.gz\"}"}
https://imagination.readthedocs.io/en/latest/getting-started/04-create-with-parameters.html	[ "# Register a new entity with parameters¶\n\nNow, since we have one entity. Let’s spice things up with more complicate entity by creating a `Report` class at `app/report.py`:\n\n```class Report(object):\ndef __init__(self,\ncalculator,\nassignment_scores:list,\nfinal_exam_score:int):\nself.calculator = calculator\nself.assignment_scores = assignment_scores # max score = 10\nself.final_exam_score = final_exam_score # max score = 100\n\ntotal_scores = sum(self.assignment_scores)\ntotal_max_scores = (10 * len(self.assignment_scores))\nassignment_part = total_scores / total_max_scores\nfinal_exam_part = final_exam_score / 100\n\nassignment_part * 0.7,\nfinal_exam_part * 0.3\n)\n\n```\n\nAt this point, again, Imagination does not know about the report. To do so, let’s register an entity of an instance of app.report.Report in `containers.xml`:\n\n```<imagination>\n<!-- ... (omitted) ... -->\n<entity id=\"report.bob\" class=\"app.report.Report\">\n\n<!-- Pass [2, 0, 1, 8, 6, 9, 7, 5] (list of integers)\nas \"assignment_scores\" -->\n<param type=\"list\" name=\"assignment_scores\">\n<item type=\"int\">2</item>\n<item type=\"int\">0</item>\n<item type=\"int\">1</item>\n<item type=\"int\">8</item>\n<item type=\"int\">6</item>\n<item type=\"int\">9</item>\n<item type=\"int\">7</item>\n<item type=\"int\">5</item>\n</param>\n\n<!-- Pass 89 (integer)\nas \"final_exam_score\" -->\n<param type=\"int\" name=\"final_exam_score\">89</param>\n\n<!-- Pass the reference of the \"calc\" entity\nas \"calculator\" -->\n<param type=\"entity\" name=\"calculator\">calc</param>\n</entity>\n</imagination>\n```\n\nwhere report.bob is the entity ID.\n\nNote\n\nIn case you are confused, here is the equivalent code.\n\n```# From the previous page...\nfrom app.util import Calculator\n\ncalc = Calculator()\n\n# Now, to work with the report class.\nfrom app.report import Report\n\nreport = Report(calc, [2, 0, 1, 8, 6, 9, 7, 5], 89)\n```\n\nThe key differences at this point are:\n\n• neither the calc entity and the report.bob entity are not instantiated immediately until the report.bob entity is requested/activated,\n• the calc is always activated before the report.bob entity as calc is a dependency of report.bob.\n• the objects are still living only in the scope of the container.\n\n## How to work with the report entity¶\n\nTo refer the report entity, for example, in `main.py`, simply use `report_bob = assembler.core.get('report.bob')`\n\nNow, to actually use the entity, let’s add something to the end of `main.py`.\n\n```# Omitted the code for main.py already shown above\nreport_bob = assembler.core.get('report.bob')\n```\n\nSo, as you can see, the entity works pretty much like a normal object, except the key differences mentioned earlier.\n\n## What can you define as parameters or items?¶\n\nType Name Data Type Example PCDATA, child nodes\nstr Unicode (default) `bamboo`\nbool Boolean (bool) `true`, `false`\nfloat Float (float) `1.2`, `2.0`\nint Integer (int) `123`\nclass Class reference `argparser.ArgumentParser`\nentity An Imagination entity `report.bob` (Entity ID)\nlist Python’s List (list) (See an example below)\ndict Python’s Dictionary (dict) (See an example below)\n\nHere is an example. From:\n\n```<imagination>\n<entity class=\"foo.Bar\" id=\"panda\">\n<param type=\"bool\" name=\"enabled\">false</param>\n<param type=\"dict\" name=\"data\">\n<item type=\"list\" key=\"collection\">\n<item type=\"str\">shiroyuki</item>\n<item type=\"str\">is</item>\n<item type=\"str\">happy</item>\n</item>\n<item type=\"str\" key=\"code\">1234</item>\n</param>\n</entity>\n</imagination>\n```\n\nThe equivalence to the Python code used to instantiate this entity will be:\n\n```panda = foo.Bar(enabled = False, data = {\n'collection': ['shiroyuki', 'is', 'happy'],\n'code': 1234,\n})\n```\n\nNote\n\nHow to define parameters and items can be used with a factorized entity, which will be mentioned in the next step.\n\nTip" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5727446,"math_prob":0.75595665,"size":4263,"snap":"2022-40-2023-06","text_gpt3_token_len":1160,"char_repetition_ratio":0.15637474,"word_repetition_ratio":0.01369863,"special_character_ratio":0.2894675,"punctuation_ratio":0.17128463,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9635437,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-01-27T23:39:25Z\",\"WARC-Record-ID\":\"<urn:uuid:79a8afbc-17f4-4a22-95ad-6bef99507696>\",\"Content-Length\":\"23779\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:6e620dca-5798-4278-89e6-05f8dcac244f>\",\"WARC-Concurrent-To\":\"<urn:uuid:c07596ef-ec0c-4e3d-8b91-2035caf6ae93>\",\"WARC-IP-Address\":\"104.17.32.82\",\"WARC-Target-URI\":\"https://imagination.readthedocs.io/en/latest/getting-started/04-create-with-parameters.html\",\"WARC-Payload-Digest\":\"sha1:MZMSDZ3RCC2UARGMMCWEUFT6CBMZHEUM\",\"WARC-Block-Digest\":\"sha1:MNVYGNUQAT3K5FJI5IO47AMWKYY4R3OO\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764499468.22_warc_CC-MAIN-20230127231443-20230128021443-00706.warc.gz\"}"}
https://forum.mango-os.com/topic/2829/chart-error-after-update	[ "# Chart error after update\n\n• Hello,\n\nI have just updated my Mango from v3.0.1 to v3.1.1. Previously I was able to take a virtual point using ma-point-values, and create a bar chart for the daily maximum for that point for the past week. The chart looked like this:", null, "Once I upgraded to 3.1.1, the chart is no longer working on the dashboard.\nI have copied the code to the play area, and it wont work there either. The code I used was:\n\n``````<ma-now update-interval=\"1 HOURS\" output=\"theTimeNow\"></ma-now>\n<ma-point-values point-xid=\"Site_Total_Real_Power\" point=\"point1\" values=\"max_val\" from=\"theTimeNow \| moment:'startOf':'day'\|moment:'subtract':7:'days'\" to=\"theTimeNow\" rollup=\"MAXIMUM\" rollup-interval=\"1 DAYS\">\n</ma-point-values>\n\n<ma-serial-chart style=\"height: 300px; width: 100%\" series-1-values=\"max_val\" series-1-point=\"point1\" default-type=\"column\">\n</ma-serial-chart>\n``````\n\nAdvice on why this output isn't working anymore would be really appreciated!\nHenry\n\n• @henryblu I think it's probably just the `moment` filters, change them to `maMoment`\n\n• Excellent! Thanks Jared, the following code worked perfectly:\n\n`````` <ma-now update-interval=\"1 HOURS\" output=\"theTimeNow\"></ma-now>\n<ma-point-values point-xid=\"Site_Total_Real_Power\" point=\"point1\" values=\"max_val\" from=\"theTimeNow \| maMoment:'startOf':'day'\|maMoment:'subtract':7:'days'\" to=\"theTimeNow\" rollup=\"MAXIMUM\" rollup-interval=\"1 DAYS\">\n</ma-point-values>\n\n<ma-serial-chart style=\"height: 300px; width: 100%\" series-1-values=\"max_val\" series-1-point=\"point1\" default-type=\"column\">\n</ma-serial-chart>\n\n``````" ]	[ null, "https://i.imgur.com/YKB04fT.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.73191243,"math_prob":0.42744464,"size":1570,"snap":"2021-43-2021-49","text_gpt3_token_len":442,"char_repetition_ratio":0.11941252,"word_repetition_ratio":0.17964073,"special_character_ratio":0.27133757,"punctuation_ratio":0.12847222,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97577983,"pos_list":[0,1,2],"im_url_duplicate_count":[null,3,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-28T18:29:10Z\",\"WARC-Record-ID\":\"<urn:uuid:33f36456-1dfd-4622-be87-bd07c365575a>\",\"Content-Length\":\"53651\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:27dac119-d3ea-438f-83c7-3b856ff8b710>\",\"WARC-Concurrent-To\":\"<urn:uuid:3ee9a1ce-3d88-43fe-9d88-b4e34932b12a>\",\"WARC-IP-Address\":\"192.241.207.234\",\"WARC-Target-URI\":\"https://forum.mango-os.com/topic/2829/chart-error-after-update\",\"WARC-Payload-Digest\":\"sha1:JWU4HGS7UNVF6VGP44AKNBIR2SEOTKE2\",\"WARC-Block-Digest\":\"sha1:BPLJAV5QL3XL7T7QMQXZX7SDQCOCI5IN\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323588398.42_warc_CC-MAIN-20211028162638-20211028192638-00480.warc.gz\"}"}
http://ixtrieve.fh-koeln.de/birds/litie/document/18886	[ "# Document (#18886)\n\nAuthor\nDuchemin, P.-Y.\nTitle\n¬La recherche d'informations sur l'internet : repertoires et moteurs de recherche\nSource\nBulletin d'informations de l'Association des Bibliothecaires Francais. 1997, no.174, S.91-96\nYear\n1997\nAbstract\nThe Internet links computer networks worldwide through the TCP/IP; in addition to electronic mail; bulleton board and news group services, files can be downloaded using the standard protocol FTP. Services have evolved to identify and facilitate access to Internet resources, e.g. Telnet, Gopher, WAIS, etc. The WWW is the most developed, using hypertext links. Search engines such as AltaVista explore Web content and create catalogues of Web pages. Gives details of the most commonly used subject guides, research tools and search engines, including URL and applications\nFootnote\nÜbers. des Titels: Information research on the Internet: indexes and search engines\nTheme\nSuchmaschinen\n\n## Similar documents (author)\n\n1. Duchemin, P.-Y.: ¬La conversion retrospective des catalogues dur Départment des Cartes et Plans de la Bibliothèque Nationale (1994) 4.85\n```4.853387 = sum of:\n4.853387 = weight(author_txt:duchemin in 334) [ClassicSimilarity], result of:\n4.853387 = fieldWeight in 334, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n9.706774 = idf(docFreq=6, maxDocs=42306)\n0.5 = fieldNorm(doc=334)\n```\n2. Duchemin, P.-Y.: Retroconversion of French cartographic material card catalogues : an overview of the situation (1993) 4.85\n```4.853387 = sum of:\n4.853387 = weight(author_txt:duchemin in 479) [ClassicSimilarity], result of:\n4.853387 = fieldWeight in 479, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n9.706774 = idf(docFreq=6, maxDocs=42306)\n0.5 = fieldNorm(doc=479)\n```\n3. Duchemin, P.-Y.: BN-OPALINE (1997) 4.85\n```4.853387 = sum of:\n4.853387 = weight(author_txt:duchemin in 1911) [ClassicSimilarity], result of:\n4.853387 = fieldWeight in 1911, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n9.706774 = idf(docFreq=6, maxDocs=42306)\n0.5 = fieldNorm(doc=1911)\n```\n4. Duchemin, P.-Y.: BN OPALINE : the map database in the Department des Cartes et Plans de la Bibliothèque Nationale: history (1993) 4.85\n```4.853387 = sum of:\n4.853387 = weight(author_txt:duchemin in 1917) [ClassicSimilarity], result of:\n4.853387 = fieldWeight in 1917, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n9.706774 = idf(docFreq=6, maxDocs=42306)\n0.5 = fieldNorm(doc=1917)\n```\n5. Duchemin, P.-Y.: ¬La nemrisation des documents graphiques (1997) 4.85\n```4.853387 = sum of:\n4.853387 = weight(author_txt:duchemin in 1956) [ClassicSimilarity], result of:\n4.853387 = fieldWeight in 1956, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n9.706774 = idf(docFreq=6, maxDocs=42306)\n0.5 = fieldNorm(doc=1956)\n```\n\n## Similar documents (content)\n\n1. Valauskas, E.J.: TurboGopher: Internet access with ease on the Macintosh (1993) 0.22\n```0.2181304 = sum of:\n0.2181304 = product of:\n0.90887666 = sum of:\n0.09399241 = weight(abstract_txt:files in 4628) [ClassicSimilarity], result of:\n0.09399241 = score(doc=4628,freq=1.0), product of:\n0.13181166 = queryWeight, product of:\n1.0116476 = boost\n5.704649 = idf(docFreq=382, maxDocs=42306)\n0.022839976 = queryNorm\n0.7130811 = fieldWeight in 4628, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n5.704649 = idf(docFreq=382, maxDocs=42306)\n0.125 = fieldNorm(doc=4628)\n0.1483584 = weight(abstract_txt:protocol in 4628) [ClassicSimilarity], result of:\n0.1483584 = score(doc=4628,freq=1.0), product of:\n0.17868993 = queryWeight, product of:\n1.1778835 = boost\n6.642049 = idf(docFreq=149, maxDocs=42306)\n0.022839976 = queryNorm\n0.8302561 = fieldWeight in 4628, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n6.642049 = idf(docFreq=149, maxDocs=42306)\n0.125 = fieldNorm(doc=4628)\n0.0494704 = weight(abstract_txt:search in 4628) [ClassicSimilarity], result of:\n0.0494704 = score(doc=4628,freq=1.0), product of:\n0.108259395 = queryWeight, product of:\n1.2965823 = boost\n3.6556938 = idf(docFreq=2971, maxDocs=42306)\n0.022839976 = queryNorm\n0.45696172 = fieldWeight in 4628, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n3.6556938 = idf(docFreq=2971, maxDocs=42306)\n0.125 = fieldNorm(doc=4628)\n0.399642 = weight(abstract_txt:gopher in 4628) [ClassicSimilarity], result of:\n0.399642 = score(doc=4628,freq=4.0), product of:\n0.21793118 = queryWeight, product of:\n1.3008045 = boost\n7.335196 = idf(docFreq=74, maxDocs=42306)\n0.022839976 = queryNorm\n1.833799 = fieldWeight in 4628, product of:\n2.0 = tf(freq=4.0), with freq of:\n4.0 = termFreq=4.0\n7.335196 = idf(docFreq=74, maxDocs=42306)\n0.125 = fieldNorm(doc=4628)\n0.0726026 = weight(abstract_txt:internet in 4628) [ClassicSimilarity], result of:\n0.0726026 = score(doc=4628,freq=2.0), product of:\n0.11096689 = queryWeight, product of:\n1.3126956 = boost\n3.701125 = idf(docFreq=2839, maxDocs=42306)\n0.022839976 = queryNorm\n0.6542726 = fieldWeight in 4628, product of:\n1.4142135 = tf(freq=2.0), with freq of:\n2.0 = termFreq=2.0\n3.701125 = idf(docFreq=2839, maxDocs=42306)\n0.125 = fieldNorm(doc=4628)\n0.14481077 = weight(abstract_txt:links in 4628) [ClassicSimilarity], result of:\n0.14481077 = score(doc=4628,freq=1.0), product of:\n0.22153169 = queryWeight, product of:\n1.8547494 = boost\n5.2294374 = idf(docFreq=615, maxDocs=42306)\n0.022839976 = queryNorm\n0.65367967 = fieldWeight in 4628, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n5.2294374 = idf(docFreq=615, maxDocs=42306)\n0.125 = fieldNorm(doc=4628)\n0.24 = coord(6/25)\n```\n2. 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https://dofnews.com/what-are-the-6-trig-ratios/	[ "# What are the 6 trig ratios?\n\n## What are the 6 trig ratios?\n\nThere are six trigonometric ratios, sine, cosine, tangent, cosecant, secant and cotangent. These six trigonometric ratios are abbreviated as sin, cos, tan, csc, sec, cot.\n\n## What are the 9 trig identities?\n\nAngle Sum and Difference Identities\n\n• Note which means you need to use plus or minus, and the.\n• sin(A B) = sin(A)cos(B) cos(A)sin(B)\n• cos(A B) = cos(A)cos(B) sin(A)sin(B)\n• tan(A B) = tan(A) tan(B)1 tan(A)tan(B)\n• cot(A B) = cot(A)cot(B) 1cot(B) cot(A)\n\n## Who is the daddy of trigonometry?\n\nHipparchus of Nicaea\n\n## What is SOH CAH TOA?\n\n“SOHCAHTOA” is a useful mnemonic for remembering the definitions of the trigonometric capabilities sine, cosine, and tangent i.e., sine equals reverse over hypotenuse, cosine equals adjoining over hypotenuse, and tangent equals reverse over adjoining, (1) (2) (3) Other mnemonics embody.\n\n## Is SOH CAH TOA just for proper triangles?\n\nQ: Is sohcahtoa just for proper triangles? A: Yes, it solely applies to proper triangles. A: They hypotenuse of a proper triangle is at all times reverse the 90 diploma angle, and is the longest aspect.\n\n180 levels\n\n## What is a forty five diploma triangle?\n\nUniv. of Wisconsin. A forty five 45 90 triangle is a particular kind of isosceles proper triangle the place the 2 legs are congruent to at least one one other and the non-right angles are each equal to 45 levels.\n\n## What is the 30-60-90 Triangle rule?\n\nTips for Remembering the 30-60-90 Rules Remembering the 30-60-90 triangle guidelines is a matter of remembering the ratio of 1: √3 : 2, and understanding that the shortest aspect size is at all times reverse the shortest angle (30°) and the longest aspect size is at all times reverse the biggest angle (90°).\n\n## What are the aspect lengths of a forty five 45 90 Triangle?