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wiki_12300_chunk_0 | Freeman Dyson | In number theory, the crank of a partition is a certain integer associated with the partition in number theory. Dyson first introduced the term without a definition in a 1944 paper in a journal published by the Mathematics Society of Cambridge University. He then gave a list of properties this yet-to-be-defined quantit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12301_chunk_0 | Crank of a partition | In number theory, the crank of a partition of an integer is a certain integer associated with the partition. The term was first introduced without a definition by Freeman Dyson in a 1944 paper published in Eureka, a journal published by the Mathematics Society of Cambridge University. Dyson then gave a list of properti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12302_chunk_0 | Diamond operator | In number theory, the diamond operators 〈d〉 are operators acting on the space of modular forms for the group Γ1(N), given by the action of a matrix (a bc δ) in Γ0(N) where δ ≈ d mod N. The diamond operators form an abelian group and commute with the Hecke operators. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12303_chunk_0 | Gaussian unitary ensemble | In number theory, the distribution of zeros of the Riemann zeta function (and other L-functions) is modeled by the distribution of eigenvalues of certain random matrices. The connection was first discovered by Hugh Montgomery and Freeman Dyson. It is connected to the Hilbert–Pólya conjecture. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12304_chunk_0 | Dirichlet's divisor problem | In number theory, the divisor summatory function is a function that is a sum over the divisor function. It frequently occurs in the study of the asymptotic behaviour of the Riemann zeta function. The various studies of the behaviour of the divisor function are sometimes called divisor problems. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12305_chunk_0 | Fundamental lemma of sieve theory | In number theory, the fundamental lemma of sieve theory is any of several results that systematize the process of applying sieve methods to particular problems. Halberstam & Richert: 92–93 write: A curious feature of sieve literature is that while there is frequent use of Brun's method there are only a few attempts to ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12306_chunk_0 | Fundamental theorem of ideal theory in number fields | In number theory, the fundamental theorem of ideal theory in number fields states that every nonzero proper ideal in the ring of integers of a number field admits unique factorization into a product of nonzero prime ideals. In other words, every ring of integers of a number field is a Dedekind domain. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12307_chunk_0 | Pillai's arithmetical function | In number theory, the gcd-sum function, also called Pillai's arithmetical function, is defined for every n {\displaystyle n} by P ( n ) = ∑ k = 1 n gcd ( k , n ) {\displaystyle P(n)=\sum _{k=1}^{n}\gcd(k,n)} or equivalently P ( n ) = ∑ d ∣ n d φ ( n / d ) {\displaystyle P(n)=\sum _{d\mid n}d\varphi (n/d)} where d {\dis... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12308_chunk_0 | Number Field Sieve | In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than 10100. Heuristically, its complexity for factoring an integer n (consisting of ⌊log2 n⌋ + 1 bits) is of the form exp ( ( ( 64 / 9 ) 1 / 3 + o ( 1 ) ) ( log n ) 1 / 3 ( log log... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12309_chunk_0 | Number Field Sieve | The general number field sieve, on the other hand, manages to search for smooth numbers that are subexponential in the size of n. Since these numbers are smaller, they are more likely to be smooth than the numbers inspected in previous algorithms. This is the key to the efficiency of the number field sieve. In order to... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12310_chunk_0 | Home prime | In number theory, the home prime HP(n) of an integer n greater than 1 is the prime number obtained by repeatedly factoring the increasing concatenation of prime factors including repetitions. The mth intermediate stage in the process of determining HP(n) is designated HPn(m). For instance, HP(10) = 773, as 10 factors a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12311_chunk_0 | Home prime | Investigations into home primes make up a minor side issue in number theory. Its questions have served as test fields for the implementation of efficient algorithms for factoring composite numbers, but the subject is really one in recreational mathematics. The outstanding computational problem as of 2016 is whether HP(... