In the field of linguistics, specifically in syntax, phonetic form (PF), also known as phonological form or the articulatory-perceptual (A-P) system, is a certain level of mental representation of a linguistic expression, derived from surface structure, and related to Logical Form. Phonetic form is the level of representation wherein expressions, or sentences, are assigned a phonetic representation, which is then pronounced by the speaker. Phonetic form takes surface structure as its input, and outputs an audible (or visual, in the case of sign languages), pronounced sentence.This is part of the Y- or T-model of grammar within minimalist grammar, wherein the syntactic structure is constructed and then transferred (called spell-out) to both the Phonetic Form and the Logical Form. Operations in this branch of the model (between spell-out and pronunciation), the syntax-phonology interface, affect the pronunciation of the utterance but not its meaning.


In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also sometimes called the double exponential distribution, because it can be thought of as two exponential distributions (with an additional location parameter) spliced together along the abscissa, although the term is also sometimes used to refer to the Gumbel distribution. The difference between two independent identically distributed exponential random variables is governed by a Laplace distribution, as is a Brownian motion evaluated at an exponentially distributed random time. Increments of Laplace motion or a variance gamma process evaluated over the time scale also have a Laplace distribution.


In the following, Exponentiation stands for repeated application of the group operation Juxtaposition stands for multiplication on the set of congruence classes or application of the group operation (as applicable) Subtraction stands for subtraction on the set of congruence classes M ∈ { 0 , 1 } ∗ {\displaystyle M\in \{0,1\}^{*}} , the set of finite bit strings s , e , e v ∈ Z q {\displaystyle s,e,e_{v}\in \mathbb {Z} _{q}} , the set of congruence classes modulo q {\displaystyle q} x , k ∈ Z q × {\displaystyle x,k\in \mathbb {Z} _{q}^{\times }} , the multiplicative group of integers modulo q {\displaystyle q} (for prime q {\displaystyle q} , Z q × = Z q ∖ 0 ¯ q {\displaystyle \mathbb {Z} _{q}^{\times }=\mathbb {Z} _{q}\setminus {\overline {0}}_{q}} ) y , r , r v ∈ G {\displaystyle y,r,r_{v}\in G} .


In software development, a leaky abstraction is an abstraction that leaks details that it is supposed to abstract away.As coined by Joel Spolsky, the Law of Leaky Abstractions states: All non-trivial abstractions, to some degree, are leaky. This statement highlights a particularly problematic cause of software defects: the reliance of the software developer on an abstraction's infallibility. Spolsky's article gives examples of an abstraction that works most of the time, but where a detail of the underlying complexity cannot be ignored, thus leaking complexity out of the abstraction back into the software that uses the abstraction.


In mathematics, least squares function approximation applies the principle of least squares to function approximation, by means of a weighted sum of other functions. The best approximation can be defined as that which minimizes the difference between the original function and the approximation; for a least-squares approach the quality of the approximation is measured in terms of the squared differences between the two.


In number theory, a pseudoprime is called an elliptic pseudoprime for (E, P), where E is an elliptic curve defined over the field of rational numbers with complex multiplication by an order in Q ( − d ) {\displaystyle \mathbb {Q} {\big (}{\sqrt {-d}}{\big )}} , having equation y2 = x3 + ax + b with a, b integers, P being a point on E and n a natural number such that the Jacobi symbol (−d | n) = −1, if (n + 1)P ≡ 0 (mod n). The number of elliptic pseudoprimes less than X is bounded above, for large X, by X / exp  ( ( 1 / 3 ) log  X log  log  log  X / log  log  X ) . {\displaystyle X/\exp((1/3)\log X\log \log \log X/\log \log X)\ .}


In telecommunication, a Hagelbarger code is a convolutional code that enables error bursts to be corrected provided that there are relatively long error-free intervals between the error bursts. In the Hagelbarger code, inserted parity check bits are spread out in time so that an error burst is not likely to affect more than one of the groups in which parity is checked.


In ring theory, a branch of mathematics, an idempotent element or simply idempotent of a ring is an element a such that a2 = a. That is, the element is idempotent under the ring's multiplication. Inductively then, one can also conclude that a = a2 = a3 = a4 = ... = an for any positive integer n. For example, an idempotent element of a matrix ring is precisely an idempotent matrix. For general rings, elements idempotent under multiplication are involved in decompositions of modules, and connected to homological properties of the ring. In Boolean algebra, the main objects of study are rings in which all elements are idempotent under both addition and multiplication.


In order to verify the monotonicity property, we set E 1 = A {\displaystyle E_{1}=A} and E 2 = B ∖ A {\displaystyle E_{2}=B\setminus A} , where A ⊆ B {\displaystyle A\subseteq B} and E i = ∅ {\displaystyle E_{i}=\varnothing } for i ≥ 3 {\displaystyle i\geq 3} . From the properties of the empty set ( ∅ {\displaystyle \varnothing } ), it is easy to see that the sets E i {\displaystyle E_{i}} are pairwise disjoint and E 1 ∪ E 2 ∪ ⋯ = B {\displaystyle E_{1}\cup E_{2}\cup \cdots =B} . Hence, we obtain from the third axiom that P ( A ) + P ( B ∖ A ) + ∑ i = 3 ∞ P ( E i ) = P ( B ) . {\displaystyle P(A)+P(B\setminus A)+\sum _{i=3}^{\infty }P(E_{i})=P(B).} Since, by the first axiom, the left-hand side of this equation is a series of non-negative numbers, and since it converges to P ( B ) {\displaystyle P(B)} which is finite, we obtain both P ( A ) ≤ P ( B ) {\displaystyle P(A)\leq P(B)} and P ( ∅ ) = 0 {\displaystyle P(\varnothing )=0} .


In number theory, the Fermat–Catalan conjecture is a generalization of Fermat's Last Theorem and of Catalan's conjecture, hence the name. The conjecture states that the equation has only finitely many solutions (a,b,c,m,n,k) with distinct triplets of values (am, bn, ck) where a, b, c are positive coprime integers and m, n, k are positive integers satisfying The inequality on m, n, and k is a necessary part of the conjecture. Without the inequality there would be infinitely many solutions, for instance with k = 1 (for any a, b, m, and n and with c = am + bn) or with m, n, and k all equal to two (for the infinitely many known Pythagorean triples).


This is, in particular, the case in graph theory, of incidence matrices, and adjacency matrices. This article focuses on matrices related to linear algebra, and, unless otherwise specified, all matrices represent linear maps or may be viewed as such. Square matrices, matrices with the same number of rows and columns, play a major role in matrix theory.


In queueing theory, a discipline within the mathematical theory of probability, Kendall's notation (or sometimes Kendall notation) is the standard system used to describe and classify a queueing node. D. G. Kendall proposed describing queueing models using three factors written A/S/c in 1953 where A denotes the time between arrivals to the queue, S the service time distribution and c the number of service channels open at the node. It has since been extended to A/S/c/K/N/D where K is the capacity of the queue, N is the size of the population of jobs to be served, and D is the queueing discipline.When the final three parameters are not specified (e.g. M/M/1 queue), it is assumed K = ∞, N = ∞ and D = FIFO.


An 8086 system, including coprocessors such as 8087 and 8089, and simpler Intel-specific system chips, was thereby described as an iAPX 86 system. There were also terms iRMX (for operating systems), iSBC (for single-board computers), and iSBX (for multimodule boards based on the 8086-architecture), all together under the heading Microsystem 80.


Thus f0 is the truth value of ψ. In order to arithmetize ψ we must use the following rules: f i ( a 1 , … , a i ) = { f i + 1 ( a 1 , … , a i , 0 ) ⋅ f i + 1 ( a 1 , … , a i , 1 ) Q i + 1 = ∀ f i + 1 ( a 1 , … , a i , 0 ) ∗ f i + 1 ( a 1 , … , a i , 1 ) Q i + 1 = ∃ {\displaystyle f_{i}(a_{1},\dots ,a_{i})={\begin{cases}f_{i+1}(a_{1},\dots ,a_{i},0)\cdot f_{i+1}(a_{1},\dots ,a_{i},1)&{\mathsf {Q}}_{i+1}=\forall \\f_{i+1}(a_{1},\dots ,a_{i},0)*f_{i+1}(a_{1},\dots ,a_{i},1)&{\mathsf {Q}}_{i+1}=\exists \end{cases}}} where as before we define x ∗ y = 1 − (1 − x)(1 − y). By using the method described in #SAT, we must face a problem that for any fi the degree of the resulting polynomial may double with each quantifier. In order to prevent this, we must introduce a new reduction operator R which will reduce the degrees of the polynomial without changing their behavior on Boolean inputs.


In public key infrastructure (PKI) systems, a certificate signing request (also CSR or certification request) is a message sent from an applicant to a certificate authority of the public key infrastructure in order to apply for a digital identity certificate. The CSR usually contains the public key for which the certificate should be issued, identifying information (such as a domain name) and a proof of authenticity including integrity protection (e.g., a digital signature). The most common format for CSRs is the PKCS #10 specification; others include the more capable CRMF and the Signed Public Key and Challenge SPKAC format generated by some web browsers.


In order to prevent such elaborate attacks, different modes of operation were introduced: tweakable narrow-block encryption (LRW and XEX) and wide-block encryption (CMC and EME). Whereas a purpose of a usual block cipher E K {\displaystyle E_{K}} is to mimic a random permutation for any secret key K {\displaystyle K} , the purpose of tweakable encryption E K T {\displaystyle E_{K}^{T}} is to mimic a random permutation for any secret key K {\displaystyle K} and any known tweak T {\displaystyle T} . The tweakable narrow-block encryption (LRW) is an instantiation of the mode of operations introduced by Liskov, Rivest, and Wagner (see Theorem 2). This mode uses two keys: K {\displaystyle K} is the key for the block cipher and F {\displaystyle F} is an additional key of the same size as block.


In the book Secretum de thesauro experimentorum ymaginationis hominum (Secret of the treasure-room of experiments in man's imagination), written ca. 1430, Fontana described mnemonic machines, written in his cypher. At least Bellicorum instrumentorum liber and this book used a cryptographic system, described as a simple, rational cipher, based on signs without letters or numbers. It has been suggested that some illustrations slightly resemble Voynich illustrations. During this time he also met with the Count of Carmagnola.