\n\nA forty five°-45°-90° triangle is a particular proper triangle that has two 45-degree angles and one 90-degree angle. The aspect lengths of this triangle are within the ratio of; Side 1: Side 2: Hypotenuse = n: n: n√2 = 1:1: √2. The 45°-45°-90° proper triangle is half of a sq..\n\n## What do you name a forty five 45 90 Triangle?\n\nA forty five – 45 – 90 diploma triangle (or isosceles proper triangle) is a triangle with angles of 45°, 45°, and 90° and sides within the ratio of. Note that it’s the form of half a sq., lower alongside the sq.’s diagonal, and that it’s additionally an isosceles triangle (each legs have the identical size).\n\n## What is the rule for a forty five 4590 Triangle?\n\nThat tells us that for each 45-45-90 triangle, the size of the hypotenuse equals the size of the leg multiplied by sq. root of two. That is the 45-45-90 Triangle Theorem.\n\n## How do you discover the world of a forty five 45 90 proper triangle?\n\nCorrect reply: Explanation: To discover the world of a triangle, multiply the bottom by the peak, then divide by 2. Since the quick legs of an isosceles triangle are the identical size, we have to know just one to know the opposite. Since, a brief aspect serves as the bottom of the triangle, the opposite quick aspect tells us the peak.\n\n## What is the connection between the legs and the hypotenuse of a forty five 45 90 Triangle?\n\nIn different phrases, in each 45-45-90 triangle, the lengths of the 2 legs are at all times equal, and the ratio of the size of the hypotenuse to the size of a leg is at all times sq. root 2 to 1.\n\n## Is the hypotenuse of a 30 60 90 triangle is 3 √ instances so long as the shortest leg of the triangle?\n\n30°-60°-90° Triangles The measures of the edges are x, x√3, and 2x. In a 30°−60°−90° triangle, the size of the hypotenuse is twice the size of the shorter leg, and the size of the longer leg is √3 instances the size of the shorter leg.\n\n## How do you discover a 30 60 90 Triangle?\n\nA Quick Guide to the 30-60-90 Degree Triangle\n\n1. Type 1: You know the quick leg (the aspect throughout from the 30-degree angle). Double its size to seek out the hypotenuse.\n2. Type 2: You know the hypotenuse. Divide the hypotenuse by 2 to seek out the quick aspect.\n3. Type 3: You know the lengthy leg (the aspect throughout from the 60-degree angle).\n\n## What is the connection between the hypotenuse and the legs of a triangle?\n\nRecall {that a} proper triangle is a triangle the place one among its angles is 90 levels. For a proper triangle, the aspect that’s reverse of the precise angle known as the hypotenuse. This aspect will at all times be the longest aspect of the precise triangle. The different two (shorter) sides are known as legs.\n\nhypotenuse\n\n## Is Side A at all times longer than Side B in a proper triangle?\n\n2 Answers. Side A and B doesn’t matter when your attempting to use this to the pythagorean theorem however aspect C should at all times be the hypotenuse. The hypotenuse is at all times the triangle’s longest aspect. It is reverse the precise angle.\n\n## Which set of aspect will make a proper triangle?\n\nAll that you just want are the lengths of the bottom and the peak. In a proper triangle, the bottom and the peak are the 2 sides which kind the precise angle.\n\n## Does 4 5 6 make proper triangles?\n\nFor a set of three numbers to be pythagorean, the sq. of the biggest quantity needs to be equal to sum of the squares of different two. Hence 4 , 5 and 6 are usually not pythagorean triple.\n\n## Does 9 12 and 15 make a proper triangle?\n\nThe three sides 9 in, 12 in, and 15 in do symbolize a proper triangle. Since the sq. of the hypotenuse is the same as the sum of the squares of the opposite two sides, it is a proper triangle.\n\n## Does 5 12 and 13 kind a proper triangle?\n\nYes, a proper triangle can have aspect lengths 5, 12, and 13. To decide if sides of size 5, 12, and 13 models could make up the edges of a proper…\n\n## Does 7/11/13 kind a proper triangle?\n\nBecause the 2 sides are equal, it is a proper triangle. NOTE: All of the lengths in Example 4 symbolize the lengths of the edges of a triangle. For instance, 4, 7 and 13 can’t be the edges of a triangle as a result of start{align}4+7end{align} isn’t better than 13.\n\n## Does 4 8 12 make proper triangles?\n\nAnswer: No, aspect lengths of 4, 8, and 12 don’t kind a proper triangle.\n\n## Does 15 36 39 kind a proper triangle?\n\nExample 1: Solve for x. Therefore, the size of the bottom x of the triangle is roughly 12.1. The Converse of the Pythagorean Theorem states that, if is true, then the given triangle is a proper triangle. Hence, the set {15, 36, 39} is a Pythagorean Triple." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.86736053,"math_prob":0.9792837,"size":6427,"snap":"2022-27-2022-33","text_gpt3_token_len":1736,"char_repetition_ratio":0.20551144,"word_repetition_ratio":0.03979678,"special_character_ratio":0.2699549,"punctuation_ratio":0.11931818,"nsfw_num_words":2,"has_unicode_error":false,"math_prob_llama3":0.99687326,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-07-01T14:31:17Z\",\"WARC-Record-ID\":\"<urn:uuid:84d27b2a-83e3-4e06-b65d-b6abb602fefd>\",\"Content-Length\":\"154822\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f3f523b7-e28d-40c9-8b7f-5cce26aee3e4>\",\"WARC-Concurrent-To\":\"<urn:uuid:a88e3efe-aebd-4fe8-aa3f-ec82ea3b1f3a>\",\"WARC-IP-Address\":\"172.67.171.130\",\"WARC-Target-URI\":\"https://dofnews.com/what-are-the-6-trig-ratios/\",\"WARC-Payload-Digest\":\"sha1:QHXUSBNEWVGFYHB4HHV5LZRXVRAKXAZ5\",\"WARC-Block-Digest\":\"sha1:VCH6ATICOJDPIPK37GVQGZSPILUJQLAI\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656103941562.52_warc_CC-MAIN-20220701125452-20220701155452-00013.warc.gz\"}"}
https://byjus.com/questions/copy-the-following-drawing-on-squared-paper-complete-each-one-of-them-such-that-the-resulting-figure-has-two-dotted-lines-as-two-lines-of-symmetry-how-did-you-go-about-completing-the-picture/	[ "", null, "# Copy the following drawing on squared paper. Complete each one of them such that the resulting figure has two dotted lines as two lines of symmetry. How did you go about completing the picture?", null, "Solutions:\n\nWe can complete the given figures by drawing similar parts as shown in these figures.\n\nLet us first draw the vertical line of symmetry and then the horizontal line of symmetry. Or first about the horizontal line of symmetry and then about the vertical line of symmetry.\n\nThe figures shown below are the completed figures the given above", null, "", null, "(0)", null, "(0)" ]	[ null, "https://www.facebook.com/tr", null, "https://cdn1.byjus.com/wp-content/uploads/2019/11/ncert-solutions-for-class-6-maths-chapter-13-exerc-48.png", null, "https://cdn1.byjus.com/wp-content/uploads/2019/11/ncert-solutions-for-class-6-maths-chapter-13-exerc-49.png", null, "https://cdn1.byjus.com/wp-content/uploads/2021/08/upvote-lineart.svg", null, "https://cdn1.byjus.com/wp-content/uploads/2021/08/downvote-lineart.svg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.88259447,"math_prob":0.97236234,"size":345,"snap":"2021-31-2021-39","text_gpt3_token_len":64,"char_repetition_ratio":0.17302053,"word_repetition_ratio":0.036363635,"special_character_ratio":0.17971015,"punctuation_ratio":0.06349207,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99843276,"pos_list":[0,1,2,3,4,5,6,7,8,9,10],"im_url_duplicate_count":[null,null,null,2,null,2,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-09-19T09:00:24Z\",\"WARC-Record-ID\":\"<urn:uuid:2e95fca1-97f4-4add-93b5-fef4015df85c>\",\"Content-Length\":\"167612\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b4a7dd4a-de5c-40a4-80e6-86028eaa1185>\",\"WARC-Concurrent-To\":\"<urn:uuid:f48de960-5df9-4a45-92db-e5f324e1a811>\",\"WARC-IP-Address\":\"162.159.129.41\",\"WARC-Target-URI\":\"https://byjus.com/questions/copy-the-following-drawing-on-squared-paper-complete-each-one-of-them-such-that-the-resulting-figure-has-two-dotted-lines-as-two-lines-of-symmetry-how-did-you-go-about-completing-the-picture/\",\"WARC-Payload-Digest\":\"sha1:DUHOUYPBRLQRYMMA3ADMT7QWVK7R7JZE\",\"WARC-Block-Digest\":\"sha1:HY6E73B2EWEIMRDIB7AOQZLQJMGDUXYU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780056752.16_warc_CC-MAIN-20210919065755-20210919095755-00376.warc.gz\"}"}
https://boundaryvalueproblems.springeropen.com/articles/10.1186/s13661-015-0368-7	[ "# Dynamics of competing systems in general heterogeneous environments\n\n## Abstract\n\nIn this paper, we address the question of the dynamics of the systems for two competing species in general heterogeneous environment with lethal boundary conditions. The existence and uniqueness of the positive steady-state solution are established under suitable conditions. Finally, we obtain global asymptotic stability of the positive steady-state solution for weak competition situation.\n\n## 1 Introduction\n\nThe model we consider has the general form\n\n$$\\left \\{ \\textstyle\\begin{array}{@{}l@{\\quad}l} \\frac{\\partial u}{\\partial t}=\\mu_{1}\\Delta u+u[a(x)-b(x)u-c(x)v] &\\mbox{in } \\Omega\\times[0,\\infty),\\\\ \\frac{\\partial v}{\\partial t}=\\mu_{2}\\Delta v+v[d(x)-e(x)u-f(x)v] &\\mbox{in } \\Omega\\times[0,\\infty),\\\\ u=v=0 &\\mbox{on } \\partial\\Omega\\times[0,\\infty),\\\\ u(x,0)=u^{0}(x),\\qquad v(x,0)=v^{0}(x) &\\mbox{in } \\partial\\Omega, \\end{array}\\displaystyle \\right .$$\n(1.1)\n\nwhere $$a(x), b(x), d(x), f(x)>0$$, $$c(x), e(x)\\geq0$$, all belong to the Hölder space $$C^{\\alpha}(\\overline{\\Omega})$$ for some constant $$0<\\alpha<1$$, and $$\\Omega\\subseteq\\mathbb{R}^{N}$$ is a bounded domain with $$C^{2+\\alpha}$$ a smooth boundary. The variables u, v represent population densities of the competing species. The boundary condition describes the situation that the boundary of Ω is lethal to the species.\n\nWhen the coefficients $$a(x)$$, $$b(x)$$, $$c(x)$$, $$d(x)$$, $$e(x)$$, $$f(x)$$ are constants (homogeneous environment), the system (1.1) has been studied extensively in past years; see, for example, and the references therein. But in the real world, the environments are usually heterogenous, and so it is more reasonable to assume that the coefficients in the system (1.1) are more general functions . Recently, He and Ni studied the dynamics of a competition model in some heterogeneous environments with Neumann boundary conditions [8, 9]. Álvarez-Caudevilla et al. considered a cooperative reaction-diffusion system in a spatiotemporally degenerate environment .\n\nTo study the dynamics of the system (1.1), we will consider its steady-state equation,\n\n$$\\left \\{ \\textstyle\\begin{array}{@{}l@{\\quad}l} \\mu_{1}\\Delta u+u[a(x)-b(x)u-c(x)v]=0 & \\mbox{in } \\Omega\\times [0,\\infty),\\\\ \\mu_{2}\\Delta v+v[d(x)-e(x)u-f(x)v]=0 & \\mbox{in } \\Omega\\times [0,\\infty),\\\\ u=v=0 & \\mbox{on } \\partial\\Omega\\times[0,\\infty). \\end{array}\\displaystyle \\right .$$\n(1.2)\n\nLet $$\\lambda_{1}>0$$ denote the principal eigenvalue for the problem\n\n$$\\left \\{ \\textstyle\\begin{array}{@{}l@{\\quad}l} \\Delta\\phi+\\lambda\\phi=0 &\\mbox{in } \\Omega,\\\\ \\phi=0 &\\mbox{on } \\partial\\Omega. \\end{array}\\displaystyle \\right .$$\n(1.3)\n\nFor any $$A(x)\\in C^{\\alpha}(\\overline{\\Omega})$$, it is well known that\n\n$$\\left \\{ \\textstyle\\begin{array}{@{}l@{\\quad}l} \\mu\\Delta\\theta+\\theta(A(x)-\\theta)=0 &\\mbox{in } \\Omega,\\\\ \\theta=0 &\\mbox{on } \\partial\\Omega, \\end{array}\\displaystyle \\right .$$\n(1.4)\n\nhas a unique positive solution $$\\theta\\in C^{2+\\alpha}(\\overline{\\Omega})$$ if $$A(x)>\\mu\\lambda_{1}$$ for all $$x\\in\\overline{\\Omega}$$ . We denote this unique positive solution by $$\\theta_{\\mu, A(x)}$$.\n\nIn the rest of the paper, we always assume that\n\n$$a(x)>\\mu_{1}\\lambda_{1}, \\qquad d(x)> \\mu_{2}\\lambda_{1} \\quad \\mbox{for all } x\\in\\overline{ \\Omega},$$\n(1.5)\n\nand denote\n\n$$\\bar{u}(x)=\\frac{\\theta_{\\mu_{1}, a(x)}}{\\min_{x\\in\\overline{\\Omega }}b(x)}, \\qquad\\bar{v}(x)=\\frac{\\theta_{\\mu_{2}, d(x)}}{\\min_{x\\in\\overline {\\Omega}}f(x)}, \\quad x\\in\\overline{\\Omega}.$$\n(1.6)\n\nNow we state the existence result for steady-state solutions.\n\n### Theorem 1.1\n\n(Existence)\n\nIf\n\n$$a(x)>\\mu_{1}\\lambda_{1}+c(x)\\bar{v}(x), \\qquad d(x)>\\mu_{2}\\lambda_{1}+e(x)\\bar{u}(x),$$\n(1.7)\n\nfor all $$x\\in\\overline{\\Omega}$$, then the system (1.2) has a positive steady-state solution $$(\\tilde{u}(x), \\tilde{v}(x))$$ with $$\\tilde{u}(x), \\tilde{v}(x)\\in C^{2+\\alpha}(\\overline{\\Omega})$$.\n\nNow we suppose further that the two competitors in (1.1) are under weak competition in the sense that $$c(x), e(x)>0$$ are small such that\n\n$$\\frac{c(x)}{\\min_{x\\in\\overline{\\Omega}}f(x)}< \\frac{a(x)}{d(x)}, \\qquad \\frac{e(x)}{\\min_{x\\in\\overline{\\Omega}}b(x)}< \\frac{d(x)}{a(x)}\\quad \\mbox{for all } x\\in\\overline{\\Omega}.$$\n(1.8)\n\nWe denote\n\n\\begin{aligned} &\\bar{\\zeta}=\\theta_{\\mu_{1}, a(x)}, \\qquad \\bar{\\eta}= \\theta_{\\mu_{2}, d(x)}, \\\\ &\\underline{\\zeta}=\\theta_{\\mu_{1}, (a(x)-c(x)\\frac{d(x)}{\\min_{x\\in \\overline{\\Omega}}f(x)})}, \\qquad \\underline{\\eta}= \\theta_{\\mu_{2}, (d(x)-e(x)\\frac{a(x)}{\\min_{x\\in \\overline{\\Omega}}b(x)})}, \\end{aligned}\n(1.9)\n\nwhich are all positive functions in Ω.\n\nThe following theorem gives a sufficient conditions for uniqueness of coexistence solution in suitable weak competition situations.\n\n### Theorem 1.2\n\n(Uniqueness)\n\nAssume that all the hypotheses of Theorems 1.1 and (1.8) are satisfied. If\n\n$$\\frac{c^{2}(x)\\bar{\\zeta}}{b^{2}(x)\\underline{\\eta}} +2\\frac{c(x)e(x)}{b(x)f(x)}+\\frac{e^{2}(x)\\bar{\\eta}}{f^{2}(x)\\underline {\\zeta}}< 4, \\quad x \\in\\overline{\\Omega},$$\n(1.10)\n\nthen the steady-state solution $$(\\tilde{u}(x), \\tilde{v}(x))$$ of (1.2) is unique.\n\n### Remark 1.1\n\nFor fixed functions $$a(x)>0$$, $$d(x)>0$$, hypothesis (1.10) will be satisfied for $$c(x), e(x)\\geq0$$ sufficiently small. This is true because $$\\underline{\\zeta}$$ (resp. $$\\underline{\\eta}$$) increases as $$c(x)$$ (resp. $$e(x)$$) decreases for $$x\\in\\Omega$$. Thus $$\\frac{e^{2}(x)\\bar{\\eta }}{f^{2}(x)\\underline{\\zeta}}$$ (resp. $$\\frac{c^{2}(x)\\bar{\\zeta }}{b^{2}(x)\\underline{\\eta}}$$) decreases as $$c(x)$$ (resp. $$e(x)$$) decreases.\n\nFinally, we state our dynamics results for the system (1.1).\n\n### Theorem 1.3\n\n(Global asymptotic stability)\n\nAssume that the hypotheses of Theorem 1.2 are satisfied. Let $$(u(x,t), v(x,t))$$ be a solution of the initial boundary value problem (1.1) with both $$u^{0}, v^{0}\\geq0,\\not\\equiv0$$ in $$C^{\\alpha}(\\overline{\\Omega })$$, $$0<\\alpha<1$$, and vanishing on Ω, then\n\n$$\\bigl(u(x,t), v(x,t)\\bigr)\\rightarrow\\bigl(\\tilde{u}(x), \\tilde{u}(x)\\bigr) \\quad \\textit{as } t\\rightarrow\\infty$$\n\nuniformly in $$\\overline{\\Omega}$$.\n\nThis paper is organized as follows: Theorem 1.1 and Theorem 1.2 are proved in Section 2. Theorem 1.3 is established in Section 3 by proving a more general theorem.\n\n## 2 Proof of Theorem 1.1 and Theorem 1.2\n\n### Proof of Theorem 1.1\n\nLet $$\\phi(x)$$ be a positive eigenfunction of the principal eigenvalue $$\\lambda_{1}$$ for the eigenvalue problem (1.3). Choose $$r_{1}>0$$ sufficiently small,\n\n\\begin{aligned} &\\mu_{1}\\Delta\\bigl(r_{1}\\phi(x)\\bigr)+r_{1} \\phi(x)\\bigl[(a(x)-b(x)r_{1}\\phi (x)-c(x)v\\bigr] \\\\ &\\quad\\geq r_{1}\\phi(x)\\bigl[(a(x)-\\mu_{1} \\lambda_{1}-b(x)r_{1}\\phi(x)-c(x)\\bar {v}\\bigr]\\geq0 \\end{aligned}\n\nand\n\n\\begin{aligned} &\\mu_{1}\\Delta\\bar{u}+\\bar{u}\\bigl[a(x)-b(x)\\bar{u}-c(x)v\\bigr] \\\\ &\\quad=\\bar{u} \\biggl[\\biggl(1-\\frac{b(x)}{\\min_{x\\in\\overline{\\Omega }}b(x)}\\biggr)\\theta_{\\mu_{1}, a(x)}-c(x)v \\biggr]\\leq0 \\end{aligned}\n\nfor all $$0\\leq v\\leq\\bar{v}$$. So, $$(\\bar{u}, r_{1}\\phi(x))$$ is a set of upper and lower solutions for u in (1.2).