In numerical analysis, the Schur complement method, named after Issai Schur, is the basic and the earliest version of non-overlapping domain decomposition method, also called iterative substructuring. A finite element problem is split into non-overlapping subdomains, and the unknowns in the interiors of the subdomains are eliminated. The remaining Schur complement system on the unknowns associated with subdomain interfaces is solved by the conjugate gradient method.


In number theory, Zsigmondy's theorem, named after Karl Zsigmondy, states that if a > b > 0 {\displaystyle a>b>0} are coprime integers, then for any integer n ≥ 1 {\displaystyle n\geq 1} , there is a prime number p (called a primitive prime divisor) that divides a n − b n {\displaystyle a^{n}-b^{n}} and does not divide a k − b k {\displaystyle a^{k}-b^{k}} for any positive integer k < n {\displaystyle k 1 {\displaystyle n>1} and n {\displaystyle n} is not equal to 6, then 2 n − 1 {\displaystyle 2^{n}-1} has a prime divisor not dividing any 2 k − 1 {\displaystyle 2^{k}-1} with k < n {\displaystyle k


In mathematics and logic, plural quantification is the theory that an individual variable x may take on plural, as well as singular, values. As well as substituting individual objects such as Alice, the number 1, the tallest building in London etc. for x, we may substitute both Alice and Bob, or all the numbers between 0 and 10, or all the buildings in London over 20 stories. The point of the theory is to give first-order logic the power of set theory, but without any "existential commitment" to such objects as sets. The classic expositions are Boolos 1984 and Lewis 1991.


In post-quantum cryptography, ring learning with errors (RLWE) is a computational problem which serves as the foundation of new cryptographic algorithms, such as NewHope, designed to protect against cryptanalysis by quantum computers and also to provide the basis for homomorphic encryption. Public-key cryptography relies on construction of mathematical problems that are believed to be hard to solve if no further information is available, but are easy to solve if some information used in the problem construction is known. Some problems of this sort that are currently used in cryptography are at risk of attack if sufficiently large quantum computers can ever be built, so resistant problems are sought.


There are two main approaches. In languages with declaration-site variance annotations (e.g., C#), the programmer annotates the definition of a generic type with the intended variance of its type parameters. With use-site variance annotations (e.g., Java), the programmer instead annotates the places where a generic type is instantiated.


The main reasons for unsuccessful call setups in mobile networks are lack of radio coverage (either in the downlink or the uplink), radio interference between different subscribers, imperfections in the functioning of the network (such as failed call setup redirect procedures), overload of the different elements of the network (such as cells), etc. The call setup success rate is one of the key performance indicators (KPIs) used by the network operators to assess the performance of their networks. It is assumed to have direct influence on the customer satisfaction with the service provided by the network and its operator. The call setup success rate is usually included, together with other technical parameters of the network, in a key performance indicator known as service accessibility. The operators of telecommunication networks aim at increasing the call setup success rate as much as practical and affordable. In mobile networks this is achieved by improving radio coverage, expanding the capacity of the network and optimising the performance of its elements, all of which may require considerable effort and significant investments on the part of the network operator.


In mathematics, the order polytope of a finite partially ordered set is a convex polytope defined from the set. The points of the order polytope are the monotonic functions from the given set to the unit interval, its vertices correspond to the upper sets of the partial order, and its dimension is the number of elements in the partial order. The order polytope is a distributive polytope, meaning that coordinatewise minima and maxima of pairs of its points remain within the polytope. The order polytope of a partial order should be distinguished from the linear ordering polytope, a polytope defined from a number n {\displaystyle n} as the convex hull of indicator vectors of the sets of edges of n {\displaystyle n} -vertex transitive tournaments.


This smaller design, the Cell Broadband Engine or Cell/BE was fabricated using a 90 nm SOI process.In March 2007, IBM announced that the 65 nm version of Cell/BE is in production at its plant (at the time, now GlobalFoundries') in East Fishkill, New York, with Bandai Namco Entertainment using the Cell/BE processor for their 357 arcade board as well as the subsequent 369. In February 2008, IBM announced that it would begin to fabricate Cell processors with the 45 nm process.In May 2008, IBM introduced the high-performance double-precision floating-point version of the Cell processor, the PowerXCell 8i, at the 65 nm feature size. In May 2008, an Opteron- and PowerXCell 8i-based supercomputer, the IBM Roadrunner system, became the world's first system to achieve one petaFLOPS, and was the fastest computer in the world until third quarter 2009. The world's three most energy-efficient supercomputers, as represented by the Green500 list, are similarly based on the PowerXCell 8i. In August 2009 the 45 nm Cell processor was introduced in concert with Sony's PlayStation 3 Slim.By November 2009, IBM had discontinued the development of a Cell processor with 32 APUs but was still developing other Cell products.


In more advanced areas of mathematics, when viewing Euclidean space as a vector space, its distance is associated with a norm called the Euclidean norm, defined as the distance of each vector from the origin. One of the important properties of this norm, relative to other norms, is that it remains unchanged under arbitrary rotations of space around the origin. By Dvoretzky's theorem, every finite-dimensional normed vector space has a high-dimensional subspace on which the norm is approximately Euclidean; the Euclidean norm is the only norm with this property.


The practice of integrity management consultancy combines numerous disciplines including established Knowledge Acquisition Methodologies, Public Relations Practices, Business Risk Management techniques, Corporate Social Responsibility (CSR), Compliance functions, Insider Threat detection and Financial expertise to help clients make responsible and ethical business decisions. Integrity management consultancies offer services such as: Identifying potential reputational and ethical threats for clients Developing practical recommendations to mitigate these threats Making recommendations for, and carrying out due diligence on local professional service suppliers Providing advice on accessing a local pool of talent and addressing cultural considerations for building an effective local workforce Conducting detailed investigations into proposed local partners and key employees Preparing community needs reviews for better targeting of CSR programmes and; Administering anti and counter-corruption trainingMany also offer general ethics audits so that companies can obtain independent verification to attest to the high standard of their corporate ethics. Of course, this is not an exhaustive list as each consultancy offers different services.


In mathematics, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element x in the set, yields x. One of the most familiar additive identities is the number 0 from elementary mathematics, but additive identities occur in other mathematical structures where addition is defined, such as in groups and rings.


In statistics, the Box–Cox distribution (also known as the power-normal distribution) is the distribution of a random variable X for which the Box–Cox transformation on X follows a truncated normal distribution. It is a continuous probability distribution having probability density function (pdf) given by f ( y ) = 1 ( 1 − I ( f < 0 ) − sgn  ( f ) Φ ( 0 , m , s ) ) 2 π s 2 exp  { − 1 2 s 2 ( y f f − m ) 2 } {\displaystyle f(y)={\frac {1}{\left(1-I(f<0)-\operatorname {sgn}(f)\Phi (0,m,{\sqrt {s}})\right){\sqrt {2\pi s^{2}}}}}\exp \left\{-{\frac {1}{2s^{2}}}\left({\frac {y^{f}}{f}}-m\right)^{2}\right\}} for y > 0, where m is the location parameter of the distribution, s is the dispersion, ƒ is the family parameter, I is the indicator function, Φ is the cumulative distribution function of the standard normal distribution, and sgn is the sign function.


In mathematics, a 2-valued morphism is a homomorphism that sends a Boolean algebra B onto the two-element Boolean algebra 2 = {0,1}. It is essentially the same thing as an ultrafilter on B, and, in a different way, also the same things as a maximal ideal of B. 2-valued morphisms have also been proposed as a tool for unifying the language of physics.


In multithreaded computing, the ABA problem occurs during synchronization, when a location is read twice, has the same value for both reads, and the read value being the same twice is used to conclude that nothing has happened in the interim; however, another thread can execute between the two reads and change the value, do other work, then change the value back, thus fooling the first thread into thinking nothing has changed even though the second thread did work that violates that assumption. The ABA problem occurs when multiple threads (or processes) accessing shared data interleave. Below is a sequence of events that illustrates the ABA problem: Process P 1 {\displaystyle P_{1}} reads value A from some shared memory location, P 1 {\displaystyle P_{1}} is preempted, allowing process P 2 {\displaystyle P_{2}} to run, P 2 {\displaystyle P_{2}} writes value B to the shared memory location P 2 {\displaystyle P_{2}} writes value A to the shared memory location P 2 {\displaystyle P_{2}} is preempted, allowing process P 1 {\displaystyle P_{1}} to run, P 1 {\displaystyle P_{1}} reads value A from the shared memory location, P 1 {\displaystyle P_{1}} determines that the shared memory value has not changed and continues.Although P 1 {\displaystyle P_{1}} can continue executing, it is possible that the behavior will not be correct due to the "hidden" modification in shared memory.


If that sum of squares is divided by n, the number of observations, the result is the mean of the squared residuals. Since this is a biased estimate of the variance of the unobserved errors, the bias is removed by dividing the sum of the squared residuals by df = n − p − 1, instead of n, where df is the number of degrees of freedom (n minus the number of parameters (excluding the intercept) p being estimated - 1). This forms an unbiased estimate of the variance of the unobserved errors, and is called the mean squared error.Another method to calculate the mean square of error when analyzing the variance of linear regression using a technique like that used in ANOVA (they are the same because ANOVA is a type of regression), the sum of squares of the residuals (aka sum of squares of the error) is divided by the degrees of freedom (where the degrees of freedom equal n − p − 1, where p is the number of parameters estimated in the model (one for each variable in the regression equation, not including the intercept)).


If neighborhood searched is limited to just one or a very small number of changes from the current solution, then it can be difficult to escape from local minima, even with additional meta-heuristic techniques such as Simulated Annealing or Tabu search. In large neighborhood search techniques, the possible changes from one solution to its neighbor may allow tens or hundreds of values to change, and this means that the size of the neighborhood may itself be sufficient to allow the search process to avoid or escape local minima, though additional meta-heuristic techniques can still improve performance. == References ==


An example from linear algebra is the multiplicative monoid of real square matrices of order n (called the full linear monoid). The map which sends a matrix to its transpose is an involution because the transpose is well defined for any matrix and obeys the law (AB)T = BTAT, which has the same form of interaction with multiplication as taking inverses has in the general linear group (which is a subgroup of the full linear monoid). However, for an arbitrary matrix, AAT does not equal the identity element (namely the diagonal matrix).