Using the Chinese remainder theorem, it suffices to evaluate f {\displaystyle f} modulo different primes p 1 , … , p ℓ {\displaystyle p_{1},\dots ,p_{\ell }} with a product at least M {\displaystyle M} . Each prime can be taken to be roughly log  M = O ( d m log  q ) {\displaystyle \log M=O(dm\log q)} , and the number of primes needed, ℓ {\displaystyle \ell } , is roughly the same. Doing this process recursively, we can get the primes as small as log  log  q {\displaystyle \log \log q} .


An example 3×3 polynomial matrix, degree 2: P = ( 1 x 2 x 0 2 x 2 3 x + 2 x 2 − 1 0 ) = ( 1 0 0 0 0 2 2 − 1 0 ) + ( 0 0 1 0 2 0 3 0 0 ) x + ( 0 1 0 0 0 0 0 1 0 ) x 2 . {\displaystyle P={\begin{pmatrix}1&x^{2}&x\\0&2x&2\\3x+2&x^{2}-1&0\end{pmatrix}}={\begin{pmatrix}1&0&0\\0&0&2\\2&-1&0\end{pmatrix}}+{\begin{pmatrix}0&0&1\\0&2&0\\3&0&0\end{pmatrix}}x+{\begin{pmatrix}0&1&0\\0&0&0\\0&1&0\end{pmatrix}}x^{2}.} We can express this by saying that for a ring R, the rings M n ( R  ) {\displaystyle M_{n}(R)} and ( M n ( R ) )  {\displaystyle (M_{n}(R))} are isomorphic.


In C#, reification is used to make parametric polymorphism implemented in the form of generics as a first-class feature of the language. In the Java programming language, there exist "reifiable types" that are "completely available at run time" (i.e. their information is not erased during compilation).


In the early years of the 21st century class browsers began to morph into modeling tools, where programmers could not only visualize their class hierarchy as a diagram, but also add classes to their code by adding them to the diagram. Most of these visualization systems have been based on some form of the Unified Modeling Language (UML).


In mathematical optimization, the active-set method is an algorithm used to identify the active constraints in a set of inequality constraints. The active constraints are then expressed as equality constraints, thereby transforming an inequality-constrained problem into a simpler equality-constrained subproblem. An optimization problem is defined using an objective function to minimize or maximize, and a set of constraints g 1 ( x ) ≥ 0 , … , g k ( x ) ≥ 0 {\displaystyle g_{1}(x)\geq 0,\dots ,g_{k}(x)\geq 0} that define the feasible region, that is, the set of all x to search for the optimal solution. Given a point x {\displaystyle x} in the feasible region, a constraint g i ( x ) ≥ 0 {\displaystyle g_{i}(x)\geq 0} is called active at x 0 {\displaystyle x_{0}} if g i ( x 0 ) = 0 {\displaystyle g_{i}(x_{0})=0} , and inactive at x {\displaystyle x} if g i ( x 0 ) > 0.


The numbers that can be represented with four bits are shown in the comparison table below. The range of numbers that can be represented is asymmetric. If the word has an even number of bits, the magnitude of the largest negative number that can be represented is twice as large as the largest positive number that can be represented, and vice versa if the word has an odd number of bits.


Then, in general, the average number at distance d {\displaystyle d} can be written as: c d = ( c 2 c 1 ) d − 1 c 1 . {\displaystyle c_{d}=\left({\frac {c_{2}}{c_{1}}}\right)^{d-1}c_{1}.} Which implies that if the ratio of c 2 c 1 {\displaystyle {\frac {c_{2}}{c_{1}}}} is larger than one, then the network can have a giant component.


The strength of a path is the strength of its weakest link. For each pair of candidates X and Y, the following table shows the strongest path from candidate X to candidate Y in red, with the weakest link underlined. Now the output of the Schulze method can be determined. For example, when comparing A and B, since ( 28 = ) p  > p  ( = 25 ) {\displaystyle (28=)p>p(=25)} , for the Schulze method candidate A is better than candidate B. Another example is that ( 31 = ) p  > p  ( = 24 ) {\displaystyle (31=)p>p(=24)} , so candidate E is better than candidate D. Continuing in this way, the result is that the Schulze ranking is E > A > C > B > D {\displaystyle E>A>C>B>D} , and E wins. In other words, E wins since p  ≥ p  {\displaystyle p\geq p} for every other candidate X.


In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod p when p is a prime number.


Bob now knows that the state of his qubit was one of the two states indicated by Alice. To determine the secret bit, Bob must distinguish between the two candidate states. For each qubit, Bob can check to see whether his measurement is consistent with either possible state.


In public policy, authorization is a feature of trusted systems used for security or social control.


Amateur radio requires the call sign to be stated at the end of a communication and every ten minutes during (some hams use countdown clocks to remind them to identify); modes such as packet radio and fast-scan television often have a provision for automatic identification, either including it as part of a digital data stream or overlaying it over an analog picture. Repeaters are often designed to automatically transmit the repeater's callsign, usually in Morse code.


In numerical analysis, catastrophic cancellation is the phenomenon that subtracting good approximations to two nearby numbers may yield a very bad approximation to the difference of the original numbers. For example, if there are two studs, one L 1 = 253.5 cm {\displaystyle L_{1}=253.5\,{\text{cm}}} long and the other L 2 = 252.5 cm {\displaystyle L_{2}=252.5\,{\text{cm}}} long, and they are measured with a ruler that is good only to the centimeter, then the approximations could come out to be L ~ 1 = 254 cm {\displaystyle {\tilde {L}}_{1}=254\,{\text{cm}}} and L ~ 2 = 252 cm {\displaystyle {\tilde {L}}_{2}=252\,{\text{cm}}} . These may be good approximations, in relative error, to the true lengths: the approximations are in error by less than 2% of the true lengths, | L 1 − L ~ 1 | / | L 1 | < 2 % {\displaystyle |L_{1}-{\tilde {L}}_{1}|/|L_{1}|<2\%} . However, if the approximate lengths are subtracted, the difference will be L ~ 1 − L ~ 2 = 254 cm − 252 cm = 2 cm {\displaystyle {\tilde {L}}_{1}-{\tilde {L}}_{2}=254\,{\text{cm}}-252\,{\text{cm}}=2\,{\text{cm}}} , even though the true difference between the lengths is L 1 − L 2 = 253.5 cm − 252.5 cm = 1 cm {\displaystyle L_{1}-L_{2}=253.5\,{\text{cm}}-252.5\,{\text{cm}}=1\,{\text{cm}}} .


In this way Fermat was able to show the non-existence of solutions in many cases of Diophantine equations of classical interest (for example, the problem of four perfect squares in arithmetic progression). In some cases, to the modern eye, his "method of infinite descent" is an exploitation of the inversion of the doubling function for rational points on an elliptic curve E. The context is of a hypothetical non-trivial rational point on E. Doubling a point on E roughly doubles the length of the numbers required to write it (as number of digits), so that a "halving" a point gives a rational with smaller terms. Since the terms are positive, they cannot decrease forever.


In mathematics, a differential field K is differentially closed if every finite system of differential equations with a solution in some differential field extending K already has a solution in K. This concept was introduced by Robinson (1959). Differentially closed fields are the analogues for differential equations of algebraically closed fields for polynomial equations.


Assuming that the data were actually generated by the model in question, they try to design algorithms that probably find the model that was used to create the data. Techniques used here include singular value decomposition (SVD) and the method of moments. In 2012 an algorithm based upon non-negative matrix factorization (NMF) was introduced that also generalizes to topic models with correlations among topics.In 2018 a new approach to topic models was proposed: it is based on stochastic block model


For example, a required bit rate, delay, delay variation, packet loss or bit error rates may be guaranteed. Quality of service is important for real-time streaming multimedia applications such as voice over IP, multiplayer online games and IPTV, since these often require fixed bit rate and are delay sensitive. Quality of service is especially important in networks where the capacity is a limited resource, for example in cellular data communication.


indegree (followers count) retweet count mention countAn initial analysis of the three aforementioned metrics showed that the users with the highest indegrees and the users with the highest retweet/mention counts were not the same. The top 1% of users by indegree are shown to have very low correlation with the same percentile of users by retweets and by mentions. This implies that follower count is not useful in determining whether a user's tweets get retweeted or whether the other users engage with them.


In principle, just what constitutes "structure" vs. non-structure can vary. In a book specifically about typography, tagging something as "italic" or "bold" may well be the whole point. For example, a discussion of when to use particular styles will likely want to give examples and counter-examples, which would no longer make sense if the rendering is not in sync with the prose. Similarly, a particular edition of a document may be of interest not only for its content but for its typographic practice as well, in which case describing that practice is not only desirable but necessary. This problem is not unique to document structure, however; it also arises in grammar when discussing grammar, and in many other cases.


In practice, the optimal number of populations or topics is not known beforehand. It can be estimated by approximation of the posterior distribution with reversible-jump Markov chain Monte Carlo.


A problem that arose in the 16th century was creating a general formula using radicals to express the solution of any polynomial equation of fixed degree k, where k ≥ 5. In the 1820s, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) showed this to be impossible, using concepts such as solvable groups from Galois theory—a new sub-field of abstract algebra. Some of the most important proofs of impossibility found in the 20th century were those related to undecidability, which showed that there are problems that cannot be solved in general by any algorithm, with one of the more prominent ones being the halting problem.


The baseline for this concept was put forward in the late 1940s, and the third era of privacy development began in the 1990s.The European Union has various privacy laws that dictate how information may be collected and used by companies. Some of those laws are written to give agency to the preferences of individuals/consumers in how their data is used.


In number theory, a Wolstenholme prime is a special type of prime number satisfying a stronger version of Wolstenholme's theorem. Wolstenholme's theorem is a congruence relation satisfied by all prime numbers greater than 3. Wolstenholme primes are named after mathematician Joseph Wolstenholme, who first described this theorem in the 19th century.


In mathematics, specifically in group theory, the direct product is an operation that takes two groups G and H and constructs a new group, usually denoted G × H. This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics. In the context of abelian groups, the direct product is sometimes referred to as the direct sum, and is denoted G ⊕ H {\displaystyle G\oplus H} . Direct sums play an important role in the classification of abelian groups: according to the fundamental theorem of finite abelian groups, every finite abelian group can be expressed as the direct sum of cyclic groups.


In the following summations, n P k {\displaystyle {}_{n}P_{k}} is the number of k-permutations of n. ∑ i = 0 n i P k ( n i ) = n P k ( 2 n − k ) {\displaystyle \sum _{i=0}^{n}{}_{i}P_{k}{n \choose i}={}_{n}P_{k}(2^{n-k})} ∑ i = 1 n i + k P k + 1 = ∑ i = 1 n ∏ j = 0 k ( i + j ) = ( n + k + 1 ) ! ( n − 1 ) ! ( k + 2 ) {\displaystyle \sum _{i=1}^{n}{}_{i+k}P_{k+1}=\sum _{i=1}^{n}\prod _{j=0}^{k}(i+j)={\frac {(n+k+1)!}{(n-1)!