\n\nSimilarly, choose $$r_{2}>0$$ sufficiently small, $$(\\bar{v}, r_{2}\\phi (x))$$ is a set of upper and lower solutions for v in (1.1). By the coupled upper and lower theorem , the system (1.2) has a steady-state solution $$(\\tilde{u}(x), \\tilde{v}(x))$$ with $$\\tilde{u}(x), \\tilde{v}(x)>0$$ for $$x\\in\\Omega$$. □\n\n### Proof of Theorem 1.2\n\nAssume that $$(\\tilde {u}_{1}(x), \\tilde{v}_{1}(x)), (\\tilde{u}_{2}(x), \\tilde{v}_{2}(x))$$ are two strictly positive steady-state solutions of the system (1.1) in Ω.\n\nLet\n\n\\begin{aligned}& p(x)=\\tilde{u}_{1}(x)-\\tilde{u}_{2}(x),\\qquad q(x)=\\tilde {v}_{1}(x)-\\tilde{v}_{2}(x), \\\\& I_{1}=b(x)\\tilde{u}_{2}(x)p(x)+c(x)\\tilde{u}_{2}(x)q(x), \\\\& I_{2}=e(x)\\tilde{v}_{1}(x)p(x)+f(x)\\tilde{v}_{1}(x)q(x), \\end{aligned}\n\nthen\n\n$$\\left \\{ \\textstyle\\begin{array}{@{}l@{\\quad}l} \\mu_{1}\\Delta p(x)+[a(x)-b(x)\\tilde{u}_{1}(x)-c(x)\\tilde {v}_{1}(x)]p(x)-I_{1}=0 &\\mbox{in } \\Omega,\\\\ \\mu_{2}\\Delta q(x)+[d(x)-e(x)\\tilde{u}_{2}(x)-f(x)\\tilde {v}_{2}(x)]q(x)-I_{2}=0 &\\mbox{in } \\Omega,\\\\ p(x)=q(x)=0 &\\mbox{on } \\partial\\Omega. \\end{array}\\displaystyle \\right .$$\n(2.1)\n\nSince $$\\tilde{u}_{1}(x)$$ is a strictly positive solution of\n\n$$\\left \\{ \\textstyle\\begin{array}{@{}l@{\\quad}l} \\mu_{1}\\Delta\\psi+[a(x)-b(x)\\tilde{u}_{1}(x)-c(x)\\tilde {v}_{1}(x)]\\psi+\\alpha\\psi=0 &\\mbox{in } \\Omega,\\\\ \\psi=0 &\\mbox{on } \\partial\\Omega, \\end{array}\\displaystyle \\right .$$\n\nwith $$\\alpha=0$$, the number $$\\alpha=0$$ must be the smallest eigenvalue of the above problem. Moreover, by the variational properties, we have\n\n$$\\int_{\\Omega}z\\bigl(-\\mu_{1}\\Delta z- \\bigl[a(x)-b(x)\\tilde{u}_{1}(x)-c(x)\\tilde {u}_{2}(x)\\bigr]z \\bigr)\\,dx\\geq0,$$\n(2.2)\n\nfor any $$z\\in C^{2}(\\overline{\\Omega})$$ which vanishes on Ω. Similarly, since $$\\tilde{v}_{2}(x)$$ is strictly positive solution of\n\n$$\\left \\{ \\textstyle\\begin{array}{@{}l@{\\quad}l} \\mu_{2}\\Delta\\psi+[d(x)-e(x)\\tilde{u}_{2}(x)-f(x)\\tilde {v}_{2}(x)]\\psi+\\alpha\\psi=0 &\\mbox{in } \\Omega,\\\\ \\psi=0 &\\mbox{on } \\partial\\Omega, \\end{array}\\displaystyle \\right .$$\n\nwith $$\\alpha=0$$, the number $$\\alpha=0$$ must be the smallest eigenvalue of the above problem. Moreover,\n\n$$\\int_{\\Omega}z\\bigl(-\\mu_{2}\\Delta z- \\bigl[d(x)-e(x)\\tilde{u}_{2}(x)-f(x)\\tilde {v}_{2}(x)\\bigr]z \\bigr)\\,dx\\geq0,$$\n(2.3)\n\nfor any $$z\\in C^{2}(\\overline{\\Omega})$$ which vanishes on Ω. Multiplying the first equation of (2.1) by $$-p(x)$$, the second by $$-q(x)$$, integrating over Ω, and adding, we deduce from (2.2) and (2.3) that\n\n$$\\int_{\\Omega}\\bigl[b(x)\\tilde{u}_{2}(x)p^{2}(x)+ \\bigl(c(x)\\tilde {u}_{2}(x)+e(x)\\tilde{v}_{1}(x) \\bigr)p(x)q(x)+f(x) \\tilde{v}_{2}(x)q^{2}(x)\\bigr]\\,dx\\leq0.$$\n(2.4)\n\nBy a comparison of scalar equations using upper and lower solutions we readily obtain, for $$x\\in\\Omega$$,\n\n\\begin{aligned} &\\frac{\\underline{\\zeta}}{b(x)}\\leq \\tilde{u}_{1}, \\tilde{u}_{2}\\leq \\frac{\\bar{\\zeta}}{b(x)}, \\\\ &\\frac{\\underline{\\eta}}{f(x)}\\leq\\tilde{v}_{1}, \\tilde{v}_{2}\\leq \\frac{\\bar{\\eta}}{f(x)}. \\end{aligned}\n(2.5)\n\nFrom (2.5), we have\n\n\\begin{aligned} &\\frac{f(x)c^{2}(x)\\bar{\\zeta}}{b(x)\\underline{\\eta}}+2c(x)e(x)+\\frac {b(x)e^{2}(x)\\bar{\\eta}}{f(x)\\underline{\\zeta}} >c^{2}(x)\\frac{\\tilde{u}_{2}(x)}{\\tilde{v}_{1}(x)}+2c(x)e(x) +e^{2}(x) \\frac{\\tilde{v}_{1}(x)}{\\tilde{u}_{2}(x)}, \\end{aligned}\n(2.6)\n\nin Ω. It follows from (1.10) that\n\n$$c^{2}(x)\\frac{\\tilde{u}_{2}(x)}{\\tilde{v}_{1}(x)}+2c(x)e(x) +e^{2}(x) \\frac{\\tilde{v}_{1}(x)}{\\tilde{u}_{2}(x)}< 4b(x)f(x) \\quad\\mbox{in } \\Omega.$$\n(2.7)\n\nThen it is easy to see that the quadratic expression in the integrand of (2.4) is positive definite for each $$x\\in\\Omega$$. Consequently, we must have $$p(x)$$ and $$q(x)$$ identically equal to zero in Ω. That is, $$(\\tilde{u}_{1}(x),\\tilde{v}_{2}(x))\\equiv(\\tilde {u}_{2}(x),\\tilde{v}_{2}(x))$$ in Ω. □\n\n## 3 Proof of Theorem 1.3\n\nNow we are in a position to prove Theorem 1.3. Note that, by Theorem 1.2 and the assumptions of Theorem 1.3, problem (1.2) has a unique positive solution $$(\\tilde{u}(x), \\tilde{u}(x))$$. Then we will establish Theorem 1.3 by proving the following theorem without the assumption (1.10).\n\n### Theorem 3.1\n\nAssume the hypotheses of Theorem 1.1 and that problem (1.2) has a unique positive solution $$(\\tilde{u}(x), \\tilde{u}(x))$$ in Ω, then $$(\\tilde{u}(x), \\tilde{u}(x))$$ is globally asymptotically stable in the following sense. Let $$(u(x,t), v(x,t))$$ be a solution of the initial boundary value problem (1.1) with both $$u^{0}, v^{0}\\geq0,\\not\\equiv0$$ in $$C^{\\alpha}(\\overline{\\Omega})$$, $$0<\\alpha<1$$, and vanishing on Ω, then\n\n$$\\bigl(u(x,t), v(x,t)\\bigr)\\rightarrow\\bigl(\\tilde{u}(x), \\tilde{u}(x)\\bigr) \\quad\\textit{as } t\\rightarrow\\infty,$$\n\nuniformly in $$\\overline{\\Omega}$$.\n\n### Proof\n\nFor convenience, we introduce the following notation: If $$w\\in C^{1}(\\overline{\\Omega})$$, $$w(x)>0$$ for all $$x\\in\\Omega$$, and $$\\partial w/\\partial\\nu<0$$ everywhere on Ω, we write $$w\\gg0$$. If $$w, z\\in C^{1}(\\overline{\\Omega})$$, we write $$w\\ll z$$ if $$z-w \\gg0$$. We first prove the theorem under the additional conditions $$u^{0}, v^{0}\\in C^{1}(\\overline{\\Omega})$$,\n\n$$u^{0}\\gg0, \\qquad v^{0}\\gg0,$$\n(3.1)\n\nand for all $$x\\in\\overline{\\Omega}$$,\n\n$$u^{0}\\leq\\bar{u}, \\qquad v^{0}\\leq\\bar{v},$$\n(3.2)\n\nwhere $$\\bar{u}$$ and $$\\bar{v}$$ are defined in (1.6).\n\nLet $$\\phi_{1}$$ be the positive eigenfunction of the principal eigenvalue in (1.3). Choose $$\\epsilon>0$$ small such that\n\n$$\\epsilon\\phi_{1}(x)\\leq u^{0}(x), \\qquad \\epsilon\\phi_{1}(x)\\leq v^{0}(x),$$\n(3.3)\n\nand\n\n\\begin{aligned} &a(x)>\\mu_{1} \\lambda_{1}+c(x)\\bar{v}+b(x)\\epsilon\\phi_{1}(x), \\\\ &d(x)>\\mu_{2}\\lambda_{1}+e(x)\\bar{u}+f(x)\\epsilon \\phi_{1}(x), \\end{aligned}\n(3.4)\n\nfor all $$x\\in\\overline{\\Omega}$$. If we let $$\\underline{u}=\\underline {v}=\\epsilon\\phi_{1}$$, then\n\n$$\\mu_{1}\\Delta\\bar{u}+\\bar{u}\\bigl[a(x)-b\\bar{u}-c(x)\\underline{v} \\bigr]=\\bar {u}\\biggl[\\biggl(1-\\frac{b(x)}{\\min_{x\\in\\overline{\\Omega}}b(x)}\\biggr)\\theta_{\\mu_{1}, a(x)}-c(x) \\underline{v}\\biggr]< 0,$$\n\nfor all $$x\\in\\Omega$$; and from (3.4), we have\n\n$$\\mu_{2}\\Delta\\underline{v}+\\underline{v}\\bigl[d(x)-e\\bar{u}-f(x) \\underline {v}\\bigr]=\\underline{v}\\bigl[d(x)-\\mu_{2} \\lambda_{1}-e(x)\\bar{u}-f(x)\\underline{v}\\bigr]>0,$$\n\non Ω. Similarly, we have\n\n\\begin{aligned}& \\mu_{1}\\Delta\\underline{u}+\\underline{u}\\bigl[a(x)-b(x) \\underline{u}-c(x)\\bar{v}\\bigr]>0, \\\\& \\mu_{2}\\Delta\\bar{v}+\\bar{v}\\bigl[d(x)-e(x)\\underline{u}-f(x)\\bar{v} \\bigr]< 0. \\end{aligned}\n\nBy Theorem 1.3 in (also see Pao , Section 10.5), the conclusion of the theorem follows from the uniqueness assumption, the inequalities $$\\underline {u}(x)\\leq u^{0}\\leq\\bar{u}(x)$$, $$\\underline{v}(x)\\leq v^{0}\\leq\\bar{v}(x)$$, $$x\\in\\overline{\\Omega}$$, and a comparison with solutions of the differential system (1.1) with initial conditions replaced at the steady-state upper lower solutions $$(\\bar{u}(x), \\underline{v}(x))$$.\n\nWe next remove condition (3.2) on the initial functions $$u^{0}(x)$$, $$v^{0}(x)$$. First, observe that there exists large $$K>1$$, such that\n\n$$u^{0}(x)\\leq K\\bar{u}, \\qquad V^{0}(x)\\leq K\\bar{v},$$\n\non Ω. Define $$(\\overline{U}(x,t), \\underline{V}(x,t))$$ to be the solution of problem (1.1) with initial conditions replaced with\n\n$$\\bigl(\\overline{U}(x,0), \\underline{V}(x,0)\\bigr)=(K\\bar{u}, 0).$$\n\nIt is clear that $$\\underline{V}\\equiv0$$, $$\\overline{U}$$ is non-negative in $$\\Omega\\times[0, \\infty)$$ and\n\n$$\\lim_{t\\rightarrow\\infty}\\overline{U}(x,t)= \\overline{U}^{\\star}(x) \\quad \\mbox{for } x\\in\\Omega,$$\n(3.5)\n\nwhere $$\\overline{U}^{\\star}(x)$$ is the unique positive solution of the problem\n\n$$\\mu_{1}\\Delta z+z\\bigl[a(x)-b(x)z\\bigr]=0 \\quad\\mbox{in } \\Omega, \\qquad z=0 \\quad\\mbox{on } \\partial\\Omega.$$\n(3.6)\n\nMoreover, the convergence above is monotone, because $$\\overline {U}(x,0)$$, $$\\underline{V}(x,0)$$ satisfies\n\n\\begin{aligned}& \\mu_{1}\\Delta\\overline{U}(x,0)+\\overline{U}(x,0)\\bigl[a(x)-b(x) \\overline {U}(x,0)-c(x)\\underline{V}(x,0)\\bigr]\\\\& \\quad=\\bigl(\\overline{U}(x,0) \\bigr)^{2}\\biggl[\\frac{\\min_{x\\in\\overline{\\Omega}}b(x)}{K}-b\\biggr]< 0, \\\\& \\mu_{2}\\Delta\\underline{V}(x,0)+\\underline{V}(x,0)\\bigl[d(x)-e(x) \\overline {U}(x,0)-f(x)\\underline{V}(x,0)\\bigr]=0. \\end{aligned}\n\nThe convergence in (3.5) is also in $$C^{1}(\\overline {\\Omega})$$ norm by using the $$W^{2,p}$$ estimates, compact embedding, and (1.1). Similarly, define $$(\\underline{U}(x,t), \\overline{V}(x,t))$$ to be the solution of problem (1.1) with initial conditions replaced with\n\n$$\\bigl(\\underline{U}(x,0), \\overline{V}(x,0)\\bigr)=(0, K\\bar{v}).$$\n\nWe have $$\\underline{U}\\equiv0$$, $$\\overline{V}$$ is non-negative in $$\\Omega\\times[0, \\infty)$$, and we have monotone $$C^{1}(\\overline{\\Omega })$$ convergence,\n\n$$\\lim_{t\\rightarrow\\infty}\\overline{V}(x,t)= \\overline{V}^{\\star}(x),$$\n(3.7)\n\nwhere $$\\overline{V}^{\\star}(x)$$ is the unique positive solution of the problem\n\n$$\\mu_{2}\\Delta z+z\\bigl[d(x)-f(x)z\\bigr] \\quad\\mbox{in } \\Omega, \\qquad z\|_{\\partial\\Omega}=0.$$\n(3.8)\n\nOn the other hand, one readily verifies that the functions $$\\underline {U}(x,t)$$, $$\\overline{U}(x,t)$$, $$\\underline{V}(x,t)$$, $$\\overline {V}(x,t)$$ satisfy\n\n\\begin{aligned} &\\mu_{1}\\Delta\\overline{U}+ \\overline{U}\\bigl[a(x)-b(x)\\overline {U}-c(x)\\underline{V}\\bigr]-\\partial \\overline{U}/\\partial t< 0, \\\\ &\\mu_{2}\\Delta\\underline{V}+\\underline{V}\\bigl[d(x)-e(x)\\overline {U}-f(x)\\underline{V}\\bigr]-\\partial\\underline{V}/\\partial t\\geq0, \\\\ &\\mu_{2}\\Delta\\overline{V}+\\overline{V}\\bigl[d(x)-e(x)\\underline {U}-f(x)\\overline{V}\\bigr]-\\partial\\overline{V}/\\partial t< 0, \\\\ &\\mu_{1}\\Delta\\underline{U}+\\underline{U}\\bigl[a(x)-b(x)\\underline {U}-c(x)\\overline{V}\\bigr]-\\partial\\underline{U}/\\partial t\\geq0, \\end{aligned}\n(3.9)\n\nfor $$(x,t)\\in\\Omega\\times(0, \\infty)$$, and\n\n\\begin{aligned} &0=\\underline{U}(x,0)\\leq u^{0}(x)\\leq \\overline{U}(x,0)=K\\bar{u}, \\\\ &0=\\underline{V}(x,0)\\leq v^{0}(x)\\leq\\overline{V}(x,0)=K\\bar{v}, \\end{aligned}\n(3.10)\n\nfor $$x\\in\\overline{\\Omega}$$. From the comparison theorems, we assert that\n\n\\begin{aligned} &0=\\underline{U}(x,t)\\leq u^{0}(x,t)\\leq\\overline{U}(x,t), \\\\ &0=\\underline{V}(x,t)\\leq v^{0}(x,t)\\leq\\overline{V}(x,t), \\end{aligned}\n(3.11)\n\nfor $$(x,t)\\in\\Omega\\times[0,\\infty)$$. We next observe that $$\\mu _{1}\\Delta\\bar{u}+\\bar{u}[a(x)-b(x)\\bar{u}]<0$$ in Ω, $$\\bar {u}\|_{\\partial\\Omega}=0$$, thus $$\\bar{u}=\\theta_{\\mu_{1},a(x)}/\\min_{x\\in\\overline{\\Omega}}b(x)$$ is a strict upper solution of the problem (3.6). Similarly, $$\\bar{v}$$ is a strict upper solution of the problem (3.8).\n\nBy monotone iteration and comparison, we obtain\n\n$$\\overline{U}^{\\star}\\ll\\bar{u}, \\qquad \\overline{V}^{\\star} \\ll\\bar{v}.$$\n(3.12)\n\nFor $$s>0$$, let $$u^{s}(x)=u(x,s)$$, $$v^{s}(x)=v(x,s)$$ for $$x\\in\\overline {\\Omega}$$. We obtain from (3.5), (3.7), (3.11), and (3.12) for $$s>0$$ sufficiently large\n\n$$u^{s}(x)\\leq\\bar{u}, \\qquad v^{s}(x)\\leq \\bar{v},$$\n(3.13)\n\nfor $$x\\in\\overline{\\Omega}$$. On the other hand for $$s>0$$, we find from the theory of parabolic equations and the strong maximum principle that $$u^{s}$$, $$v^{s}$$ are in $$C^{1}(\\overline{\\Omega})$$ and\n\n$$u^{s}(x)\\gg0,\\qquad v^{s}(x)\\gg0.$$\n(3.14)\n\nComparing (3.13) and (3.14), respectively, with (3.2) and (3.1), we obtain the conclusion of this theorem by using the first part of the proof. □\n\n## References\n\n1. Cosner, C, Laser, C: Stable coexistence states in the Volterra-Lotka competition model with diffusion. SIAM J. Appl. Math. 44, 1112-1132 (1984)\n\n2. Dancer, E: On the existence and uniqueness of positive solutions for competing species models with diffusion. Trans. Am. Math. Soc. 36, 829-859 (1991)\n\n3. Dancer, E, Guo, Z: Uniqueness and stability for solutions of competing species equations with large interactions. Commun. Appl. Nonlinear Anal. 1, 19-45 (1994)\n\n4. Leung, A: Nonlinear Systems of Partial Differential Equations: Applications to Life and Physical Sciences. World Scientific, Singapore (2009)\n\n5. Leung, A: Systems of Nonlinear Partial Equations. Applications to Biology and Engineering. Kluwer Academic, New York (1989)\n\n6. Ruan, W, Pao, C: Positive steady-state solutions of competing reaction diffusion systems. J. Differ. Equ. 117, 411-427 (1995)\n\n7. Cantrell, R, Cosner, C: Spatial Ecology via Reaction-Diffusion Equations. Wiley, New York (2003)\n\n8. He, X, Ni, W: The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system I: heterogeneity vs. homogeneity. J. Differ. Equ. 254, 528-546 (2013)\n\n9. He, X, Ni, W: The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system II: the general case. J. Differ. Equ. 254, 4088-4108 (2013)\n\n10. Álvarez-Caudevilla, P, Du, Y, Peng, R: Qualitative analysis of a cooperative reaction-diffusion system in a spatiotemporally degenerate environment. SIAM J. Math. Anal. 46, 499-531 (2014)\n\n11. Pao, C: Nonlinear Parabolic and Elliptic Equations. Plenum, New York (1992)\n\n## Acknowledgements\n\nThe authors are very grateful to the anonymous referees for their valuable comments and suggestions. This work is supported by the Natural Science Foundation of Shanghai, China (No. 13ZR1430100).\n\n## Author information\n\nAuthors\n\n### Corresponding author\n\nCorrespondence to Benlong Xu.\n\n### Competing interests\n\nThe authors declare that they have no competing interests.\n\n### Authors’ contributions\n\nAll the authors read and approved the final manuscript.\n\n## Rights and permissions", null, "" ]	[ null, "https://boundaryvalueproblems.springeropen.com/track/article/10.1186/s13661-015-0368-7", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5488178,"math_prob":1.0000004,"size":7451,"snap":"2022-40-2023-06","text_gpt3_token_len":3043,"char_repetition_ratio":0.18893515,"word_repetition_ratio":0.06056338,"special_character_ratio":0.40679103,"punctuation_ratio":0.13822116,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000045,"pos_list":[0,1,2],"im_url_duplicate_count":[null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-02-02T17:48:51Z\",\"WARC-Record-ID\":\"<urn:uuid:6546199c-57ef-4324-916d-27ef36668d8d>\",\"Content-Length\":\"233522\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:63dda7c7-0d21-4967-83f7-70549d18d2c1>\",\"WARC-Concurrent-To\":\"<urn:uuid:a767c067-333a-4aa9-bec4-823832d7ed2b>\",\"WARC-IP-Address\":\"146.75.32.95\",\"WARC-Target-URI\":\"https://boundaryvalueproblems.springeropen.com/articles/10.1186/s13661-015-0368-7\",\"WARC-Payload-Digest\":\"sha1:AXCKAIDCNJFSFR7IDNTEVM4DEXHUZPST\",\"WARC-Block-Digest\":\"sha1:GPAIBEMXFBGVXANL6UKSAHWFF3Y3G7P5\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764500035.14_warc_CC-MAIN-20230202165041-20230202195041-00010.warc.gz\"}"}