For each unit, its duties and responsibilities in response to each command are specified. This is analogous to establishing a work breakdown structure for a development project, or defining an organizational chart for a business or military operation. Step 3 specifies the processing that is triggered within each unit upon receipt of an input command.


In military munitions, a fuze (sometimes fuse) is the part of the device that initiates function. In some applications, such as torpedoes, a fuze may be identified by function as the exploder. The relative complexity of even the earliest fuze designs can be seen in cutaway diagrams.


Are the preconditions each necessary and collectively sufficient to reach the long-term outcomes and ultimate impact? Are there gaps in the logic?


In the decision phase the following approaches are usually used: Fixed Threshold – In this approach, the scores are compared to a threshold which was set previously and if the score is higher than the threshold a cut is declared. Adaptive Threshold – In this approach, the scores are compared to a threshold which considers various scores in the video to adapt the threshold to the properties of the current video. Like in the previous case, if the score is higher than the corresponding threshold a cut is declared. Machine Learning - Machine learning techniques can be applied also to the decision process.


: ch1 Four-vectors describe, for instance, position xμ in spacetime modeled as Minkowski space, a particle's four-momentum pμ, the amplitude of the electromagnetic four-potential Aμ(x) at a point x in spacetime, and the elements of the subspace spanned by the gamma matrices inside the Dirac algebra. The Lorentz group may be represented by 4×4 matrices Λ. The action of a Lorentz transformation on a general contravariant four-vector X (like the examples above), regarded as a column vector with Cartesian coordinates with respect to an inertial frame in the entries, is given by (matrix multiplication) where the components of the primed object refer to the new frame. Related to the examples above that are given as contravariant vectors, there are also the corresponding covariant vectors xμ, pμ and Aμ(x).


In quantum physics, the mathematical notion is usually applied to representations of the gauge group. For example, an SU ( 2 ) {\displaystyle {\text{SU}}(2)} gauge theory will have multiplets which are fields whose representation of SU ( 2 ) {\displaystyle {\text{SU}}(2)} is determined by the single half-integer number s =: n / 2 {\displaystyle s=:n/2} , the isospin. Since irreducible SU ( 2 ) {\displaystyle {\text{SU}}(2)} representations are isomorphic to the n {\displaystyle n} th symmetric power of the fundamental representation, every field has n {\displaystyle n} symmetrized internal indices. Fields also transform under representations of the Lorentz group SO ( 1 , 3 ) {\displaystyle {\text{SO}}(1,3)} , or more generally its spin group Spin ( 1 , 3 ) {\displaystyle {\text{Spin}}(1,3)} which can be identified with SL ( 2 , C ) {\displaystyle {\text{SL}}(2,\mathbb {C} )} due to an exceptional isomorphism.


In order to transmit information, the continuous wave must be turned off and on with a telegraph key to produce the different length pulses, "dots" and "dashes", that spell out text messages in Morse code, so a "continuous wave" radiotelegraphy signal consists of pulses of sine waves with a constant amplitude interspersed with gaps of no signal. In on-off carrier keying, if the carrier wave is turned on or off abruptly, communications theory can show that the bandwidth will be large; if the carrier turns on and off more gradually, the bandwidth will be smaller. The bandwidth of an on-off keyed signal is related to the data transmission rate as: B n = B K {\displaystyle B_{n}=BK} where B n {\displaystyle B_{n}} is the necessary bandwidth in hertz, B {\displaystyle B} is the keying rate in signal changes per second (baud rate), and K {\displaystyle K} is a constant related to the expected radio propagation conditions; K=1 is difficult for a human ear to decode, K=3 or K=5 is used when fading or multipath propagation is expected.The spurious noise emitted by a transmitter which abruptly switches a carrier on and off is called key clicks.


In mathematics, a group G is called the direct sum of two normal subgroups with trivial intersection if it is generated by the subgroups. In abstract algebra, this method of construction of groups can be generalized to direct sums of vector spaces, modules, and other structures; see the article direct sum of modules for more information. A group which can be expressed as a direct sum of non-trivial subgroups is called decomposable, and if a group cannot be expressed as such a direct sum then it is called indecomposable.


This theorem is one of the main reasons why 1 is not considered a prime number: if 1 were prime, then factorization into primes would not be unique; for example, 2 = 2 ⋅ 1 = 2 ⋅ 1 ⋅ 1 = … {\displaystyle 2=2\cdot 1=2\cdot 1\cdot 1=\ldots } The theorem generalizes to other algebraic structures that are called unique factorization domains and include principal ideal domains, Euclidean domains, and polynomial rings over a field. However, the theorem does not hold for algebraic integers. This failure of unique factorization is one of the reasons for the difficulty of the proof of Fermat's Last Theorem. The implicit use of unique factorization in rings of algebraic integers is behind the error of many of the numerous false proofs that have been written during the 358 years between Fermat's statement and Wiles's proof.


Common proof rules used are modus ponens and universal instantiation.In contrast, an indirect proof may begin with certain hypothetical scenarios and then proceed to eliminate the uncertainties in each of these scenarios until an inescapable conclusion is forced. For example, instead of showing directly p ⇒ q, one proves its contrapositive ~q ⇒ ~p (one assumes ~q and shows that it leads to ~p). Since p ⇒ q and ~q ⇒ ~p are equivalent by the principle of transposition (see law of excluded middle), p ⇒ q is indirectly proved. Proof methods that are not direct include proof by contradiction, including proof by infinite descent. Direct proof methods include proof by exhaustion and proof by induction.


In mathematics, a Cantor cube is a topological group of the form {0, 1}A for some index set A. Its algebraic and topological structures are the group direct product and product topology over the cyclic group of order 2 (which is itself given the discrete topology). If A is a countably infinite set, the corresponding Cantor cube is a Cantor space. Cantor cubes are special among compact groups because every compact group is a continuous image of one, although usually not a homomorphic image.


In set theory and Boolean algebra, it is often stated as "union and intersection interchange under complementation", which can be formally expressed as: A ∪ B ¯ = A ¯ ∩ B ¯ , A ∩ B ¯ = A ¯ ∪ B ¯ , {\displaystyle {\begin{aligned}{\overline {A\cup B}}&={\overline {A}}\cap {\overline {B}},\\{\overline {A\cap B}}&={\overline {A}}\cup {\overline {B}},\end{aligned}}} where: A ¯ {\displaystyle {\overline {A}}} is the negation of A {\displaystyle A} , the overline being written above the terms to be negated, ∩ {\displaystyle \cap } is the intersection operator (AND), ∪ {\displaystyle \cup } is the union operator (OR).


In telecommunication, a burst transmission or data burst is the broadcast of a relatively high-bandwidth transmission over a short period. Burst transmission can be intentional, broadcasting a compressed message at a very high data signaling rate within a very short transmission time. In the 1980s, the term "data burst" (and "info burst") was used for a technique used by some United Kingdom and South African TV programmes to transmit large amounts of primarily textual information.


In mathematics, a Multibrot set is the set of values in the complex plane whose absolute value remains below some finite value throughout iterations by a member of the general monic univariate polynomial family of recursions. The name is a portmanteau of multiple and Mandelbrot set. The same can be applied to the Julia set, this being called Multijulia set.


For simple patterns, summation of long sequences may be represented with most summands replaced by ellipses. For example, summation of the first 100 natural numbers may be written as 1 + 2 + 3 + 4 + ⋯ + 99 + 100. Otherwise, summation is denoted by using Σ notation, where ∑ {\textstyle \sum } is an enlarged capital Greek letter sigma.


Of these, the addresses of the 16 general-purpose registers (R0-R15) are not fixed. Instead, the R0 register is located at the address pointed to by the "Context Pointer" (CP) register, and the remaining 15 registers follow sequentially thereafter.Register windows also provide an easy upgrade path. Since the additional registers are invisible to the programs, additional windows can be added at any time.


Near set theory provides a formal basis for the observation, comparison, and classification of elements in sets based on their closeness, either spatially or descriptively. Near sets offer a framework for solving problems based on human perception that arise in areas such as image processing, computer vision as well as engineering and science problems. Near sets have a variety of applications in areas such as topology, pattern detection and classification, abstract algebra, mathematics in computer science, and solving a variety of problems based on human perception that arise in areas such as image analysis, image processing, face recognition, ethology, as well as engineering and science problems.


In the field of selection principles, Tsaban devised the method of omission of intervals for establishing covering properties of sets of real numbers that have certain combinatorial structures. In nonabelian cryptology he devised the algebraic span method that solved a number of computational problems that underlie a number of proposals for nonabelian public-key cryptographic schemes (such as the commutator key exchange).


This allowed the STAR to use vectors of length not limited by the length of registers, making it highly flexible. Unfortunately, the pipeline had to be very long in order to allow it to have enough instructions in flight to make up for the slow memory. That meant the machine incurred a high cost when switching from processing vectors to performing operations on non-vector operands. Additionally, the low scalar performance of the machine meant that after the switch had taken place and the machine was running scalar instructions, the performance was quite poor. The result was rather disappointing real-world performance, something that could, perhaps, have been forecast by Amdahl's law.


While it is possible to write computer programs as long lists of numbers (machine language) and while this technique was used with many early computers, it is extremely tedious and potentially error-prone to do so in practice, especially for complicated programs. Instead, each basic instruction can be given a short name that is indicative of its function and easy to remember – a mnemonic such as ADD, SUB, MULT or JUMP. These mnemonics are collectively known as a computer's assembly language. Converting programs written in assembly language into something the computer can actually understand (machine language) is usually done by a computer program called an assembler.


In secure computing terms, because access to a resource via a handle is mediated by another system, a handle functions as a capability: it not only identifies an object, but also associates access rights. For example, while a filename is forgeable (it is just a guessable identifier), a handle is given to a user by an external system, and thus represents not just identity, but also granted access. For example, if a program wishes to read the system password file (/etc/passwd) in read/write mode (O_RDWR), it could try to open the file via the following call: This call asks the operating system to open the specified file with the specified access rights. If the OS allows this, then it opens the file (creates an entry in the per-process file descriptor table) and returns a handle (file descriptor, index into this table) to the user: the actual access is controlled by the OS, and the handle is a token of that.