In the 1920s the German military began using a 3-rotor Enigma, whose security was increased in 1930 by the addition of a plugboard. The Polish Cipher Bureau sought to break it due to the threat that Poland faced from Germany, but its early attempts did not succeed. Near the beginning of 1929, the Polish Cipher Bureau realized that mathematicians may make good codebreakers; the bureau invited math students at Poznań University to take a class on cryptology. After the class, the Bureau recruited some students to work part-time at a Bureau branch set up in Poznań for the students.


In order to recognize a pedestrian, the computational system uses AI pattern recognition technology that typically uses machine learning and deep convolutional neural networks based on millions of images. In a simplified description, images from the car's camera and radar are compared to the prototypes stored in the computer. If a match is made and confirmed, the other systems in the PCAM are invoked. PCAM technologies can be improved with additional information from connected vehicles. A thorough description of the processes for pedestrian detection in about 2010 is provided in . AI technologies have improved dramatically since then, as can be seen in an update in May 2016.


Under computer control, dots would be activated to create tactile patterns of highs and lows representing the information to be read. Visual and tactile impressions of a virtual surface are displayed by a high resolution tactile display, a so-called "artificial skin" (Fig. 6).


In most object-oriented languages, objects can be referred to using references. Some examples of such languages are Java, C++, C#, VB.NET, and many scripting languages, such as Perl, Python, and Ruby. In this case, it matters whether the state of an object can vary when objects are shared via references.


In multilinear algebra, the higher-order singular value decomposition (HOSVD) of a tensor is a specific orthogonal Tucker decomposition. It may be regarded as one type of generalization of the matrix singular value decomposition. It has applications in computer vision, computer graphics, machine learning, scientific computing, and signal processing.


In order to take advantage of the above described representations of elements with their traces and furthermore ensure sufficient security, that will be discussed below, we need to find primes p {\displaystyle p} and q {\displaystyle q} , where p {\displaystyle p} denotes the characteristic of the field G F ( p 6 ) {\displaystyle GF(p^{6})} with p ≡ 2 mod 3 {\displaystyle p\equiv 2\ {\text{mod}}\ 3} and q {\displaystyle q} is the size of the subgroup, such that q {\displaystyle q} divides p 2 − p + 1 {\displaystyle p^{2}-p+1} . We denote with P {\displaystyle P} and Q {\displaystyle Q} the sizes of p {\displaystyle p} and q {\displaystyle q} in bits. To achieve security comparable to 1024-bit RSA, we should choose 6 P {\displaystyle 6P} about 1024, i.e. P ≈ 170 {\displaystyle P\approx 170} and Q {\displaystyle Q} can be around 160. A first easy algorithm to compute such primes p {\displaystyle p} and q {\displaystyle q} is the next Algorithm A: Algorithm A Find r ∈ Z {\displaystyle r\in \mathbb {Z} } such that q = r 2 − r + 1 {\displaystyle q=r^{2}-r+1} is a Q {\displaystyle Q} -bit prime.


In routers and switches, active queue management (AQM) is the policy of dropping packets inside a buffer associated with a network interface controller (NIC) before that buffer becomes full, often with the goal of reducing network congestion or improving end-to-end latency. This task is performed by the network scheduler, which for this purpose uses various algorithms such as random early detection (RED), Explicit Congestion Notification (ECN), or controlled delay (CoDel). RFC 7567 recommends active queue management as a best practice.


In mathematics and computer science, computational number theory, also known as algorithmic number theory, is the study of computational methods for investigating and solving problems in number theory and arithmetic geometry, including algorithms for primality testing and integer factorization, finding solutions to diophantine equations, and explicit methods in arithmetic geometry. Computational number theory has applications to cryptography, including RSA, elliptic curve cryptography and post-quantum cryptography, and is used to investigate conjectures and open problems in number theory, including the Riemann hypothesis, the Birch and Swinnerton-Dyer conjecture, the ABC conjecture, the modularity conjecture, the Sato-Tate conjecture, and explicit aspects of the Langlands program.


In multi-agent reinforcement learning experiments, researchers try to optimize the performance of a learning agent on a given task, in cooperation or competition with one or more agents. These agents learn by trial-and-error, and researchers may choose to have the learning algorithm play the role of two or more of the different agents. When successfully executed, this technique has a double advantage: It provides a straightforward way to determine the actions of the other agents, resulting in a meaningful challenge. It increases the amount of experience that can be used to improve the policy, by a factor of two or more, since the viewpoints of each of the different agents can be used for learning.


The case (x, y, z) = (2l, 2m, n) and all its permutations were proven for l, m ≥ 5 primes and n = 3, 5, 7, 11 by Anni and Siksek. The case (x, y, z) = (2l, 2m, 13) and all its permutations were proven for l, m ≥ 5 primes by Billerey, Chen, Dembélé, Dieulefait, Freitas. The case (x, y, z) = (3l, 3m, n) is direct for l, m ≥ 2 and n ≥ 3 from work by Kraus.


In 2007, AMD added the AMD Athlon, AMD Turion, and Mobile AMD Sempron processors to its embedded product line. Leveraging the same 64-bit instruction set and Direct Connect Architecture as the AMD Opteron but at lower power levels, these processors were well suited to a variety of traditional embedded applications. Throughout 2007 and into 2008, AMD has continued to add both single-core Mobile AMD Sempron and AMD Athlon processors and dual-core AMD Athlon X2 and AMD Turion processors to its embedded product line and now offers embedded 64-bit solutions starting with 8 W TDP Mobile AMD Sempron and AMD Athlon processors for fan-less designs up to multi-processor systems leveraging multi-core AMD Opteron processors all supporting longer than standard availability.The ATI acquisition in 2006 included the Imageon and Xilleon product lines.


The entropy of a closed system, determined relative to this zero point, is then the absolute entropy of that system. Mathematically, the absolute entropy of any system at zero temperature is the natural log of the number of ground states times the Boltzmann constant kB = 1.38×10−23 J K−1.


Some payphones were altered to accept tokens, while others have been designed to use telephone cards. For example, in St Petersburg, payment for payphones can be made with metro tokens. In some regions, calls from public phones are free of charge.


In mathematics, the Hermite constant, named after Charles Hermite, determines how long a shortest element of a lattice in Euclidean space can be. The constant γn for integers n > 0 is defined as follows. For a lattice L in Euclidean space Rn with unit covolume, i.e. vol(Rn/L) = 1, let λ1(L) denote the least length of a nonzero element of L. Then √γn is the maximum of λ1(L) over all such lattices L. The square root in the definition of the Hermite constant is a matter of historical convention. Alternatively, the Hermite constant γn can be defined as the square of the maximal systole of a flat n-dimensional torus of unit volume.


In statistics, logistic regression is a type of regression analysis used for predicting the outcome of a categorical dependent variable (with a limited number of categories) or dichotomic dependent variable based on one or more predictor variables. The probabilities describing the possible outcome of a single trial are modeled, as a function of explanatory (independent) variables, using a logistic function or multinomial distribution. Logistic regression measures the relationship between a categorical or dichotomic dependent variable and usually a continuous independent variable (or several), by converting the dependent variable to probability scores. The probabilities can be retrieved using the logistic function or the multinomial distribution, while those probabilities, like in probability theory, takes on values between zero and one: So the model tested can be defined by: whereas yi is the category of the dependent variable for the i-th observation and xij is the j independent variable (j=1,2,...k) for that observation, βj is the j-th coefficient of xij and indicates its influence on and expected from the fitted model . Note: independent variables in logistic regression can also be continuous.


If there is branching, so multiple future revisions are based on a past revision, or undoing, so a revision can depend on a revision older than its immediate predecessor, then the resulting graph is instead a directed tree (each node can have more than one child), and has multiple tips, corresponding to the revisions without children ("latest revision on each branch"). In principle the resulting tree need not have a preferred tip ("main" latest revision) – just various different revisions – but in practice one tip is generally identified as HEAD. When a new revision is based on HEAD, it is either identified as the new HEAD, or considered a new branch.


Kernel methods have the advantage of having convex loss functions, with no local minima, and of being only moderately complex to implement. Because high-dimensional feature space is linear, kernel adaptive filters can be thought of as a generalization of linear adaptive filters. As with linear adaptive filters, there are two general approaches to adapting a filter: the least mean squares filter (LMS) and the recursive least squares filter (RLS).Self organising kernel adaptive filters that use iteration to achieve convex LMS error minimisation address some of the statistical and practical issues of non-linear models that do not arise in the linear case.


{\displaystyle p_{t}-p_{t-1}=0\quad {\text{for all present and future }}t.} The concept of a steady state has relevance in many fields, in particular thermodynamics, economics, and engineering. If a system is in a steady state, then the recently observed behavior of the system will continue into the future.


In operating systems that use virtual memory, every process is given the impression that it is working with large, contiguous sections of memory. Physically, the memory of each process may be dispersed across different areas of physical memory, or may have been moved (paged out) to secondary storage, typically to a hard disk drive (HDD) or solid-state drive (SSD). When a process requests access to data in its memory, it is the responsibility of the operating system to map the virtual address provided by the process to the physical address of the actual memory where that data is stored. The page table is where the operating system stores its mappings of virtual addresses to physical addresses, with each mapping also known as a page table entry (PTE).


This suggests that the nucleoprotein assembly is sequential in the PriA, DnaD, DnaB order. The preferred DNA substrate mimics an arrested DNA replication fork with unreplicated lagging strand, structurally identical to a product of recombinational repair of a stalled replication fork. == References ==


In statistics, sequential analysis or sequential hypothesis testing is statistical analysis where the sample size is not fixed in advance. Instead data are evaluated as they are collected, and further sampling is stopped in accordance with a pre-defined stopping rule as soon as significant results are observed. Thus a conclusion may sometimes be reached at a much earlier stage than would be possible with more classical hypothesis testing or estimation, at consequently lower financial and/or human cost.


In statistics, the relationship square is a graphical representation for use in the factorial analysis of a table individuals x variables. This representation completes classical representations provided by principal component analysis (PCA) or multiple correspondence analysis (MCA), namely those of individuals, of quantitative variables (correlation circle) and of the categories of qualitative variables (at the centroid of the individuals who possess them). It is especially important in factor analysis of mixed data (FAMD) and in multiple factor analysis (MFA).


