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In numerical analysis, different decompositions are used to implement efficient matrix algorithms. For instance, when solving a system of linear equations A x = b {\displaystyle A\mathbf {x} =\mathbf {b} } , the matrix A can be decomposed via the LU decomposition. The LU decomposition factorizes a matrix into a lower triangular matrix L and an upper triangular matrix U. The systems L ( U x ) = b {\displaystyle L(U\mathbf {x} )=\mathbf {b} } and U x = L − 1 b {\displaystyle U\mathbf {x} =L^{-1}\mathbf {b} } require fewer additions and multiplications to solve, compared with the original system A x = b {\displaystyle A\mathbf {x} =\mathbf {b} } , though one might require significantly more digits in inexact arithmetic such as floating point. Similarly, the QR decomposition expresses A as QR with Q an orthogonal matrix and R an upper triangular matrix. The system Q(Rx) = b is solved by Rx = QTb = c, and the system Rx = c is solved by 'back substitution'. The number of additions and multiplications required is about twice that of using the LU solver, but no more digits are required in inexact arithmetic because the QR decomposition is numerically stable.
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In the context of network theory a scale-free ideal network is a random network with a degree distribution following the scale-free ideal gas density distribution. These networks are able to reproduce city-size distributions and electoral results by unraveling the size distribution of social groups with information theory on complex networks when a competitive cluster growth process is applied to the network. In models of scale-free ideal networks it is possible to demonstrate that Dunbar's number is the cause of the phenomenon known as the 'six degrees of separation'.
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In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs. Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed.
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Due to this superposition, measurement of the qubit will "collapse" it into one of its basis states with a given probability. Because of the entanglement, measurement of one qubit will "collapse" the other qubit to a state whose measurement will yield one of two possible values, where the value depends on which Bell's state the two qubits are in initially. Bell's states can be generalized to certain quantum states of multi-qubit systems, such as the GHZ state for 3 or more subsystems. Understanding of Bell's states is useful in analysis of quantum communication, such as superdense coding and quantum teleportation. The no-communication theorem prevents this behavior from transmitting information faster than the speed of light.
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In contrast to the steady incremental improvements of the past few decades, the application of deep learning decreased word error rate by 30%. This innovation was quickly adopted across the field.
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In metric graph theory, a convex subgraph of an undirected graph G is a subgraph that includes every shortest path in G between two of its vertices. Thus, it is analogous to the definition of a convex set in geometry, a set that contains the line segment between every pair of its points. Convex subgraphs play an important role in the theory of partial cubes and median graphs. In particular, in median graphs, the convex subgraphs have the Helly property: if a family of convex subgraphs has the property that all pairwise intersections are nonempty, then the whole family has a nonempty intersection.
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In principle, a 64-bit microprocessor can address 16 EiB (16 × 10246 = 264 = 18,446,744,073,709,551,616 bytes, or about 18.4 exabytes) of memory. However, not all instruction sets, and not all processors implementing those instruction sets, support a full 64-bit virtual or physical address space. The x86-64 architecture (as of 2016) allows 48 bits for virtual memory and, for any given processor, up to 52 bits for physical memory. These limits allow memory sizes of 256 TiB (256 × 10244 bytes) and 4 PiB (4 × 10245 bytes), respectively.
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In mathematics, pentation (or hyper-5) is the next hyperoperation after tetration and before hexation. It is defined as iterated (repeated) tetration (assuming right-associativity), just as tetration is iterated right-associative exponentiation. It is a binary operation defined with two numbers a and b, where a is tetrated to itself b-1 times. For instance, using hyperoperation notation for pentation and tetration, 2 3 {\displaystyle 23} means tetrating 2 to itself 2 times, or 2 ( 2 2 ) {\displaystyle 2(22)} . This can then be reduced to 2 ( 2 2 ) = 2 4 = 2 2 2 2 = 2 2 4 = 2 16 = 65 , 536. {\displaystyle 2(2^{2})=24=2^{2^{2^{2}}}=2^{2^{4}}=2^{16}=65,536.}
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In sociology, command hierarchy is seen as the most visible element of a "power network." In this model, social capital is viewed as being mobilized in response to orders that move through the hierarchy leading to the phrase "command and control".
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Later it was demonstrated that all basic mathematical structures either are some kinds of named sets or are built of named sets. According to Anellis, Burgin & Kaloujnine introduced set-theoretical named sets in 1983 and Burgin introduced named sets in the most general form in 1990. Since then Burgin continued to develop this theory in a series of papers and a book. In 2011, Zellweger applied the theory of named sets to model data relations in the relational database for an end-user interface.
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Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets. In the case of finite sets, this agrees with the intuitive notion of number of elements. In the case of infinite sets, the behavior is more complex.
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In the area of modern algebra known as group theory, the Suzuki groups, denoted by Sz(22n+1), 2B2(22n+1), Suz(22n+1), or G(22n+1), form an infinite family of groups of Lie type found by Suzuki (1960), that are simple for n ≥ 1. These simple groups are the only finite non-abelian ones with orders not divisible by 3.
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This eliminates noncredible threats, which are threats that a player would not carry out if they were ever called upon to do so. For example, consider a dynamic game with an incumbent firm and a potential entrant to the industry. The incumbent has a monopoly and wants to maintain its market share.
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In nonstandard analysis, a branch of mathematics, overspill (referred to as overflow by Goldblatt (1998, p. 129)) is a widely used proof technique. It is based on the fact that the set of standard natural numbers N is not an internal subset of the internal set *N of hypernatural numbers. By applying the induction principle for the standard integers N and the transfer principle we get the principle of internal induction: For any internal subset A of *N, if 1 is an element of A, and for every element n of A, n + 1 also belongs to A,then A = *NIf N were an internal set, then instantiating the internal induction principle with N, it would follow N = *N which is known not to be the case.
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In this case, the voting circuit can output the correct result, and discard the erroneous version. After this, the internal state of the erroneous replication is assumed to be different from that of the other two, and the voting circuit can switch to a DMR mode. This model can be applied to any larger number of replications.