https://stacks.math.columbia.edu/tag/04DC	[ "## 29.45 Universal homeomorphisms\n\nThe following definition is really superfluous since a universal homeomorphism is really just an integral, universally injective and surjective morphism, see Lemma 29.45.5.\n\nDefinition 29.45.1. A morphisms $f : X \\to Y$ of schemes is called a universal homeomorphism if the base change $f' : Y' \\times _ Y X \\to Y'$ is a homeomorphism for every morphism $Y' \\to Y$.\n\nFirst we state the obligatory lemmas.\n\nLemma 29.45.2. The base change of a universal homeomorphism of schemes by any morphism of schemes is a universal homeomorphism.\n\nProof. This is immediate from the definition. $\\square$\n\nLemma 29.45.3. The composition of a pair of universal homeomorphisms of schemes is a universal homeomorphism.\n\nProof. Omitted. $\\square$\n\nThe following simple lemma is the key to characterizing universal homeomorphisms.\n\nLemma 29.45.4. Let $f : X \\to Y$ be a morphism of schemes. If $f$ is a homeomorphism onto a closed subset of $Y$ then $f$ is affine.\n\nProof. Let $y \\in Y$ be a point. If $y \\not\\in f(X)$, then there exists an affine neighbourhood of $y$ which is disjoint from $f(X)$. If $y \\in f(X)$, let $x \\in X$ be the unique point of $X$ mapping to $y$. Let $y \\in V$ be an affine open neighbourhood. Let $U \\subset X$ be an affine open neighbourhood of $x$ which maps into $V$. Since $f(U) \\subset V \\cap f(X)$ is open in the induced topology by our assumption on $f$ we may choose a $h \\in \\Gamma (V, \\mathcal{O}_ Y)$ such that $y \\in D(h)$ and $D(h) \\cap f(X) \\subset f(U)$. Denote $h' \\in \\Gamma (U, \\mathcal{O}_ X)$ the restriction of $f^\\sharp (h)$ to $U$. Then we see that $D(h') \\subset U$ is equal to $f^{-1}(D(h))$. In other words, every point of $Y$ has an open neighbourhood whose inverse image is affine. Thus $f$ is affine, see Lemma 29.11.3. $\\square$\n\nLemma 29.45.5. Let $f : X \\to Y$ be a morphism of schemes. The following are equivalent:\n\n1. $f$ is a universal homeomorphism, and\n\n2. $f$ is integral, universally injective and surjective.\n\nProof. Assume $f$ is a universal homeomorphism. By Lemma 29.45.4 we see that $f$ is affine. Since $f$ is clearly universally closed we see that $f$ is integral by Lemma 29.44.7. It is also clear that $f$ is universally injective and surjective.\n\nAssume $f$ is integral, universally injective and surjective. By Lemma 29.44.7 $f$ is universally closed. Since it is also universally bijective (see Lemma 29.9.4) we see that it is a universal homeomorphism. $\\square$\n\nLemma 29.45.6. Let $X$ be a scheme. The canonical closed immersion $X_{red} \\to X$ (see Schemes, Definition 26.12.5) is a universal homeomorphism.\n\nProof. Omitted. $\\square$\n\nLemma 29.45.7. Let $f : X \\to S$ and $S' \\to S$ be morphisms of schemes. Assume\n\n1. $S' \\to S$ is a closed immersion,\n\n2. $S' \\to S$ is bijective on points,\n\n3. $X \\times _ S S' \\to S'$ is a closed immersion, and\n\n4. $X \\to S$ is of finite type or $S' \\to S$ is of finite presentation.\n\nThen $f : X \\to S$ is a closed immersion.\n\nProof. Assumptions (1) and (2) imply that $S' \\to S$ is a universal homeomorphism (for example because $S_{red} = S'_{red}$ and using Lemma 29.45.6). Hence (3) implies that $X \\to S$ is homeomorphism onto a closed subset of $S$. Then $X \\to S$ is affine by Lemma 29.45.4. Let $U \\subset S$ be an affine open, say $U = \\mathop{\\mathrm{Spec}}(A)$. Then $S' = \\mathop{\\mathrm{Spec}}(A/I)$ by (1) for a locally nilpotent ideal $I$ by (2). As $f$ is affine we see that $f^{-1}(U) = \\mathop{\\mathrm{Spec}}(B)$. Assumption (4) tells us $B$ is a finite type $A$-algebra (Lemma 29.15.2) or that $I$ is finitely generated (Lemma 29.21.7). Assumption (3) is that $A/I \\to B/IB$ is surjective. From Algebra, Lemma 10.126.9 if $A \\to B$ is of finite type or Algebra, Lemma 10.20.1 if $I$ is finitely generated and hence nilpotent we deduce that $A \\to B$ is surjective. This means that $f$ is a closed immersion, see Lemma 29.2.1. $\\square$\n\n## Post a comment\n\nYour email address will not be published. Required fields are marked.\n\nIn your comment you can use Markdown and LaTeX style mathematics (enclose it like $\\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).\n\nUnfortunately JavaScript is disabled in your browser, so the comment preview function will not work.\n\nAll contributions are licensed under the GNU Free Documentation License.\n\nIn order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 04DC. Beware of the difference between the letter 'O' and the digit '0'." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8478967,"math_prob":0.9983587,"size":5186,"snap":"2021-21-2021-25","text_gpt3_token_len":1615,"char_repetition_ratio":0.19895793,"word_repetition_ratio":0.3585746,"special_character_ratio":0.31237948,"punctuation_ratio":0.14891775,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9999734,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-24T05:39:39Z\",\"WARC-Record-ID\":\"<urn:uuid:f797be5d-4a42-4916-b849-321a6deee31a>\",\"Content-Length\":\"19167\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:865bf59d-63cc-428d-88b0-dd25b59dbd77>\",\"WARC-Concurrent-To\":\"<urn:uuid:2ac532d1-0247-4d0a-a7c9-a071485569f0>\",\"WARC-IP-Address\":\"128.59.222.85\",\"WARC-Target-URI\":\"https://stacks.math.columbia.edu/tag/04DC\",\"WARC-Payload-Digest\":\"sha1:FYDDYLBDDIJ5YFNS7R6ICDFNFZPK6FAK\",\"WARC-Block-Digest\":\"sha1:J3CX5XUY4ASRGQRUGSFXXXYUYKAFU5OE\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623488551052.94_warc_CC-MAIN-20210624045834-20210624075834-00330.warc.gz\"}"}
https://appscrawl.com/microsoft-math-solver-for-pc-windows-7-8-10-mac-free-download/	[ "# Microsoft Math Solver For PC / Windows 7/8/10 / Mac – Free Download", null, "You can now play Microsoft Math Solver for PC on a desktop/laptop running Windows XP, Windows 7, Windows 8, Windows 8.1, Windows 10 and MacOS/OS X. This can easily be done with the help of BlueStacks or Andy OS Android emulator.\n\nMicrosoft Math solver app provides help with a variety of problems including arithmetic, algebra, trigonometry, calculus, statistics, and other topics using an advanced AI powered math solver. Simply write a math problem on screen or use the camera to snap a math photo. Microsoft Math problem solver instantly recognizes the problem and helps you to solve it with ⚡FREE Step-By-Step Explanations⚡, interactive graphs, similar problems from the web and online video lectures. Quickly look up related math concepts. Get help with your homework problems and gain confidence in mastering the techniques 💯 with Microsoft Math. ⚡It is absolutely FREE and No Ads!⚡\n\nHIGHLIGHTS\n● ✏️ Write a math equation on screen as you naturally do on paper\n● 📷 Scan printed or handwritten math photo\n● ⌨️ Type and edit using advanced scientific math calculator\n● Get interactive Step-by-Step explanations 💡 & Graphing calculator 📈\n● Import images with math equations from gallery 🖼️\n● Scan and Solve Math Worksheets with multiple problems\n● 🔎Search the web for similar problems and video lectures\n● Try math word problems\n● Scan and plot x-y data tables for linear/non-linear functions\n● Learn math in your language – supports Chinese, French, German, Hindi, Italian, Japanese, Portuguese, Russian, Spanish, and many more\n\nSUPPORTED PROBLEMS\n● Elementary: arithmetic, real, complex numbers, LCM, GCD, factors, roman numerals\n● Pre-Algebra: radicals and exponents, fractions, matrices, determinants\n● Algebra: quadratic equations, system of equations, inequalities, rational expressions, linear, quadratic and exponential graphs\n● Word problems on math concepts, number theory, probability, volume, surface area\n● Basic Calculus: Summations, Limits, derivatives, integrals\n● Statistics: Mean, Median, Mode, Standard Deviation, permutations, combinations\n\nFind out more about Microsoft Math Solver app on our website 👉 Link: https://math.microsoft.com\n\nMicrosoft Math Solver For PC can be easily installed and used on a desktop computer or laptop running Windows XP, Windows 7, Windows 8, Windows 8.1, Windows 10 and a Macbook, iMac running Mac OS X. This will be done using an Android emulator. To install Microsoft Math Solver For PC, we will use BlueStacks app player. The method listed below is set to help you get Microsoft Math Solver For PC. Go ahead and get it done now.", null, "Download and use Microsoft Math Solver on your PC & Mac using an Android Emulator.\n\n### Step to Step Guide / Microsoft Math Solver For PC:\n\nMicrosoft Math Solver by Microsoft Corporation,", null, "" ]	[ null, "http://lh3.googleusercontent.com/IynxIYziGff1xaEiuAEZirWVMrL5W0QpSf0d6favDk3IeOB7Cg8PmsX40DSFgtR7Dw=w300", null, "http://lh3.googleusercontent.com/IynxIYziGff1xaEiuAEZirWVMrL5W0QpSf0d6favDk3IeOB7Cg8PmsX40DSFgtR7Dw=w300", null, "http://appscrawl.com/wp-content/uploads/2017/03/gp_logo-1.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8122217,"math_prob":0.7409282,"size":3749,"snap":"2020-45-2020-50","text_gpt3_token_len":858,"char_repetition_ratio":0.15006675,"word_repetition_ratio":0.05901639,"special_character_ratio":0.21259002,"punctuation_ratio":0.14265129,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9815612,"pos_list":[0,1,2,3,4,5,6],"im_url_duplicate_count":[null,2,null,2,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-11-24T23:37:24Z\",\"WARC-Record-ID\":\"<urn:uuid:5130a00b-cad3-423f-b1ab-067236e01b54>\",\"Content-Length\":\"41655\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0e8c02d6-30cf-480c-9ddf-2fc2f67a677f>\",\"WARC-Concurrent-To\":\"<urn:uuid:333c8071-35e9-42ad-abe6-9eb4751905ca>\",\"WARC-IP-Address\":\"199.188.204.205\",\"WARC-Target-URI\":\"https://appscrawl.com/microsoft-math-solver-for-pc-windows-7-8-10-mac-free-download/\",\"WARC-Payload-Digest\":\"sha1:X4CJ2DTMDMVPW5VS26OF2ABVEMIJFWBR\",\"WARC-Block-Digest\":\"sha1:AM6UWKYLUYORDQEKHREKC6BHQK3V22Q2\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-50/CC-MAIN-2020-50_segments_1606141177607.13_warc_CC-MAIN-20201124224124-20201125014124-00610.warc.gz\"}"}
https://transform.softwareunderground.org/2022-gstools/binary	[ "Tutorials", null, "# binary fields\n\n%matplotlib widget\nimport matplotlib.pyplot as plt\nplt.ioff()\n# turn of warnings\nimport warnings\nwarnings.filterwarnings('ignore')\n\nHere we transform a field to a binary field with only two values. The dividing value is the mean by default and the upper and lower values are derived to preserve the variance.\n\nSee transform.binary\n\nimport gstools as gs\n\n# structured field with a size of 100x100 and a grid-size of 1x1\nx = y = range(100)\nmodel = gs.Gaussian(dim=2, var=1, len_scale=10)\nsrf = gs.SRF(model, seed=20170519)\nsrf.structured([x, y])\nsrf.transform(\"binary\")\nsrf.plot()" ]	[ null, "https://transform.softwareunderground.org/build/_assets/jupyter-7WANWHPN.svg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.69593465,"math_prob":0.97600514,"size":588,"snap":"2022-40-2023-06","text_gpt3_token_len":157,"char_repetition_ratio":0.12842466,"word_repetition_ratio":0.0,"special_character_ratio":0.26870748,"punctuation_ratio":0.13043478,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9931765,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-01-30T22:12:37Z\",\"WARC-Record-ID\":\"<urn:uuid:f5547ab4-18d9-48cf-857f-8069b35044e4>\",\"Content-Length\":\"83309\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:69239852-32ea-4b4b-aac0-3ba867649bbf>\",\"WARC-Concurrent-To\":\"<urn:uuid:c94663fd-bd4c-4d78-abd1-30dc805b1278>\",\"WARC-IP-Address\":\"76.76.21.98\",\"WARC-Target-URI\":\"https://transform.softwareunderground.org/2022-gstools/binary\",\"WARC-Payload-Digest\":\"sha1:X4TL6KSBE2UWRPD46YMRHOATZBAR4RIF\",\"WARC-Block-Digest\":\"sha1:DYASVLIXSTA2JDONIT3K5GBPPHC5OZWU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764499829.29_warc_CC-MAIN-20230130201044-20230130231044-00813.warc.gz\"}"}
http://entercad.ru/acadauto.en/ex_majorradius.htm	[ "```Sub Example_MajorRadius()\n' This example creates an Ellipse in model space and displays\n' both the Major radius and the Minor radius of the new Ellipse\n\nDim majAxis(0 To 2) As Double, center(0 To 2) As Double\n\n' Create an ellipse in model space\ncenter(0) = 5: center(1) = 5: center(2) = 0\nmajAxis(0) = 10: majAxis(1) = 20: majAxis(2) = 0" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5215217,"math_prob":0.97707987,"size":766,"snap":"2020-45-2020-50","text_gpt3_token_len":213,"char_repetition_ratio":0.15223098,"word_repetition_ratio":0.075630255,"special_character_ratio":0.26501307,"punctuation_ratio":0.120567374,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98504937,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-11-24T23:37:37Z\",\"WARC-Record-ID\":\"<urn:uuid:dcd4584d-fcfd-4fde-b6d4-6a3839c6f8fc>\",\"Content-Length\":\"9205\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:741cd6f0-e64c-4687-8aaa-9b86eab39c3a>\",\"WARC-Concurrent-To\":\"<urn:uuid:13050e50-48f2-41a6-96f1-c481296dc74f>\",\"WARC-IP-Address\":\"45.89.69.168\",\"WARC-Target-URI\":\"http://entercad.ru/acadauto.en/ex_majorradius.htm\",\"WARC-Payload-Digest\":\"sha1:V64HJJH5FYIW4LH2CTASVUCYSVG26BZV\",\"WARC-Block-Digest\":\"sha1:CRCM4OWBP3YFHHC2C5CFNOH2JT3DC5EK\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-50/CC-MAIN-2020-50_segments_1606141177607.13_warc_CC-MAIN-20201124224124-20201125014124-00299.warc.gz\"}"}