Beyond that, operations like the converse of a relation and the composition of relations are available, satisfying the laws of a calculus of relations, for which there are textbooks by Ernst Schröder, Clarence Lewis, and Gunther Schmidt. A deeper analysis of relations involves decomposing them into subsets called concepts, and placing them in a complete lattice. In some systems of axiomatic set theory, relations are extended to classes, which are generalizations of sets. This extension is needed for, among other things, modeling the concepts of "is an element of" or "is a subset of" in set theory, without running into logical inconsistencies such as Russell's paradox. The terms correspondence, dyadic relation and two-place relation are synonyms for binary relation, though some authors use the term "binary relation" for any subset of a Cartesian product X × Y {\displaystyle X\times Y} without reference to X and Y, and reserve the term "correspondence" for a binary relation with reference to X and Y.


In the following, let G = ( V , E ) {\displaystyle G=(V,E)} be an arbitrary graph.


In mathematics, more specifically in functional analysis, a K-space is an F-space V {\displaystyle V} such that every extension of F-spaces (or twisted sum) of the form is equivalent to the trivial one where R {\displaystyle \mathbb {R} } is the real line.


It has also been reported that Stuxnet and associated variants have infected more than 30,000 systems and had a lasting presence which was extremely difficult to eradicate and purify. Both malicious programs exploited Zero-Day attacks on Windows-based operating systems.As computing crosses the cyber-physical barrier, there is significant effort spent on 'smart' systems, for instance smart cities, smart homes, smart manufacturing and smart vehicles.


{\displaystyle M(x)=M(\lfloor x\rfloor ).} Less formally, M ( x ) {\displaystyle M(x)} is the count of square-free integers up to x that have an even number of prime factors, minus the count of those that have an odd number. The first 143 M(n) values are (sequence A002321 in the OEIS) The Mertens function slowly grows in positive and negative directions both on average and in peak value, oscillating in an apparently chaotic manner passing through zero when n has the values 2, 39, 40, 58, 65, 93, 101, 145, 149, 150, 159, 160, 163, 164, 166, 214, 231, 232, 235, 236, 238, 254, 329, 331, 332, 333, 353, 355, 356, 358, 362, 363, 364, 366, 393, 401, 403, 404, 405, 407, 408, 413, 414, 419, 420, 422, 423, 424, 425, 427, 428, ... (sequence A028442 in the OEIS).Because the Möbius function only takes the values −1, 0, and +1, the Mertens function moves slowly, and there is no x such that |M(x)| > x. H. Davenport demonstrated that, for any fixed h, ∑ n = 1 x μ ( n ) exp  ( i 2 π n θ ) = O ( x log h  x ) {\displaystyle \sum _{n=1}^{x}\mu (n)\exp(i2\pi n\theta )=O\left({\frac {x}{\log ^{h}


In mathematics and theoretical computer science, a semiautomaton is a deterministic finite automaton having inputs but no output. It consists of a set Q of states, a set Σ called the input alphabet, and a function T: Q × Σ → Q called the transition function. Associated with any semiautomaton is a monoid called the characteristic monoid, input monoid, transition monoid or transition system of the semiautomaton, which acts on the set of states Q. This may be viewed either as an action of the free monoid of strings in the input alphabet Σ, or as the induced transformation semigroup of Q. In older books like Clifford and Preston (1967) semigroup actions are called "operands". In category theory, semiautomata essentially are functors.


In mathematics, a unimodular matrix M is a square integer matrix having determinant +1 or −1. Equivalently, it is an integer matrix that is invertible over the integers: there is an integer matrix N that is its inverse (these are equivalent under Cramer's rule). Thus every equation Mx = b, where M and b both have integer components and M is unimodular, has an integer solution. The n × n unimodular matrices form a group called the n × n general linear group over Z {\displaystyle \mathbb {Z} } , which is denoted GL n  ( Z ) {\displaystyle \operatorname {GL} _{n}(\mathbb {Z} )} .


In probability theory, an exponentially modified Gaussian distribution (EMG, also known as exGaussian distribution) describes the sum of independent normal and exponential random variables. An exGaussian random variable Z may be expressed as Z = X + Y, where X and Y are independent, X is Gaussian with mean μ and variance σ2, and Y is exponential of rate λ. It has a characteristic positive skew from the exponential component. It may also be regarded as a weighted function of a shifted exponential with the weight being a function of the normal distribution.


In some cases, this can occur recursively, with processing objects calling higher-up processing objects with commands that attempt to solve some smaller part of the problem; in this case recursion continues until the command is processed, or the entire tree has been explored. An XML interpreter might work in this manner. This pattern promotes the idea of loose coupling.


The outcome of this approach was revolutionary at the time; it suggested that the search terms be left in their original format, what would today be known as a natural language query.Another major change was how the results were judged. In the original tests, a success occurred only if the index returned the exact document that had been used to generate the search. However, this was not typical of an actual query; a user looking for information on aircraft landing gear might be happy with any of the collection's many papers on the topic, but Cranfield 1 would consider such a result a failure in spite of returning relevant materials. In the second series, the results were judged by 3rd parties who gave a qualitative answer on whether the query generated a relevant set of papers, as opposed to returning a specified original document.


In mathematics, a preordered class is a class equipped with a preorder.


In mathematics, a group is called boundedly generated if it can be expressed as a finite product of cyclic subgroups. The property of bounded generation is also closely related with the congruence subgroup problem (see Lubotzky & Segal 2003).


It also had a special mode where each of the operations could be executed sequentially. It implemented register rotation to aid in software pipelining of loops. There was an instruction cache only, since it was felt that a data cache would be inefficient on sparse array operations.


In telecommunication, time code ambiguity is the shortest interval between successive repetitions of the same time code value. For example, in a time code in which year-of-century (the '72' in 10/04/72) is the most slowly changing field, the time code ambiguity would be 100 years; it is ambiguous whether this value refers to a date in 1872, 1972 or some other century. For a digital clock in which hours and minutes up to a maximum of 11:59 are displayed, the time code ambiguity would be 12 hours. The Year 2000 problem is an example of the pitfalls of time code ambiguity. Very often dates are now recorded with 4 digit years (10/04/1972). Assuming that the use of a 4-digit year field would continue, even in the far future, this would change the time code ambiguity from 100 years to 10 000 years.


The file size distribution of publicly available audio and video data files (MIME types) follows a log-normal distribution over five orders of magnitude. File sizes of 140 million files on personal computers running the Windows OS, collected in 1999. Sizes of text-based emails (1990s) and multimedia-based emails (2000s).


Conversely, in the languages of Western Europe, the dot on a lower-case ⟨i⟩ is not a glyph in because it does not convey any distinction, and an ⟨ı⟩ in which the dot has been accidentally omitted is still likely to be recognized correctly. However, in Turkish and adjacent languages, this dot is a glyph because that language has two distinct versions of the letter i, with and without a dot. In Japanese syllabaries, some of the characters are made up of more than one separate mark, but in general these separate marks are not glyphs because they have no meaning by themselves.


In most designs the 56000 is dedicated to one single task, because digital signal processing using special hardware is mostly real-time and does not allow any interruption. For less demanding tasks which are not time-critical, designers normally use a separate CPU or MCU. The 56000 can execute a 1024-point complex Fast Fourier transform (FFT) in 59,898 clock cycles, taking 1.8 ms at 33 MHz, or a rate of just over 555 operations per second, allowing both realtime decoding and encoding of reasonably advanced audio codecs such as MP3 for direct-to-disc recording purposes.The addition of SIMD instructions to most desktop computer CPUs have meant that dedicated DSP chips like the 56000 have partly disappeared from some application fields, but they continue to be used widely in communications and other professional uses.


Common practical digital signals are represented as 8-bit (256 levels), 16-bit (65,536 levels), 24-bit (16.8 million levels), and 32-bit (4.3 billion levels) using pulse-code modulation where the number of quantization levels is not necessarily limited to powers of two. A floating point representation is used in many DSP applications. == References ==


In the TV series Person of Interest, bluesnarfing, often mistakenly referred to as bluejacking in the show and at other times forced pairing and phone cloning, is a common element in the show used to spy on and track the people the main characters are trying to save or stop.


In number theory, the Davenport–Erdős theorem states that, for sets of multiples of integers, several different notions of density are equivalent.Let A = a 1 , a 2 , … {\displaystyle A=a_{1},a_{2},\dots } be a sequence of positive integers. Then the multiples of A {\displaystyle A} are another set M ( A ) {\displaystyle M(A)} that can be defined as the set M ( A ) = { k a ∣ k ∈ N , a ∈ A } {\displaystyle M(A)=\{ka\mid k\in \mathbb {N} ,a\in A\}} of numbers formed by multiplying members of A {\displaystyle A} by arbitrary positive integers.According to the Davenport–Erdős theorem, for a set M ( A ) {\displaystyle M(A)} , the following notions of density are equivalent, in the sense that they all produce the same number as each other for the density of M ( A ) {\displaystyle M(A)}: The lower natural density, the inferior limit as n {\displaystyle n} goes to infinity of the proportion of members of M ( A ) {\displaystyle M(A)} in the interval  {\displaystyle } . The logarithmic density or multiplicative density, the weighted proportion of members of M ( A ) {\displaystyle M(A)} in the interval  {\displaystyle } , again in the limit, where the weight of an element a {\displaystyle a} is 1 / a {\displaystyle 1/a} .


For this reason, in a quantum algorithm there is no way to deterministically put bits in a specific prescribed state unless one is given access to bits whose original state is known in advance. Such bits, whose values are known a priori, are known as ancilla bits in a quantum or reversible computing task. A trivial use for ancilla bits is downgrading complicated quantum gates into simple gates.


Salt baths can be used as an antiseptic and fungicide, and will not damage beneficial bacteria, though ordinary table salt may contain additives which can harm fish. Alternatives include aquarium salt, Kosher salt or rock salt. Gradually raising the temperature of the tank may kill certain parasites, though some diseased fish may be harmed and certain species can not tolerate high temperatures.


In terms of size measure, the two most common cases are the largest-area empty rectangle and largest-perimeter empty rectangle.Another major classification is whether the rectangle is sought among axis-oriented or arbitrarily oriented rectangles.


It is conjectured that MLPT has the same approximation ratio for more general cardinality constraints (c>3). Currently, it is known that the approximation ratio of MLPT for general c>3 is at most 2. Chen, He and Lin show that, for the same problem, MLPT attains at least ( 3 m − 1 ) / ( 4 m − 2 ) {\displaystyle (3m-1)/(4m-2)} of the maximum smallest sum, which is again the same ratio that LPT attains for the unconstrained problem.Another constraint is that the number of jobs on all machines should be n / m {\displaystyle n/m} rounded either up or down.