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12312_chunk_0 | Home prime | Details of the history of this search, as well as the sequences leading to home primes for all other numbers through 100, are maintained at Patrick De Geest's worldofnumbers website. A wiki primarily associated with the Great Internet Mersenne Prime Search maintains the complete known data through 1000 in base 10 and a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12313_chunk_0 | Ideal class group | In number theory, the ideal class group (or class group) of an algebraic number field K is the quotient group JK /PK where JK is the group of fractional ideals of the ring of integers of K, and PK is its subgroup of principal ideals. The class group is a measure of the extent to which unique factorization fails in the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12314_chunk_0 | Integer complexity | In number theory, the integer complexity of an integer is the smallest number of ones that can be used to represent it using ones and any number of additions, multiplications, and parentheses. It is always within a constant factor of the logarithm of the given integer. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12315_chunk_0 | Integer square root | In number theory, the integer square root (isqrt) of a non-negative integer n is the non-negative integer m which is the greatest integer less than or equal to the square root of n, For example, isqrt ( 27 ) = ⌊ 27 ⌋ = ⌊ 5.19615242270663... ⌋ = 5. {\displaystyle \operatorname {isqrt} (27)=\lfloor {\sqrt {27}}\rfloor ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12316_chunk_0 | Larger sieve | In number theory, the larger sieve is a sieve invented by Patrick X. Gallagher. The name denotes a heightening of the large sieve. Combinatorial sieves like the Selberg sieve are strongest, when only a few residue classes are removed, while the term large sieve means that this sieve can take advantage of the removal of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12317_chunk_0 | Quadratic Reciprocity | In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard statement is: This law, together with its supplements, allows the easy calc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12318_chunk_0 | Quadratic Reciprocity | {\displaystyle \left(\pm a^{\frac {p+1}{4}}\right)^{2}=a^{\frac {p+1}{2}}=a\cdot a^{\frac {p-1}{2}}\equiv a\left({\frac {a}{p}}\right)=a{\bmod {p}}.} This formula only works if it is known in advance that a {\displaystyle a} is a quadratic residue, which can be checked using the law of quadratic reciprocity. The quadra... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12319_chunk_0 | Quadratic Reciprocity | (Art. 151)Privately, Gauss referred to it as the "golden theorem". | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12320_chunk_0 | Quadratic Reciprocity | He published six proofs for it, and two more were found in his posthumous papers. There are now over 240 published proofs. The shortest known proof is included below, together with short proofs of the law's supplements (the Legendre symbols of −1 and 2). Generalizing the reciprocity law to higher powers has been a lead... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12321_chunk_0 | Proofs of quadratic reciprocity | In number theory, the law of quadratic reciprocity, like the Pythagorean theorem, has lent itself to an unusually large number of proofs. Several hundred proofs of the law of quadratic reciprocity have been published. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12322_chunk_0 | Local zeta-function | In number theory, the local zeta function Z(V, s) (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as Z ( V , s ) = exp ( ∑ m = 1 ∞ N m m ( q − s ) m ) {\displaystyle Z(V,s)=\exp \left(\sum _{m=1}^{\infty }{\frac {N_{m}}{m}}(q^{-s})^{m}\right)} where V is a non-singular n-dime... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12323_chunk_0 | Multiplicative persistence | In number theory, the multiplicative digital root of a natural number n {\displaystyle n} in a given number base b {\displaystyle b} is found by multiplying the digits of n {\displaystyle n} together, then repeating this operation until only a single-digit remains, which is called the multiplicative digital root of n {... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12324_chunk_0 | Pisano period | In number theory, the nth Pisano period, written as π(n), is the period with which the sequence of Fibonacci numbers taken modulo n repeats. Pisano periods are named after Leonardo Pisano, better known as Fibonacci. The existence of periodic functions in Fibonacci numbers was noted by Joseph Louis Lagrange in 1774. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12325_chunk_0 | Löschian number | In number theory, the numbers of the form x2 + xy + y2 for integer x, y are called the Löschian numbers (or Loeschian numbers). These numbers are named after August Lösch. They are the norms of the Eisenstein integers. They are a set of whole numbers, including zero, and having prime factorization in which all primes c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12326_chunk_0 | Odd greedy expansion | In number theory, the odd greedy expansion problem asks whether a greedy algorithm for finding Egyptian fractions with odd denominators always succeeds. As of 2021, it remains unsolved. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12327_chunk_0 | Optic equation | In number theory, the optic equation is an equation that requires the sum of the reciprocals of two positive integers a and b to equal the reciprocal of a third positive integer c: 1 a + 1 b = 1 c . {\displaystyle {\frac {1}{a}}+{\frac {1}{b}}={\frac {1}{c}}.} Multiplying both sides by abc shows that the optic equation... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12328_chunk_0 | P-adic order | In number theory, the p-adic valuation or p-adic order of an integer n is the exponent of the highest power of the prime number p that divides n. It is denoted ν p ( n ) {\displaystyle \nu _{p}(n)} . Equivalently, ν p ( n ) {\displaystyle \nu _{p}(n)} is the exponent to which p {\displaystyle p} appears in the prime fa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12329_chunk_0 | Parity problem (sieve theory) | In number theory, the parity problem refers to a limitation in sieve theory that prevents sieves from giving good estimates in many kinds of prime-counting problems. The problem was identified and named by Atle Selberg in 1949. Beginning around 1996, John Friedlander and Henryk Iwaniec developed some parity-sensitive s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12330_chunk_0 | Partition function (number theory) | In number theory, the partition function p(n) represents the number of possible partitions of a non-negative integer n. For instance, p(4) = 5 because the integer 4 has the five partitions 1 + 1 + 1 + 1, 1 + 1 + 2, 1 + 3, 2 + 2, and 4. No closed-form expression for the partition function is known, but it has both asymp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12331_chunk_0 | Partition function (number theory) | The multiplicative inverse of its generating function is the Euler function; by Euler's pentagonal number theorem this function is an alternating sum of pentagonal number powers of its argument. Srinivasa Ramanujan first discovered that the partition function has nontrivial patterns in modular arithmetic, now known as ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12332_chunk_0 | Big Omega function (prime factor) | In number theory, the prime omega functions ω ( n ) {\displaystyle \omega (n)} and Ω ( n ) {\displaystyle \Omega (n)} count the number of prime factors of a natural number n . {\displaystyle n.} Thereby ω ( n ) {\displaystyle \omega (n)} (little omega) counts each distinct prime factor, whereas the related function Ω (... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12333_chunk_0 | Radical of an integer | In number theory, the radical of a positive integer n is defined as the product of the distinct prime numbers dividing n. Each prime factor of n occurs exactly once as a factor of this product: The radical plays a central role in the statement of the abc conjecture. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12334_chunk_0 | Ruler function | In number theory, the ruler function of an integer n {\displaystyle n} can be either of two closely related functions. One of these functions counts the number of times n {\displaystyle n} can be evenly divided by two, which for the numbers 1, 2, 3, ... is Alternatively, the ruler function can be defined as the same nu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12335_chunk_0 | Ruler function | In advanced mathematics, the 0-based ruler function is the 2-adic valuation of the number, and the lexicographically earliest infinite square-free word over the natural numbers. It also gives the position of the bit that changes at each step of the Gray code.In the Tower of Hanoi puzzle, with the disks of the puzzle nu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12336_chunk_0 | Second Hardy–Littlewood conjecture | In number theory, the second Hardy–Littlewood conjecture concerns the number of primes in intervals. Along with the first Hardy–Littlewood conjecture, the second Hardy–Littlewood conjecture was proposed by G. H. Hardy and John Edensor Littlewood in 1923. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12337_chunk_0 | Diophantine approximations | In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated by rational numbers. For this problem, a rational number a/b is a "good" approxi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12338_chunk_0 | Diophantine approximations | This problem was solved during the 18th century by means of continued fractions. Knowing the "best" approximations of a given number, the main problem of the field is to find sharp upper and lower bounds of the above difference, expressed as a function of the denominator. It appears that these bounds depend on the natu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12339_chunk_0 | Diophantine approximations | Diophantine approximations and transcendental number theory are very close areas that share many theorems and methods. Diophantine approximations also have important applications in the study of Diophantine equations. The 2022 Fields Medal was awarded to James Maynard for his work on Diophantine approximation. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12340_chunk_0 | Sum of squares function | In number theory, the sum of squares function is an arithmetic function that gives the number of representations for a given positive integer n as the sum of k squares, where representations that differ only in the order of the summands or in the signs of the numbers being squared are counted as different, and is denot... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12341_chunk_0 | Nicomachus's theorem | In number theory, the sum of the first n cubes is the square of the nth triangular number. That is, 1 3 + 2 3 + 3 3 + ⋯ + n 3 = ( 1 + 2 + 3 + ⋯ + n ) 2 . {\displaystyle 1^{3}+2^{3}+3^{3}+\cdots +n^{3}=\left(1+2+3+\cdots +n\right)^{2}.} The same equation may be written more compactly using the mathematical notation for ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12342_chunk_0 | Nicomachus's theorem | {\displaystyle \sum _{k=1}^{n}k^{3}={\bigg (}\sum _{k=1}^{n}k{\bigg )}^{2}.} This identity is sometimes called Nicomachus's theorem, after Nicomachus of Gerasa (c. 60 – c. 120 CE). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12343_chunk_0 | Sum of two squares theorem | In number theory, the sum of two squares theorem relates the prime decomposition of any integer n > 1 to whether it can be written as a sum of two squares, such that n = a2 + b2 for some integers a, b. An integer greater than one can be written as a sum of two squares if and only if its prime decomposition contains no ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12344_chunk_0 | Landau's totient constant | In number theory, the totient summatory function Φ ( n ) {\displaystyle \Phi (n)} is a summatory function of Euler's totient function defined by: Φ ( n ) := ∑ k = 1 n φ ( k ) , n ∈ N {\displaystyle \Phi (n):=\sum _{k=1}^{n}\varphi (k),\quad n\in \mathbf {N} } It is the number of coprime integer pairs {p, q}, 1 ≤ p ≤ q ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12345_chunk_0 | Unit function | In number theory, the unit function is a completely multiplicative function on the positive integers defined as: ε ( n ) = { 1 , if n = 1 0 , if n ≠ 1 {\displaystyle \varepsilon (n)={\begin{cases}1,&{\mbox{if }}n=1\\0,&{\mbox{if }}n\neq 1\end{cases}}} It is called the unit function because it is the identity element fo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12346_chunk_0 | Von Staudt–Clausen theorem | In number theory, the von Staudt–Clausen theorem is a result determining the fractional part of Bernoulli numbers, found independently by Karl von Staudt (1840) and Thomas Clausen (1840). Specifically, if n is a positive integer and we add 1/p to the Bernoulli number B2n for every prime p such that p − 1 divides 2n, we... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12347_chunk_0 | Integer factorisation | In number theory, there are many integer factoring algorithms that heuristically have expected running time L n = e ( 1 + o ( 1 ) ) ( log n ) ( log log n ) {\displaystyle L_{n}\left=e^{(1+o(1)){\sqrt {(\log n)(\log \log n)}}}} in little-o and L-notation. Some examples of those algorithms are the elliptic curve m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12348_chunk_0 | Coprime integers | In number theory, two integers a and b are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides a does not divide b, and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12349_chunk_0 | Multiplicative independence | In number theory, two positive integers a and b are said to be multiplicatively independent if their only common integer power is 1. That is, for integers n and m, a n = b m {\displaystyle a^{n}=b^{m}} implies n = m = 0 {\displaystyle n=m=0} . Two integers which are not multiplicatively independent are said to be multi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12350_chunk_0 | Prime geodesic | In number theory, various "prime geodesic theorems" have been proved which are very similar in spirit to the prime number theorem. To be specific, we let π(x) denote the number of closed geodesics whose norm (a function related to length) is less than or equal to x; then π(x) ∼ x/ln(x). This result is usually