https://www.colorhexa.com/00e823	[ "# #00e823 Color Information\n\nIn a RGB color space, hex #00e823 is composed of 0% red, 91% green and 13.7% blue. Whereas in a CMYK color space, it is composed of 100% cyan, 0% magenta, 84.9% yellow and 9% black. It has a hue angle of 129.1 degrees, a saturation of 100% and a lightness of 45.5%. #00e823 color hex could be obtained by blending #00ff46 with #00d100. Closest websafe color is: #00ff33.\n\n• R 0\n• G 91\n• B 14\nRGB color chart\n• C 100\n• M 0\n• Y 85\n• K 9\nCMYK color chart\n\n#00e823 color description : Pure (or mostly pure) lime green.\n\n# #00e823 Color Conversion\n\nThe hexadecimal color #00e823 has RGB values of R:0, G:232, B:35 and CMYK values of C:1, M:0, Y:0.85, K:0.09. Its decimal value is 59427.\n\nHex triplet RGB Decimal 00e823 `#00e823` 0, 232, 35 `rgb(0,232,35)` 0, 91, 13.7 `rgb(0%,91%,13.7%)` 100, 0, 85, 9 129.1°, 100, 45.5 `hsl(129.1,100%,45.5%)` 129.1°, 100, 91 00ff33 `#00ff33`\nCIE-LAB 80.645, -79.354, 72.876 29.158, 57.831, 11.216 0.297, 0.589, 57.831 80.645, 107.74, 137.437 80.645, -75.971, 95.566 76.047, -64.641, 44.488 00000000, 11101000, 00100011\n\n# Color Schemes with #00e823\n\n• #00e823\n``#00e823` `rgb(0,232,35)``\n• #e800c5\n``#e800c5` `rgb(232,0,197)``\nComplementary Color\n• #51e800\n``#51e800` `rgb(81,232,0)``\n• #00e823\n``#00e823` `rgb(0,232,35)``\n• #00e897\n``#00e897` `rgb(0,232,151)``\nAnalogous Color\n• #e80051\n``#e80051` `rgb(232,0,81)``\n• #00e823\n``#00e823` `rgb(0,232,35)``\n• #9700e8\n``#9700e8` `rgb(151,0,232)``\nSplit Complementary Color\n• #e82300\n``#e82300` `rgb(232,35,0)``\n• #00e823\n``#00e823` `rgb(0,232,35)``\n• #2300e8\n``#2300e8` `rgb(35,0,232)``\n• #c5e800\n``#c5e800` `rgb(197,232,0)``\n• #00e823\n``#00e823` `rgb(0,232,35)``\n• #2300e8\n``#2300e8` `rgb(35,0,232)``\n• #e800c5\n``#e800c5` `rgb(232,0,197)``\n• #009c17\n``#009c17` `rgb(0,156,23)``\n• #00b51b\n``#00b51b` `rgb(0,181,27)``\n• #00cf1f\n``#00cf1f` `rgb(0,207,31)``\n• #00e823\n``#00e823` `rgb(0,232,35)``\n• #02ff29\n``#02ff29` `rgb(2,255,41)``\n• #1cff3e\n``#1cff3e` `rgb(28,255,62)``\n• #36ff54\n``#36ff54` `rgb(54,255,84)``\nMonochromatic Color\n\n# Alternatives to #00e823\n\nBelow, you can see some colors close to #00e823. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #17e800\n``#17e800` `rgb(23,232,0)``\n• #04e800\n``#04e800` `rgb(4,232,0)``\n• #00e810\n``#00e810` `rgb(0,232,16)``\n• #00e823\n``#00e823` `rgb(0,232,35)``\n• #00e836\n``#00e836` `rgb(0,232,54)``\n• #00e84a\n``#00e84a` `rgb(0,232,74)``\n• #00e85d\n``#00e85d` `rgb(0,232,93)``\nSimilar Colors\n\n# #00e823 Preview\n\nThis text has a font color of #00e823.\n\n``<span style=\"color:#00e823;\">Text here</span>``\n#00e823 background color\n\nThis paragraph has a background color of #00e823.\n\n``<p style=\"background-color:#00e823;\">Content here</p>``\n#00e823 border color\n\nThis element has a border color of #00e823.\n\n``<div style=\"border:1px solid #00e823;\">Content here</div>``\nCSS codes\n``.text {color:#00e823;}``\n``.background {background-color:#00e823;}``\n``.border {border:1px solid #00e823;}``\n\n# Shades and Tints of #00e823\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, #001002 is the darkest color, while #fcfffc is the lightest one.\n\n• #001002\n``#001002` `rgb(0,16,2)``\n• #002405\n``#002405` `rgb(0,36,5)``\n• #003708\n``#003708` `rgb(0,55,8)``\n• #004b0b\n``#004b0b` `rgb(0,75,11)``\n• #005f0e\n``#005f0e` `rgb(0,95,14)``\n• #007211\n``#007211` `rgb(0,114,17)``\n• #008614\n``#008614` `rgb(0,134,20)``\n• #009a17\n``#009a17` `rgb(0,154,23)``\n``#00ad1a` `rgb(0,173,26)``\n• #00c11d\n``#00c11d` `rgb(0,193,29)``\n• #00d420\n``#00d420` `rgb(0,212,32)``\n• #00e823\n``#00e823` `rgb(0,232,35)``\n• #00fc26\n``#00fc26` `rgb(0,252,38)``\n• #10ff34\n``#10ff34` `rgb(16,255,52)``\n• #24ff45\n``#24ff45` `rgb(36,255,69)``\n• #37ff56\n``#37ff56` `rgb(55,255,86)``\n• #4bff66\n``#4bff66` `rgb(75,255,102)``\n• #5fff77\n``#5fff77` `rgb(95,255,119)``\n• #72ff88\n``#72ff88` `rgb(114,255,136)``\n• #86ff98\n``#86ff98` `rgb(134,255,152)``\n• #9affa9\n``#9affa9` `rgb(154,255,169)``\n``#adffba` `rgb(173,255,186)``\n• #c1ffca\n``#c1ffca` `rgb(193,255,202)``\n• #d4ffdb\n``#d4ffdb` `rgb(212,255,219)``\n• #e8ffeb\n``#e8ffeb` `rgb(232,255,235)``\n• #fcfffc\n``#fcfffc` `rgb(252,255,252)``\nTint Color Variation\n\n# Tones of #00e823\n\nA tone is produced by adding gray to any pure hue. In this case, #6b7d6e is the less saturated color, while #00e823 is the most saturated one.\n\n• #6b7d6e\n``#6b7d6e` `rgb(107,125,110)``\n• #628668\n``#628668` `rgb(98,134,104)``\n• #598f61\n``#598f61` `rgb(89,143,97)``\n• #50985b\n``#50985b` `rgb(80,152,91)``\n• #47a155\n``#47a155` `rgb(71,161,85)``\n• #3eaa4f\n``#3eaa4f` `rgb(62,170,79)``\n• #36b248\n``#36b248` `rgb(54,178,72)``\n• #2dbb42\n``#2dbb42` `rgb(45,187,66)``\n• #24c43c\n``#24c43c` `rgb(36,196,60)``\n• #1bcd36\n``#1bcd36` `rgb(27,205,54)``\n• #12d62f\n``#12d62f` `rgb(18,214,47)``\n• #09df29\n``#09df29` `rgb(9,223,41)``\n• #00e823\n``#00e823` `rgb(0,232,35)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #00e823 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":0.50097644,"math_prob":0.8838481,"size":3666,"snap":"2023-40-2023-50","text_gpt3_token_len":1618,"char_repetition_ratio":0.13517204,"word_repetition_ratio":0.007352941,"special_character_ratio":0.55591923,"punctuation_ratio":0.22866894,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9924742,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-11-28T17:00:04Z\",\"WARC-Record-ID\":\"<urn:uuid:4b34eeed-b0da-4098-8fb9-629f27d1a6cf>\",\"Content-Length\":\"36162\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0df36871-c404-469d-9ec9-61d759c256dc>\",\"WARC-Concurrent-To\":\"<urn:uuid:7e722ac7-4cfa-4c83-b013-e80acb84a5d1>\",\"WARC-IP-Address\":\"178.32.117.56\",\"WARC-Target-URI\":\"https://www.colorhexa.com/00e823\",\"WARC-Payload-Digest\":\"sha1:5TTUA7UXWW327TB6EE6VA5UTEEHDOK3L\",\"WARC-Block-Digest\":\"sha1:T5JRQKHV5HICJ4D3DKSJTMUCYYBYEMDM\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679099892.46_warc_CC-MAIN-20231128151412-20231128181412-00498.warc.gz\"}"}
https://www.onlinemath4all.com/trigonometry-word-problems-with-solutions.html	[ "# TRIGONOMETRY WORD PROBLEMS WITH SOLUTIONS\n\nTrigonometry Word Problems with Solutions :\n\nIn this section, you will learn how to solve word problems in trigonometry step by step.\n\n## Trigonometry Word Problems with Solutions\n\nProblem 1 :\n\nThe angle of elevation of the top of the building at a distance of 50 m from its foot on a horizontal plane is found to be 60 degree. Find the height of the building.\n\nSolution :\n\nDraw a sketch.", null, "Here, AB represents height of the building, BC represents distance of the building from the point of observation.\n\nIn the right triangle ABC, the side which is opposite to the angle 60 degree is known as opposite side (AB), the side which is opposite to 90 degree is called hypotenuse side (AC) and the remaining side is called adjacent side (BC).\n\nNow we need to find the length of the side AB.\n\ntan 60° = AB/BC\n\n√3 = AB/50\n\n√3 x 50 = AB\n\nAB = 50√3\n\nApproximate value of √3 is 1.732\n\nAB = 50 (1.732)\n\nAB = 86.6 m\n\nSo, the height of the building is 86.6 m.\n\nProblem 2 :\n\nA ladder placed against a wall such that it reaches the top of the wall of height 6 m and the ladder is inclined at an angle of 60 degree. Find how far the ladder is from the foot of the wall.\n\nSolution :\n\nDraw a sketch.", null, "Here AB represents height of the wall, BC represents the distance between the wall and the foot of the ladder and AC represents the length of the ladder.\n\nIn the right triangle ABC, the side which is opposite to angle 60 degree is known as opposite side (AB), the side which is opposite to 90 degree is called hypotenuse side (AC) and remaining side is called adjacent side (BC).\n\nNow, we need to find the distance between foot of the ladder and the wall. That is, we have to find the length of BC.\n\ntan θ = Opposite side/Adjacent side\n\ntan60° = AB/BC\n\n√3 = 6/BC\n\nBC = 6/√3\n\nBC = (6/√3) x (√3/√3)\n\nBC = (6√3)/3\n\nBC = 2√3\n\nApproximate value of √3 is 1.732\n\nBC = 2 (1.732)\n\nBC = 3.464 m\n\nSo, the distance between foot of the ladder and the wall is 3.464 m.\n\nProblem 3 :\n\nA string of a kite is 100 meters long and it makes an angle of 60° with horizontal. Find the height of the kite,assuming that there is no slack in the string.\n\nSolution :\n\nDraw a sketch.", null, "Here AB represents height of kite from the ground, BC represents the distance of kite from the point of observation.\n\nIn the right triangle ABC the side which is opposite to angle 60 degree is known as opposite side (AB), the side which is opposite to 90 degree is called hypotenuse side (AC) and remaining side is called adjacent side (BC).\n\nNow we need to find the height of the side AB.\n\nSin θ = Opposite side/Hypotenuse side\n\nsinθ = AB/AC\n\nsin 60° = AB/100\n\n√3/2 = AB/100\n\n(√3/2) x 100 = AB\n\nAB = 50 √3 m\n\nSo, the height of kite from the ground 50 √3 m.\n\nProblem 4 :\n\nFrom the top of the tower 30 m height a man is observing the base of a tree at an angle of depression measuring 30 degree. Find the distance between the tree and the tower.\n\nSolution :\n\nDraw a sketch.", null, "Here AB represents height of the tower, BC represents the distance between foot of the tower and the foot of the tree.\n\nNow we need to find the distance between foot of the tower and the foot of the tree (BC).\n\ntan θ = Opposite side/Adjacent side\n\ntan 30° = AB/BC\n\n1/√3 = 30/BC\n\nBC = 30√3\n\nApproximate value of √3 is 1.732\n\nBC = 30 (1.732)\n\nBC = 81.96 m\n\nSo, the distance between the tree and the tower is 51.96 m.\n\nProblem 5 :\n\nA man wants to determine the height of a light house. He measured the angle at A and found that tan A = 3/4. What is the height of the light house if A is 40 m from the base?\n\nSolution :\n\nDraw a sketch.", null, "Here BC represents height of the light house, AB represents the distance between the light house from the point of observation.\n\nIn the right triangle ABC the side which is opposite to the angle A is known as opposite side (BC), the side which is opposite to 90 degree is called hypotenuse side (AC) and remaining side is called adjacent side (AB).\n\nNow we need to find the height of the light house (BC).\n\ntanA = BC/AB\n\nGiven : tanA = 3/4\n\n3/4 = BC/40\n\n3 x 40 = BC x 4\n\nBC = (3 x 40)/4\n\nBC = (3 x 10)\n\nBC = 30 m\n\nSo, the height of the light house is 30 m.\n\nProblem 6 :\n\nA man wants to determine the height of a light house. He A ladder is leaning against a vertical wall makes an angle of 20° with the ground. The foot of the ladder is 3 m from the wall.Find the length of ladder.\n\nSolution :\n\nDraw a sketch.", null, "Here AB represents height of the wall, BC represents the distance of the wall from the foot of the ladder.\n\nIn the right triangle ABC the side which is opposite to the angle 20 degree is known as opposite side (AB),the side which is opposite to 90 degree is called hypotenuse side (AC) and remaining side is called adjacent side (BC).\n\nNow we need to find the length of the ladder (AC).\n\nCos θ = Adjacent side/Hypotenuse side\n\nCos θ = BC/AC\n\nCos 20° = 3/AC\n\n0.9396 = 3/AC\n\nAC = 3/0.9396\n\nAC = 3.192\n\nSo, the length of the ladder is 3.192 m.\n\nProblem 7 :\n\nA kite flying at a height of 65 m is attached to a string inclined at 31° to the horizontal. What is the length of string ?\n\nSolution :\n\nDraw a sketch.", null, "Here AB represents height of the kite. In the right triangle ABC the side which is opposite to angle 31 degree is known as opposite side (AB), the side which is opposite to 90 degree is called hypotenuse side (AC) and the remaining side is called adjacent side (BC).\n\nNow we need to find the length of the string AC.\n\nSin θ = Opposite side/Hypotenuse side\n\nSin θ = AB/AC\n\nSin 31° = AB/AC\n\n0.5150 = 65/AC\n\nAC = 65/0.5150\n\nAC = 126.2 m\n\nHence, the length of the string is 126.2 m.\n\nProblem 8 :\n\nThe length of a string between a kite and a point on the ground is 90 m. If the string is making an angle θ with the level ground such that tan θ = 15/8, how high will the kite be ?\n\nSolution :\n\nDraw a sketch.", null, "Here AB represents height of the balloon from the ground. In the right triangle ABC the side which is opposite to angle θ is known as opposite side (AB), the side which is opposite to 90 degree is called hypotenuse side (AC) and remaining side is called adjacent side (BC).\n\nNow we need to find the length of the side AB.\n\nTan θ = 15/8 --------> Cot θ = 8/15\n\nCsc θ = √(1+ cot²θ)\n\nCsc θ = √(1 + 64/225)\n\nCsc θ = √(225 + 64)/225\n\nCsc θ = √289/225\n\nCsc θ = 17/15 -------> Sin θ = 15/17\n\nBut, sin θ = Opposite side/Hypotenuse side = AB/AC\n\nAB/AC = 15/17\n\nAB/90 = 15/17\n\nAB = (15 x 90)/17\n\nAB = 79.41\n\nSo, the height of the tower is 79.41 m.\n\nProblem 9 :\n\nAn aeroplane is observed to be approaching the airpoint. It is at a distance of 12 km from the point of observation and makes an angle of elevation of 50 degree. Find the height above the ground.\n\nSolution :\n\nDraw a sketch.", null, "Here AB represents height of the airplane from the ground.In the right triangle ABC the side which is opposite to angle 50 degree is known as opposite side (AB), the side which is opposite to 90 degree is called hypotenuse side (AC) and remaining side is called adjacent side (BC).\n\nNow we need to find the length of the side AB.\n\nFrom the figure given above, AB stands for the height of the aeroplane above the ground.\n\nsin θ = Opposite side/Hypotenuse side\n\nsin 50° = AB/AC\n\n0.7660 = h/12\n\n0.7660 x 12 = h\n\nh = 9.192 km\n\nSo, the height of the aeroplane above the ground is 9.192 km.\n\nProblem 10 :\n\nA balloon is connected to a meteorological station by a cable of length 200 m inclined at 60 degree angle . Find the height of the balloon from the ground. (Imagine that there is no slack in the cable)\n\nSolution :\n\nDraw a sketch.", null, "Here AB represents height of the balloon from the ground. In the right triangle ABC the side which is opposite to angle 60 degree is known as opposite side (AB), the side which is opposite to 90 degree is called hypotenuse (AC) and the remaining side is called as adjacent side (BC).\n\nNow we need to find the length of the side AB.\n\nFrom the figure given above, AB stands for the height of the balloon above the ground.\n\nsin θ = Opposite side/Hypotenuse side\n\nsin θ = AB/AC\n\nsin 60° = AB/200\n\n√3/2 = AB/200\n\nAB = (√3/2) x 200\n\nAB = 100√3\n\nApproximate value of √3 is 1.732\n\nAB = 100 (1.732)\n\nAB = 173.2 m\n\nSo, the height of the balloon from the ground is 173.2 m.", null, "After having gone through the stuff given above, we hope that the students would have understood, how to solve word problems in trigonometry.\n\nApart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.\n\nYou can also visit our following web pages on different stuff in math.\n\nWORD PROBLEMS\n\nWord problems on simple equations\n\nWord problems on linear equations\n\nAlgebra word problems\n\nWord problems on trains\n\nArea and perimeter word problems\n\nWord problems on direct variation and inverse variation\n\nWord problems on unit price\n\nWord problems on unit rate\n\nWord problems on comparing rates\n\nConverting customary units word problems\n\nConverting metric units word problems\n\nWord problems on simple interest\n\nWord problems on compound interest\n\nWord problems on types of angles\n\nComplementary and supplementary angles word problems\n\nDouble facts word problems\n\nTrigonometry word problems\n\nPercentage word problems\n\nProfit and loss word problems\n\nMarkup and markdown word problems\n\nDecimal word problems\n\nWord problems on fractions\n\nWord problems on mixed fractrions\n\nOne step equation word problems\n\nLinear inequalities word problems\n\nRatio and proportion word problems\n\nTime and work word problems\n\nWord problems on sets and venn diagrams\n\nWord problems on ages\n\nPythagorean theorem word problems\n\nPercent of a number word problems\n\nWord problems on constant speed\n\nWord problems on average speed\n\nWord problems on sum of the angles of a triangle is 180 degree\n\nOTHER TOPICS\n\nProfit and loss shortcuts\n\nPercentage shortcuts\n\nTimes table shortcuts\n\nTime, speed and distance shortcuts\n\nRatio and proportion shortcuts\n\nDomain and range of rational functions\n\nDomain and range of rational functions with holes\n\nGraphing rational functions\n\nGraphing rational functions with holes\n\nConverting repeating decimals in to fractions\n\nDecimal representation of rational numbers\n\nFinding square root using long division\n\nL.C.M method to solve time and work problems\n\nTranslating the word problems in to algebraic expressions\n\nRemainder when 2 power 256 is divided by 17\n\nRemainder when 17 power 23 is divided by 16\n\nSum of all three digit numbers divisible by 6\n\nSum of all three digit numbers divisible by 7\n\nSum of all three digit numbers divisible by 8\n\nSum of all three digit numbers formed using 1, 3, 4\n\nSum of all three four digit numbers formed with non zero digits\n\nSum of all three four digit numbers formed using 0, 1, 2, 3\n\nSum of all three four digit numbers formed using 1, 2, 5, 6", null, "" ]	[ null, "https://www.onlinemath4all.com/images/xpracprob1.png.pagespeed.ic.FbmQvIb53p.png", null, "https://www.onlinemath4all.com/images/xpractprob2.png.pagespeed.ic.MqttwmxAKK.png", null, "https://www.onlinemath4all.com/images/xpracprob12.png.pagespeed.ic._k0G1WH6nY.png", null, "https://www.onlinemath4all.com/images/xpracprob3.png.pagespeed.ic.fepYjjPXRF.png", null, "https://www.onlinemath4all.com/images/xpracprob4.png.pagespeed.ic.AqeJBx0GZG.png", null, "https://www.onlinemath4all.com/images/xpractprob14.png.pagespeed.ic.H8WK1_jrFK.png", null, "https://www.onlinemath4all.com/images/xpractprob15.png.pagespeed.ic.74_B2DO3As.png", null, 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https://programming.arora-aditya.com/leetcode/tree/701.