The notation a ≤T b indicates there is a set in degree b that computes a set in degree a. Equivalently, a ≤T b holds if and only if every set in b computes every set in a. A function f from the natural numbers to the natural numbers is said to be diagonally nonrecursive (DNR) if, for all n, f ( n ) ≠ ϕ n ( n ) {\displaystyle f(n)\not =\phi _{n}(n)} (here inequality holds by definition if ϕ n ( n ) {\displaystyle \phi _{n}(n)} is undefined). If the range of f is the set {0,1} then f is a DNR2 function. It is known that there are DNR functions that do not compute any DNR2 function.


In order to implement the MinHash scheme as described above, one needs the hash function h to define a random permutation on n elements, where n is the total number of distinct elements in the union of all of the sets to be compared. But because there are n! different permutations, it would require Ω(n log n) bits just to specify a truly random permutation, an infeasibly large number for even moderate values of n. Because of this fact, by analogy to the theory of universal hashing, there has been significant work on finding a family of permutations that is "min-wise independent", meaning that for any subset of the domain, any element is equally likely to be the minimum. It has been established that a min-wise independent family of permutations must include at least lcm  ( 1 , 2 , ⋯ , n ) ≥ e n − o ( n ) {\displaystyle \operatorname {lcm} (1,2,\cdots ,n)\geq e^{n-o(n)}} different permutations, and therefore that it needs Ω(n) bits to specify a single permutation, still infeasibly large.


In semantics, research into linguistic universals has taken place in a number of ways. Some linguists, starting with Gottfried Leibniz, have pursued the search for a hypothetic irreducible semantic core of all languages. A modern variant of this approach can be found in the natural semantic metalanguage of Anna Wierzbicka and associates. See, for example, and Other lines of research suggest cross-linguistic tendencies to use body part terms metaphorically as adpositions, or tendencies to have morphologically simple words for cognitively salient concepts.


In mathematics, the Heawood number of a surface is an upper bound for the number of colors that suffice to color any graph embedded in the surface. In 1890 Heawood proved for all surfaces except the sphere that no more than H ( S ) = ⌊ 7 + 49 − 24 e ( S ) 2 ⌋ = ⌊ 7 + 1 + 48 g ( S ) 2 ⌋ {\displaystyle H(S)=\left\lfloor {\frac {7+{\sqrt {49-24e(S)}}}{2}}\right\rfloor =\left\lfloor {\frac {7+{\sqrt {1+48g(S)}}}{2}}\right\rfloor } colors are needed to color any graph embedded in a surface of Euler characteristic e ( S ) {\displaystyle e(S)} , or genus g ( S ) {\displaystyle g(S)} for an orientable surface. The number H ( S ) {\displaystyle H(S)} became known as Heawood number in 1976.


In the Japanese kantenji braille, the standard 8-dot Braille patterns 678, 1678, 4678, and 14678 are the patterns related to Braille pattern dots-356, since the two additional dots of kantenji patterns 0356, 3567, and 03567 are placed above the base 6-dot cell, instead of below, as in standard 8-dot braille.


In mathematics, a palindromic prime (sometimes called a palprime) is a prime number that is also a palindromic number. Palindromicity depends on the base of the number system and its notational conventions, while primality is independent of such concerns. The first few decimal palindromic primes are: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, … (sequence A002385 in the OEIS)Except for 11, all palindromic primes have an odd number of digits, because the divisibility test for 11 tells us that every palindromic number with an even number of digits is a multiple of 11. It is not known if there are infinitely many palindromic primes in base 10. The largest known as of October 2021 is 101888529 - 10944264 - 1.which has 1,888,529 digits, and was found on 18 October 2021 by Ryan Propper and Serge Batalov. On the other hand, it is known that, for any base, almost all palindromic numbers are composite, i.e. the ratio between palindromic composites and all palindromes less than n tends to 1.


The sum of distances is widely used in applications such as the facility location problem. The proposed algorithm uses Lloyd-style iteration which alternates between an expectation (E) and maximization (M) step, making this an expectation–maximization algorithm. In the E step, all objects are assigned to their nearest median. In the M step, the medians are recomputed by using the median in each single dimension.


In programming language semantics, normalisation by evaluation (NBE) is a style of obtaining the normal form of terms in the λ-calculus by appealing to their denotational semantics. A term is first interpreted into a denotational model of the λ-term structure, and then a canonical (β-normal and η-long) representative is extracted by reifying the denotation. Such an essentially semantic, reduction-free, approach differs from the more traditional syntactic, reduction-based, description of normalisation as reductions in a term rewrite system where β-reductions are allowed deep inside λ-terms. NBE was first described for the simply typed lambda calculus. It has since been extended both to weaker type systems such as the untyped lambda calculus using a domain theoretic approach, and to richer type systems such as several variants of Martin-Löf type theory.


If N {\displaystyle N} is V, then κ {\displaystyle \kappa } (the critical point of j {\displaystyle j} ) is always a measurable cardinal, i.e. an uncountable cardinal number κ such that there exists a κ {\displaystyle \kappa } -complete, non-principal ultrafilter over κ {\displaystyle \kappa } . Specifically, one may take the filter to be { A ∣ A ⊆ κ ∧ κ ∈ j ( A ) } {\displaystyle \{A\mid A\subseteq \kappa \land \kappa \in j(A)\}} . Generally, there will be many other <κ-complete, non-principal ultrafilters over κ {\displaystyle \kappa } .


As part of Schmidt's proof, he proves that a finite p-group is a modular group if and only if every subgroup is permutable, by (Schmidt 1994, Lemma 2.3.2, p. 55). Every subgroup of a finite p-group is subnormal, and those finite groups in which subnormality and permutability coincide are called PT-groups. In other words, a finite p-group is an Iwasawa group if and only if it is a PT-group.


In statistics, groups of individual data points may be classified as belonging to any of various statistical data types, e.g. categorical ("red", "blue", "green"), real number (1.68, -5, 1.7e+6), odd number (1,3,5) etc. The data type is a fundamental component of the semantic content of the variable, and controls which sorts of probability distributions can logically be used to describe the variable, the permissible operations on the variable, the type of regression analysis used to predict the variable, etc. The concept of data type is similar to the concept of level of measurement, but more specific: For example, count data require a different distribution (e.g. a Poisson distribution or binomial distribution) than non-negative real-valued data require, but both fall under the same level of measurement (a ratio scale). Various attempts have been made to produce a taxonomy of levels of measurement. The psychophysicist Stanley Smith Stevens defined nominal, ordinal, interval, and ratio scales. Nominal measurements do not have meaningful rank order among values, and permit any one-to-one transformation.


In mathematics, specifically in group theory, the Prüfer p-group or the p-quasicyclic group or p∞-group, Z(p∞), for a prime number p is the unique p-group in which every element has p different p-th roots. The Prüfer p-groups are countable abelian groups that are important in the classification of infinite abelian groups: they (along with the group of rational numbers) form the smallest building blocks of all divisible groups. The groups are named after Heinz Prüfer, a German mathematician of the early 20th century.


In statistics, generalized iterative scaling (GIS) and improved iterative scaling (IIS) are two early algorithms used to fit log-linear models, notably multinomial logistic regression (MaxEnt) classifiers and extensions of it such as MaxEnt Markov models and conditional random fields. These algorithms have been largely surpassed by gradient-based methods such as L-BFGS and coordinate descent algorithms.


In the EOQ model, it is assumed that the orders are received all at once. However, in the EBQ model, this assumption is relaxed.There are two types of costs: those which increase with the batch size such as working capital investment in materials and labor, cost of handling and storing materials, insurance and tax charges, interest on capital investment, etc., and those which decrease with the batch size such as cost (per unit) of setting up machines, cost of preparing paper work that enters and controls the production of the order, etc. These costs, i.e., (a) and (b) are plotted and added graphically (figure). The figure graphs the holding cost and ordering cost per year equations. The third line is the addition of these two equations, which generates the total inventory cost per year.


The functional dependencies, along with the attribute domains, are selected so as to generate constraints that would exclude as much data inappropriate to the user domain from the system as possible. A notion of logical implication is defined for functional dependencies in the following way: a set of functional dependencies Σ {\displaystyle \Sigma } logically implies another set of dependencies Γ {\displaystyle \Gamma } , if any relation R satisfying all dependencies from Σ {\displaystyle \Sigma } also satisfies all dependencies from Γ {\displaystyle \Gamma } ; this is usually written Σ ⊨ Γ {\displaystyle \Sigma \models \Gamma } . The notion of logical implication for functional dependencies admits a sound and complete finite axiomatization, known as Armstrong's axioms.


Another proof, which is a simplification of Lambert's proof, is due to Miklós Laczkovich. Many of these are proofs by contradiction. In 1882, Ferdinand von Lindemann proved that π {\displaystyle \pi } is not just irrational, but transcendental as well.


In mathematics and related subjects, understanding a mathematical expression depends on an understanding of symbols of grouping, such as parentheses (), brackets , and braces {}. These same symbols are also used in ways where they are not symbols of grouping. For example, in the expression 3(x+y) the parentheses are symbols of grouping, but in the expression (3, 5) the parentheses may indicate an open interval. The most common symbols of grouping are the parentheses and the brackets, and the brackets are usually used to avoid too many repeated parentheses.


The radical of an ideal in a commutative ring. In geometry, the convex hull of a set S of points is the smallest convex set of which S is a subset. In formal languages, the Kleene closure of a language can be described as the set of strings that can be made by concatenating zero or more strings from that language. In group theory, the conjugate closure or normal closure of a set of group elements is the smallest normal subgroup containing the set. In mathematical analysis and in probability theory, the closure of a collection of subsets of X under countably many set operations is called the σ-algebra generated by the collection.


The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrices, which allows computing in vector spaces.


In mathematics, a Zorn ring is an alternative ring in which for every non-nilpotent x there exists an element y such that xy is a non-zero idempotent (Kaplansky 1968, pages 19, 25). Kaplansky (1951) named them after Max August Zorn, who studied a similar condition in (Zorn 1941). For associative rings, the definition of Zorn ring can be restated as follows: the Jacobson radical J(R) is a nil ideal and every right ideal of R which is not contained in J(R) contains a nonzero idempotent. Replacing "right ideal" with "left ideal" yields an equivalent definition. Left or right Artinian rings, left or right perfect rings, semiprimary rings and von Neumann regular rings are all examples of associative Zorn rings.