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In statistics, binary data is a statistical data type consisting of categorical data that can take exactly two possible values, such as "A" and "B", or "heads" and "tails". It is also called dichotomous data, and an older term is quantal data. The two values are often referred to generically as "success" and "failure". As a form of categorical data, binary data is nominal data, meaning the values are qualitatively different and cannot be compared numerically.
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In network theory, the Wiener connector is a means of maximizing efficiency in connecting specified "query vertices" in a network. Given a connected, undirected graph and a set of query vertices in a graph, the minimum Wiener connector is an induced subgraph that connects the query vertices and minimizes the sum of shortest path distances among all pairs of vertices in the subgraph. In combinatorial optimization, the minimum Wiener connector problem is the problem of finding the minimum Wiener connector. It can be thought of as a version of the classic Steiner tree problem (one of Karp's 21 NP-complete problems), where instead of minimizing the size of the tree, the objective is to minimize the distances in the subgraph.The minimum Wiener connector was first presented by Ruchansky et al. in 2015.The minimum Wiener connector has applications in many domains where there is a graph structure and an interest in learning about connections between sets of individuals.
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Early scrambling or encryption methods required a hard line for authorization of receive sites. Today, a digital cellular telephone is sufficient for most situations. C-Band transportable service remains a prevalent source of long-haul transmission because of its immunity to the "rain fade" that Ku band experiences in significant rainstorms.
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In principle, there can be more than one such code for a given word length, but the term Gray code was first applied to a particular binary code for non-negative integers, the binary-reflected Gray code, or BRGC. Bell Labs researcher George R. Stibitz described such a code in a 1941 patent application, granted in 1943. Frank Gray introduced the term reflected binary code in his 1947 patent application, remarking that the code had "as yet no recognized name". He derived the name from the fact that it "may be built up from the conventional binary code by a sort of reflection process".
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In mathematics, a surjective function (also known as surjection, or onto function ) is a function f such that every element y can be mapped from some element x such that f(x) = y. In other words, every element of the function's codomain is the image of at least one element of its domain. It is not required that x be unique; the function f may map one or more elements of X to the same element of Y. The term surjective and the related terms injective and bijective were introduced by Nicolas Bourbaki, a group of mainly French 20th-century mathematicians who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French word sur means over or above, and relates to the fact that the image of the domain of a surjective function completely covers the function's codomain. Any function induces a surjection by restricting its codomain to the image of its domain.
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. . , n } → { 0 , 1 } {\displaystyle f:\{0,1,...,n\}\rightarrow \{0,1\}} . Symmetric Boolean functions are used to classify Boolean satisfiability problems.
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In mathematics, in the areas of group theory and combinatorics, Hall words provide a unique monoid factorisation of the free monoid. They are also totally ordered, and thus provide a total order on the monoid. This is analogous to the better-known case of Lyndon words; in fact, the Lyndon words are a special case, and almost all properties possessed by Lyndon words carry over to Hall words. Hall words are in one-to-one correspondence with Hall trees.
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As told by Hardy: I remember once going to see him when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one, and that I hoped it was not an unfavourable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."
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In physical theories, a test particle, or test charge, is an idealized model of an object whose physical properties (usually mass, charge, or size) are assumed to be negligible except for the property being studied, which is considered to be insufficient to alter the behavior of the rest of the system. The concept of a test particle often simplifies problems, and can provide a good approximation for physical phenomena. In addition to its uses in the simplification of the dynamics of a system in particular limits, it is also used as a diagnostic in computer simulations of physical processes.
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Kernel adaptive filters implement a nonlinear transfer function using kernel methods. In these methods, the signal is mapped to a high-dimensional linear feature space and a nonlinear function is approximated as a sum over kernels, whose domain is the feature space. If this is done in a reproducing kernel Hilbert space, a kernel method can be a universal approximator for a nonlinear function.
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A semigroup without an identity element can be easily turned into a monoid by just adding an identity element. Consequently, monoids are studied in the theory of semigroups rather than in group theory. Semigroups should not be confused with quasigroups, which are a generalization of groups in a different direction; the operation in a quasigroup need not be associative but quasigroups preserve from groups a notion of division.
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In mathematics, probabilistic metric spaces are a generalization of metric spaces where the distance no longer takes values in the non-negative real numbers R ≥ 0, but in distribution functions.Let D+ be the set of all probability distribution functions F such that F(0) = 0 (F is a nondecreasing, left continuous mapping from R into such that max(F) = 1). Then given a non-empty set S and a function F: S × S → D+ where we denote F(p, q) by Fp,q for every (p, q) ∈ S × S, the ordered pair (S, F) is said to be a probabilistic metric space if: For all u and v in S, u = v if and only if Fu,v(x) = 1 for all x > 0. For all u and v in S, Fu,v = Fv,u. For all u, v and w in S, Fu,v(x) = 1 and Fv,w(y) = 1 ⇒ Fu,w(x + y) = 1 for x, y > 0.
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In modern news, social media has become a major source of amateur reporting. The Arab Spring is believed to have been aided by citizen journalism that was reported using and disseminated by Facebook and Twitter. The Top 15 Most Popular Social Media Sites range from 15 million users to 900 million users, and as their user base grows stronger, amateur reporting's base expands as well.
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In particular, this problem arises when we attempt to build a so-called RASP, a "universal machine" (see more at Universal Turing machine) that uses its finite-state machine to interpret a "program of instructions" located in its registers – i.e. we are building what is nowadays called a computer with the von Neumann architecture.Observe that the counter machine's finite state machine must call out a register explicitly (directly) by its name/number: INC (65,356) calls out register number "65,365" explicitly. If the number of registers exceeds the capability of the finite state machine to address them, then registers outside the bounds will be unreachable. For example, if the finite state machine can only reach 65,536 = 216 registers then how can it reach the 65,537th?So how do we address a register beyond the bounds of the finite state machine?