In order-theoretic mathematics, a series-parallel partial order is a partially ordered set built up from smaller series-parallel partial orders by two simple composition operations.The series-parallel partial orders may be characterized as the N-free finite partial orders; they have order dimension at most two. They include weak orders and the reachability relationship in directed trees and directed series–parallel graphs. The comparability graphs of series-parallel partial orders are cographs.Series-parallel partial orders have been applied in job shop scheduling, machine learning of event sequencing in time series data, transmission sequencing of multimedia data, and throughput maximization in dataflow programming.Series-parallel partial orders have also been called multitrees; however, that name is ambiguous: multitrees also refer to partial orders with no four-element diamond suborder and to other structures formed from multiple trees.


It is common in economic models that involve decision-making over time to assume that decision-makers are exponential discounters. Exponential discounting posits that the decision maker assigns future utility of any good according to the formula U ( t ) = U ( 0 ) e − ρ t {\displaystyle U(t)=U(0)e^{-\rho t}} where t = 0 {\displaystyle t=0} is the present, U ( 0 ) {\displaystyle U(0)} is the utility assigned to the good if it were consumed immediately, and ρ {\displaystyle \rho } is the "discount factor", which is the same for all goods and constant over time. Mathematically, it is the unique continuous function that satisfies the equation U ( t 1 ) / U ( t 2 ) = U ( t 1 + c ) / U ( t 2 + c ) ; {\displaystyle U(t_{1})/U(t_{2})=U(t_{1}+c)/U(t_{2}+c);} that is, the ratio of utility values for a good at two different moments of time only depends on the interval between these times, but not on their choice.


Working in the opposite direction, the second expression asserts that A is false and B is false (or equivalently that "not A" and "not B" are true). Knowing this, a disjunction of A and B must be false also. The negation of said disjunction must thus be true, and the result is identical to the first claim.


In mathematics, a cyclic graph may mean a graph that contains a cycle, or a graph that is a cycle, with varying definitions of cycles. See: Cycle (graph theory), a cycle in a graph Forest (graph theory), an undirected graph with no cycles Biconnected graph, an undirected graph in which every edge belongs to a cycle Directed acyclic graph, a directed graph with no cycles Strongly connected graph, a directed graph in which every edge belongs to a cycle Aperiodic graph, a directed graph in which the cycle lengths have no nontrivial common divisor Pseudoforest, a directed or undirected graph in which every connected component includes at most one cycle Cycle graph, a graph that has the structure of a single cycle Pancyclic graph, a graph that has cycles of all possible lengths Cycle detection (graph theory), the algorithmic problem of finding cycles in graphsOther similarly-named concepts include Cycle graph (algebra), a graph that illustrates the cyclic subgroups of a group Circulant graph, a graph with an automorphism which permutes its vertices cyclically.


In the 1890s Michelson built a mechanical device called the harmonic analyzer, for computing coefficients of Fourier series and drawing graphs of their partial sums. He and S. W. Stratton published a paper about this machine in the American Journal of Science in 1898.


Prices of $60 for keyboards were achieved, and Key Tronic rapidly became the largest independent keyboard manufacturer. Meanwhile, IBM made their own keyboards, using their own patented technology: Keys on older IBM keyboards were made with a "buckling spring" mechanism, in which a coil spring under the key buckles under pressure from the user's finger, triggering a hammer that presses two plastic sheets (membranes) with conductive traces together, completing a circuit. This produces a clicking sound and gives physical feedback for the typist, indicating that the key has been depressed.The first electronic keyboards had a typewriter key travel distance of 0.187 inches (4.75 mm), keytops were a half-inch (12.7 mm) high, and keyboards were about two inches (5 cm) thick.


Transportation modes are an essential component of transport systems since they are the means by which mobility is supported. Geographers consider a wide range of modes that may be grouped into three broad categories based on the medium they exploit: land, water and air. Each mode has its own requirements and features, and is adapted to serve the specific demands of freight and passenger traffic.


In the 17th century, European mathematicians Isaac Barrow, René Descartes, Pierre de Fermat, Blaise Pascal, John Wallis and others discussed the idea of a derivative. In particular, in Methodus ad disquirendam maximam et minima and in De tangentibus linearum curvarum distributed in 1636, Fermat introduced the concept of adequality, which represented equality up to an infinitesimal error term. This method could be used to determine the maxima, minima, and tangents to various curves and was closely related to differentiation.Isaac Newton would later write that his own early ideas about calculus came directly from "Fermat's way of drawing tangents."


In regular expressions, the period (., also called "dot") is the wildcard pattern which matches any single character. Combined with the asterisk operator . * it will match any number of any characters. In this case, the asterisk is also known as the Kleene star.


In telecommunications, falsing is a signaling error condition when a signal decoder detects a valid input although the implied protocol function was not intended. This is also known as a false decode. Other forms are referred to as talk-off.


In some computer programming languages, the size in bits of certain data types 16-bit computing A 16-bit integer can represent up to 65,536 values.


Declare a new variable in the procedure (called L for reference). On each remaining unconnected exit path, add a statement that sets L to the label value on that path. Combine the resulting programs into a selection statement that executes the program with the entry path label indicated by L Construct a loop that executes this selection statement as long as L is not 0. Construct a sequence that initializes L to 1 and executes the loop.Note that this construction can be improved by converting some cases of the selection statement into subprocedures.


In mathematics, list edge-coloring is a type of graph coloring that combines list coloring and edge coloring. An instance of a list edge-coloring problem consists of a graph together with a list of allowed colors for each edge. A list edge-coloring is a choice of a color for each edge, from its list of allowed colors; a coloring is proper if no two adjacent edges receive the same color.


In punctuation, a word divider is a form of glyph which separates written words. In languages which use the Latin, Cyrillic, and Arabic alphabets, as well as other scripts of Europe and West Asia, the word divider is a blank space, or whitespace. This convention is spreading, along with other aspects of European punctuation, to Asia and Africa, where words are usually written without word separation.In computing, the word "delimiter" refers to a character that separates two words. In character encoding, word segmentation depends on which characters are defined as word dividers.


In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers). Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise "Éléments de géométrie algébrique"; one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne). Strongly based on commutative algebra, scheme theory allows a systematic use of methods of topology and homological algebra.


In the Japanese kantenji braille, the standard 8-dot Braille patterns 23578, 123578, 234578, and 1234578 are the patterns related to Braille pattern dots-12346, since the two additional dots of kantenji patterns 012346, 123467, and 0123467 are placed above the base 6-dot cell, instead of below, as in standard 8-dot braille.


In stream ciphers, IVs are loaded into the keyed internal secret state of the cipher, after which a number of cipher rounds are executed prior to releasing the first bit of output. For performance reasons, designers of stream ciphers try to keep that number of rounds as small as possible, but because determining the minimal secure number of rounds for stream ciphers is not a trivial task, and considering other issues such as entropy loss, unique to each cipher construction, related-IVs and other IV-related attacks are a known security issue for stream ciphers, which makes IV loading in stream ciphers a serious concern and a subject of ongoing research.


where F× is the multiplicative group of F (that is, F excluding 0). These elements are "special" in that they form an algebraic subvariety of the general linear group – they satisfy a polynomial equation (since the determinant is polynomial in the entries). When F is a finite field of order q, the notation SL(n, q) is sometimes used.


In mathematics, a primary cyclic group is a group that is both a cyclic group and a p-primary group for some prime number p. That is, it is a cyclic group of order pm, Cpm, for some prime number p, and natural number m. Every finite abelian group G may be written as a finite direct sum of primary cyclic groups, as stated in the fundamental theorem of finite abelian groups: G = ⨁ 1 ≤ i ≤ n C p i m i . {\displaystyle G=\bigoplus _{1\leq i\leq n}\mathrm {C} _{{p_{i}}^{m_{i}}}.} This expression is essentially unique: there is a bijection between the sets of groups in two such expressions, which maps each group to one that is isomorphic.


The question then becomes: on which level of iterated logarithms do to compare two numbers? There is a sense in which one may want to consider 10 10 10 {\displaystyle 10^{10^{10}}} and 10 10 9 {\displaystyle 10^{10^{9}}} to be "close in magnitude". The relative error between these two numbers is large, and the relative error between their logarithms is still large; however, the relative error in their second-iterated logarithms is small: log 10  ( log 10  ( 10 10 10 ) ) = 10 {\displaystyle \log _{10}(\log _{10}(10^{10^{10}}))=10} and log 10  ( log 10  ( 10 10 9 ) ) = 9 {\displaystyle \log _{10}(\log _{10}(10^{10^{9}}))=9} Such comparisons of iterated logarithms are common, e.g., in analytic number theory.


Like for conic sections, a line cuts this oval at, at most, two points. A non-singular plane cubic defines an elliptic curve, over any field K for which it has a point defined. Elliptic curves are now normally studied in some variant of Weierstrass's elliptic functions, defining a quadratic extension of the field of rational functions made by extracting the square root of a cubic.


Column 3 - F = Furniture Vans, Carriages, Motor Cars, Portable Engines and Machines on Wheels. Column 4 - L = Livestock Column 5 - H = Horse Boxes and Prize Cattle Vans. Column 6 - C = Carriages and Motor Cars by Passenger or Parcels Train. == References ==


In random graphs the sizes of components are given by a random variable, which, in turn, depends on the specific model of how random graphs are chosen. In the G ( n , p ) {\displaystyle G(n,p)} version of the Erdős–Rényi–Gilbert model, a graph on n {\displaystyle n} vertices is generated by choosing randomly and independently for each pair of vertices whether to include an edge connecting that pair, with probability p {\displaystyle p} of including an edge and probability 1 − p {\displaystyle 1-p} of leaving those two vertices without an edge connecting them. The connectivity of this model depends on p {\displaystyle p} , and there are three different ranges of p {\displaystyle p} with very different behavior from each other. In the analysis below, all outcomes occur with high probability, meaning that the probability of the outcome is arbitrarily close to one for sufficiently large values of n {\displaystyle n} .


In mathematics, a hereditary property is a property of an object that is inherited by all of its subobjects, where the meaning of subobject depends on the context. These properties are particularly considered in topology and graph theory, but also in set theory.