-insert-into-a-binary-search-tree	[ "# Approach 1: Recursively insert into left or right parts\n\n`# Definition for a binary tree node.# class TreeNode(object):# def __init__(self, x):# self.val = x# self.left = None# self.right = Noneclass Solution(object): def insertIntoBST(self, root, val): \"\"\" :type root: TreeNode :type val: int :rtype: TreeNode \"\"\" if root == None: return TreeNode(val) if val > root.val: root.right = self.insertIntoBST(root.right, val) else: root.left = self.insertIntoBST(root.left, val) return root`\n\nTime Complexity: O(logN) assuming balanced BST (Otherwise O(N) worst case where N is number of nodes)\n\nSpace Complexity: O(M) height of the stack (recursive call)\n\n# Approach 2: Iterative Approach 1\n\n`# Definition for a binary tree node.# class TreeNode(object):# def __init__(self, x):# self.val = x# self.left = None# self.right = Noneclass Solution(object): def insertIntoBST(self, root, val): \"\"\" https://leetcode.com/problems/insert-into-a-binary-search-tree/description/ :type root: TreeNode :type val: int :rtype: TreeNode \"\"\" parser = root prevParser = root while parser: if val > parser.val: prevParser = parser parser = parser.right else: prevParser = parser parser = parser.left if val > prevParser.val: prevParser.right = TreeNode(val) else: prevParser.left = TreeNode(val) return root`\n\nWe iterate through the tree and each time eliminate half the tree based on the value of the current node we are on. If the value to be inserted is greater than current node's value we move to the right tree and vice versa. By the time we finish iterating like this we reach the position where the node is supposed to inserted. We insert it in it's position using the `TreeNode prevParser` which stores the position of the parser one iteration behind i.e. the parent node\n\nTime Complexity: O(logN) assuming balanced BST (Otherwise O(N) worst case where N is number of nodes)\n\nSpace Complexity: O(1)" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.56060016,"math_prob":0.9184231,"size":1747,"snap":"2020-10-2020-16","text_gpt3_token_len":413,"char_repetition_ratio":0.14056225,"word_repetition_ratio":0.30088496,"special_character_ratio":0.23354322,"punctuation_ratio":0.1954023,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9931992,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-02-18T01:14:20Z\",\"WARC-Record-ID\":\"<urn:uuid:be8e368c-a252-4d1c-8afe-cebf002670d6>\",\"Content-Length\":\"231531\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f0ac488a-5839-422b-a57f-0c8d5fd7808b>\",\"WARC-Concurrent-To\":\"<urn:uuid:5b0e535b-c721-43a7-812d-78890a363fc5>\",\"WARC-IP-Address\":\"45.55.238.214\",\"WARC-Target-URI\":\"https://programming.arora-aditya.com/leetcode/tree/701.-insert-into-a-binary-search-tree\",\"WARC-Payload-Digest\":\"sha1:AK4PA6IN2GJMTGVVCZADITNVZT5QKYM4\",\"WARC-Block-Digest\":\"sha1:RMU3JDFZERYVHJ7UI45F4T7CP3UNUY3N\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-10/CC-MAIN-2020-10_segments_1581875143455.25_warc_CC-MAIN-20200217235417-20200218025417-00091.warc.gz\"}"}
https://math.answers.com/other-math/What_is_half_of_28	[ "", null, "", null, "", null, "0\n\n# What is half of 28?\n\nHalf of 28 Is 14. Or do 14 + 14 = 28 plus, 2x14", null, "Study guides\n\n20 cards\n\n## A number a power of a variable or a product of the two is a monomial while a polynomial is the of monomials\n\n➡️\nSee all cards\n3.75\n859 Reviews\n\nno idea what it is im just 9 but solved the hardest question about if there are 600 dollars then how much pennies are there", null, "2x14 = 28 or 14+14=28", null, "14", null, "14", null, "14", null, "14", null, "fourteen", null, "20", null, "", null, "Earn +20 pts", null, "", null, "" ]	[ null, "https://math.answers.com/icons/searchIcon.svg", null, "https://math.answers.com/icons/notificationBellIcon.svg", null, "https://math.answers.com/icons/coinIcon.svg", null, "https://math.answers.com/icons/sendIcon.svg", null, "https://math.answers.com/icons/sendIcon.svg", null, "https://math.answers.com/icons/sendIcon.svg", null, "https://math.answers.com/icons/sendIcon.svg", null, "https://math.answers.com/icons/sendIcon.svg", null, "https://math.answers.com/icons/sendIcon.svg", null, "https://math.answers.com/icons/sendIcon.svg", null, "https://math.answers.com/icons/sendIcon.svg", null, "https://math.answers.com/icons/sendIcon.svg", null, "https://math.answers.com/icons/coinIcon.svg", null, "https://math.answers.com/icons/searchIcon.svg", null, "https://st.answers.com/html_test_assets/imp_-_pixel.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9470267,"math_prob":0.87535083,"size":274,"snap":"2022-05-2022-21","text_gpt3_token_len":111,"char_repetition_ratio":0.22222222,"word_repetition_ratio":0.0625,"special_character_ratio":0.49635038,"punctuation_ratio":0.077922076,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95410985,"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],"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],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-05-27T18:41:09Z\",\"WARC-Record-ID\":\"<urn:uuid:1b90979e-7281-490a-89bf-4415be6feefb>\",\"Content-Length\":\"237485\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d14e117c-80a3-4e79-bea2-aef45e87be32>\",\"WARC-Concurrent-To\":\"<urn:uuid:0cbe9467-b3ab-4288-8f29-eb7262971ccd>\",\"WARC-IP-Address\":\"146.75.32.203\",\"WARC-Target-URI\":\"https://math.answers.com/other-math/What_is_half_of_28\",\"WARC-Payload-Digest\":\"sha1:4FAVRHILXA3RBU6NT3JTIVCIAA7S7XLM\",\"WARC-Block-Digest\":\"sha1:N572TBKTYO3ZD2VIFNDYGOUOW7DUNIV6\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-21/CC-MAIN-2022-21_segments_1652662675072.99_warc_CC-MAIN-20220527174336-20220527204336-00233.warc.gz\"}"}
https://www.degruyter.com/view/j/gmj.ahead-of-print/gmj-2017-0062/gmj-2017-0062.xml	[ "Show Summary Details\nMore options …", null, "Georgian Mathematical Journal\n\nEditor-in-Chief: Kiguradze, Ivan / Buchukuri, T.\n\nEditorial Board: Kvinikadze, M. / Bantsuri, R. / Baues, Hans-Joachim / Besov, O.V. / Bojarski, B. / Duduchava, R. / Engelbert, Hans-Jürgen / Gamkrelidze, R. / Gubeladze, J. / Hirzebruch, F. / Inassaridze, Hvedri / Jibladze, M. / Kadeishvili, T. / Kegel, Otto H. / Kharazishvili, Alexander / Kharibegashvili, S. / Khmaladze, E. / Kiguradze, Tariel / Kokilashvili, V. / Krushkal, S. I. / Kurzweil, J. / Kwapien, S. / Lerche, Hans Rudolf / Mawhin, Jean / Ricci, P.E. / Tarieladze, V. / Triebel, Hans / Vakhania, N. / Zanolin, Fabio\n\nIMPACT FACTOR 2018: 0.551\n\nCiteScore 2018: 0.52\n\nSCImago Journal Rank (SJR) 2018: 0.320\nSource Normalized Impact per Paper (SNIP) 2018: 0.711\n\nMathematical Citation Quotient (MCQ) 2018: 0.27\n\nOnline\nISSN\n1572-9176\nSee all formats and pricing\nMore options …\n\nA Tauberian theorem for the generalized Nörlund summability method\n\nİbrahim Çanak", null, "/ Naim L. Braha\n• Department of Computer Sciences and Applied Mathematics, College Vizioni per Arsim, Rr, Ahmet Kaciku, Ferizaj, 70000, Kosovo\n• Email\n• Other articles by this author:\n• De Gruyter OnlineGoogle Scholar\n/ Ümit Totur", null, "Published Online: 2018-01-10 \| DOI: https://doi.org/10.1515/gmj-2017-0062\n\nAbstract\n\nLet $\\left({p}_{n}\\right)$ and $\\left({q}_{n}\\right)$ be any two non-negative real sequences, with ${R}_{n}:={\\sum }_{k=0}^{n}{p}_{k}{q}_{n-k}\\ne 0$ ($n\\in ℕ$). Let ${\\sum }_{k=0}^{\\mathrm{\\infty }}{a}_{k}$ be a series of real or complex numbers with partial sums $\\left({s}_{n}\\right)$, and set ${t}_{n}^{p,q}:=\\frac{1}{{R}_{n}}{\\sum }_{k=0}^{n}{p}_{k}{q}_{n-k}{s}_{k}$ for $n\\in ℕ$. In this paper, we present the necessary and sufficient conditions under which the existence of the limit ${lim}_{n\\to \\mathrm{\\infty }}{s}_{n}=L$ follows from that of ${lim}_{n\\to \\mathrm{\\infty }}{t}_{n}^{p,q}=L$. These conditions are one-sided or two-sided if $\\left({s}_{n}\\right)$ is a sequence of real or complex numbers, respectively.\n\nMSC 2010: 40G15; 41A36\n\nReferences\n\n• \n\nD. Borwein, On products of sequences, J. Lond. Math. Soc. 33 (1958), 352–357. Google Scholar\n\n• \n\nI. Çanak and Y. Erdem, On Tauberian theorems for $\\left(A\\right)\\left(C,\\alpha \\right)$ summability method, Appl. Math. Comput. 218 (2011), no. 6, 2829–2836.\n\n• \n\nI. Çanak and U. Totur, Extended Tauberian theorem for the weighted mean method of summability, Ukrainian Math. J. 65 (2013), no. 7, 1032–1041.\n\n• \n\nI. Çanak and U. Totur, A theorem for the $\\left(J,p\\right)$ summability method, Acta Math. Hungar. 145 (2015), no. 1, 220–228. Google Scholar\n\n• \n\nY. Erdem, I. Çanak and B. P. Allahverdiev, Two theorems on the product of Abel and Cesàro summability methods, C. R. Acad. Bulgare Sci. 68 (2015), no. 3, 287–294. Google Scholar\n\n• \n\nG. H. Hardy, Divergent Series, Clarendon Press, Oxford, 1949. Google Scholar\n\n• \n\nR. Kiesel, General Nörlund transforms and power series methods, Math. Z. 214 (1993), no. 2, 273–286.\n\n• \n\nR. Kiesel and U. Stadtmüller, Tauberian- and convexity theorems for certain $\\left(N,p,q\\right)$-means, Canad. J. Math. 46 (1994), no. 5, 982–994. Google Scholar\n\n• \n\nE. Landau, Über die Bedeutung einiger neuen Grenzwertsätze der Herren Hardy und Axel, Prac. Mat.-Fiz. 21 (1910), 97–177. Google Scholar\n\n• \n\nF. Móricz and B. E. Rhoades, Necessary and sufficient Tauberian conditions for certain weighted mean methods of summability, Acta Math. Hungar. 66 (1995), no. 1–2, 105–111.\n\n• \n\nR. Schmidt, Über divergente Folgen und lineare Mittelbildungen, Math. Z. 22 (1925), no. 1, 89–152.\n\n• \n\nU. Stadtmüller and A. Tali, On certain families of generalized Nörlund methods and power series methods, J. Math. Anal. Appl. 238 (1999), no. 1, 44–66.\n\n• \n\nU. Totur and I. Çanak, Some general Tauberian conditions for the weighted mean summability method, Comput. Math. Appl. 63 (2012), no. 5, 999–1006.\n\nAccepted: 2016-05-23\n\nPublished Online: 2018-01-10\n\nCitation Information: Georgian Mathematical Journal, ISSN (Online) 1572-9176, ISSN (Print) 1072-947X,\n\nExport Citation\n\n© 2018 Walter de Gruyter GmbH, Berlin/Boston.", null, "" ]	[ null, "https://www.degruyter.com/doc/cover/s1072947X.jpg", null, "https://www.degruyter.com/staticfiles/images/orcid_iD_icon.svg", null, "https://www.degruyter.com/staticfiles/images/orcid_iD_icon.svg", null, "https://www.copyright.com/wp-content/themes/copyrightclearancecenter/img/link-generator/get-permission-btn-big.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5024569,"math_prob":0.9458759,"size":2205,"snap":"2019-43-2019-47","text_gpt3_token_len":779,"char_repetition_ratio":0.16356201,"word_repetition_ratio":0.0062111802,"special_character_ratio":0.36462584,"punctuation_ratio":0.29,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9877834,"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\":\"2019-10-15T13:56:59Z\",\"WARC-Record-ID\":\"<urn:uuid:eeda31bf-43fe-48e0-98cf-4f458875e7cf>\",\"Content-Length\":\"152971\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:18b81b50-742d-41e2-8328-64ccd7e31afc>\",\"WARC-Concurrent-To\":\"<urn:uuid:bca238c9-3cdd-4098-8324-fa26a48123f1>\",\"WARC-IP-Address\":\"91.203.202.198\",\"WARC-Target-URI\":\"https://www.degruyter.com/view/j/gmj.ahead-of-print/gmj-2017-0062/gmj-2017-0062.xml\",\"WARC-Payload-Digest\":\"sha1:YQJNJPMTDZK4GUPCCMZDZDXRPBP5JRRG\",\"WARC-Block-Digest\":\"sha1:2UCI2AOVPWRUSLKAY4DLQEN6K6OJ5MCL\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570986659097.10_warc_CC-MAIN-20191015131723-20191015155223-00077.warc.gz\"}"}
https://projecteuclid.org/euclid.agt/1511895851	[ "## Algebraic & Geometric Topology\n\n### Near-symplectic $2n$–manifolds\n\nRamón Vera\n\n#### Abstract\n\nWe give a generalization of the concept of near-symplectic structures to $2n$ dimensions. According to our definition, a closed $2$–form on a $2n$–manifold $M$ is near-symplectic if it is symplectic outside a submanifold $Z$ of codimension $3$ where $ωn−1$ vanishes. We depict how this notion relates to near-symplectic $4$–manifolds and broken Lefschetz fibrations via some examples. We define a generalized broken Lefschetz fibration as a singular map with indefinite folds and Lefschetz-type singularities. We show that, given such a map on a $2n$–manifold over a symplectic base of codimension $2$, the total space carries such a near-symplectic structure whose singular locus corresponds precisely to the singularity set of the fibration. A second part studies the geometry around the codimension-$3$ singular locus $Z$. We describe a splitting property of the normal bundle $NZ$ that is also present in dimension four. A tubular neighbourhood theorem for $Z$ is provided, which has a Darboux-type theorem for near-symplectic forms as a corollary.\n\n#### Article information\n\nSource\nAlgebr. Geom. Topol., Volume 16, Number 3 (2016), 1403-1426.\n\nDates\nReceived: 12 August 2014\nRevised: 19 August 2015\nAccepted: 3 October 2015\nFirst available in Project Euclid: 28 November 2017\n\nPermanent link to this document\nhttps://projecteuclid.org/euclid.agt/1511895851\n\nDigital Object Identifier\ndoi:10.2140/agt.2016.16.1403\n\nMathematical Reviews number (MathSciNet)\nMR3523044\n\nZentralblatt MATH identifier\n1342.53110\n\n#### Citation\n\nVera, Ramón. Near-symplectic $2n$–manifolds. Algebr. Geom. Topol. 16 (2016), no. 3, 1403--1426. doi:10.2140/agt.2016.16.1403. https://projecteuclid.org/euclid.agt/1511895851" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.79918325,"math_prob":0.92257786,"size":2085,"snap":"2019-35-2019-39","text_gpt3_token_len":593,"char_repetition_ratio":0.13214801,"word_repetition_ratio":0.014814815,"special_character_ratio":0.25659472,"punctuation_ratio":0.16802168,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9859371,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-08-25T12:22:44Z\",\"WARC-Record-ID\":\"<urn:uuid:ff657cd6-5273-44c7-9854-ced8a181118f>\",\"Content-Length\":\"37672\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7988e4e9-ff03-482b-a231-e9ff62b0ce78>\",\"WARC-Concurrent-To\":\"<urn:uuid:d11308ac-2ab7-4272-b0a5-aadc52e5f5e6>\",\"WARC-IP-Address\":\"132.236.27.47\",\"WARC-Target-URI\":\"https://projecteuclid.org/euclid.agt/1511895851\",\"WARC-Payload-Digest\":\"sha1:AWWJ4SKTNEZEQRMGCS4NUYEZOXXY4NCD\",\"WARC-Block-Digest\":\"sha1:XIUAGJMLRJFW3LTHV6UYWHYVAWCAYXHH\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-35/CC-MAIN-2019-35_segments_1566027323328.16_warc_CC-MAIN-20190825105643-20190825131643-00067.warc.gz\"}"}
https://www.physicsforums.com/threads/this-is-so-simple-why-cant-i-figure-it-out.373801/	[ "# This is SO simple, why can't I figure it out?\n\nThis is SO simple, why can't I figure it out??\n\nI need to figure out the length of 2 strings and L1+L2 = 4\n\nI get to the point, using other parts of the problem that are not pertinent to this, of:\n1.11L2 + L2 = 4\n\nThe answer is L2 = 1.9\n\nHOW in the world does L2 work out to be 1.9?\nHow do you solve the above equation for L2?\n\nCan someone please explain to me the algebra or some rule that I have forgotten that solves L2 as 1.9.\n\nThank you!!!!\n\nRelated Precalculus Mathematics Homework Help News on Phys.org\nMark44\nMentor\n\n1.11 L2 + L2 = 4\n==> 2.11 L2 = 4\n==> L2 = 1.8957 (approx.)\n\nHallsofIvy" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9619172,"math_prob":0.9629829,"size":848,"snap":"2020-24-2020-29","text_gpt3_token_len":266,"char_repetition_ratio":0.116113745,"word_repetition_ratio":0.8852459,"special_character_ratio":0.32665095,"punctuation_ratio":0.14351852,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99393255,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-13T06:11:28Z\",\"WARC-Record-ID\":\"<urn:uuid:5bd947a5-f9cf-420c-a952-05e054fcaff1>\",\"Content-Length\":\"68758\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b31d33d0-afb6-468d-a756-ce4d2db38cf0>\",\"WARC-Concurrent-To\":\"<urn:uuid:a7eaaede-0e5f-4afe-b943-94555542e1be>\",\"WARC-IP-Address\":\"23.111.143.85\",\"WARC-Target-URI\":\"https://www.physicsforums.com/threads/this-is-so-simple-why-cant-i-figure-it-out.373801/\",\"WARC-Payload-Digest\":\"sha1:HITUYYA5BM5ZKIMWX6OBQHTRGQ3YJ2WY\",\"WARC-Block-Digest\":\"sha1:DJGBVFPZXP4TQMU52YEF27ODD43AG2KB\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593657142589.93_warc_CC-MAIN-20200713033803-20200713063803-00010.warc.gz\"}"}