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In order to achieve a precise and effective typestate analysis, it is necessary to address the problem of aliasing. Aliasing occurs when an object has more than one reference or pointer that points to it. For the analysis to be correct, state changes to a given object must be reflected in all references that point to that object, but in general it is a difficult problem to track all such references. This becomes especially hard if the analysis needs to be modular, that is, applicable to each part of a large program separately without taking the rest of the program into account.
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The Coordinating Committee admitted that they "wrestled with articulating a business case for implementing RDA", nevertheless the report recommended that RDA be adopted by the three national libraries, contingent on several improvements being made. The earliest possible date for implementation was given as January 2013, as the consensus emerging from the analysis of the test data showed that while there were discernible benefits to implementing RDA, these benefits would not be realized without further changes to current cataloging practices, including developing a successor to the MARC format.Several other institutions were involved in the RDA test. Many of these institutions documented their findings in a special issue of Cataloging & Classification Quarterly.
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In supervised learning, a random sub-sample of all records is taken and manually classified as either 'fraudulent' or 'non-fraudulent' (task can be decomposed on more classes to meet algorithm requirements). Relatively rare events such as fraud may need to be over sampled to get a big enough sample size. These manually classified records are then used to train a supervised machine learning algorithm. After building a model using this training data, the algorithm should be able to classify new records as either fraudulent or non-fraudulent.
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In the cellular phone industry, mobile phones and their networks sometimes support concatenated short message service (or concatenated SMS) to overcome the limitation on the number of characters that can be sent in a single SMS text message transmission (which is usually 160). Using this method, long messages are split into smaller messages by the sending device and recombined at the receiving end. Each message is then billed separately.
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In set theory, functions are identified with their function graphs. Using set builder notation, a collection of pairs may be characterized, f := { ⟨ x , y ⟩ ∣ x ∈ A ∧ ψ ( x , y ) } . {\displaystyle f:={\big \{}\langle x,y\rangle \mid x\in A\land \psi (x,y){\big \}}.} The axiom of replacement in Zermelo–Fraenkel set theory implies that this is actually a set and a function in the above sense.
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This notation has been retained in operating systems that were directly or indirectly derived from CP/M, including DR-DOS, MS-DOS, OS/2 and Windows. On Linux systems, the command hexcat produces this classic output format too.
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In the fields of databases and transaction processing (transaction management), a schedule (or history) of a system is an abstract model to describe execution of transactions running in the system. Often it is a list of operations (actions) ordered by time, performed by a set of transactions that are executed together in the system. If the order in time between certain operations is not determined by the system, then a partial order is used. Examples of such operations are requesting a read operation, reading, writing, aborting, committing, requesting a lock, locking, etc. Not all transaction operation types should be included in a schedule, and typically only selected operation types (e.g., data access operations) are included, as needed to reason about and describe certain phenomena. Schedules and schedule properties are fundamental concepts in database concurrency control theory.
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In order of increasing strength, i.e., decreasing sets of pairs, three of the possible partial orders on the Cartesian product of two partially ordered sets are (see Fig.4): the lexicographical order: (a, b) ≤ (c, d) if a < c or (a = c and b ≤ d); the product order: (a, b) ≤ (c, d) if a ≤ c and b ≤ d; the reflexive closure of the direct product of the corresponding strict orders: (a, b) ≤ (c, d) if (a < c and b < d) or (a = c and b = d).All three can similarly be defined for the Cartesian product of more than two sets. Applied to ordered vector spaces over the same field, the result is in each case also an ordered vector space. See also orders on the Cartesian product of totally ordered sets.
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In mathematics, particularly in functional analysis and topology, closed graph is a property of functions. A function f: X → Y between topological spaces has a closed graph if its graph is a closed subset of the product space X × Y. A related property is open graph.This property is studied because there are many theorems, known as closed graph theorems, giving conditions under which a function with a closed graph is necessarily continuous. One particularly well-known class of closed graph theorems are the closed graph theorems in functional analysis.
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Attack - An attempt by a player to win a point by hitting the ball over the net. Attack line - In indoor volleyball, a line three metres from the net which marks the limit for where a back-row player may advance to hit a ball from above the net. Back-row player - In indoor volleyball, any of three players positioned at the back of the court.
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Army used in for tactical group messaging in its Force Battle Command Brigade and Below (FBCB2) system.Several other approaches to reliable multicast were being developed at approximately the same time, and in April 1999, the IETF chartered the Reliable Multicast Transport Working Group (RMTWG) to standardize reliable multicast transport.The RMTWG pursued the strategy of developing building blocks and protocol instantiations. This strategy avoided a "one size fits all" protocol, which in turn could accommodate the large number of applications and types of applications that reliable multicast could support. Building blocks were defined as “a set of easily-separable coarse-grained modular components that are common to multiple protocols along with abstract APIs that define a building block's access methods and their arguments.” Initial building blocks included negative acknowledgments, forward error correction, a generic signaling mechanism for router assist, and transport protection Protocol instantiations were defined as “specifications that define the necessary gluing logic and minimal additional functionality required to realize a working protocol from one or more building blocks.” Those specifications would also include an abstract API that defined the interface between the protocol implementation and an application. Two protocol instantiations were chosen: A NACK-based protocol An Asynchronous Layered Coding protocolIn July 2005 the NACK-based protocol building blocks and protocol instantiation were submitted as “Experimental” in RFC 3940, and in November 2009 “NACK-Oriented Reliable Multicast (NORM) Transport Protocol” was approved in RFC 5740.The RMTWG was disestablished in September 2013.
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In probability and statistics, Student's t-distribution (or simply the t-distribution) t ν {\displaystyle t_{\nu }} is a continuous probability distribution that generalizes the standard normal distribution. Like the latter, it is symmetric around zero and bell-shaped. However, t ν {\displaystyle t_{\nu }} has heavier tails and the amount of probability mass in the tails is controlled by the parameter ν {\displaystyle \nu } .