In telecommunications, average bitrate (ABR) refers to the average amount of data transferred per unit of time, usually measured per second, commonly for digital music or video. An MP3 file, for example, that has an average bit rate of 128 kbit/s transfers, on average, 128,000 bits every second. It can have higher bitrate and lower bitrate parts, and the average bitrate for a certain timeframe is obtained by dividing the number of bits used during the timeframe by the number of seconds in the timeframe. Bitrate is not reliable as a standalone measure of audio or video quality, since more efficient compression methods use lower bitrates to encode material at a similar quality.


In number theory, he proved there is no odd perfect number with fewer than four prime factors. In algebra, he was notable for the study of associative algebras. He first introduced the terms idempotent and nilpotent in 1870 to describe elements of these algebras, and he also introduced the Peirce decomposition. In the philosophy of mathematics, he became known for the statement that "Mathematics is the science that draws necessary conclusions".


In systems theory, it involves the description of a network whose components are compartments that represent a population of elements that are equivalent with respect to the manner in which they process input signals to the compartment. Instant homogeneous distribution of materials or energies within a "compartment." The exchange rate of materials or energies among the compartments is related to the densities of these compartments. Usually, it is desirable that the materials do not undergo chemical reactions while transmitting among the compartments. When concentration of the cell is of interest, typically the volume is assumed to be constant over time, though this may not be totally true in reality.


In short, it is a cartesian product and it corresponds to a product in the category of types. Most functional programming languages have a primitive notion of product type.


In the CS reconstruction models using constrained ℓ 1 {\displaystyle \ell _{1}} minimization, larger coefficients are penalized heavily in the ℓ 1 {\displaystyle \ell _{1}} norm. It was proposed to have a weighted formulation of ℓ 1 {\displaystyle \ell _{1}} minimization designed to more democratically penalize nonzero coefficients. An iterative algorithm is used for constructing the appropriate weights. Each iteration requires solving one ℓ 1 {\displaystyle \ell _{1}} minimization problem by finding the local minimum of a concave penalty function that more closely resembles the ℓ 0 {\displaystyle \ell _{0}} norm. An additional parameter, usually to avoid any sharp transitions in the penalty function curve, is introduced into the iterative equation to ensure stability and so that a zero estimate in one iteration does not necessarily lead to a zero estimate in the next iteration. The method essentially involves using the current solution for computing the weights to be used in the next iteration.


In order to capture sufficient thermoacoustic data to form an accurate 3D map of electromagnetic absorption, it is necessary to surround the anatomy being imaged with a 2D array of transducers. The world's first 3D thermoacoustic animal scanner (Fig. 8: left panel) accomplished this by combining a cylindrical array of 128 transducers (Fig. 8: center panel) with rotation of the animal being imaged about the vertical axis. The net result was to capture thermoacoustic data over the surface of a sphere surrounding the animal being imaged (Fig. 8: right panel). This device was capable of visualizing structures as small as 1/3 millimeter.


Applied to a function with one variable, the Y combinator usually does not terminate. More interesting results are obtained by applying the Y combinator to functions of two or more variables. The additional variables may be used as a counter, or index.


New centers were then chosen to better represent the cluster. This process was repeated until the centers at the start matched the centers at the end. When the cluster centers have not changed, it could be interpreted that this means proper clusters have been chosen.


In the diagrams below, P is used for N’s parent, S for the sibling of N, C (meaning close nephew) for S’s child in the same direction as N, and D (meaning distant nephew) for S’s other child (S cannot be a NIL node in the first iteration, because it must have black height one, which was the black height of N before its deletion, but C and D may be NIL nodes). The diagrams show the current node N as the left child of its parent P even though it is possible for N to be on either side. The code samples cover both possibilities by means of the side variable dir. At the beginning (in the first iteration) of removal, N is the NIL node replacing the node to be deleted.


Testers debug materials and procedures. The team reviews and revises the project according to feedback. After completing the development of the course material, the designers should conduct an imperative pilot test; this can be carried out by involving key stakeholders and rehearsing the course material.


In mathematics and computer science, in the field of coding theory, the Hamming bound is a limit on the parameters of an arbitrary block code: it is also known as the sphere-packing bound or the volume bound from an interpretation in terms of packing balls in the Hamming metric into the space of all possible words. It gives an important limitation on the efficiency with which any error-correcting code can utilize the space in which its code words are embedded. A code that attains the Hamming bound is said to be a perfect code.


In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special cases of a preorder: an antisymmetric (or skeletal) preorder is a partial order, and a symmetric preorder is an equivalence relation. The name preorder comes from the idea that preorders (that are not partial orders) are 'almost' (partial) orders, but not quite; they are neither necessarily antisymmetric nor asymmetric. Because a preorder is a binary relation, the symbol ≤ {\displaystyle \,\leq \,} can be used as the notational device for the relation.


Whether or not a name is well-formed depends on the type of computer system being used. Early computers permitted only a few letters or digits in the name of a file, but modern computers allow long names (some up to 255 characters) containing almost any combination of unicode letters or unicode digits, making it easier to understand the purpose of a file at a glance. Some computer systems allow file names to contain spaces; others do not.


We may assume by probability amplification that both circuit families have success probability at least 5/6. Then we create a composite algorithm where the circuits for L 1 {\displaystyle L_{1}} and L 2 {\displaystyle L_{2}} are run independently, and we set P to the conjunction of P 1 {\displaystyle P_{1}} and P 2 {\displaystyle P_{2}} , and Q to the conjunction of Q 1 {\displaystyle Q_{1}} and Q 2 {\displaystyle Q_{2}} . It is not hard to see by a union bound that this composite algorithm correctly decides membership in L 1 ∩ L 2 {\displaystyle L_{1}\cap L_{2}} with (conditional) probability at least 2/3.More generally, combinations of these ideas show that PostBQP is closed under union and BQP truth-table reductions.


In number theory, Tunnell's theorem gives a partial resolution to the congruent number problem, and under the Birch and Swinnerton-Dyer conjecture, a full resolution.


In the 1992 article "Extending and Formalizing the Framework for Information Systems Architecture" John F. Sowa and John Zachman present the framework and its recent extensions and show how it can be formalized in the notation of conceptual graphs. Also in 1992: John Zachman's co-author John Sowa proposed the additions of the Scope perspective of the ‘planner’ (bounding lists common to the enterprise and its environment) and the Detailed Representation perspective of the ‘sub-contractor’ (being the out-of-context vendor solution components). The Who, When and Why columns were brought into public view, the notion of the four levels of metaframeworks and a depiction of integration associations across the perspectives were all outlined in the paper. Keri Anderson Healey assisted by creating a model of the models (the framework metamodel) which was also included in the article. Later during the 1990s Methodologists like Clive Finkelstein refocused on the top two framework rows which he labeled Enterprise Engineering and has one of the most successful methods for converging the business needs with information technology engineering implementation, and determining a logical build sequence of the pieces.


The Piotrowski law is a case of the so-called logistic model (cf. logistic equation). It was shown that it covers also language acquisition processes (cf. language acquisition law). Text block law: Linguistic units (e.g. words, letters, syntactic functions and constructions) show a specific frequency distribution in equally large text blocks.


A symplectomorphism is an isomorphism of symplectic manifolds. A permutation is an automorphism of a set. In geometry, isomorphisms and automorphisms are often called transformations, for example rigid transformations, affine transformations, projective transformations.Category theory, which can be viewed as a formalization of the concept of mapping between structures, provides a language that may be used to unify the approach to these different aspects of the basic idea.


In number theory, the prime omega functions ω ( n ) {\displaystyle \omega (n)} and Ω ( n ) {\displaystyle \Omega (n)} count the number of prime factors of a natural number n . {\displaystyle n.} Thereby ω ( n ) {\displaystyle \omega (n)} (little omega) counts each distinct prime factor, whereas the related function Ω ( n ) {\displaystyle \Omega (n)} (big omega) counts the total number of prime factors of n , {\displaystyle n,} honoring their multiplicity (see arithmetic function). That is, if we have a prime factorization of n {\displaystyle n} of the form n = p 1 α 1 p 2 α 2 ⋯ p k α k {\displaystyle n=p_{1}^{\alpha _{1}}p_{2}^{\alpha _{2}}\cdots p_{k}^{\alpha _{k}}} for distinct primes p i {\displaystyle p_{i}} ( 1 ≤ i ≤ k {\displaystyle 1\leq i\leq k} ), then the respective prime omega functions are given by ω ( n ) = k {\displaystyle \omega (n)=k} and Ω ( n ) = α 1 + α 2 + ⋯ + α k {\displaystyle \Omega (n)=\alpha _{1}+\alpha _{2}+\cdots +\alpha _{k}} . These prime factor counting functions have many important number theoretic relations.


In software development, a sanity test (a form of software testing which offers "quick, broad, and shallow testing") evaluates the result of a subset of application functionality to determine whether it is possible and reasonable to proceed with further testing of the entire application. Sanity tests may sometimes be used interchangeably with smoke tests insofar as both terms denote tests which determine whether it is possible and reasonable to continue testing further. On the other hand, a distinction is sometimes made that a smoke test is a non-exhaustive test that ascertains whether the most crucial functions of a programme work before proceeding with further testing whereas a sanity test refers to whether specific functionality such as a particular bug fix works as expected without testing the wider functionality of the software. In other words, a sanity test determines whether the intended result of a code change works correctly while a smoke test ensures that nothing else important was broken in the process.


In probability theory, regular conditional probability is a concept that formalizes the notion of conditioning on the outcome of a random variable. The resulting conditional probability distribution is a parametrized family of probability measures called a Markov kernel.


In model theory, a branch of mathematical logic, and in algebra, the reduced product is a construction that generalizes both direct product and ultraproduct. Let {Si | i ∈ I} be a family of structures of the same signature σ indexed by a set I, and let U be a filter on I. The domain of the reduced product is the quotient of the Cartesian product ∏ i ∈ I S i {\displaystyle \prod _{i\in I}S_{i}} by a certain equivalence relation ~: two elements (ai) and (bi) of the Cartesian product are equivalent if { i ∈ I: a i = b i } ∈ U {\displaystyle \left\{i\in I:a_{i}=b_{i}\right\}\in U} If U only contains I as an element, the equivalence relation is trivial, and the reduced product is just the original Cartesian product. If U is an ultrafilter, the reduced product is an ultraproduct. Operations from σ are interpreted on the reduced product by applying the operation pointwise.