In mathematical finance, if security weights maximize the expected geometric growth rate (which is equivalent to maximizing log wealth), then a portfolio is growth optimal. Computations of growth optimal portfolios can suffer tremendous garbage in, garbage out problems. For example, the cases below take as given the expected return and covariance structure of assets, but these parameters are at best estimates or models that have significant uncertainty.


For example, during marine transfer operations when flammable products are transferred between the marine terminal and tanker ships or barges, two-way radio communication needs to be constantly maintained in case the transfer needs to stop for unforeseen reasons such as a spill. The United States Coast Guard requires that the two way radio must be certified as intrinsically safe. Another example is intrinsically safe or explosion-proof mobile phones used in explosive atmospheres, such as refineries.


In probability theory — specifically, in stochastic analysis — a killed process is a stochastic process that is forced to assume an undefined or "killed" state at some (possibly random) time.


It is used as an optimality criterion in parameter selection and model selection. In general, total sum of squares = explained sum of squares + residual sum of squares. For a proof of this in the multivariate ordinary least squares (OLS) case, see partitioning in the general OLS model.


b  {\displaystyle z\neq x\ \to (\lambda z.b)=\lambda z.b} This is to stop bound variables with the same name being substituted. This would not have occurred in a canonically renamed lambda expression. For example the previous rules would have wrongly translated, ( λ x .


In matrix theory, Sylvester's determinant identity is an identity useful for evaluating certain types of determinants. It is named after James Joseph Sylvester, who stated this identity without proof in 1851.Given an n-by-n matrix A {\displaystyle A} , let det ( A ) {\displaystyle \det(A)} denote its determinant. Choose a pair u = ( u 1 , … , u m ) , v = ( v 1 , … , v m ) ⊂ ( 1 , … , n ) {\displaystyle u=(u_{1},\dots ,u_{m}),v=(v_{1},\dots ,v_{m})\subset (1,\dots ,n)} of m-element ordered subsets of ( 1 , … , n ) {\displaystyle (1,\dots ,n)} , where m ≤ n. Let A v u {\displaystyle A_{v}^{u}} denote the (n−m)-by-(n−m) submatrix of A {\displaystyle A} obtained by deleting the rows in u {\displaystyle u} and the columns in v {\displaystyle v} . Define the auxiliary m-by-m matrix A ~ v u {\displaystyle {\tilde {A}}_{v}^{u}} whose elements are equal to the following determinants ( A ~ v u ) i j := det ( A v  u  ) , {\displaystyle ({\tilde {A}}_{v}^{u})_{ij}:=\det(A_{v}^{u}),} where u  {\displaystyle u} , v  {\displaystyle v} denote the m−1 element subsets of u {\displaystyle u} and v {\displaystyle v} obtained by deleting the elements u i {\displaystyle u_{i}} and v j {\displaystyle v_{j}} , respectively.


This trick is mentioned by Kleene (1952) p. 223. The problem arises when the number to be created exhausts the number of instructions available to the finite state machine; there is always a bigger constant than the number of instructions available to the finite state machine.Unbounded numbers of registers versus bounded state-machine instructions: This is more severe than the first problem.


The number 1 is then appended to the end of c {\displaystyle c} , and the extrapolated limit is s = X c ∑ i = 1 k c i , {\displaystyle s={Xc \over \sum _{i=1}^{k}c_{i}},} where X = ( x 2 , x 3 , . . . , x k + 1 ) {\displaystyle X=(x_{2},x_{3},...,x_{k+1})} is the matrix whose columns are the k {\displaystyle k} iterates starting at 2. The following 4 line MATLAB code segment implements the MPE algorithm: == References ==


In sequential search, a consumer looks for a product or service one at a time until they find it, McCall J.J. introduced this type of search to economics. In economics, the sequential search model is used to examine how consumers choose which goods or services to purchase when they have asymmetrical information (incomplete)of those goods' quality. Consumers in sequential search models must choose whether to stop looking for a better good or service or to buy what they have found so far.


In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the gamma distribution. There are two equivalent parameterizations in common use: With a shape parameter k {\displaystyle k} and a scale parameter θ {\displaystyle \theta } . With a shape parameter α = k {\displaystyle \alpha =k} and an inverse scale parameter β = 1 / θ {\displaystyle \beta =1/\theta } , called a rate parameter.In each of these forms, both parameters are positive real numbers. The gamma distribution is the maximum entropy probability distribution (both with respect to a uniform base measure and a 1 / x {\displaystyle 1/x} base measure) for a random variable X {\displaystyle X} for which E = kθ = α/β is fixed and greater than zero, and E = ψ(k) + ln(θ) = ψ(α) − ln(β) is fixed (ψ is the digamma function).


The value associated with the largest identifier in that majority is V2, so it must propose it. This Proposer then gets all Acceptors to accept V2, achieving consensus. Proposer Acceptor Learner | | | | | | | | | | | X--------------->|->|->|->|->| | | Prepare(1) |<---------------X--X--X--X--X | | Promise(1,{null,null,null,null,null}) x--------------->| | | | | | | Accept!


This fact is called weak duality. In general, the optimal values of the primal and dual problems need not be equal. Their difference is called the duality gap. For convex optimization problems, the duality gap is zero under a constraint qualification condition. This fact is called strong duality.


There is a standard, ETSI ES 202 130, that covers European languages and other languages used in Europe, published by the independent ETSI organisation in 2003 and updated in 2007. Work describing some principles of the standard is available.Since many newer smartphones, such as the Palm Treo and BlackBerry, have full alphanumeric keyboards instead of the traditional telephone keypads, the user must execute additional steps to dial a number containing convenience letters. On certain BlackBerry devices, a user can press the Alt key, followed by the desired letter, and the device will generate the appropriate DTMF tone.


Egalitarian zero The index I(x) is zero in the egalitarian case, when all values xi are equal. Bounded above by maximum inequality The index I(x) attains its maximum value for maximum inequality. (all xi are zero except one) This value is usually unity as the number of agents n approaches infinity. Subgroup decomposabilityThis property states that if a set of agents x is divided into two disjoint subsets (y and z) then the I(x) is expressible as: where μ(x) and μ(y) are the mean incomes of x and y. and the w functions are scalar weighting function of the sets y and z. In a stronger statement, wy = μy / μx and wz = μz / μx.


But they will not be maintained. Another issue is that an XML document can be transcoded from one encoding to another during transport. When the XML document is converted to a more limited character set, such as ASCII, characters that can no longer be represented are converted to nnn; character references for a lossless conversion. But within a CDATA section, these characters can not be represented at all, and have to be removed or converted to some equivalent, altering the content of the CDATA section.


In most wireless ad hoc networks, the nodes compete for access to shared wireless medium, often resulting in collisions (interference). Collisions can be handled using centralized scheduling or distributed contention access protocols. Using cooperative wireless communications improves immunity to interference by having the destination node combine self-interference and other-node interference to improve decoding of the desired signals.


In order to adapt to the changes, the Linuxtag focused on the core issue of the professional use of open source software in 2014. Therefore, LinuxTag started a strategic partnership with the droidcon. The 20th LinuxTag took place between 8 and 10 May 2014 in the STATION Berlin. In spatial and temporal proximity were the Media Convention Berlin (May 6 to 7), the re: publica (6 to 8 May) and the droidcon (8 to 10 May 2014). All events aimed towards a close relationship in order to achieve a high level of appreciation of the combined effort.


The framework is developed and supported by the Science of High-Performance Computing (SHPC) group of the Oden Institute for Computational Engineering and Sciences at The University of Texas at Austin and the Matthews Research Group at Southern Methodist University. BLIS yields high performance on many current CPU microarchitectures in both single-threaded and multithreaded modes of execution. BLIS also offers competitive performance for some cases of matrix multiplication in which one or more matrix operands are unusually skinny and/or small.The framework achieves high performance by employing specialized kernels (typically written in GNU extended inline assembly syntax) along with cache and register blocking through matrix operands. BLIS also works on processors for which custom kernels have not yet been written; in those cases, the framework relies upon portable kernel implementations that perform at a lower rate of computation. BLIS is sometimes described as a refactoring of GotoBLAS2, which was created by Kazushige Goto at the Texas Advanced Computing Center.


In statistical mechanics, bootstrap percolation is a percolation process in which a random initial configuration of active cells is selected from a lattice or other space, and then cells with few active neighbors are successively removed from the active set until the system stabilizes. The order in which this removal occurs makes no difference to the final stable state.When the threshold of active neighbors needed for an active cell to survive is high enough (depending on the lattice), the only stable states are states with no active cells, or states in which every cluster of active cells is infinitely large. For instance, on the square lattice with the von Neumann neighborhood, there are finite clusters with at least two active neighbors per cluster cell, but when three or four active neighbors are required, any stable cluster must be infinite. With three active neighbors needed to stay active, an infinite cluster must stretch infinitely in three or four of the possible cardinal directions, and any finite holes it contains will necessarily be rectangular.


Most commonly however, these white spaces exist naturally between used channels, since assigning nearby transmissions to immediately adjacent channels will cause destructive interference to both. In addition to white space assigned for technical reasons, there is also unused radio spectrum which has either never been used, or is becoming free as a result of technical changes. In particular, the switchover to digital television frees up large areas between about 50 MHz and 700 MHz.


In order to achieve a linear improvement in usefulness over time it is necessary to have an exponential increase in technology over time. Moore's law delivers an exponential increase in technology, and so when Claasen's law is combined with Moore's law it implies a linear improvement in usefulness over time.


In mathematical optimization, Zadeh's rule (also known as the least-entered rule) is an algorithmic refinement of the simplex method for linear optimization. The rule was proposed around 1980 by Norman Zadeh (son of Lotfi A. Zadeh), and has entered the folklore of convex optimization since then.Zadeh offered a reward of $1,000 to anyone who can show that the rule admits polynomially many iterations or to prove that there is a family of linear programs on which the pivoting rule requires subexponentially many iterations to find the optimum.


In principle, any arbitrary boolean function, including addition, multiplication, and other mathematical functions, can be built up from a functionally complete set of logic operators. In 1987, Conway's Game of Life became one of the first examples of general-purpose computing using an early stream processor called a blitter to invoke a special sequence of logical operations on bit vectors.General-purpose computing on GPUs became more practical and popular after about 2001, with the advent of both programmable shaders and floating point support on graphics processors. Notably, problems involving matrices and/or vectors – especially two-, three-, or four-dimensional vectors – were easy to translate to a GPU, which acts with native speed and support on those types. A significant milestone for GPGPU was the year 2003 when two research groups independently discovered GPU-based approaches for the solution of general linear algebra problems on GPUs that ran faster than on CPUs.