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In number theory, the Fermat quotient of an integer a with respect to an odd prime p is defined as q p ( a ) = a p − 1 − 1 p , {\displaystyle q_{p}(a)={\frac {a^{p-1}-1}{p}},} or δ p ( a ) = a − a p p {\displaystyle \delta _{p}(a)={\frac {a-a^{p}}{p}}} .This article is about the former; for the latter see p-derivation. The quotient is named after Pierre de Fermat. If the base a is coprime to the exponent p then Fermat's little theorem says that qp(a) will be an integer. If the base a is also a generator of the multiplicative group of integers modulo p, then qp(a) will be a cyclic number, and p will be a full reptend prime.
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The challenge is to reduce the number of colors from n to, e.g., Δ + 1. The more colors are employed, e.g. O(Δ) instead of Δ + 1, the fewer communication rounds are required.A straightforward distributed version of the greedy algorithm for (Δ + 1)-coloring requires Θ(n) communication rounds in the worst case − information may need to be propagated from one side of the network to another side. The simplest interesting case is an n-cycle.
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In the 1993 film Jurassic Park, Connection Machines (non-functioning dummies) are visible in the park's control room, programmer Dennis Nedry mentions "eight Connection Machines" and a video about dinosaur cloning mentions "Thinking Machines supercomputers". In the 1996 film Mission Impossible, Luther Stickell asks Franz Krieger for "Thinking Machine laptops" to help hack into the CIA's Langley supercomputer.Tom Clancy's novel Rainbow Six speaks of the NSA's "star machine from a company gone bankrupt, the Super-Connector from Thinking Machines, Inc., of Cambridge, Massachusetts" in the NSA's basement. In addition, in The Bear and the Dragon says the National Security Agency could crack nearly any book or cipher with one of three custom operating systems designed for a Thinking Machines supercomputer. In the 2008 video game Fallout 3, it is mentioned that the pre-war firm that made the computer systems for Vaults is called Think Machine.
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In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers R {\displaystyle \mathbb {R} } , sometimes called the continuum. It is an infinite cardinal number and is denoted by c {\displaystyle {\mathfrak {c}}} (lowercase Fraktur "c") or | R | {\displaystyle |\mathbb {R} |} .The real numbers R {\displaystyle \mathbb {R} } are more numerous than the natural numbers N {\displaystyle \mathbb {N} } . Moreover, R {\displaystyle \mathbb {R} } has the same number of elements as the power set of N . {\displaystyle \mathbb {N} .}
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In mathematics, a Gödel numbering for sequences provides an effective way to represent each finite sequence of natural numbers as a single natural number. While a set theoretical embedding is surely possible, the emphasis is on the effectiveness of the functions manipulating such representations of sequences: the operations on sequences (accessing individual members, concatenation) can be "implemented" using total recursive functions, and in fact by primitive recursive functions. It is usually used to build sequential “data types” in arithmetic-based formalizations of some fundamental notions of mathematics. It is a specific case of the more general idea of Gödel numbering. For example, recursive function theory can be regarded as a formalization of the notion of an algorithm, and can be regarded as a programming language to mimic lists by encoding a sequence of natural numbers in a single natural number.
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In mathematical finite group theory, the uniqueness case is one of the three possibilities for groups of characteristic 2 type given by the trichotomy theorem. The uniqueness case covers groups G of characteristic 2 type with e(G) ≥ 3 that have an almost strongly p-embedded maximal 2-local subgroup for all primes p whose 2-local p-rank is sufficiently large (usually at least 3). Aschbacher (1983a, 1983b) proved that there are no finite simple groups in the uniqueness case.
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However, since only Alice knows b {\displaystyle b} , it makes it virtually impossible for either Bob or Eve to distinguish the states of the qubits. Also, after Bob has received the qubits, we know that Eve cannot be in possession of a copy of the qubits sent to Bob, by the no-cloning theorem, unless she has made measurements. Her measurements, however, risk disturbing a particular qubit with probability ½ if she guesses the wrong basis.
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In regards to issues with connectivity, it is important to consider the musical score of composite films. Much like visual repetition or thematic similarities, the score offers another medium through which stories can be linked. Rather than leaving viewers with content to process, decode, intellectualize and then react to, the film score can produce emotional reactions immediately, before viewers have a chance to analyze their own responses. There are three main categories of composite film scores:
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In other words, a "sum" s 1 {\displaystyle s_{1}} with only one term evaluates to that one term, while a "sum" s 0 {\displaystyle s_{0}} with no terms evaluates to 0. Allowing a "sum" with only 1 or 0 terms reduces the number of cases to be considered in many mathematical formulas.
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In mathematics, a weighing matrix of order n {\displaystyle n} and weight w {\displaystyle w} is a matrix W {\displaystyle W} with entries from the set { 0 , 1 , − 1 } {\displaystyle \{0,1,-1\}} such that: W W T = w I n {\displaystyle WW^{\mathsf {T}}=wI_{n}} Where W T {\displaystyle W^{\mathsf {T}}} is the transpose of W {\displaystyle W} and I n {\displaystyle I_{n}} is the identity matrix of order n {\displaystyle n} . The weight w {\displaystyle w} is also called the degree of the matrix. For convenience, a weighing matrix of order n {\displaystyle n} and weight w {\displaystyle w} is often denoted by W ( n , w ) {\displaystyle W(n,w)} .Weighing matrices are so called because of their use in optimally measuring the individual weights of multiple objects. When the weighing device is a balance scale, the statistical variance of the measurement can be minimized by weighing multiple objects at once, including some objects in the opposite pan of the scale where they subtract from the measurement.
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For example, a model might be selected by maximizing its performance on some set of training data, and yet its suitability might be determined by its ability to perform well on unseen data; then over-fitting occurs when a model begins to "memorize" training data rather than "learning" to generalize from a trend. As an extreme example, if the number of parameters is the same as or greater than the number of observations, then a model can perfectly predict the training data simply by memorizing the data in its entirety. (For an illustration, see Figure 2.)