In queueing theory, Bartlett's theorem gives the distribution of the number of customers in a given part of a system at a fixed time.


The Euclidean algorithm for computing the greatest common divisor of two integers is one example. Given two integers a {\displaystyle a} and b {\displaystyle b} , the algorithm performs O ( log  a + log  b ) {\displaystyle O(\log a+\log b)} arithmetic operations on numbers with at most O ( log  a + log  b ) {\displaystyle O(\log a+\log b)} bits. At the same time, the number of arithmetic operations cannot be bounded by the number of integers in the input (which is constant in this case, there are always only two integers in the input).


The computation of the USPCA and CSPCA solutions is demanding when the data matrix is large. With Projection SPCA (PSPCA) approximate CSPCA solutions can computed much more efficiently by simply projecting the current first PC onto subsets of the variables. This means that the solutions can be computed with efficient linear regression routines. PSPCA is fast and can be shown to explain a proportion of the variance of the dataset comparable with that explained by CSPCA.


Cash prizes of varying size, up to US$200,000 (and prizes up to $20,000 awarded), were offered for factorization of some of them. The smallest RSA number was factored in a few days. Most of the numbers have still not been factored and many of them are expected to remain unfactored for many years to come.


In mathematics, Hooley's delta function ( Δ ( n ) {\displaystyle \Delta (n)} ), also called Erdős--Hooley delta-function, defines the maximum number of divisors of n {\displaystyle n} in  {\displaystyle } for all u {\displaystyle u} , where e {\displaystyle e} is the Euler's number. The first few terms of this sequence are 1 , 2 , 1 , 2 , 1 , 2 , 1 , 2 , 1 , 2 , 1 , 3 , 1 , 2 , 2 , 2 , 1 , 2 , 1 , 3 , 2 , 2 , 1 , 4 {\displaystyle 1,2,1,2,1,2,1,2,1,2,1,3,1,2,2,2,1,2,1,3,2,2,1,4} (sequence A226898 in the OEIS).


Hans van der Laan: Modern Primitive. The sequence was described by Ian Stewart in his Scientific American column Mathematical Recreations in June 1996.


In pseudocode the algorithm can be expressed as (0-based array):


Specialized hobbyist websites (like Geocaching.com) FAQ and its Benchmark Hunting forum can provide more information. Some of these marks have precise "adjusted" coordinates (latitude and longitude). The adjusted coordinates are precise to sub-centimeter accuracy, while others, typically true elevation Bench Marks, have only coordinates scaled from a map.


C variables are used directly while register names are quoted as string literals. __asm in Microsoft Visual C++ (MSVC), Borland/Embarcadero C compiler, and descendents. This syntax is not based on ISO rules at all; programmers simply write ASM inside a block without needing to conform to C syntax.


The dispatch has all these three services combined into one dispatch for the best multi-coordinated response to an incident or an emergency. And also facilitates in information management, emergency communication and care of citizens. These services are the main structure for a response to an emergency. It can happen that, for a specific emergency, the co-operation with another service is needed, for instance the Ministry of Defence, water board(s) or Rijkswaterstaat. The safety region can integrate these other services into their structure by adding them to specific conferences on operational or administrative level.All regions operate according to the Coordinated Regional Incident Management system.


The statement that two sets A and B are equinumerous is usually denoted A ≈ B {\displaystyle A\approx B\,} or A ∼ B {\displaystyle A\sim B} , or | A | = | B | . {\displaystyle |A|=|B|.} The definition of equinumerosity using bijections can be applied to both finite and infinite sets, and allows one to state whether two sets have the same size even if they are infinite.


It is possible and sometimes desirable for the condition to always evaluate to be true. This creates an infinite loop. When an infinite loop is created intentionally there is usually another control structure that allows termination of the loop.


Note that the four points x {\displaystyle x} , y {\displaystyle y} , x ∧ y {\displaystyle x\wedge y} , and x ∨ y {\displaystyle x\vee y} form a rectangle in Euclidean space (in the sense that x ∧ y − x = y − x ∨ y {\displaystyle x\wedge y-x=y-x\vee y} , x − x ∨ y = x ∧ y − y {\displaystyle x-x\vee y=x\wedge y-y} , and x ∧ y − x {\displaystyle x\wedge y-x} and x − x ∨ y {\displaystyle x-x\vee y} are orthogonal). On the other hand, supermodularity and concavity together guarantee that u ( x ∨ y − λ v ) − u ( y ) ≥ u ( x ) − u ( x ∧ y + λ v ) . {\displaystyle u(x\vee y-\lambda v)-u(y)\geq u(x)-u(x\wedge y+\lambda v).} for any λ ∈  {\displaystyle \lambda \in } , where v = y − x ∧ y = x ∨ y − x {\displaystyle v=y-x\wedge y=x\vee y-x} . In this case, crucially, the four points x {\displaystyle x} , y {\displaystyle y} , x ∨ y − λ v {\displaystyle x\vee y-\lambda v} , and x ∧ y + λ v {\displaystyle x\wedge y+\lambda v} form a backward-leaning parallelogram in Euclidean space.


The standard specifies that the correct encoding of a code point uses only the minimum number of bytes required to hold the significant bits of the code point. Longer encodings are called overlong and are not valid UTF-8 representations of the code point. This rule maintains a one-to-one correspondence between code points and their valid encodings, so that there is a unique valid encoding for each code point. This ensures that string comparisons and searches are well-defined.


In morphology, a null morpheme or zero morpheme is a morpheme that has no phonetic form. In simpler terms, a null morpheme is an "invisible" affix. It is a concept useful for analysis, by contrasting null morphemes with alternatives that do have some phonetic realization.


That cannot be your task. There is no such thing as valid jury nullification. You would violate your oath and the law if you willfully brought a verdict contrary to the law given to you in this case." The Ninth Circuit upheld the first three sentences of the jury's instruction and overruled the remainder but deemed that instruction a harmless error and affirmed the conviction.


Again, it depends on the context how exactly the word "filtration" is to be understood. Descending filtrations are not to be confused with the dual notion of cofiltrations (which consist of quotient objects rather than subobjects). Filtrations are widely used in abstract algebra, homological algebra (where they are related in an important way to spectral sequences), and in measure theory and probability theory for nested sequences of σ-algebras. In functional analysis and numerical analysis, other terminology is usually used, such as scale of spaces or nested spaces.


In BNPF encoding, a single byte (8 bits) would be represented by a highly redundant character framing sequence starting with a single ASCII "B", eight ASCII characters where a "0" would be represented by a "N" and a "1" would be represented by a "P", followed by an ending ASCII "F". These ten-character ASCII sequences were separated by one or more whitespace characters, therefore using at least eleven ASCII characters for each byte stored (9% efficiency). The ASCII "N" and "P" characters differed in four bit positions, providing excellent protection from single punch errors. Alternative schemes were also available where "L" and "H" or "0" and "1" were also available to represent data bits, but in both of these encoding schemes, the two data-bearing ASCII characters differ in only one bit position, providing very poor single punch error detection.


In the following definitions, x T {\displaystyle \mathbf {x} ^{\textsf {T}}} is the transpose of x {\displaystyle \mathbf {x} } , x ∗ {\displaystyle \mathbf {x} ^{*}} is the conjugate transpose of x {\displaystyle \mathbf {x} } and 0 {\displaystyle \mathbf {0} } denotes the n-dimensional zero-vector.


Alice sends | ψ ⟩ {\displaystyle |\psi \rangle } over a public and authenticated quantum channel E {\displaystyle {\mathcal {E}}} to Bob. Bob receives a state E ( ρ ) = E ( | ψ ⟩ ⟨ ψ | ) {\displaystyle {\mathcal {E}}(\rho )={\mathcal {E}}(|\psi \rangle \langle \psi |)} , where E {\displaystyle {\mathcal {E}}} represents both the effects of noise in the channel and eavesdropping by a third party we'll call Eve. After Bob receives the string of qubits, both Bob and Eve have their own states.


In the early days of computing the total capacity of HDDs was specified in seven to nine decimal digits frequently truncated with the idiom millions. By the 1970s, the total capacity of HDDs was given by manufacturers using SI decimal prefixes such as megabytes (1 MB = 1,000,000 bytes), gigabytes (1 GB = 1,000,000,000 bytes) and terabytes (1 TB = 1,000,000,000,000 bytes). However, capacities of memory are usually quoted using a binary interpretation of the prefixes, i.e. using powers of 1024 instead of 1000.


The formula is used in calculating the frequency of each note in the piece. The values are then added together and divided by the number of notes. This is the average frequency of those notes. It is said that such techniques were used by classical composers, especially those who involved mathematics heavily in their music.


The general number field sieve, on the other hand, manages to search for smooth numbers that are subexponential in the size of n. Since these numbers are smaller, they are more likely to be smooth than the numbers inspected in previous algorithms. This is the key to the efficiency of the number field sieve. In order to achieve this speed-up, the number field sieve has to perform computations and factorizations in number fields. This results in many rather complicated aspects of the algorithm, as compared to the simpler rational sieve. The size of the input to the algorithm is log2 n or the number of bits in the binary representation of n. Any element of the order nc for a constant c is exponential in log n. The running time of the number field sieve is super-polynomial but sub-exponential in the size of the input.


In modular arithmetic, any unit fraction can be converted into an equivalent whole number using the extended Euclidean algorithm. This conversion can be used to perform modular division: dividing by a number x {\displaystyle x} , modulo y {\displaystyle y} , can be performed by converting the unit fraction 1 / x {\displaystyle 1/x} into an equivalent whole number modulo y {\displaystyle y} , and then multiplying by that number.In more detail, suppose that x {\displaystyle x} is relatively prime to y {\displaystyle y} (otherwise, division by x {\displaystyle x} is not defined modulo y {\displaystyle y} ). The extended Euclidean algorithm for the greatest common divisor can be used to find integers a {\displaystyle a} and b {\displaystyle b} such that Bézout's identity is satisfied: In modulo- y {\displaystyle y} arithmetic, the term b y {\displaystyle by} can be eliminated as it is zero modulo y {\displaystyle y} . This leaves That is, a {\displaystyle a} is the modular inverse of x {\displaystyle x} , the number that when multiplied by x {\displaystyle x} produces one. Equivalently, Thus division by x {\displaystyle x} (modulo y {\displaystyle y} ) can instead be performed by multiplying by the integer a {\displaystyle a} .