https://www.jpost.com/israel/first-indictment-served-in-tel-aviv-pedophile-ring-case	[ "(function (a, d, o, r, i, c, u, p, w, m) { m = d.getElementsByTagName(o), a[c] = a[c] \|\| {}, a[c].trigger = a[c].trigger \|\| function () { (a[c].trigger.arg = a[c].trigger.arg \|\| []).push(arguments)}, a[c].on = a[c].on \|\| function () {(a[c].on.arg = a[c].on.arg \|\| []).push(arguments)}, a[c].off = a[c].off \|\| function () {(a[c].off.arg = a[c].off.arg \|\| []).push(arguments) }, w = d.createElement(o), w.id = i, w.src = r, w.async = 1, w.setAttribute(p, u), m.parentNode.insertBefore(w, m), w = null} )(window, document, \"script\", \"https://95662602.adoric-om.com/adoric.js\", \"Adoric_Script\", \"adoric\",\"9cc40a7455aa779b8031bd738f77ccf1\", \"data-key\");\nvar domain=window.location.hostname; var params_totm = \"\"; (new URLSearchParams(window.location.search)).forEach(function(value, key) {if (key.startsWith('totm')) { params_totm = params_totm +\"&\"+key.replace('totm','')+\"=\"+value}}); var rand=Math.floor(10*Math.random()); var script=document.createElement(\"script\"); script.src=`https://stag-core.tfla.xyz/pre_onetag?pub_id=34&domain=\\${domain}&rand=\\${rand}&min_ugl=0\\${params_totm}`; document.head.append(script);" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9491984,"math_prob":0.96919554,"size":623,"snap":"2023-40-2023-50","text_gpt3_token_len":133,"char_repetition_ratio":0.08077545,"word_repetition_ratio":0.0,"special_character_ratio":0.19903691,"punctuation_ratio":0.06140351,"nsfw_num_words":4,"has_unicode_error":false,"math_prob_llama3":0.9789482,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-10-03T12:09:06Z\",\"WARC-Record-ID\":\"<urn:uuid:00ac906d-9509-49ab-851e-2be04c5d2b4d>\",\"Content-Length\":\"78711\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:796a0d67-6300-4190-b9f2-2f473bf8bea3>\",\"WARC-Concurrent-To\":\"<urn:uuid:3d25e6be-d4da-4404-896c-59add187dd2d>\",\"WARC-IP-Address\":\"159.60.130.79\",\"WARC-Target-URI\":\"https://www.jpost.com/israel/first-indictment-served-in-tel-aviv-pedophile-ring-case\",\"WARC-Payload-Digest\":\"sha1:HSS5F3XBFZXL4AUDMHOFZQOTAVQTRG2A\",\"WARC-Block-Digest\":\"sha1:PHVQPCZL43LAVIQZOSCLX7ZJNNYRWX7R\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233511075.63_warc_CC-MAIN-20231003092549-20231003122549-00810.warc.gz\"}"}
https://trends.fbm.vutbr.cz/index.php/trends/article/view/135	[ "# Estimation of the Behavioral Equilibrium Real Exchange Rate of the Czech Koruna\n\n• Vít Pošta\n\n## Keywords:\n\nBEER, equilibrium real exchange rate, productivity differential, real exchange rate, VEC\n\n## Abstract\n\nPurpose of the article: The paper examines the behavior of the real exchange rate in the Czech Republic. It focuses on the analysis of its driving forces with the emphasis on the turbulences which have been lately seen in the financial and real sector of the economy. Methodology/methods: Real equilibrium exchange rate can be estimated using various approaches ranging from purely statistical to fully structural models. In this paper it is estimated using the BEER methodology, i.e. behavioral equilibrium exchange rate. The BEER approach as applied here rests on building vector error correction models which relate the behavior of the actual real exchange rate to various economic fundamentals from both the real and financial sector of the economy. Scientific aim: The estimated behavioral equilibrium exchange rate serves as a benchmark to which the actual behavior of real exchange rate is compared. The paper also points to various problems that are faced when estimating the real equilibrium exchange rate in a posttransitive economy. Findings: Three variants of the model, which differ in the respective fundamental variables inluded in the estimation, are estimated in the paper. The gap between the estimated real equilibrium exchange rate and real exchange rate as well as the key determinants of the real equilibrium exchange rate are analyzed and compared. The models show that the misalignment between the real exchange rate and fundamentals have narrowed in the recession and post recession period. The key drivers of the real equilibrium exchange rate are the productivity differential, real interest rate differential and net foreign assets. Conclusions: (limits, implications etc) The relatively short time series for the Czech economy, especially for some of the variables, do not enable to make reliable estimation of models which would include all of the variables discussed in this paper. This paper is a part of a research project financed by IGA University of Economics, Prague" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.91623425,"math_prob":0.8866121,"size":2262,"snap":"2023-40-2023-50","text_gpt3_token_len":414,"char_repetition_ratio":0.18644819,"word_repetition_ratio":0.020710059,"special_character_ratio":0.17064545,"punctuation_ratio":0.08443272,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9700914,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-24T11:09:48Z\",\"WARC-Record-ID\":\"<urn:uuid:26249f8d-b8f2-49be-b563-3efdfd0e9582>\",\"Content-Length\":\"21057\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e03c791d-b413-4f0d-bb3c-2e963ee7336a>\",\"WARC-Concurrent-To\":\"<urn:uuid:4962e8a4-a34f-4ca1-a7ba-faeeb02a6ca4>\",\"WARC-IP-Address\":\"147.229.124.25\",\"WARC-Target-URI\":\"https://trends.fbm.vutbr.cz/index.php/trends/article/view/135\",\"WARC-Payload-Digest\":\"sha1:N4VVQC3X5CZ7CBNKULVI6C7UPVBKOPVA\",\"WARC-Block-Digest\":\"sha1:C4QBA63DA3V2MHNATW6VFVFHVWEIG7OY\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233506632.31_warc_CC-MAIN-20230924091344-20230924121344-00578.warc.gz\"}"}
https://dba.stackexchange.com/questions/257844/sql-query-returning-same-value-while-using-where-1-condition	[ "# SQL Query returning same value while using where 1 condition\n\nI have created a temp table and inserted the values as given below.\n\n``````create table #temp( val int );\n\ninsert into #temp values(333);\ninsert into #temp values(222);\ninsert into #temp values(111);\n``````\n\nOn querying the below select statement I got 333 as the answer.\n\n``````Select \nfrom #temp a\nWhere 1 =(\nSelect COUNT(VAL)\nfrom #temp b\nwhere a.val <= b.val\n);\n``````\n\nResult:\n\n``````val\n\n333\n``````\n\n• SQL server returned exactly what you requested. What is your expected result? Jan 21, 2020 at 10:00\n• @DenisRubashkin, Can you please explain how did query return 333? Jan 21, 2020 at 10:03\n• You ask server to find the record for which there exists only one record with the value of `val` equal or greater (so this 'only one' is this record itself) if it is alone (count=1, not duplicated). And you obtain strictly the record which matches this conditions. Why you're surprised? Jan 21, 2020 at 10:15\n\nRewrite your query such a way:\n\n``````SELECT a., x.cnt\nFROM #temp a\nCROSS APPLY (\nSELECT COUNT(VAL) AS cnt\nFROM #temp b\nWHERE a.val <= b.val\n) x\n--WHERE x.cnt = 1\n``````\n\nIf you uncomment the where clause you would get `333 \| 1` as a result. You request a row from the outer table which doesn't have duplicates or bigger values.\n\nYou could see your inner count query\n\n``````Select COUNT(VAL) from #temp b\nwhere a.val <= b.val;\n``````\n\nas\n\n``````(Select COUNT(VAL) from #temp b\nwhere 333 <= 333,222,111) = 1\n\n(Select COUNT(VAL) from #temp b\nwhere 222 <= 333,222,111) = 2\n\n(Select COUNT(VAL) from #temp b\nwhere 111 <= 333,222,111) = 3\n``````\n\nShowing that only `333` from `#temp a` has one match, as it has only one equals to or smaller than match with the 3 values in `#temp b`.\n\nThis returns a count of `1` and is why the value `333` from `#temp a` is returned." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.83228546,"math_prob":0.92270625,"size":624,"snap":"2022-27-2022-33","text_gpt3_token_len":154,"char_repetition_ratio":0.13387097,"word_repetition_ratio":0.0,"special_character_ratio":0.27083334,"punctuation_ratio":0.12,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98399216,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-06-27T03:17:03Z\",\"WARC-Record-ID\":\"<urn:uuid:eb7d6079-39db-446b-8f1e-ae914ea6a3e9>\",\"Content-Length\":\"239857\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:06b5f35f-7527-4ce9-98c7-3e496f5930d6>\",\"WARC-Concurrent-To\":\"<urn:uuid:056121d6-39bd-4ef9-bada-13f0443641c7>\",\"WARC-IP-Address\":\"151.101.193.69\",\"WARC-Target-URI\":\"https://dba.stackexchange.com/questions/257844/sql-query-returning-same-value-while-using-where-1-condition\",\"WARC-Payload-Digest\":\"sha1:ZDXHXQKBMXFAOLJEI4LTLAKJZ64KOZQK\",\"WARC-Block-Digest\":\"sha1:RWXFD7RPJV7EEV56KSXEVXBLDOVK26VF\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656103324665.17_warc_CC-MAIN-20220627012807-20220627042807-00717.warc.gz\"}"}
https://gameplay.tips/guides/1124-dishonored-death-of-the-outsider.html	[ "# Dishonored: Death of the Outsider – Bank Vault Code and Fibonacci\n\nOther Dishonored: Death of the Outsider Guides:\n\n## Fibonacci\n\n“In mathematics, the Fibonacci numbers are the numbers in the following integer sequence, called the Fibonacci sequence, and characterized by the fact that every number after the first two is the sum of the two preceding ones” – Wikipedia description of the Fibonacci numbers.\n\nThat means starting with the numbers 0, 1 and we write down the next numbers in the sequence we will be getting an infinite amount of numbers.\n\nWhat it has to do with DotO? – the chalkboard you find shows the depiction and the first numbers in the Fibonacci sequence. We get:\n\n0, 1, 1, 2, 3, 5, 8, 13\n\n## Calculation\n\nUsing the rules for the Fibonacci numbers we can calculate:\n\n• 0\n• 1\n• 1 + 1 = 2\n• 1 + 2 = 3\n• 2 + 3 = 5\n• 3 + 5 = 8\n• 5 + 8 = 13\n\n## The Vault\n\nIn the vault we have 6 safes which need – if we count the safe with key locks as a normal safe – 6 x 3 = 18 numbers. Since we only have 8 (counting any number from 10 up as 2 numbers since the pads only feature numbers from 0 – 9) which means we have to add numbers to our list.\n\nDone right following the Fibonacci numbers we would get a list with the numbers\n0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144\n\nTo open all safes we can now just pack the numbers into packets of 3 to fit into the locks:\n\n• 1 – 011\n• 2 – 235\n• 3 – 813\n• 4 – 213 This one can be skipped as it is outfitted with different locks.\n• 5 – 455\n• 6 – 891" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8608997,"math_prob":0.9965943,"size":1919,"snap":"2022-40-2023-06","text_gpt3_token_len":560,"char_repetition_ratio":0.1535248,"word_repetition_ratio":0.07235142,"special_character_ratio":0.30536738,"punctuation_ratio":0.114285715,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9934742,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-09-27T05:37:13Z\",\"WARC-Record-ID\":\"<urn:uuid:327f401d-653e-43b9-b321-80e30cee8a31>\",\"Content-Length\":\"135310\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2f18d262-cf0d-4059-be4c-403f8974c949>\",\"WARC-Concurrent-To\":\"<urn:uuid:f28e2630-c004-4b17-98ba-a8c5a45c7776>\",\"WARC-IP-Address\":\"172.67.72.139\",\"WARC-Target-URI\":\"https://gameplay.tips/guides/1124-dishonored-death-of-the-outsider.html\",\"WARC-Payload-Digest\":\"sha1:C7BBNU24M67ATN5AGRKY56MZQPL6ETYD\",\"WARC-Block-Digest\":\"sha1:N3JC7ZZ2GPK66V43O6V4WIOZUODBSPKQ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030334987.39_warc_CC-MAIN-20220927033539-20220927063539-00147.warc.gz\"}"}
http://doc.51windows.net/jscript5/html/jsobjActiveXObject.htm	[ "ActiveXObject 对象\n\n`newObj = new ActiveXObject(servername.typename[, location])`\n\nActiveXObject 对象语法有这些部分：\n\nnewObj\n\nservername\n\ntypename\n\nlocation\n\n说明\n\nAutomation 服务器至少提供一类对象。例如，字处理应用程序可能提供应用程序对象、文档对象和工具栏对象。\n\n``````var ExcelSheet;\nExcelApp = new ActiveXObject(\"Excel.Application\");\nExcelSheet = new ActiveXObject(\"Excel.Sheet\");``````\n\n````// `使` Excel `通过` Application `对象可见。\n`ExcelSheet.Application.Visible = true;`\n`// `将一些文本放置到表格的第一格中。\n`ExcelSheet.ActiveSheet.Cells(1,1).Value = \"This is column A, row 1\";`\n`// `保存表格。\n`ExcelSheet.SaveAs(\"C:\\\\TEST.XLS\");`\n`// `用` Application `对象用` Quit `方法关闭` Excel`。\n`ExcelSheet.Application.Quit();````\n\n``````function GetAppVersion() {\nvar XLApp = new ActiveXObject(\"Excel.Application\", \"MyServer\");\nreturn(XLApp.Version);\n```}```\n\n请参阅\n\nGetObject 函数\n\nwww.51windows.Net" ]	[ null ]	{"ft_lang_label":"__label__zh","ft_lang_prob":0.8214831,"math_prob":0.4064026,"size":1292,"snap":"2019-26-2019-30","text_gpt3_token_len":572,"char_repetition_ratio":0.185559,"word_repetition_ratio":0.0,"special_character_ratio":0.19117647,"punctuation_ratio":0.16477273,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9638064,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-06-20T18:01:33Z\",\"WARC-Record-ID\":\"<urn:uuid:82335149-c438-4391-b9e5-2919807f8e65>\",\"Content-Length\":\"4085\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2a3384cf-fab5-469e-81e3-3aebb4ff4821>\",\"WARC-Concurrent-To\":\"<urn:uuid:58afe6b0-df55-4541-8496-ae331dc37398>\",\"WARC-IP-Address\":\"120.92.10.42\",\"WARC-Target-URI\":\"http://doc.51windows.net/jscript5/html/jsobjActiveXObject.htm\",\"WARC-Payload-Digest\":\"sha1:MLEOC3JWMOXALV34UIW62RQ5RFZUPG4V\",\"WARC-Block-Digest\":\"sha1:ZIQLEM6MTHRHYQCP6ZDKXMBGBOAUYOHY\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-26/CC-MAIN-2019-26_segments_1560627999263.6_warc_CC-MAIN-20190620165805-20190620191805-00427.warc.gz\"}"}