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In number theory, Carmichael's theorem, named after the American mathematician R. D. Carmichael, states that, for any nondegenerate Lucas sequence of the first kind Un(P, Q) with relatively prime parameters P, Q and positive discriminant, an element Un with n ≠ 1, 2, 6 has at least one prime divisor that does not divide any earlier one except the 12th Fibonacci number F(12) = U12(1, −1) = 144 and its equivalent U12(−1, −1) = −144. In particular, for n greater than 12, the nth Fibonacci number F(n) has at least one prime divisor that does not divide any earlier Fibonacci number. Carmichael (1913, Theorem 21) proved this theorem. Recently, Yabuta (2001) gave a simple proof.
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In the context of primitive recursive functions, it is convenient to have a means to represent finite sequences of natural numbers as single natural numbers. One such method, Gödel's encoding, represents a sequence of positive integers ⟨ n 0 , n 1 , n 2 , … , n k ⟩ {\displaystyle \langle n_{0},n_{1},n_{2},\ldots ,n_{k}\rangle } as ∏ i = 0 k p i n i {\displaystyle \prod _{i=0}^{k}p_{i}^{n_{i}}} ,where pi represent the ith prime. It can be shown that, with this representation, the ordinary operations on sequences are all primitive recursive. These operations include Determining the length of a sequence, Extracting an element from a sequence given its index, Concatenating two sequences.Using this representation of sequences, it can be seen that if h(m) is primitive recursive then the function f ( n ) = h ( ⟨ f ( 0 ) , f ( 1 ) , f ( 2 ) , … , f ( n − 1 ) ⟩ ) {\displaystyle f(n)=h(\langle f(0),f(1),f(2),\ldots ,f(n-1)\rangle )} .is also primitive recursive. When the sequence ⟨ n 0 , n 1 , n 2 , … , n k ⟩ {\displaystyle \langle n_{0},n_{1},n_{2},\ldots ,n_{k}\rangle } is allowed to include zeros, it is instead represented as ∏ i = 0 k p i ( n i + 1 ) {\displaystyle \prod _{i=0}^{k}p_{i}^{(n_{i}+1)}} ,which makes it possible to distinguish the codes for the sequences ⟨ 0 ⟩ {\displaystyle \langle 0\rangle } and ⟨ 0 , 0 ⟩ {\displaystyle \langle 0,0\rangle } .
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In the following example, statement b explicitly negates statement a: Statements can also be mutually exclusive, without explicitly negating each other as in the following example:
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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In the classic problem the fuel in the jeep and at fuel dumps is treated as a continuous quantity. More complex variations on the problem have been proposed in which the fuel can only be left or collected in discrete amounts.In the camel and bananas problem, the merchant has n units of bananas. The camel can carry at most 1 unit of bananas at any time, and can travel 1 unit of distance on 1 unit of bananas. The market is at m units of distance away.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Another square-difference-free set is obtained by doubling the Moser–de Bruijn sequence. The best known upper bound on the size of a square-difference-free set of numbers up to n {\displaystyle n} is only slightly sublinear, but the largest known sets of this form are significantly smaller, of size ≈ n 0.733412 {\displaystyle \approx n^{0.733412}} . Closing the gap between these upper and lower bounds remains an open problem. The sublinear size bounds on square-difference-free sets can be generalized to sets where certain other polynomials are forbidden as differences between pairs of elements.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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In signal processing, pre-emphasis is a technique to protect against anticipated noise. The idea is to boost (and hence distort) the frequency range that is most susceptible to noise beforehand, so that after a noisy process (transmission over cable, tape recording...) more information can be recovered from that frequency range. Removal of the distortion caused by pre-emphasis is called de-emphasis, making the output accurately reproduce the original input. Emphasis is commonly used in FM broadcasting (preemphasis improvement) and vinyl (e.g. LP) records. For example, high-frequency signal components may be emphasized to produce a more equal modulation index for a transmitted frequency spectrum, and therefore a better signal-to-noise ratio for the entire frequency range.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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In protected mode, the segment_part is replaced by a 16-bit selector, in which the 13 upper bits (bit 3 to bit 15) contain the index of an entry inside a descriptor table. The next bit (bit 2) specifies whether the operation is used with the GDT or the LDT. The lowest two bits (bit 1 and bit 0) of the selector are combined to define the privilege of the request, where the values of 0 and 3 represent the highest and the lowest privilege, respectively. This means that the byte offset of descriptors in the descriptor table is the same as the 16-bit selector, provided the lower three bits are zeroed. The descriptor table entry defines the real linear address of the segment, a limit value for the segment size, and some attribute bits (flags).
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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{\displaystyle x|(y\mid (x\mid z))=((z\mid y)\mid y)\mid x.} In 1973, Padmanabhan and Quackenbush demonstrated a method that, in principle, would yield a 1-basis for Boolean algebra.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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In mathematics, the rational sieve is a general algorithm for factoring integers into prime factors. It is a special case of the general number field sieve. While it is less efficient than the general algorithm, it is conceptually simpler. It serves as a helpful first step in understanding how the general number field sieve works.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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In mathematics, numerical analysis, and numerical partial differential equations, domain decomposition methods solve a boundary value problem by splitting it into smaller boundary value problems on subdomains and iterating to coordinate the solution between adjacent subdomains. A coarse problem with one or few unknowns per subdomain is used to further coordinate the solution between the subdomains globally. The problems on the subdomains are independent, which makes domain decomposition methods suitable for parallel computing. Domain decomposition methods are typically used as preconditioners for Krylov space iterative methods, such as the conjugate gradient method, GMRES, and LOBPCG.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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In neural networks, each neuron receives input from some number of locations in the previous layer. In a convolutional layer, each neuron receives input from only a restricted area of the previous layer called the neuron's receptive field. Typically the area is a square (e.g. 5 by 5 neurons). Whereas, in a fully connected layer, the receptive field is the entire previous layer.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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According to Kunegis, Blattner, and Moser several online networks follow a non-linear preferential attachment model. Communication networks and online contact networks are sub-linear while interaction networks are super-linear. The co-author network among scientists also shows the signs of sub-linear preferential attachment.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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In social systems, deterministic chaos is infrequent, because the elements of the system include individuals whose values, awareness, will, foresight, and fallibility, affect the dynamic behavior of the system. However, this does not completely exclude any notional possibility of deterministic chaos in social systems. In fact some authorities argue an increase in the development of nonlinear dynamics and instabilities of social systems.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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The classification efficiency is usually indicated by Receiver operating characteristics. In the original SIMCA method, the ends of the hyper-plane of each class are closed off by setting statistical control limits along the retained principal components axes (i.e., score value between plus and minus 0.5 times score standard deviation). More recent adaptations of the SIMCA method close off the hyper-plane by construction of ellipsoids (e.g. Hotelling's T2 or Mahalanobis distance). With such modified SIMCA methods, classification of an object requires both that its orthogonal distance from the model and its projection within the model (i.e. score value within the region defined by the ellipsoid) are not significant.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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This is not to say that all software data ought to be manipulated as graphs, but rather that they can be exchanged as graphs. It can be used to represent instance data as well as schemas for describing the structure of the data. Moreover, the schema can be explicitly stated along with instance data.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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In the Hindley–Milner type system, expressions can be given multiple types through parametric polymorphism. But naively giving multiple types to references breaks type safety. The following are typing rules for references and related operators in ML-like languages. r e f: ∀ α .