In the 1760s, Johann Heinrich Lambert was the first to prove that the number π is irrational, meaning it cannot be expressed as a fraction a / b {\displaystyle a/b} , where a {\displaystyle a} and b {\displaystyle b} are both integers. In the 19th century, Charles Hermite found a proof that requires no prerequisite knowledge beyond basic calculus. Three simplifications of Hermite's proof are due to Mary Cartwright, Ivan Niven, and Nicolas Bourbaki.


With some luck it will be the target of the search. If the target is not found by the time the search pattern has reached maximum convenient radius, the centre point may be shifted and another search started. This can be repeated as often as necessary, but the positions of the centre points must be chosen to allow the full search area to be covered.


(where φ ( β / α ) {\displaystyle \varphi (\beta /\alpha )} denotes the formula formed by substituting all free occurrences of the variable α {\displaystyle \alpha } in φ {\displaystyle \varphi } by β {\displaystyle \beta } .) Likewise, finding a counterexample disproves (proves the negation of) a universal conclusion. This is used in a proof by contradiction.


Even the "system specific" code needs modification only slightly from implementation to implementation. The result is great amounts of implementation work saved while porting the system. Finally, it was an error to enter into a second round of optimization before bootstrapping the compiler.


In mathematics, a differential invariant is an invariant for the action of a Lie group on a space that involves the derivatives of graphs of functions in the space. Differential invariants are fundamental in projective differential geometry, and the curvature is often studied from this point of view. Differential invariants were introduced in special cases by Sophus Lie in the early 1880s and studied by Georges Henri Halphen at the same time. Lie (1884) was the first general work on differential invariants, and established the relationship between differential invariants, invariant differential equations, and invariant differential operators.


In order to understand how sounds are made, experimental procedures are often adopted. Palatography is one of the oldest instrumental phonetic techniques used to record data regarding articulators. In traditional, static palatography, a speaker's palate is coated with a dark powder. The speaker then produces a word, usually with a single consonant.


In the 1960s, the term double dabble was also used for a different mental algorithm, used by programmers to convert a binary number to decimal. It is performed by reading the binary number from left to right, doubling if the next bit is zero, and doubling and adding one if the next bit is one. In the example above, 11110011, the thought process would be: "one, three, seven, fifteen, thirty, sixty, one hundred twenty-one, two hundred forty-three", the same result as that obtained above.


Using matrices, this formula can be written x o l d = A x n e w , {\displaystyle \mathbf {x} _{\mathrm {old} }=A\,\mathbf {x} _{\mathrm {new} },} where "old" and "new" refer respectively to the firstly defined basis and the other basis, x o l d {\displaystyle \mathbf {x} _{\mathrm {old} }} and x n e w {\displaystyle \mathbf {x} _{\mathrm {new} }} are the column vectors of the coordinates of the same vector on the two bases, and A {\displaystyle A} is the change-of-basis matrix (also called transition matrix), which is the matrix whose columns are the coordinate vectors of the new basis vectors on the old basis. This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces.


In mathematical analysis, Agmon's inequalities, named after Shmuel Agmon, consist of two closely related interpolation inequalities between the Lebesgue space L ∞ {\displaystyle L^{\infty }} and the Sobolev spaces H s {\displaystyle H^{s}} . It is useful in the study of partial differential equations. Let u ∈ H 2 ( Ω ) ∩ H 0 1 ( Ω ) {\displaystyle u\in H^{2}(\Omega )\cap H_{0}^{1}(\Omega )} where Ω ⊂ R 3 {\displaystyle \Omega \subset \mathbb {R} ^{3}} . Then Agmon's inequalities in 3D state that there exists a constant C {\displaystyle C} such that ‖ u ‖ L ∞ ( Ω ) ≤ C ‖ u ‖ H 1 ( Ω ) 1 / 2 ‖ u ‖ H 2 ( Ω ) 1 / 2 , {\displaystyle \displaystyle \|u\|_{L^{\infty }(\Omega )}\leq C\|u\|_{H^{1}(\Omega )}^{1/2}\|u\|_{H^{2}(\Omega )}^{1/2},} and ‖ u ‖ L ∞ ( Ω ) ≤ C ‖ u ‖ L 2 ( Ω ) 1 / 4 ‖ u ‖ H 2 ( Ω ) 3 / 4 .


In number theory, the numbers of the form x2 + xy + y2 for integer x, y are called the Löschian numbers (or Loeschian numbers). These numbers are named after August Lösch. They are the norms of the Eisenstein integers. They are a set of whole numbers, including zero, and having prime factorization in which all primes congruent to 2 mod 3 have even powers (there is no restriction of primes congruent to 0 or 1 mod 3).


Heyting implication, Y ⇒ Z {\displaystyle Y\Rightarrow Z} , is an alternative notation for Z Y {\displaystyle Z^{Y}} . The above adjunction results translate to implication ( ⇒: H × H → H {\displaystyle \Rightarrow :H\times H\to H} ) being right adjoint to meet ( ∧: H × H → H {\displaystyle \wedge :H\times H\to H} ). This adjunction can be written as ( − ∧ Y ) ⊣ ( Y ⇒ − ) {\displaystyle (-\wedge Y)\dashv (Y\Rightarrow -)} , or more fully as: In the category of topological spaces, the exponential object Z Y {\displaystyle Z^{Y}} exists provided that Y {\displaystyle Y} is a locally compact Hausdorff space.


In the European Union, RRM data analysis showed that authentic information was "de-contextualized", "manipulated" and "distorted", then used by "questionable" writers on "untrustworthy" sites to "seed conversations", which were then "framed using a divisive and inflammatory narrative." These would reference the original authentic source. In a "coordinated fashion" the information would be "amplified" with the most "susceptible communities" targeted.


In elliptic geometry, an elliptic rectangle is a figure in the elliptic plane whose four edges are elliptic arcs which meet at equal angles greater than 90°. Opposite arcs are equal in length. In hyperbolic geometry, a hyperbolic rectangle is a figure in the hyperbolic plane whose four edges are hyperbolic arcs which meet at equal angles less than 90°. Opposite arcs are equal in length.


In telecommunications, transmission is the process of sending or propagating an analog or digital signal via a medium that is wired, wireless, or fiber-optic.Transmission system technologies typically refer to physical layer protocol duties such as modulation, demodulation, line coding, equalization, error control, bit synchronization and multiplexing, but it may also involve higher-layer protocol duties, for example, digitizing an analog signal, and data compression. Transmission of a digital message, or of a digitized analog signal, is known as data transmission. Examples of transmission are the sending of signals with limited duration, for example, a block or packet of data, a phone call, or an email.


In mathematics, the binary octahedral group, name as 2O or ⟨2,3,4⟩ is a certain nonabelian group of order 48. It is an extension of the chiral octahedral group O or (2,3,4) of order 24 by a cyclic group of order 2, and is the preimage of the octahedral group under the 2:1 covering homomorphism Spin  ( 3 ) → SO  ( 3 ) {\displaystyle \operatorname {Spin} (3)\to \operatorname {SO} (3)} of the special orthogonal group by the spin group. It follows that the binary octahedral group is a discrete subgroup of Spin(3) of order 48. The binary octahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, under the isomorphism Spin  ( 3 ) ≅ Sp  ( 1 ) {\displaystyle \operatorname {Spin} (3)\cong \operatorname {Sp} (1)} where Sp(1) is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on quaternions and spatial rotations.)


The optional Selector Channel also enabled the use of other high speed devices such as the 1200 lpm 0776 printer or the 2000 lpm 0770 printer. The machine had either 4K or 16K memory chips, and typical machines had between 128 and 512 KiB memory. It ran an OS called OS/3, and could run up to 7 jobs at one time, not counting various OS extensions such as the print spooler and telecommunications access (ICAM).


Lastly, the difference between the two highest values of FEV1 should also be within 150 mL. The highest FVC and FEV1 may be used from each different test. Until the results of three tests meet the criteria of reproducibility, the test can be repeated up to eight times. If it is still not possible to get accurate results, the best three tests are used.


For example, right triangles, isosceles triangles and equilateral triangles are non-generic and non-degenerate. In fact, degenerate cases often correspond to singularities, either in the object or in some configuration space. For example, a conic section is degenerate if and only if it has singular points (e.g., point, line, intersecting lines).


These incorrect responses often matched the incorrect response of the majority group (i.e., actors). Overall, 75% of participants gave at least one incorrect answer out of the 12 critical trials. In his opinion regarding the study results, Asch put it this way: "That intelligent, well-meaning, young people are willing to call white black is a matter of concern."


397-398).Cook and Reckhow (1973) say it most succinctly: The indirect instructions are necessary in order for a fixed program to access an unbounded number of registers as the inputs vary." (p. 73)Unbounded capacities of registers versus bounded capacities of state-machine instructions: The so-called finite state part of the machine is supposed to be – by the normal definition of algorithm – very finite both in the number of "states" (instructions) and the instructions' sizes (their capacity to hold symbols/signs).


In statistics, the generalized Dirichlet distribution (GD) is a generalization of the Dirichlet distribution with a more general covariance structure and almost twice the number of parameters. Random vectors with a GD distribution are completely neutral.The density function of p 1 , … , p k − 1 {\displaystyle p_{1},\ldots ,p_{k-1}} is  − 1 p k b k − 1 − 1 ∏ i = 1 k − 1  {\displaystyle \left^{-1}p_{k}^{b_{k-1}-1}\prod _{i=1}^{k-1}\left} where we define p k = 1 − ∑ i = 1 k − 1 p i {\textstyle p_{k}=1-\sum _{i=1}^{k-1}p_{i}} . Here B ( x , y ) {\displaystyle B(x,y)} denotes the Beta function. This reduces to the standard Dirichlet distribution if b i − 1 = a i + b i {\displaystyle b_{i-1}=a_{i}+b_{i}} for 2 ⩽ i ⩽ k − 1 {\displaystyle 2\leqslant i\leqslant k-1} ( b 0 {\displaystyle b_{0}} is arbitrary).