https://www.mathematicalway.com/mathematics/trigonometry/cotangent/	[ "# Cotangent", null, "", null, "", null, "", null, "", null, "(No Ratings Yet)", null, "Loading...", null, "The cotangent is the reciprocal trigonometric ratio of the tangent. It is the reciprocal or multiplicative inverse of the tangent, that is, tan θ · cotθ = 1.\n\nIn a right triangle, the cotangent of the angle θ is defined as the ratio of the adjacent leg (b) to the opposite leg (a).", null, "Like other trig functions, it is usually abbreviated. So, in a formula, the cotangent is abbreviated as cot (cotangent-, cotangens: from co ”mutually” + Latin tangens, that means “to touch” (Latin verb: tangere)).\n\n## Cotangent for Special Common Angles\n\nThe following table gives the values of cotangent for common angles:", null, "", null, "## Properties of Cotangent\n\n• Domain:", null, "(all real numbers), except n · π, where n is an integer. Or this casuistry: x ≠ ±π; ±2π; ±3π;… (that is, odd multiples of π).\n• Range:", null, "(all real numbers)\n• Symmetry: since cot (-x) = -cot (x) then cot (x) is an odd function and its graph is symmetric with respect to the origin (0, 0).\n• Increasing-decreasing behaviour: over one period and from 0 to 2π, sec (x) is decreasing.\n• End behaviour: The limits as x approaches π+ k · π do not exist since the function values oscillate between -∞ and +∞. Hence, there are no horizontal asymptotes. This is a periodic function with period π.\n• The derivative of cotangent function:", null, "• The integral of cotangent function:", null, "## Graphical Representation of the Cotangent Function", null, "The cotangent is a periodic function with period π radians (180 degrees), so this section of the graph will be repeated in different periods.\n\n## Geometric Representation of the Cotangent", null, "## Relationship Between Cotangent and Other Trigonometric Functions\n\nThere are some basic trigonometric identities involving cotangent:\n\n• Relationship between cotangent and sine:", null, "• Relationship between cotangent and cosine:", null, "• Relationship between cotangent and tangent:", null, "• Relationship between cotangent and secant:", null, "• Relationship between cotangent and cosecant:", null, "(1) Note: the sign depends on the quadrant of the original angle.\n\n## Trigonometric Identities Involving the Cotangent Function\n\n### Cotangent of Complementary, Supplementary and Conjugate Angles\n\n• Cotangent of a Complementary Angle:", null, "• Cotangent of a Supplementary Angle:", null, "• Cotangent of a Conjugate Angle:", null, "### Cotangent of Negative Angles\n\n• Cotangent of a Negative Angle:", null, "### Cotangent of Angles that Differs by 90º or 180º\n\n• Cotangent of an Angle that Differs by 90º:", null, "• Cotangent of an Angle that Differs by 180º:", null, "## Other Reciprocal Trigonometric Ratios\n\nReciprocal trigonometric ratios are the multiplicative inverses of trigonometric ratios. These are:\n\n• Secant (sec): It is the reciprocal ratio of the cosine. That is, sec θ · cos θ = 1.", null, "• Cosecant (csc): It is the reciprocal ratio of the sine. That is, csc θ · sen θ = 1 or csc θ = 1/sen θ.", null, "## Reciprocal Trigonometric Ratios for Special Common Angles\n\nThe inverse trigonometric ratios for the most common angles (0º, 30º, 45º, 60º, 90º, 180º and 270º) are:", null, "## Relationships Between Trigonometric Functions\n\nAny trigonometric ratio can be expressed in terms of any other. The following table shows the formulas with which each one is expressed as a function of the other:", null, "Note: the sign + or – depends on the quadrant of the original angle.\n\n## Trigonometric Ratios of Angle θ", null, "If θ is one of the acute angles in a right triangle ABC, then:\n\n• The sine is the ratio between the length of the oppisite leg (a) divided by the length of the hypotenuse (c). In a formula, it is written simply as sin:", null, "• The cosine is the ratio between the length of the adjacent leg (b) divided by the length of the hypotenuse (c). In a formula, it is written simply as cos:", null, "• The tangent is the ratio between the length of the opposite leg (a) divided by the length of the adjacent leg (b). In a formula, it is written simply as tan:", null, "## Trigonometric Functions\n\nTrigonometric functions are also called circular functions. The reason is that in a right triangle that has been drawn on the coordinate axis, with the vertex of the angle θ at the center of a circle (O), that is, in standard position, its vertex B can go through all the points of this circle.", null, "Trigonometric functions and reciprocal trigonometric functions can be graphically represented in a right triangle on a circle of r=1.\n\nAUTHOR: Bernat Requena Serra\n\nYEAR: 2021" ]	[ null, "https://www.mathematicalway.com/wp-content/plugins/wp-postratings/images/stars_crystal/rating_off.gif", null, "https://www.mathematicalway.com/wp-content/plugins/wp-postratings/images/stars_crystal/rating_off.gif", null, "https://www.mathematicalway.com/wp-content/plugins/wp-postratings/images/stars_crystal/rating_off.gif", null, "https://www.mathematicalway.com/wp-content/plugins/wp-postratings/images/stars_crystal/rating_off.gif", null, "https://www.mathematicalway.com/wp-content/plugins/wp-postratings/images/stars_crystal/rating_off.gif", null, 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https://rdrr.io/github/danheck/RRreg/man/RRsimu.html	[ "# RRsimu: Monte Carlo simulation for one or two RR variables In danheck/RRreg: Correlation and Regression Analyses for Randomized Response Data\n\n RRsimu R Documentation\n\n## Monte Carlo simulation for one or two RR variables\n\n### Description\n\nSimulate and analyse bivariate data including either one RR variable (either correlation, logistic, or linear regression model) or two RR variables (only correlations). Useful for power analysis, parametric bootstraps or for testing the effects of noncompliance on the stability of estimates.\n\n### Usage\n\n```RRsimu(\nnumRep,\nn,\npi,\nmodel,\np,\ncor = 0,\nb.log = 0,\ncomplyRates = c(1, 1),\nsysBias = c(0, 0),\nmethod = c(\"RRuni\", \"RRcor\", \"RRlog\", \"RRlin\"),\nalpha = 0.05,\ngroupRatio = 0.5,\nMLest = FALSE,\ngetPower = TRUE,\nnCPU = 1\n)\n```\n\n### Arguments\n\n `numRep` number of replications `n` sample size `pi` true proportion of carriers of sensitive attribute (for 2 RR variables: `vector`) `model` either one or two RR model (as `vector`), see `RRuni` `p` randomization probability (for 2 RR variables: a `list`). See `RRuni` for details. `cor` true Pearson-correlation used for data generation (for `RRcor`). Can also be used to generate data with two dichotomous RR variables. `b.log` true regression coefficient in logistic regression (for `RRlog`) `complyRates` vector with two values giving the proportions of participants who adhere to the instructions in the subset with or without the sensitive attribute, respectively (for 2 RR variables: a `list`) `sysBias` probability of responding 'yes' (coded as 1 in the RR variable) in case of non-compliance for carriers and non-carriers, respectively. See `RRgen` `method` vector specifying which RR methods to be used in each replication. For a single RR variable, the methods `RRuni`, `RRcor`,`RRlog`, and `RRlin` are available. For 2 RR variables, only `RRcor` is available. `alpha` significance threshold for testing the logistic regression parameter `beta` `groupRatio` proportion of participants in group 1. Only for two-group models (e.g., `\"SLD\"`) (for 2 RR variables: `vector`) `MLest` concerns `RRuni`: whether to use `optim` to get ML instead of moment estimates (only relevant if pi is outside of [0,1]) `getPower` whether to compute power for `method=\"RRcor\"` (performs an additional bootstrap assuming independence) `nCPU` either the number of CPU cores or a cluster initialized via `makeCluster`.\n\n### Details\n\nFor a single RR variable:\n\nThe parameter `b.log` is the slope-coefficient for the true, latent values in a logistic regression model that is used for data generation.\n\nThe argument `cor` is used for data generation for linear models. The directly measured covariate is sampled from a normal distribution with shifted means, depending on the true state on the sensitive attribute (i.e., the true, underlying values on the RR variable). For dichotomous RR variables, this corresponds to the assumption of an ordinary t-test, where the dependent variable is normally distributed within groups with equal variance. The difference in means is chosen in a way, to obtain the point-biserial correlation defined by `cor`.\n\nFor two RR variables:\n\n`cor` has to be used. In case of two dichotomous RR variables, the true group membership of individuals is sampled from a 2x2 cross table. Within this table, probabilities are chosen in a way, to obtain the point-tetrachoric correlation defined by `cor`\n\nNote, that for the FR model with multiple response categories (e.g., from 0 to 4), the specified `cor` is not the exact target of the sampling procedure. It assumes a normal distribution for each true state, with constant differences between the groups (i.e., it assumes an interval scaled variable).\n\n### Value\n\nA list containing\n\n `parEsts` matrix containing the estimated parameters `results` matrix with mean parameters, standard errors, and number of samples to which the respective method could not be fitted `power` vector with the estimated power of the selected randomized response procedures\n\n### Examples\n\n```# Not run: Simulate data according to the Warner model\n# mcsim <- RRsimu(numRep=100, n=300, pi=.4,\n# model=\"Warner\", p=.2, cor=.3)\n# print(mcsim)\n```\n\ndanheck/RRreg documentation built on Dec. 3, 2022, 7:50 p.m." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7908918,"math_prob":0.97471577,"size":3487,"snap":"2023-40-2023-50","text_gpt3_token_len":858,"char_repetition_ratio":0.106804475,"word_repetition_ratio":0.017605634,"special_character_ratio":0.2291368,"punctuation_ratio":0.13700788,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9962614,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-03T10:51:53Z\",\"WARC-Record-ID\":\"<urn:uuid:6e1d75b7-9e6b-4730-b37e-45c2af8e3f16>\",\"Content-Length\":\"28506\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b1c857a7-1f85-4d65-b69f-c534ccb9567c>\",\"WARC-Concurrent-To\":\"<urn:uuid:bb5a83f0-d188-43ea-b785-218d128a7c93>\",\"WARC-IP-Address\":\"51.81.83.12\",\"WARC-Target-URI\":\"https://rdrr.io/github/danheck/RRreg/man/RRsimu.html\",\"WARC-Payload-Digest\":\"sha1:TNNZUY7VGFWZMUUFYXOM7SHA626FVNTF\",\"WARC-Block-Digest\":\"sha1:HCJACTVNGGTDS7MEOCHNUGBBINGGCNLZ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100499.43_warc_CC-MAIN-20231203094028-20231203124028-00095.warc.gz\"}"}
http://export.arxiv.org/abs/2011.11822	[ "physics.plasm-ph\n\n# Title: Simple magnetic reconnection example\n\nAbstract: In laboratory and natural plasmas of practical interest, the spatial scale $\\Delta_d$ at which magnetic field lines lose distinguishability on the time scale set by an ideal evolution differs enormously from the scale $a$ of magnetic reconnection across the field lines. In the solar corona, plasma resistivity gives $a/\\Delta_d\\sim10^{12}$, which is the magnetic Reynold number $R_m$. The standard resolution of the paradox of disparate scales is for the current density $j$ associated with the reconnecting field $B_{rec}$ to be concentrated by the ideal evolution, so $j\\sim B_{rec}/\\mu_0\\Delta_d$, an amplification by a factor $R_m$. A second resolution is for the ideal evolution to increase the ratio of the maximum to minimum separation between pairs of arbitrarily chosen magnetic field lines, $\\Delta_{max}/\\Delta_{min}$, when calculated at various points in time. Reconnection becomes inevitable when $\\Delta_{max}/\\Delta_{min}\\sim R_m$. As demonstrated using a simple model of the solar corona, the natural rate of increase in time is linear for the current density but exponential for $\\Delta_{max}/\\Delta_{min}$. Reconnection occurs on a time scale and with a current density enhanced by only $\\ln(a/\\Delta_d)$ from the ideal evolution time and from the current density $B_{rec}/\\mu_0a$. In both resolutions of the paradox, once a sufficient region has undergone reconnection, the magnetic field loses force balance and evolves ideally on an Alfv\\'en transit time. This ideal evolution generally expands the region in which $\\Delta_{max}/\\Delta_{min}$ is large.\n Subjects: Plasma Physics (physics.plasm-ph); Astrophysics of Galaxies (astro-ph.GA); Solar and Stellar Astrophysics (astro-ph.SR) Cite as: arXiv:2011.11822 [physics.plasm-ph] (or arXiv:2011.11822v2 [physics.plasm-ph] for this version)\n\n## Submission history\n\nFrom: Allen Boozer [view email]\n[v1] Tue, 24 Nov 2020 00:57:51 GMT (619kb,D)\n[v2] Mon, 7 Dec 2020 01:32:12 GMT (743kb,D)\n\nLink back to: arXiv, form interface, contact." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8290992,"math_prob":0.99182963,"size":2051,"snap":"2021-04-2021-17","text_gpt3_token_len":497,"char_repetition_ratio":0.12945774,"word_repetition_ratio":0.0,"special_character_ratio":0.24963433,"punctuation_ratio":0.11290322,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98644865,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-01-19T19:15:22Z\",\"WARC-Record-ID\":\"<urn:uuid:cd88188a-1e31-42f3-8f6a-ebc49c443cde>\",\"Content-Length\":\"16802\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:73fd9d47-58a3-4bd2-8b26-12c71fb01e1a>\",\"WARC-Concurrent-To\":\"<urn:uuid:7c5ef1ea-2482-4384-824e-3ae68c7beec2>\",\"WARC-IP-Address\":\"128.84.21.203\",\"WARC-Target-URI\":\"http://export.arxiv.org/abs/2011.11822\",\"WARC-Payload-Digest\":\"sha1:LRGAFDE6QRYINPZ3ULAW7AU7VJKA3VEV\",\"WARC-Block-Digest\":\"sha1:ERWOMUPMQDTHJ2KBXHBLTTUPGXM3ZFBX\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-04/CC-MAIN-2021-04_segments_1610703519600.31_warc_CC-MAIN-20210119170058-20210119200058-00627.warc.gz\"}"}
https://www.onlinemathlearning.com/exterior-angle-triangle.html	[ "", null, "# Exterior Angle Theorem\n\nRelated Topics:\nMore Lessons for Geometry\nMath Worksheets\n\nExamples, solutions, Videos, worksheets, games and activities to help Geometry students learn about the Exterior Angle Theorem.\nHow to define the interior and exterior angles of a triangle?\nHow to solve problems related to the exterior angle theorem using Algebra?\n\nExterior Angles\nIf one side of a triangle is extended beyond the vertex, an exterior angle is formed. This exterior angle is supplementary with its adjacent, linear angle. Since the angle sum in a triangle is also 180 degrees, the exterior angle must have a measure equal to the sum of the remaining angles, called the remote interior angles.\n\nHow to define exterior angles and their remote interior angles and how to prove their properties.\nIntroduction to the Interior and Exterior Angles of a Triangle\nThis video will define the interior and exterior angles of a triangle and then state several theorems involving the interior and exterior angles of a triangle.\nExterior Angle Theorem\nThis video explains the exterior angle theorem of geometry. The theorem is used to work out some applications in finding angles of a triangle. The Exterior Angle Theorem\nStudents learn the exterior angle theorem, which states that the exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. Students are then asked to solve problems related to the exterior angle theorem using Algebra.\n\nRotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.\n\nYou can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.", null, "" ]	[ null, "https://www.onlinemathlearning.com/objects/default_image.gif", null, "https://www.onlinemathlearning.com/objects/default_image.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8733709,"math_prob":0.89515907,"size":1972,"snap":"2019-51-2020-05","text_gpt3_token_len":372,"char_repetition_ratio":0.21087398,"word_repetition_ratio":0.085443035,"special_character_ratio":0.18052739,"punctuation_ratio":0.084745765,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9985529,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-01-22T19:20:44Z\",\"WARC-Record-ID\":\"<urn:uuid:7f68c088-8fc1-4c87-b39e-f9aa5b3e1408>\",\"Content-Length\":\"45690\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:42adf900-3789-45d8-a035-6f960f1c2b55>\",\"WARC-Concurrent-To\":\"<urn:uuid:7981c0ae-98c4-45f0-b7f0-235e56059ca0>\",\"WARC-IP-Address\":\"173.247.219.45\",\"WARC-Target-URI\":\"https://www.onlinemathlearning.com/exterior-angle-triangle.html\",\"WARC-Payload-Digest\":\"sha1:UDRZNGT5OOVY2X4ZT4IP2YP7N6DB5UCV\",\"WARC-Block-Digest\":\"sha1:DETGDV6F2GZFGSQXV3KU26KOKMQGEHPH\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-05/CC-MAIN-2020-05_segments_1579250607407.48_warc_CC-MAIN-20200122191620-20200122220620-00180.warc.gz\"}"}