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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In markup languages and the digital humanities, overlap occurs when a document has two or more structures that interact in a non-hierarchical manner. A document with overlapping markup cannot be represented as a tree. This is also known as concurrent markup. Overlap happens, for instance, in poetry, where there may be a metrical structure of feet and lines; a linguistic structure of sentences and quotations; and a physical structure of volumes and pages and editorial annotations.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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In number theory, a kth root of unity modulo n for positive integers k, n ≥ 2, is a root of unity in the ring of integers modulo n; that is, a solution x to the equation (or congruence) x k ≡ 1 ( mod n ) {\displaystyle x^{k}\equiv 1{\pmod {n}}} . If k is the smallest such exponent for x, then x is called a primitive kth root of unity modulo n. See modular arithmetic for notation and terminology. The roots of unity modulo n are exactly the integers that are coprime with n. In fact, these integers are roots of unity modulo n by Euler's theorem, and the other integers cannot be roots of unity modulo n, because they are zero divisors modulo n. A primitive root modulo n, is a generator of the group of units of the ring of integers modulo n. There exist primitive roots modulo n if and only if λ ( n ) = φ ( n ) , {\displaystyle \lambda (n)=\varphi (n),} where λ {\displaystyle \lambda } and φ {\displaystyle \varphi } are respectively the Carmichael function and Euler's totient function. A root of unity modulo n is a primitive kth root of unity modulo n for some divisor k of λ ( n ) , {\displaystyle \lambda (n),} and, conversely, there are primitive kth roots of unity modulo n if and only if k is a divisor of λ ( n ) . {\displaystyle \lambda (n).}
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In number theory, a Wieferich prime is a prime number p such that p2 divides 2p − 1 − 1, therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides 2p − 1 − 1. Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's Last Theorem, at which time both of Fermat's theorems were already well known to mathematicians.Since then, connections between Wieferich primes and various other topics in mathematics have been discovered, including other types of numbers and primes, such as Mersenne and Fermat numbers, specific types of pseudoprimes and some types of numbers generalized from the original definition of a Wieferich prime. Over time, those connections discovered have extended to cover more properties of certain prime numbers as well as more general subjects such as number fields and the abc conjecture. As of April 2023, the only known Wieferich primes are 1093 and 3511 (sequence A001220 in the OEIS).
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Very often, and in this article, the coefficients of the equations are real or complex numbers and the solutions are searched in the same set of numbers, but the theory and the algorithms apply for coefficients and solutions in any field. For solutions in an integral domain like the ring of the integers, or in other algebraic structures, other theories have been developed, see Linear equation over a ring. Integer linear programming is a collection of methods for finding the "best" integer solution (when there are many). Gröbner basis theory provides algorithms when coefficients and unknowns are polynomials. Also tropical geometry is an example of linear algebra in a more exotic structure.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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In simple terms, software verification is: "Assuming we should build X, does our software achieve its goals without any bugs or gaps?" On the other hand, software validation is: "Was X what we should have built? Does X meet the high-level requirements?"
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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A bijective proof. Two sets are shown to have the same number of members by exhibiting a bijection, i.e. a one-to-one correspondence, between them.The term "combinatorial proof" may also be used more broadly to refer to any kind of elementary proof in combinatorics. However, as Glass (2003) writes in his review of Benjamin & Quinn (2003) (a book about combinatorial proofs), these two simple techniques are enough to prove many theorems in combinatorics and number theory.
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Supercritical p > ( 1 + ε ) / n {\displaystyle p>(1+\varepsilon )/n} There is a single giant component containing a linear number of vertices. For large values of p {\displaystyle p} its size approaches the whole graph: | C 1 | ≈ y n {\displaystyle |C_{1}|\approx yn} where y {\displaystyle y} is the positive solution to the equation e − p n y = 1 − y {\displaystyle e^{-pny}=1-y} . The remaining components are small, with logarithmic size.In the same model of random graphs, there will exist multiple connected components with high probability for values of p {\displaystyle p} below a significantly higher threshold, p < ( 1 − ε ) ( log n ) / n {\displaystyle p<(1-\varepsilon )(\log n)/n} , and a single connected component for values above the threshold, p > ( 1 + ε ) ( log n ) / n {\displaystyle p>(1+\varepsilon )(\log n)/n} .