credited ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12351_chunk_0 | Prime geodesic | thesis, Grigory Margulis proved a similar result for surfaces of variable negative curvature, while in his 1980 Ph.D. thesis, Peter Sarnak proved an analogue of Chebotarev's density theorem. There are other similarities to number theory — error estimates are improved upon, in much the same way that error estimates of t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12352_chunk_0 | Prime geodesic | Also, there is a Selberg zeta function which is formally similar to the usual Riemann zeta function and shares many of its properties. Algebraically, prime geodesics can be lifted to higher surfaces in much the same way that prime ideals in the ring of integers of a number field can be split (factored) in a Galois exte... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12353_chunk_0 | Mertens conjecture | In number theory, we define the Mertens function as M ( n ) = ∑ 1 ≤ k ≤ n μ ( k ) , {\displaystyle M(n)=\sum _{1\leq k\leq n}\mu (k),} where μ(k) is the Möbius function; the Mertens conjecture is that for all n > 1, | M ( n ) | < n . {\displaystyle |M(n)|<{\sqrt {n}}.} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12354_chunk_0 | Zero-sum problem | In number theory, zero-sum problems are certain kinds of combinatorial problems about the structure of a finite abelian group. Concretely, given a finite abelian group G and a positive integer n, one asks for the smallest value of k such that every sequence of elements of G of size k contains n terms that sum to 0. The... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12355_chunk_0 | Zero-sum problem | (Indeed, the lower bound is easy to see: the multiset containing n − 1 copies of 0 and n − 1 copies of 1 contains no n-subset summing to a multiple of n.) This result is known as the Erdős–Ginzburg–Ziv theorem after its discoverers. It may also be deduced from the Cauchy–Davenport theorem.More general results than this... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12356_chunk_0 | Gauss-Laguerre quadrature | In numerical analysis Gauss–Laguerre quadrature (named after Carl Friedrich Gauss and Edmond Laguerre) is an extension of the Gaussian quadrature method for approximating the value of integrals of the following kind: ∫ 0 + ∞ e − x f ( x ) d x . {\displaystyle \int _{0}^{+\infty }e^{-x}f(x)\,dx.} In this case ∫ 0 + ∞ e ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12357_chunk_0 | Sinc numerical methods | In numerical analysis and applied mathematics, sinc numerical methods are numerical techniques for finding approximate solutions of partial differential equations and integral equations based on the translates of sinc function and Cardinal function C(f,h) which is an expansion of f defined by C ( f , h ) ( x ) = ∑ k = ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12358_chunk_0 | Godunov method | In numerical analysis and computational fluid dynamics, Godunov's scheme is a conservative numerical scheme, suggested by Sergei Godunov in 1959, for solving partial differential equations. One can think of this method as a conservative finite volume method which solves exact, or approximate Riemann problems at each in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12359_chunk_0 | Godunov's theorem | In numerical analysis and computational fluid dynamics, Godunov's theorem — also known as Godunov's order barrier theorem — is a mathematical theorem important in the development of the theory of high-resolution schemes for the numerical solution of partial differential equations. The theorem states that: Linear numeri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12360_chunk_0 | Adaptive rejection sampling | In numerical analysis and computational statistics, rejection sampling is a basic technique used to generate observations from a distribution. It is also commonly called the acceptance-rejection method or "accept-reject algorithm" and is a type of exact simulation method. The method works for any distribution in R m {\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12361_chunk_0 | Digital Signal Processing | In numerical analysis and functional analysis, a discrete wavelet transform is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key advantage it has over Fourier transforms is temporal resolution: it captures both frequency and location information. The accuracy o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12362_chunk_0 | LDU decomposition | In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix decomposition). The product sometimes includes a permutation matrix as well. LU decomposition can be viewed as the matrix form... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12363_chunk_0 | LDU decomposition | Computers usually solve square systems of linear equations using LU decomposition, and it is also a key step when inverting a matrix or computing the determinant of a matrix. The LU decomposition was introduced by the Polish astronomer Tadeusz Banachiewicz in 1938. To quote: "It appears that Gauss and Doolittle applied... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12364_chunk_0 | Symmetric sparse matrix | In numerical analysis and scientific computing, a sparse matrix or sparse array is a matrix in which most of the elements are zero. There is no strict definition regarding the proportion of zero-value elements for a matrix to qualify as sparse but a common criterion is that the number of non-zero elements is roughly eq... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12365_chunk_0 | Symmetric sparse matrix | Conceptually, sparsity corresponds to systems with few pairwise interactions. For example, consider a line of balls connected by springs from one to the next: this is a sparse system as only adjacent balls are coupled. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12366_chunk_0 | Symmetric sparse matrix | By contrast, if the same line of balls were to have springs connecting each ball to all other balls, the system would correspond to a dense matrix. The concept of sparsity is useful in combinatorics and application areas such as network theory and numerical analysis, which typically have a low density of significant da... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12367_chunk_0 | Symmetric sparse matrix | When storing and manipulating sparse matrices on a computer, it is beneficial and often necessary to use specialized algorithms and data structures that take advantage of the sparse structure of the matrix. Specialized computers have been made for sparse matrices, as they are common in the machine learning field. Opera... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12368_chunk_0 | Diffuse element method | In numerical analysis the diffuse element method (DEM) or simply diffuse approximation is a meshfree method. The diffuse element method was developed by B. Nayroles, G. Touzot and Pierre Villon at the Universite de Technologie de Compiegne, in 1992. It is in concept rather similar to the much older smoothed particle hy... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12369_chunk_0 | Diffuse element method | In the paper they describe a "diffuse approximation method", a method for function approximation from a given set of points. In fact the method boils down to the well-known moving least squares for the particular case of a global approximation (using all available data points). Using this function approximation method,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12370_chunk_0 | Aitken extrapolation | In numerical analysis, Aitken's delta-squared process or Aitken extrapolation is a series acceleration method, used for accelerating the rate of convergence of a sequence. It is named after Alexander Aitken, who introduced this method in 1926. Its early form was known to Seki Kōwa (end of 17th century) and was found fo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12371_chunk_0 | Bairstow's method | In numerical analysis, Bairstow's method is an efficient algorithm for finding the roots of a real polynomial of arbitrary degree. The algorithm first appeared in the appendix of the 1920 book Applied Aerodynamics by Leonard Bairstow. The algorithm finds the roots in complex conjugate pairs using only real arithmetic. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12372_chunk_0 | Brent's method | In numerical analysis, Brent's method is a hybrid root-finding algorithm combining the bisection method, the secant method and inverse quadratic interpolation. It has the reliability of bisection but it can be as quick as some of the less-reliable methods. The algorithm tries to use the potentially fast-converging seca... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12373_chunk_0 | Brent's method | Brent's method is due to Richard Brent and builds on an earlier algorithm by Theodorus Dekker. Consequently, the method is also known as the Brent–Dekker method. Modern improvements on Brent's method include Chandrupatla's method, which is simpler and faster for functions that are flat around their roots; Ridders' meth... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12374_chunk_0 | Broyden's method | In numerical analysis, Broyden's method is a quasi-Newton method for finding roots in k variables. It was originally described by C. G. Broyden in 1965.Newton's method for solving f(x) = 0 uses the Jacobian matrix, J, at every iteration. However, computing this Jacobian is a difficult and expensive operation. The idea ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12375_chunk_0 | Estrin's scheme | In numerical analysis, Estrin's scheme (after Gerald Estrin), also known as Estrin's method, is an algorithm for numerical evaluation of polynomials. Horner's method for evaluation of polynomials is one of the most commonly used algorithms for this purpose, and unlike Estrin's scheme it is optimal in the sense that it ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12376_chunk_0 | Gauss–Legendre quadrature | In numerical analysis, Gauss–Legendre quadrature is a form of Gaussian quadrature for approximating the definite integral of a function. For integrating over the interval , the rule takes the form: ∫ − 1 1 f ( x ) d x ≈ ∑ i = 1 n w i f ( x i ) {\displaystyle \int _{-1}^{1}f(x)\,dx\approx \sum _{i=1}^{n}w_{i}f(x_{i})} w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12377_chunk_0 | Gauss–Legendre quadrature | This algorithm was popular, but significantly more efficient algorithms exist. Algorithms based on the Newton–Raphson method are able to compute quadrature rules for significantly larger problem sizes. In 2014, Ignace Bogaert presented explicit asymptotic formulas for the Gauss–Legendre quadrature weights and nodes, wh... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12378_chunk_0 | Bailey's method (root finding) | In numerical analysis, Halley's method is a root-finding algorithm used for functions of one real variable with a continuous second derivative. It is named after its inventor Edmond Halley. The algorithm is second in the class of Householder's methods, after Newton's method. Like the latter, it iteratively produces a s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12379_chunk_0 | Hermite interpolation | In numerical analysis, Hermite interpolation, named after Charles Hermite, is a method of polynomial interpolation, which generalizes Lagrange interpolation. Lagrange interpolation allows computing a polynomial of degree less than n that takes the same value at n given points as a given function. Instead, Hermite inter... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12380_chunk_0 | Hermite interpolation | However, there are other methods for computing a Hermite interpolating polynomial. One can use linear algebra, by taking the coefficients of the interpolating polynomial as unknowns, and writing as linear equations the constraints that the interpolating polynomial must satisfy. For another method, see Chinese remainder... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12381_chunk_0 | Laguerre's method | In numerical analysis, Laguerre's method is a root-finding algorithm tailored to polynomials. In other words, Laguerre's method can be used to numerically solve the equation p(x) = 0 for a given polynomial p(x). One of the most useful properties of this method is that it is, from extensive empirical study, very close t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12382_chunk_0 | Lebedev grid | In numerical analysis, Lebedev quadrature, named after Vyacheslav Ivanovich Lebedev, is an approximation to the surface integral of a function over a three-dimensional sphere. The grid is constructed so to have octahedral rotation and inversion symmetry. The number and location of the grid points together with a corres... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12383_chunk_0 | Newton method | In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function. The most basic version starts with a single-variable function f ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12384_chunk_0 | Newton method | The number of correct digits roughly doubles with each step. This algorithm is first in the class of Householder's methods, succeeded by Halley's method. The method can also be extended to complex functions and to systems of equations. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12385_chunk_0 | Richardson extrapolation | In numerical analysis, Richardson extrapolation is a sequence acceleration method used to improve the rate of convergence of a sequence of estimates of some value A ∗ = lim h → 0 A ( h ) {\displaystyle A^{\ast }=\lim _{h\to 0}A(h)} . In essence, given the value of A ( h ) {\displaystyle A(h)} for several values of h {\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12386_chunk_0 | Ridders' method | In numerical analysis, Ridders' method is a root-finding algorithm based on the false position method and the use of an exponential function to successively approximate a root of a continuous function f ( x ) {\displaystyle f(x)} . The method is due to C. Ridders.Ridders' method is simpler than Muller's method or Brent... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12387_chunk_0 | Romberg integration | In numerical analysis, Romberg's method is used to estimate the definite integral by applying Richardson extrapolation repeatedly on the trapezium rule or the rectangle rule (midpoint rule). The estimates generate a triangular array. Romberg's method is a Newton–Cotes formula – it evaluates the integrand at equally spa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12388_chunk_0 | Stone method | In numerical analysis, Stone's method, also known as the strongly implicit procedure or SIP, is an algorithm for solving a sparse linear system of equations. The method uses an incomplete LU decomposition, which approximates the exact LU decomposition, to get an iterative solution of the problem. The method is named af... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12389_chunk_0 | Stone method | The biggest disadvantage is that it fails to take advantage of coefficient matrix to be a sparse matrix. The LU decomposition of a sparse matrix is usually not sparse, thus, for a large system of equations, LU decomposition may require a prohibitive amount of memory and number of arithmetical operations. In the precond... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12390_chunk_0 | Stone method | This brings one to idea of using approximate factorization LU of A as the iteration matrix M. A version of incomplete lower-upper decomposition method was proposed by Stone in 1968. This method is designed for equation system arising from discretisation of partial differential equations and was firstly used for a penta... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12391_chunk_0 | Blossom (functional) | In numerical analysis, a blossom is a functional that can be applied to any polynomial, but is mostly used for Bézier and spline curves and surfaces. The blossom of a polynomial ƒ, often denoted B , {\displaystyle {\mathcal {B}},} is completely characterised by the three properties: It is a symmetric function of its a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12392_chunk_0 | Midpoint method | In numerical analysis, a branch of applied mathematics, the midpoint method is a one-step method for numerically solving the differential equation, y ′ ( t ) = f ( t , y ( t ) ) , y ( t 0 ) = y 0 . {\displaystyle y'(t)=f(t,y(t)),\quad y(t_{0})=y_{0}.} The explicit midpoint method is given by the formula the implicit mi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12393_chunk_0 | Midpoint method | The explicit midpoint method is sometimes also known as the modified Euler method, the implicit method is the most simple collocation method, and, applied to Hamiltonian dynamics, a symplectic integrator. Note that the modified Euler method can refer to Heun's method, for further clarity see List of Runge–Kutta methods... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12394_chunk_0 | Midpoint method | However, if the second derivative is only positive between t n {\displaystyle t_{n}} and t n + 1 {\displaystyle t_{n+1}} , or only negative (as in the diagram), the curve will increasingly veer away from the tangent, leading to larger errors as h {\displaystyle h} increases. The diagram illustrates that the tangent at ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12395_chunk_0 | Hermite curve | In numerical analysis, a cubic Hermite spline or cubic Hermite interpolator is a spline where each piece is a third-degree polynomial specified in Hermite form, that is, by its values and first derivatives at the end points of the corresponding domain interval.Cubic Hermite splines are typically used for interpolation ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12396_chunk_0 | Hermite curve | The resulting spline will be continuous and will have continuous first derivative. Cubic polynomial splines can be specified in other ways, the Bezier cubic being the most common. However, these two methods provide the same set of splines, and data can be easily converted between the Bézier and Hermite forms; so the na... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12397_chunk_0 | Hermite curve | Cubic polynomial splines are extensively used in computer graphics and geometric modeling to obtain curves or motion trajectories that pass through specified points of the plane or three-dimensional space. In these applications, each coordinate of the plane or space is separately interpolated by a cubic spline function... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12398_chunk_0 | Multigrid methods | In numerical analysis, a multigrid method (MG method) is an algorithm for solving differential equations using a hierarchy of discretizations. They are an example of a class of techniques called multiresolution methods, very useful in problems exhibiting multiple scales of behavior. For example, many basic relaxation m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_12399_chunk_0 | Multigrid methods | For example, the finite element method may be recast as a multigrid method. In these cases, multigrid methods are among the fastest solution techniques known today. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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