In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set. For example, {1, 2, 3} and {4, 5, 6} are disjoint sets, while {1, 2, 3} and {3, 4, 5} are not disjoint. A collection of two or more sets is called disjoint if any two distinct sets of the collection are disjoint.


The child element is within the parent element, such as in a Venn diagram. This structure is most effective in representing simple hierarchical relationships. For example, when directing someone to open a file on a computer desktop, one may first direct them towards the main folder, then the subfolders within the main folder.


In statistics, modes of variation are a continuously indexed set of vectors or functions that are centered at a mean and are used to depict the variation in a population or sample. Typically, variation patterns in the data can be decomposed in descending order of eigenvalues with the directions represented by the corresponding eigenvectors or eigenfunctions. Modes of variation provide a visualization of this decomposition and an efficient description of variation around the mean. Both in principal component analysis (PCA) and in functional principal component analysis (FPCA), modes of variation play an important role in visualizing and describing the variation in the data contributed by each eigencomponent. In real-world applications, the eigencomponents and associated modes of variation aid to interpret complex data, especially in exploratory data analysis (EDA).


In the case of the leaky bucket algorithm as a queue, the only defined limit for this algorithm is the bandwidth of its output.The bandwidth limit for the connection may be specified in a traffic contract. A bandwidth limit may be specified as a packet or frame rate, a byte or bit rate, or as an emission interval between the packets.


In modern mathematics, one often studies spaces whose points are themselves mathematical objects. A distance function on such a space generally aims to measure the dissimilarity between two objects. Here are some examples: Functions to a metric space. If X is any set and M is a metric space, then the set of all bounded functions f: X → M {\displaystyle f\colon X\to M} (i.e. those functions whose image is a bounded subset of M {\displaystyle M} ) can be turned into a metric space by defining the distance between two bounded functions f and g to be This metric is called the uniform metric or supremum metric.


In the United Kingdom, the Official Charts Company compiles a weekly compilation albums chart, limited to various artists compilations and soundtrack compilations.


In probability theory, the law (or formula) of total probability is a fundamental rule relating marginal probabilities to conditional probabilities. It expresses the total probability of an outcome which can be realized via several distinct events, hence the name.


Here, the word "broadcast" is used in the sense of conveying the state to two or more recipients. For multiple recipients to each receive the state, there must be, in some sense, a way of duplicating the state. The no-broadcast theorem generalizes the no-cloning theorem for mixed states.


Later releases added the ability to write unformatted dumps, called at that time core image dumps. In modern operating systems, a process address space may contain gaps, and it may share pages with other processes or files, so more elaborate representations are used; they may also include other information about the state of the program at the time of the dump. In Unix-like systems, core dumps generally use the standard executable image-format: a.out in older versions of Unix, ELF in modern Linux, System V, Solaris, and BSD systems, Mach-O in macOS, etc.


Various multi-compartment models can be used in the areas of pharmacokinetics and pharmacology, in the support of efforts in drug discovery, and in environmental science. In humans and related organisms, there are five major body compartments: the blood plasma, interstitial fluids, fat tissues, intracellular fluids, and transcellular fluids, the latter of which includes fluids in the pleural (peritoneal) cavity. The relative percents of body mass of these are included in the following table.


In statistics, intra-rater reliability is the degree of agreement among repeated administrations of a diagnostic test performed by a single rater. Intra-rater reliability and inter-rater reliability are aspects of test validity.


It not only breaks the isolation that operating systems should provide between programs and hardware, raising both stability and security concerns, but also could leave the graphics hardware in an inconsistent state if two or more user space programs try to do the mode-setting at the same time. To avoid these conflicts, the X Server became in practice the only user space program that performed mode-setting operations; the remainder user space programs relied on the X Server to set the appropriate mode and to handle any other operation involving mode-setting. Initially the mode-setting was performed exclusively during the X Server startup process, but later the X Server gained the ability to do it while running.


{\displaystyle {\frac {\sum _{v}d(v)^{2}}{2|E|}}={\frac {|V|}{2|E|}}(\mu ^{2}+\sigma ^{2})={\frac {\mu ^{2}+\sigma ^{2}}{\mu }}=\mu +{\frac {\sigma ^{2}}{\mu }}.} For a graph that has vertices of varying degrees (as is typical for social networks), σ 2 {\displaystyle {\sigma }^{2}} is strictly positive, which implies that the average degree of a friend is strictly greater than the average degree of a random node.


The consumer has income I {\displaystyle I} , and hence a budget set of affordable packages B ( p , I ) = { x: p ⋅ x ≤ I } , {\displaystyle B(p,I)=\{x:p\cdot x\leq I\},} where p ⋅ x = ∑ i L p i x i {\displaystyle p\cdot x=\sum _{i}^{L}p_{i}x_{i}} is the dot product of the price and quantity vectors. The consumer has a utility function u: R + L → R . {\displaystyle u:\mathbb {R} _{+}^{L}\rightarrow \mathbb {R} .} The consumer's Marshallian demand correspondence is defined to be x ∗ ( p , I ) = argmax x ∈ B ( p , I )  u ( x ) {\displaystyle x^{*}(p,I)=\operatorname {argmax} _{x\in B(p,I)}u(x)}


In business telephony, a telephone extension may refer to a phone on an internal telephone line attached to a private branch exchange (PBX) or Centrex system. The PBX operates much as a community switchboard does for a geographic telephone numbering plan and allows multiple lines inside the office to connect without each phone requiring a separate outside line. In these systems, one usually has to dial a number (typically 9 in North America, 0 in Europe) to tell the PBX to connect with an outside landline (also called DDCO, or Direct Dial Central Office) to dial an external number.


In order to use a hurricane tracking chart, one needs access to latitude/longitude pairs of the cyclone's center and maximum sustained wind information in order to know which symbol to depict. New tropical cyclone information is available at least every twelve hours in the Southern Hemisphere and at least every six hours in the Northern Hemisphere from Regional Specialized Meteorological Centers and Tropical Cyclone Warning Centers.In decades past, newspaper, television, and radio (including weather radio) were primary sources for this information. Local television stations within threatened markets would advertise tropical cyclone positions within the morning, evening, and nightly news during their weather segments. The Weather Channel includes the information within their tropical updates every hour during the Atlantic and Pacific hurricane seasons. Starting in the mid 1990s, the World Wide Web allowed for the development of ftp and web sites by the Bureau of Meteorology in Australia, Canadian Hurricane Centre, Central Pacific Hurricane Center, the Nadi Tropical Cyclone Centre/Fiji Meteorological Service, Japan Meteorological Agency, Joint Typhoon Warning Center, Météo-France La Réunion, National Hurricane Center, and the Philippine Atmospheric, Geophysical and Astronomical Services Administration which allows the end user to get their information from their official products.


In modern usage, metric is used almost exclusively in commercial transactions. These units are mostly historical, although they are still used in some limited contexts and in Maltese idioms and set phrases. Many of these terms are directly related to Arabic units and some to Sicilian units. The Weights and Measures Ordinance of 1921 established uniformity in the conversion of such weights and measures. All these measures were defined as simple multiples of the Imperial units then in use in Britain.


In mathematics, the Chung–Fuchs theorem, named after Chung Kai-lai and Wolfgang Heinrich Johannes Fuchs, states that for a particle undergoing a random walk in m-dimensions, it is certain to come back infinitely often to any neighborhood of the origin on a one-dimensional line (m = 1) or two-dimensional plane (m = 2), but in three or more dimensional spaces it will leave to infinity. Specifically, if a position of the particle is described by the vector X n {\displaystyle X_{n}}: where Z 1 , Z 2 , … , Z n {\displaystyle Z_{1},Z_{2},\dots ,Z_{n}} are independent m-dimensional vectors with a given multivariate distribution, then if m = 1 {\displaystyle m=1} , E ( | Z i | ) < ∞ {\displaystyle E(|Z_{i}|)<\infty } and E ( Z i ) = 0 {\displaystyle E(Z_{i})=0} , or if m = 2 {\displaystyle m=2} E ( | Z i 2 | ) < ∞ {\displaystyle E(|Z_{i}^{2}|)<\infty } and E ( Z i ) = 0 {\displaystyle E(Z_{i})=0} , the following holds: However, for m ≥ 3 {\displaystyle m\geq 3} ,


That means we can compute and store f {\displaystyle f} on all the possible values in T = ( log  log  q ) m {\displaystyle T=(\log \log q)^{m}} time and space. If we take d = log  q {\displaystyle d=\log q} , we get m = log  n log  log  q {\displaystyle m={\tfrac {\log n}{\log \log q}}} , so the time/space requirement is just n log  log  q log  log  log  q . {\displaystyle n^{\frac {\log \log q}{\log \log \log q}}.} Kedlaya and Umans further show how to combine this preprocessing with fast (FFT) multipoint evaluation. This allows optimal algorithms for many important algebraic problems, such as polynomial modular composition.


Bot is a natural type for the "null pointer" value (a pointer which does not point to any object) of languages like Java: in Java, the null type is the universal subtype of reference types. null is the only value of the null type; and it can be cast to any reference type. However, the null type is not a bottom type as described above, it is not a subtype of int and other primitive types. A type system including both Top and Bot seems to be a natural target for type inference, allowing the constraints on an omitted type parameter to be captured by a pair of bounds: we write S<:X<:T to mean "the value of X must lie somewhere between S and T." In such a scheme, a completely unconstrained parameter is bounded below by Bot and above by Top.


In the remainder of this article, base ten is assumed. The single-digit final state reached in the process of calculating an integer's additive persistence is its digital root. Put another way, a number's additive persistence counts how many times we must sum its digits to arrive at its digital root.


In the board game industry, playtesting applies both to feedback gathered during the early design process as well as late stage exposure to the target audience by a game's publisher. Major types of boardgame testing include local testing — where a designer, developer, or publisher representative moderates the test in person, and remote testing, where groups receive copies of the game or files to assemble their own version.Wizards of the Coast ran a public playtest of their new Dungeon Command miniatures game. In this case, they used the feedback generated on the rules to improve the game but also used feedback on the playtest itself to improve logistics on the D&D Next playtest.Steve Jackson Games uses Munchkin players from the area around their offices to test new cards and expansions, as well as distributing playtest packages at conventions. According to the SJG website, this is done "so we  can observe carefully which cards work well, which jokes aren't as funny as we thought, and so on."