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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For example, if x 5 {\displaystyle x_{5}} is non-basic and its coefficient in r {\displaystyle r} is positive, then increasing it above 0 may make z {\displaystyle z} larger. If it is possible to do so without violating other constraints, then the increased variable becomes basic (it "enters the basis"), while some basic variable is decreased to 0 to keep the equality constraints and thus becomes non-basic (it "exits the basis"). If this process is done carefully, then it is possible to guarantee that z {\displaystyle z} increases until it reaches an optimal BFS.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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The concept is not unlike the limited licensing approach for computer software, which places rigid restrictions on resale and reproduction. The intent is to make users understand that the content of any textbook is the intellectual property of the author and/or the publisher, and that as such, subject to copyright. Obviously, this idea is completely opposed to the millennia-old tradition of the sale of used books, and would make that entire industry illegal.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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{\displaystyle M^{\text{T}}\Omega M=\Omega .} Under a change of basis, represented by a matrix A, we have Ω ↦ A T Ω A {\displaystyle \Omega \mapsto A^{\text{T}}\Omega A} M ↦ A − 1 M A . {\displaystyle M\mapsto A^{-1}MA.} One can always bring Ω {\displaystyle \Omega } to either the standard form given in the introduction or the block diagonal form described below by a suitable choice of A.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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In personality pathology, dimensional models of personality disorders (also known as the dimensional approach to personality disorders, dimensional classification, and dimensional assessments) conceptualize personality disorders as quantitatively rather than qualitatively different from normal personality. They consist of extreme, maladaptive levels of certain personality characteristics (these characteristics are commonly described as facets within broader personality factors or traits). Within the context of personality psychology, a "dimension" refers to a continuum on which an individual can have various levels of a characteristic, in contrast to the dichotomous categorical approach in which an individual does or does not possess a characteristic.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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In mathematics, a map or mapping is a function in its general sense. These terms may have originated as from the process of making a geographical map: mapping the Earth surface to a sheet of paper.The term map may be used to distinguish some special types of functions, such as homomorphisms. For example, a linear map is a homomorphism of vector spaces, while the term linear function may have this meaning or it may mean a linear polynomial.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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In mathematics, Euclid numbers are integers of the form En = pn # + 1, where pn # is the nth primorial, i.e. the product of the first n prime numbers. They are named after the ancient Greek mathematician Euclid, in connection with Euclid's theorem that there are infinitely many prime numbers.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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In the Japanese encodings ISO 646-JP (a 7-bit code based on ASCII), JIS X 0201 (an 8-bit code), and Shift JIS (a multi-byte encoding which is 8-bit for ASCII), the code point 0x5C that would be used for backslash in ASCII is instead rendered as a yen sign ¥. Due to extensive use of the 005C code point to represent the yen sign, even today some fonts such as MS Mincho render the backslash character as a ¥, so the characters at Unicode code points 00A5 (¥) and 005C (\) both render as ¥ when these fonts are selected. Computer programs still treat 005C as a backslash in these environments but display it as a yen sign, causing confusion, especially in MS-DOS filenames.Several other ISO 646 versions also replace backslash with other characters, including ₩ (Korean), Ö (German, Swedish), Ø (Danish, Norwegian), ç (French) and Ñ (Spanish), leading to similar problems, though with less lasting impact compared to the yen sign. In 1991, RFC 1345 suggested // as a unique two-character mnemonic that might be used in internet standards as "a practical way of identifying character, without reference to a coded character set and its code in coded character set". Consequently, this style may be seen in early Internet Engineering Task Force documents.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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In most computer programming languages a do while loop is a control flow statement that executes a block of code and then either repeats the block or exits the loop depending on a given boolean condition. The do while construct consists of a process symbol and a condition. First the code within the block is executed. Then the condition is evaluated.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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In the 1980s, the Graphical Kernel System (GKS) library, based on a 1970s specification with a similar basic geometry and command structure to NAPLPS, was widely implemented on microcomputers, and became the basis of Digital Research's GSX graphics system used in their GEM GUI. GKS was later extended into a 3D version, and additions to this resulted in PHIGS (Programmer's Hierarchical Interactive Graphics System), a competitor to OpenGL.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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As a result, it is often considered to be a more intuitive, but a less systematic approach to divisions – where the efficiency is highly dependent upon one's numeracy skills. To calculate the whole number quotient of dividing a large number by a small number, the student repeatedly takes away "chunks" of the large number, where each "chunk" is an easy multiple (for example 100×, 10×, 5× 2×, etc.) of the small number, until the large number has been reduced to zero – or the remainder is less than the small number itself.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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And symmetrically (when Xk is a sub-martingale): P ( X N − X 0 ≤ − ϵ ) ≤ exp ( − ϵ 2 2 ∑ k = 1 N c k 2 ) . {\displaystyle {\text{P}}(X_{N}-X_{0}\leq -\epsilon )\leq \exp \left({-\epsilon ^{2} \over 2\sum _{k=1}^{N}c_{k}^{2}}\right).} If X is a martingale, using both inequalities above and applying the union bound allows one to obtain a two-sided bound: P ( | X N − X 0 | ≥ ϵ ) ≤ 2 exp ( − ϵ 2 2 ∑ k = 1 N c k 2 ) . {\displaystyle {\text{P}}(|X_{N}-X_{0}|\geq \epsilon )\leq 2\exp \left({-\epsilon ^{2} \over 2\sum _{k=1}^{N}c_{k}^{2}}\right).}
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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As in discrete percolation, a common research focus of continuum percolation is studying the conditions of occurrence for infinite or giant components. Other shared concepts and analysis techniques exist in these two types of percolation theory as well as the study of random graphs and random geometric graphs. Continuum percolation arose from an early mathematical model for wireless networks, which, with the rise of several wireless network technologies in recent years, has been generalized and studied in order to determine the theoretical bounds of information capacity and performance in wireless networks. In addition to this setting, continuum percolation has gained application in other disciplines including biology, geology, and physics, such as the study of porous material and semiconductors, while becoming a subject of mathematical interest in its own right.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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In mathematics, Light's associativity test is a procedure invented by F. W. Light for testing whether a binary operation defined in a finite set by a Cayley multiplication table is associative. The naive procedure for verification of the associativity of a binary operation specified by a Cayley table, which compares the two products that can be formed from each triple of elements, is cumbersome. Light's associativity test simplifies the task in some instances (although it does not improve the worst-case runtime of the naive algorithm, namely O ( n 3 ) {\displaystyle {\mathcal {O}}\left(n^{3}\right)} for sets of size n {\displaystyle n} ).