The notation J represents the part of the correlations that can be attributed to classical correlations and varies in dependence on the chosen eigenbasis; therefore, in order for the quantum discord to reflect the purely nonclassical correlations independently of basis, it is necessary that J first be maximized over the set of all possible projective measurements onto the eigenbasis: D A ( ρ ) = I ( ρ ) − max { Π j A } J { Π j A } ( ρ ) = S ( ρ A ) − S ( ρ ) + min { Π j A } S ( ρ B | { Π j A } ) {\displaystyle {\mathcal {D}}_{A}(\rho )=I(\rho )-\max _{\{\Pi _{j}^{A}\}}J_{\{\Pi _{j}^{A}\}}(\rho )=S(\rho _{A})-S(\rho )+\min _{\{\Pi _{j}^{A}\}}S(\rho _{B|\{\Pi _{j}^{A}\}})} Nonzero quantum discord indicates the presence of correlations that are due to noncommutativity of quantum operators. For pure states, the quantum discord becomes a measure of quantum entanglement, more specifically, in that case it equals the entropy of entanglement.Vanishing quantum discord is a criterion for the pointer states, which constitute preferred effectively classical states of a system. It could be shown that quantum discord must be non-negative and that states with vanishing quantum discord can in fact be identified with pointer states.


In the PKCS#1 standard, the random oracles are identical. The PKCS#1 standard further requires that the random oracles be MGF1 with an appropriate hash function.


In statistics, the absolute deviation of an element of a data set is the absolute difference between that element and a given point. Typically the deviation is reckoned from the central value, being construed as some type of average, most often the median or sometimes the mean of the data set: where Di is the absolute deviation, xi is the data element, m(X) is the chosen measure of central tendency of the data set—sometimes the mean ( x ¯ {\displaystyle {\overline {x}}} ), but most often the median.


At a minimum, each profile specification contains information on the following topics: Dependencies on other formats Suggested user interface formats Specific parts of the Bluetooth protocol stack used by the profile. To perform its task, each profile uses particular options and parameters at each layer of the stack. This may include an outline of the required service record, if appropriate.This article summarizes the current definitions of profiles defined and adopted by the Bluetooth SIG and possible applications of each profile.


Social networks like Facebook, Twitter and Ning allow learners, once they move beyond the personal connections, to embrace a community where they can learn from each other. Social interaction is an important part of the learning process. As technology has grown and become an integral part of the lives of children, teens, and young adults, older adults have been forced to adapt.


The B5000 was designed in 1961, long before the term RISC was invented. The architecture puts six 8-bit instructions in a 48-bit word, and was a precursor to very long instruction word (VLIW) design (see below: 1990 to today). The Burroughs architecture was one of the inspirations for Charles H. Moore's programming language Forth, which in turn inspired his later MISC chip designs.


In mathematical optimization and related fields, relaxation is a modeling strategy. A relaxation is an approximation of a difficult problem by a nearby problem that is easier to solve. A solution of the relaxed problem provides information about the original problem. For example, a linear programming relaxation of an integer programming problem removes the integrality constraint and so allows non-integer rational solutions.


The distribution was first introduced in a more general context and is an example of the vine copula, an approach to constrained high-dimensional probability distributions. It has been implemented as part of the Stan probabilistic programming language and as a library linked to the Turing.jl probabilistic programming library in Julia. The distribution has a single shape parameter η {\displaystyle \eta } and the probability density function for a d × d {\displaystyle d\times d} matrix R {\displaystyle \mathbf {R} } is p ( R ; η ) = C ×  η − 1 {\displaystyle p(\mathbf {R} ;\eta )=C\times ^{\eta -1}} with normalizing constant C = 2 ∑ k = 1 d ( 2 η − 2 + d − k ) ( d − k ) ∏ k = 1 d − 1  d − k {\displaystyle C=2^{\sum _{k=1}^{d}(2\eta -2+d-k)(d-k)}\prod _{k=1}^{d-1}\left^{d-k}} , a complicated expression including a product over Beta functions. For η = 1 {\displaystyle \eta =1} , the distribution is uniform over the space of all correlation matrices; i.e. the space of positive definite matrices with unit diagonal.


This design enabled more efficient program execution, as the Program Counter and status flags could be saved and restored with a single operation. This resulted in faster subroutine calls and interrupt response than traditional designs, which would have to do two register loads or saves when calling or returning from a subroutine. Despite having a 32-bit ALU and word-length, processors based on ARM architecture version 1 and 2 had only a 26-bit PC and address bus, and were consequently limited to 64 MiB of addressable memory.


This leads to faster convergence in terms of basis set size than pure gaussian-type basis set, but requires calculation of more complex integrals. To simplify them, interelectron distances are expanded into a series making for simpler integrals. The idea of R12 methods is quite old, but practical implementations begun to appear only recently.


Swapping conflict: a swapping conflict is the case in which two agents exchange their position, passing on the same edge at the same time in two different directions. It is expressed as π i  = π j  {\displaystyle \pi _{i}=\pi _{j}} and π j  = π i  {\displaystyle \pi _{j}=\pi _{i}} .When formalizing a MAPF problem, it is possible to decide which conflicts are allowed and which are forbidden. A unified standard about permitted and denied conflicts does not exist, however vertex and edge conflicts are usually not allowed.


The dispute had several levels, some technical (sockets vs. streams, BSD tty vs. System V termio) and some cultural. The divide was roughly between longhairs and shorthairs; programmers and technical people tended to line up with Berkeley and BSD, more business-oriented types with AT&T and System V. While HP, IBM and others chose System V as the basis for their Unix offerings, other vendors such as Sun Microsystems and DEC extended BSD. Throughout its development, though, System V was infused with features from BSD, while BSD variants such as DEC's Ultrix received System V features. AT&T and Sun Microsystems worked together to merge System V with BSD-based SunOS to produce Solaris, one of the primary System V descendants still in use today. Since the early 1990s, due to standardization efforts such as POSIX and the success of Linux, the division between System V and BSD has become less important.


The rank of a free abelian group is the cardinality of a basis; every two bases for the same group give the same rank, and every two free abelian groups with the same rank are isomorphic. Every subgroup of a free abelian group is itself free abelian; this fact allows a general abelian group to be understood as a quotient of a free abelian group by "relations", or as a cokernel of an injective homomorphism between free abelian groups. The only free abelian groups that are free groups are the trivial group and the infinite cyclic group.


In the late 1980s, the European standardization group GSM who worked on the Pan-European Digital mobile communication system GSM greatly expanded the use of ALOHA channels for access to radio channels in mobile telephony. In addition, SMS message texting was implemented in 2G mobile phones. In the early 2000s additional ALOHA channels were added to 2.5G and 3G mobile phones with the widespread introduction of General Packet Radio Service (GPRS), using a slotted-ALOHA random-access channel combined with a version of the Reservation ALOHA scheme first analyzed by a group at BBN Technologies.


This leads to the notion of "genetic distance", which is a measure of recombination frequency averaged over a (suitably large) sample of pedigrees. Loosely speaking, one may say that this is because recombination is greatly influenced by the proximity of one gene to another. If two genes are located close together on a chromosome, the likelihood that a recombination event will separate these two genes is less than if they were farther apart.


In the present context, our system is assumed to have the energy levels ε i {\displaystyle \varepsilon _{i}} with degeneracies g i {\displaystyle g_{i}} . As before, we would like to calculate the probability that our system has energy ε i {\displaystyle \varepsilon _{i}} . If our system is in state s 1 {\displaystyle \;s_{1}} , then there would be a corresponding number of microstates available to the reservoir.


In mathematics, particularly computational algebra, Berlekamp's algorithm is a well-known method for factoring polynomials over finite fields (also known as Galois fields). The algorithm consists mainly of matrix reduction and polynomial GCD computations. It was invented by Elwyn Berlekamp in 1967. It was the dominant algorithm for solving the problem until the Cantor–Zassenhaus algorithm of 1981. It is currently implemented in many well-known computer algebra systems.


This theorem is useful in searching for cycles of the permutation, since an efficient search can look only at multiples of divisors of MN−1 (Brenner, 1973). Laflin & Brebner (1970) pointed out that the cycles often come in pairs, which is exploited by several algorithms that permute pairs of cycles at a time. In particular, let s be the smallest element of some cycle C of length k. It follows that MN−1−s is also an element of a cycle of length k (possibly the same cycle).


In the discrete formulation of the pursuit–evasion problem, the environment is modeled as a graph.


Musser reported that on a median-of-3 killer sequence of 100,000 elements, introsort's running time was 1/200 that of median-of-3 quicksort. Musser also considered the effect on caches of Sedgewick's delayed small sorting, where small ranges are sorted at the end in a single pass of insertion sort. He reported that it could double the number of cache misses, but that its performance with double-ended queues was significantly better and should be retained for template libraries, in part because the gain in other cases from doing the sorts immediately was not great.


This definition encompasses most curves that are studied in mathematics; notable exceptions are level curves (which are unions of curves and isolated points), and algebraic curves (see below). Level curves and algebraic curves are sometimes called implicit curves, since they are generally defined by implicit equations. Nevertheless, the class of topological curves is very broad, and contains some curves that do not look as one may expect for a curve, or even cannot be drawn.


The nines' complement of a number given in decimal representation is formed by replacing each digit with nine minus that digit. To subtract a decimal number y (the subtrahend) from another number x (the minuend) two methods may be used: In the first method the nines' complement of x is added to y. Then the nines' complement of the result obtained is formed to produce the desired result. In the second method the nines' complement of y is added to x and one is added to the sum.


In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied to it, that result in the creation of a larger collection of objects, called the generated set. The larger set is then said to be generated by the smaller set. It is commonly the case that the generating set has a simpler set of properties than the generated set, thus making it easier to discuss and examine. It is usually the case that properties of the generating set are in some way preserved by the act of generation; likewise, the properties of the generated set are often reflected in the generating set.


A 6600/7600 Peripheral Processor system was used for I/O, largely unchanged. Some effort was made to help compatibility between the older machines and the 8600, but the change in word length made this difficult. Instead, floating point formats were retained, allowing Fortran code to port directly.