For instance, the XML document and the ASCII tree have the same structure. XML Trees do not show the content in an Instance document, only the structure of the document. In this example Product is the Root Element of the tree and the two child nodes of Product are Name and Details. Details contains two child nodes, Description and Price. The tree command in Windows and *nix also produce a similar tree structure and path.


In order for a table to be in sixth normal form, it has to be in fifth normal form first and then it requires that each table satisfies only trivial join dependencies. Let’s take a simple example with a table already in 5NF: Here, in the users table, every attribute is non null and the primary key is the username: Users_table This table is in 5NF because each join dependency is implied by the unique candidate key of the table (Username). More specifically, the only possible join dependencies are: {username, status}, {username, department}. The 6NF version would look like this: Users Users_dept So, from one table in 5NF, 6NF produces two tables.


In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a totally ordered set. There are several variants and generalizations of the lexicographical ordering. One variant applies to sequences of different lengths by comparing the lengths of the sequences before considering their elements. Another variant, widely used in combinatorics, orders subsets of a given finite set by assigning a total order to the finite set, and converting subsets into increasing sequences, to which the lexicographical order is applied. A generalization defines an order on a Cartesian product of partially ordered sets; this order is a total order if and only if all factors of the Cartesian product are totally ordered.


By 1995, the Internet was fully commercialized in the U.S. when the NSFNet was decommissioned, removing the last restrictions on use of the Internet to carry commercial traffic. As technology advanced and commercial opportunities fueled reciprocal growth, the volume of Internet traffic started experiencing similar characteristics as that of the scaling of MOS transistors, exemplified by Moore's law, doubling every 18 months.


In standard truth-functional propositional logic, distribution in logical proofs uses two valid rules of replacement to expand individual occurrences of certain logical connectives, within some formula, into separate applications of those connectives across subformulas of the given formula. The rules are where " ⇔ {\displaystyle \Leftrightarrow } ", also written ≡ , {\displaystyle \,\equiv ,\,} is a metalogical symbol representing "can be replaced in a proof with" or "is logically equivalent to".


In mathematics, in the area of quantum information geometry, the Bures metric (named after Donald Bures) or Helstrom metric (named after Carl W. Helstrom) defines an infinitesimal distance between density matrix operators defining quantum states. It is a quantum generalization of the Fisher information metric, and is identical to the Fubini–Study metric when restricted to the pure states alone.


The cylinder was rotated every 24 hours, providing an approximate time for a given quake.Luigi Palmieri, influenced by Mallet's 1848 paper, invented a seismometer in 1856 that could record the time of an earthquake. This device used metallic pendulums which closed an electric circuit with vibration, which then powered an electromagnet to stop a clock.


(e.g. in a claim Bob instance of Human, Bob is the subject and an Instance, while the object, Human, is an ordinary class; but a further claim that Human instance of Animal species makes "Animal species" a metaclass because it has a member, "Human", that is also a Class). OWL 2 DL supports metaclasses by a feature called punning, in which one entity is interpreted as two different types of thing—a class and an individual—depending on its syntactic context. For example, through punning, an ontology could have a concept hierarchy such as Harry the eagle instance of golden eagle, golden eagle subclass of bird, and golden eagle instance of species.


In mathematics and its applications, the root mean square of a set of numbers x i {\displaystyle x_{i}} (abbreviated as RMS, RMS or rms and denoted in formulas as either x R M S {\displaystyle x_{\mathrm {RMS} }} or R M S x {\displaystyle \mathrm {RMS} _{x}} ) is defined as the square root of the mean square (the arithmetic mean of the squares) of the set. The RMS is also known as the quadratic mean (denoted M 2 {\displaystyle M_{2}} ) and is a particular case of the generalized mean. The RMS of a continuously varying function (denoted f R M S {\displaystyle f_{\mathrm {RMS} }} ) can be defined in terms of an integral of the squares of the instantaneous values during a cycle. For alternating electric current, RMS is equal to the value of the constant direct current that would produce the same power dissipation in a resistive load. In estimation theory, the root-mean-square deviation of an estimator is a measure of the imperfection of the fit of the estimator to the data.


This could be confused with the Greek letter Theta on a badly focused display, but in practice there was no confusion because theta was not (then) a displayable character and very little used anyway. An alternative, the slashed zero (looking similar to the letter O except for the slash), was primarily used in hand-written coding sheets before transcription to punched cards or tape, and is also used in old-style ASCII graphic sets descended from the default typewheel on the Teletype Model 33 ASR. This form is similar to the symbol ∅ {\displaystyle \emptyset } , or "∅" (Unicode character U+2205), representing the empty set, as well as to the letter Ø used in several Scandinavian languages.


In processor design, there are two ways to increase on-chip parallelism with fewer resource requirements: one is superscalar technique which tries to exploit instruction-level parallelism (ILP); the other is multithreading approach exploiting thread-level parallelism (TLP). Superscalar means executing multiple instructions at the same time while thread-level parallelism (TLP) executes instructions from multiple threads within one processor chip at the same time. There are many ways to support more than one thread within a chip, namely: Interleaved multithreading: Interleaved issue of multiple instructions from different threads, also referred to as temporal multithreading. It can be further divided into fine-grained multithreading or coarse-grained multithreading depending on the frequency of interleaved issues.


Consequently, an irreducible ideal of a Noetherian ring is primary. Various methods of generalizing primary ideals to noncommutative rings exist, but the topic is most often studied for commutative rings. Therefore, the rings in this article are assumed to be commutative rings with identity.


In mathematics, a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a set defines a partition of this set, and every partition defines an equivalence relation. A set equipped with an equivalence relation or a partition is sometimes called a setoid, typically in type theory and proof theory.


In statistics, a sampling frame is the source material or device from which a sample is drawn. It is a list of all those within a population who can be sampled, and may include individuals, households or institutions.Importance of the sampling frame is stressed by Jessen and Salant and Dillman. In many practical situations the frame is a matter of choice to the survey planner, and sometimes a critical one. Some very worthwhile investigations are not undertaken at all because of the lack of an apparent frame; others, because of faulty frames, have ended in a disaster or in cloud of doubt.


In the Kochen–Specker article the possibility is discussed that the value attribution v ( A ) {\displaystyle v(\mathbf {A} )} may be context-dependent, i.e. observables corresponding to equal vectors in different columns of the table need not have equal values because different columns correspond to different measurement arrangements. Since subquantum reality (as described by the hidden-variable theory) may be dependent on the measurement context, it is possible that relations between quantum-mechanical observables and hidden variables are just homomorphic rather than isomorphic. This would make obsolete the requirement of a context-independent value attribution. Hence, the KS theorem only excludes noncontextual hidden-variable theories. The possibility of contextuality has given rise to the so-called modal interpretations of quantum mechanics.


In mathematics, finite-dimensional distributions are a tool in the study of measures and stochastic processes. A lot of information can be gained by studying the "projection" of a measure (or process) onto a finite-dimensional vector space (or finite collection of times).


In the SubBytes step, each byte a i , j {\displaystyle a_{i,j}} in the state array is replaced with a SubByte S ( a i , j ) {\displaystyle S(a_{i,j})} using an 8-bit substitution box. Note that before round 0, the state array is simply the plaintext/input. This operation provides the non-linearity in the cipher.


In number theory, zero-sum problems are certain kinds of combinatorial problems about the structure of a finite abelian group. Concretely, given a finite abelian group G and a positive integer n, one asks for the smallest value of k such that every sequence of elements of G of size k contains n terms that sum to 0. The classic result in this area is the 1961 theorem of Paul Erdős, Abraham Ginzburg, and Abraham Ziv. They proved that for the group Z / n Z {\displaystyle \mathbb {Z} /n\mathbb {Z} } of integers modulo n, Explicitly this says that any multiset of 2n − 1 integers has a subset of size n the sum of whose elements is a multiple of n, but that the same is not true of multisets of size 2n − 2.


In physics, Averroes did not adopt the inductive method that was being developed by Al-Biruni in the Islamic world and is closer to today's physics. Rather, he was—in the words of historian of science Ruth Glasner—a "exegetical" scientist who produced new theses about nature through discussions of previous texts, especially the writings of Aristotle. because of this approach, he was often depicted as an unimaginative follower of Aristotle, but Glasner argues that Averroes's work introduced highly original theories of physics, especially his elaboration of Aristotle's minima naturalia and on motion as forma fluens, which were taken up in the west and are important to the overall development of physics. Averroes also proposed a definition of force as "the rate at which work is done in changing the kinetic condition of a material body"—a definition close to that of power in today's physics.


For example, a typical server runs at 425 W and VMware estimates a hardware reduction ratio of up to 15:1. A virtual machine (VM) can be more easily controlled and inspected from a remote site than a physical machine, and the configuration of a VM is more flexible. This is very useful in kernel development and for teaching operating system courses, including running legacy operating systems that do not support modern hardware.


In multi-stage job scheduling problems, there are other options for the machine environments: O: Open-shop problem. Every job j {\displaystyle j} consists of m {\displaystyle m} operations O i j {\displaystyle O_{ij}} for i = 1 , … , m {\displaystyle i=1,\ldots ,m} .


In response, the field of integrity management has been born to help clients conduct business, even in the most challenging of markets, without compromising their ethics. With the help of expert advice, companies can go beyond abiding by the law to taking a voluntary, proactive approach to ensuring a company's activities promote behaving responsibly, with fairness, sustainability, and cultural sensitivity in the communities in which they operate. It is very different in this respect from the more reactive field of risk management, although some risk management companies have attempted to embrace ethical risk as an area of specialism. Many risk management consultancies are adjuncts to private security companies.


{\displaystyle ~-2\ln()=-2\;\sum _{i=1}^{k}x_{i}\ln \left({\frac {\pi _{i}}{p_{i}}}\right)~.} (The factor − 2 {\displaystyle ~-2~} is chosen to make the statistic asymptotically chi-squared distributed, for convenient comparison to a familiar statistic commonly used for the same application.) If the null hypothesis is true, then as N {\displaystyle ~N~} increases, the distribution of − 2 ln  (  ) {\displaystyle ~-2\ln()~} converges to that of chi-squared with k − 1 {\displaystyle ~k-1~} degrees of freedom.