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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For example, the proton spin puzzle, the EMC effect, the distributions of electric charges inside the nucleons, as found by Hofstadter in 1956, and the ad hoc CKM matrix elements. When the term "preon" was coined, it was primarily to explain the two families of spin-1/2 fermions: quarks and leptons. More recent preon models also account for spin-1 bosons, and are still called "preons". Each of the preon models postulates a set of fewer fundamental particles than those of the Standard Model, together with the rules governing how those fundamental particles combine and interact. Based on these rules, the preon models try to explain the Standard Model, often predicting small discrepancies with this model and generating new particles and certain phenomena which do not belong to the Standard Model.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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In the 20th century, following the development of formal logic, the ampersand became a commonly used logical notation for the binary operator or sentential connective AND. This usage was adopted in computing. Many languages with syntax derived from C, including C++, Perl, and more differentiate between: & for bitwise AND. (4 & 2) is zero, (4 & 5) is 4.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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In the US, the CDC recommends essential components of AMS programs (ASP) for acute care hospitals, small and critical access hospitals, resource-limited facilities, long-term care facilities, and outpatient facilities.As of 2014, thirteen internet-based institutional ASP resources in US academic medical centers had been published. An ASP has the following tasks, in line with quality improvement theory:
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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An arrow from the node representing a candidate X to the one representing a candidate Y is labelled with d. To avoid cluttering the diagram, an arrow has only been drawn from X to Y when d > d (i.e. the table cells with light green background), omitting the one in the opposite direction (the table cells with light red background). One example of computing the strongest path strength is p = 33: the strongest path from B to D is the direct path (B, D) which has strength 33. But when computing p, the strongest path from A to C is not the direct path (A, C) of strength 26, rather the strongest path is the indirect path (A, D, C) which has strength min(30, 28) = 28.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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In mathematical logic, a formal theory is a set of sentences expressed in a formal language. A formal system (also called a logical calculus, or a logical system) consists of a formal language together with a deductive apparatus (also called a deductive system). The deductive apparatus may consist of a set of transformation rules, which may be interpreted as valid rules of inference, or a set of axioms, or have both.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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These directories contain files with names such as "ABCD1234.JPG" that consist of four alphanumeric characters (often "100_", "DSC0", "DSCF", "IMG_", "MOV_", or "P000"), followed by a number. Handling of directories with possibly user-created duplicate numbers may vary among camera firmwares. DCF 2.0 adds support for DCF optional files recorded in an optional color space (that is, Adobe RGB rather than sRGB). Such files must be indicated by a leading "_" (as in "_DSC" instead of "100_" or "DSC0").
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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In structure mining, a graph kernel is a kernel function that computes an inner product on graphs. Graph kernels can be intuitively understood as functions measuring the similarity of pairs of graphs. They allow kernelized learning algorithms such as support vector machines to work directly on graphs, without having to do feature extraction to transform them to fixed-length, real-valued feature vectors. They find applications in bioinformatics, in chemoinformatics (as a type of molecule kernels), and in social network analysis.Concepts of graph kernels have been around since the 1999, when D. Haussler introduced convolutional kernels on discrete structures.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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In statistical analysis of binary classification, the F-score or F-measure is a measure of a test's accuracy. It is calculated from the precision and recall of the test, where the precision is the number of true positive results divided by the number of all positive results, including those not identified correctly, and the recall is the number of true positive results divided by the number of all samples that should have been identified as positive. Precision is also known as positive predictive value, and recall is also known as sensitivity in diagnostic binary classification.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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In order to measure the information of a string relative to another there is the need to rely on relative semi-distances (NRC). These are measures that do not need to respect symmetry and triangle inequality distance properties. Although the NCD and the NRC seem very similar, they address different questions. The NCD measures how similar both strings are, mostly using the information content, while the NRC indicates the fraction of a target string that cannot be constructed using information from another string. For a comparison, with application to the evolution of primate genomes, see.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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According to the Curry–Howard isomorphism, lambda calculus on its own can express theorems in intuitionistic logic only, and several classical logical theorems can't be written at all. However with these new operators one is able to write terms that have the type of, for example, Peirce's law. Semantically these operators correspond to continuations, found in some functional programming languages.
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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In mathematics, a binary relation associates elements of one set, called the domain, with elements of another set, called the codomain. A binary relation over sets X and Y is a new set of ordered pairs (x, y) consisting of elements x in X and y in Y. It is a generalization of the more widely understood idea of a unary function. It encodes the common concept of relation: an element x is related to an element y, if and only if the pair (x, y) belongs to the set of ordered pairs that defines the binary relation. A binary relation is the most studied special case n = 2 of an n-ary relation over sets X1, ..., Xn, which is a subset of the Cartesian product X 1 × ⋯ × X n .
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Unlike Schoof's algorithm, the SEA algorithm is typically implemented as a probabilistic algorithm (of the Las Vegas type), so that root-finding and other operations can be performed more efficiently. Its computational complexity is dominated by the cost of computing the modular polynomials Ψ ℓ ( X , Y ) {\displaystyle \Psi _{\ell }(X,Y)} , but as these do not depend on E {\displaystyle E} , they may be computed once and reused. Under the heuristic assumption that there are sufficiently many small Elkies primes, and excluding the cost of computing modular polynomials, the asymptotic running time of the SEA algorithm is O ( n 2 M ( n 2 ) / log n ) = O ( n 4 + o ( 1 ) ) {\displaystyle O(n^{2}M(n^{2})/\log {n})=O(n^{4+o(1)})} , where n = log q {\displaystyle n=\log {q}} . Its space complexity is O ( n 3 log n ) {\displaystyle O(n^{3}\log {n})} , but when precomputed modular polynomials are used this increases to O ( n 4 ) {\displaystyle O(n^{4})} .
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https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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