In multivariate statistics, random matrices were introduced by John Wishart, who sought to estimate covariance matrices of large samples. Chernoff-, Bernstein-, and Hoeffding-type inequalities can typically be strengthened when applied to the maximal eigenvalue (i.e. the eigenvalue of largest magnitude) of a finite sum of random Hermitian matrices. Random matrix theory is used to study the spectral properties of random matrices—such as sample covariance matrices—which is of particular interest in high-dimensional statistics. Random matrix theory also saw applications in neuronal networks and deep learning, with recent work utilizing random matrices to show that hyper-parameter tunings can be cheaply transferred between large neural networks without the need for re-training.In numerical analysis, random matrices have been used since the work of John von Neumann and Herman Goldstine to describe computation errors in operations such as matrix multiplication. Although random entries are traditional "generic" inputs to an algorithm, the concentration of measure associated with random matrix distributions implies that random matrices will not test large portions of an algorithm's input space.


The second phase requires choosing an ontology by which to organize categories of things. Temporal expressions and some numerical expressions (e.g., money, percentages, etc.) may also be considered as named entities in the context of the NER task. While some instances of these types are good examples of rigid designators (e.g., the year 2001) there are also many invalid ones (e.g., I take my vacations in “June”).


In the balanced partition problem, there are constraints on the number of jobs that can be assigned to each machine. A simple constraint is that each machine can process at most c jobs. The LPT rule assigns each job to the machine with the smallest load from among those with fewer than c jobs. This rule is called modified LPT or MLPT.


In mathematics, Porter's constant C arises in the study of the efficiency of the Euclidean algorithm. It is named after J. W. Porter of University College, Cardiff. Euclid's algorithm finds the greatest common divisor of two positive integers m and n. Hans Heilbronn proved that the average number of iterations of Euclid's algorithm, for fixed n and averaged over all choices of relatively prime integers m < n, is 12 ln  2 π 2 ln  n + o ( ln  n ) . {\displaystyle {\frac {12\ln 2}{\pi ^{2}}}\ln n+o(\ln n).} Porter showed that the error term in this estimate is a constant, plus a polynomially-small correction, and Donald Knuth evaluated this constant to high accuracy. It is: C = 6 ln  2 π 2  − 1 2 = 6 ln  2 ( ( 48 ln  A ) − ( ln  2 ) − ( 4 ln  π ) − 2 ) π 2 − 1 2 = 1.4670780794 … {\displaystyle {\begin{aligned}C&={{6\ln 2} \over {\pi ^{2}}}\left-{{1} \over {2}}\\&={{{6\ln 2}((48\ln A)-(\ln 2)-(4\ln \pi )-2)} \over {\pi ^{2}}}-{{1} \over {2}}\\&=1.4670780794\ldots \end{aligned}}} where γ {\displaystyle \gamma } is the Euler–Mascheroni constant ζ {\displaystyle \zeta } is the Riemann zeta function A {\displaystyle A} is the Glaisher–Kinkelin constant(sequence A086237 in the OEIS) − ζ ′ ( 2 ) = π 2 6  = ∑ k = 2 ∞ ln  k k 2 {\displaystyle -\zeta ^{\prime }(2)={{\pi ^{2}} \over 6}\left=\sum _{k=2}^{\infty }{{\ln k} \over {k^{2}}}} {\displaystyle }


As with the analogous operation on the real numbers, a fundamental use of this operation is in solving, when possible, linear congruences of the form a x ≡ b ( mod m ) . {\displaystyle ax\equiv b{\pmod {m}}.} Finding modular multiplicative inverses also has practical applications in the field of cryptography, e.g. public-key cryptography and the RSA algorithm. A benefit for the computer implementation of these applications is that there exists a very fast algorithm (the extended Euclidean algorithm) that can be used for the calculation of modular multiplicative inverses.


In order to expose an application's geographic data with GML, a community or organization creates an XML schema specific to the application domain of interest (the application schema). This schema describes the object types whose data the community is interested in and which community applications must expose. For example, an application for tourism may define object types including monuments, places of interest, museums, road exits, and viewpoints in its application schema.


For example, reads by Illumina’s sequencing technology capture reads of 100-mers. However, the problem with the sequencing is that only small fractions out of all the possible 100-mers that are present in the genome are actually generated. This is due to read errors, but more importantly, just simple coverage holes that occur during sequencing.


Similarly, a Highway Net whose gates are opened through strongly positive bias weights behaves like a ResNet. The skip connections used in modern neural networks (e.g., Transformers) are dominantly identity mappings. DenseNets in 2016 were designed as deep neural networks that attempt to connect each layer to every other layer.


In previous decades, multilevel security (MLS) technologies were developed and implemented that enabled objective and deterministic security, but left little wiggle room for subjective and discretionary interpretation. These enforced mandatory access control (MAC) with near certainty. This rigidity prevented simpler solutions that would seem acceptable on the surface. Automated information systems have enabled extensive information sharing that is sometimes contrary to the need to avoid sharing secrets with adversaries.


The orders of the full symmetry groups are twice as much again (24, 48, and 120). See (Coxeter 1973) for a derivation of these facts.


Connecting elements into a pipeline is analogous to function composition. Narrowly speaking, a pipeline is linear and one-directional, though sometimes the term is applied to more general flows. For example, a primarily one-directional pipeline may have some communication in the other direction, known as a return channel or backchannel, as in the lexer hack, or a pipeline may be fully bi-directional. Flows with one-directional tree and directed acyclic graph topologies behave similarly to (linear) pipelines – the lack of cycles makes them simple – and thus may be loosely referred to as "pipelines".


In mathematics, a colored matroid is a matroid whose elements are labeled from a set of colors, which can be any set that suits the purpose, for instance the set of the first n positive integers, or the sign set {+, −}. The interest in colored matroids is through their invariants, especially the colored Tutte polynomial, which generalizes the Tutte polynomial of a signed graph of Kauffman (1989).There has also been study of optimization problems on matroids where the objective function of the optimization depends on the set of colors chosen as part of a matroid basis.


In the 1998 book Vectorproducts and Applications, Szép presented a new approach to system theory that contained a summary of his results from research between 1990 and 1995 as well as the applications of multiplicative structures in coding theory, game theory, and distribution vectors.


Algorithms that push content based on user search histories, frequent clicks and paid advertising leads to unbalanced, poorly sourced, and actively misleading information. It is also highly profitable.


Philips, LSI Logic and Integrated Device Technology (IDT) have since joined them. Today, the MIPS cores are one of the most-used "heavyweight" cores in the market for computer-like devices: handheld PCs, set-top boxes, etc. Since the MIPS architecture is licensable, it has attracted several processor start-up companies over the years. One of the first start-ups to design MIPS processors was Quantum Effect Devices (see next section).


The variables ε i {\displaystyle \varepsilon _{i}} are random variables, which in standard linear regression are distributed according to a standard normal distribution; they express the influence of any unknown factors on the outcome. This makes it possible to find optimal coefficients through the method of least squares using simple matrix operations. In particular, the optimal coefficients β ^ {\displaystyle {\boldsymbol {\hat {\beta }}}} as estimated by least squares can be written as follows: β ^ = ( X T X ) − 1 X T y .


In the animated short cartoon The Bowling Alley-Cat (1942), cat and mouse Tom and Jerry do battle inside a bowling center.In Dreamer (1979), Tim Matheson plays a man aspiring to be a professional bowler who faces a challenger played by Dick Weber.In Greedy (1994), Michael J. Fox plays an "honest but luckless pro bowler with a bad wrist and a good woman. "The Farrelly brothers' comedy Kingpin (1996) is a bowling comedy about which Randy Quaid said in an interview, "If we can't laugh at bowling, what can we laugh at? "In the Coen Brothers' The Big Lebowski (1998), "the Dude" (Jeff Bridges), a "slacker's slacker", hangs out with his buddies at a bowling alley, in which John Goodman's character pulls out a gun to threaten a competitor who stepped over the foul line and refused to accept the mandatory zero score for the shot.In the Disney Channel's Alley Cats Strike (2000), high school students engage in a bowling rivalry.


For example, the original subroutine call instructions BAL, Branch and Link, and its register-register equivalent, BALR, Branch and Link Register, store certain status information, the instruction length code, the condition code and the program mask, in the top byte of the return address. A BAS, Branch and Save, instruction was added to allow 31-bit return addresses. BAS, and its register-register equivalent, BASR, Branch and Save Register, was part of the instruction set of the 360/67, which was the only System/360 model to allow addresses longer than 24 bits.


In mathematics, the Schwartz–Zippel lemma (also called the DeMillo–Lipton–Schwartz–Zippel lemma) is a tool commonly used in probabilistic polynomial identity testing, i.e. in the problem of determining whether a given multivariate polynomial is the 0-polynomial (or identically equal to 0). It was discovered independently by Jack Schwartz, Richard Zippel, and Richard DeMillo and Richard J. Lipton, although DeMillo and Lipton's version was shown a year prior to Schwartz and Zippel's result. The finite field version of this bound was proved by Øystein Ore in 1922.


(We use the transitive closure of {x} rather than of x itself to avoid confusing the elements of x with elements of its elements or whatever.) A code includes that information identifying x and also information about the particular injection from X into ω which was used to create E. The extra information about the injection is non-essential, so there are many codes for the same set which are equally useful. So codes are a way of mapping H ℵ 1 {\displaystyle H_{\aleph _{1}}} into the powerset of ω×ω.


{\displaystyle \nabla \cdot \mathbf {H} (x)=0.} as H {\displaystyle \mathbf {H} } contains similar terms. The expression ∇ × E ( x ) {\displaystyle \nabla \times \mathbf {E} (x)} contains terms of the form p × ϵ 1 ( p ) {\displaystyle \mathbf {p} \times \mathbf {\epsilon ^{1}} (\mathbf {p} )} while ∂ H ( x ) / ∂ t {\displaystyle \partial \mathbf {H} (x)/\partial t} contains terms of form i p 0 ϵ 1 ( p ) {\displaystyle ip_{0}\mathbf {\epsilon ^{1}} (\mathbf {p} )} .


In quantum error correction, CSS codes, named after their inventors, Robert Calderbank, Peter Shor and Andrew Steane, are a special type of stabilizer code constructed from classical codes with some special properties. An example of a CSS code is the Steane code.


This is exactly analogous to Güntsch's system, designed as a means to improve performance, rather than to solve the problems involved in multi-programming. The first true virtual memory system was that implemented at the University of Manchester to create a one-level storage system as part of the Atlas Computer. It used a paging mechanism to map the virtual addresses available to the programmer onto the real memory that consisted of 16,384 words of primary core memory with an additional 98,304 words of secondary drum memory.