In 1964, IBM introduced its System/360 series which used microcode to allow a single expansive instruction set architecture (ISA) to run across a wide variety of machines by implementing more or less instructions in hardware depending on the need. This allowed the ISA to be expansive, and this became the paragon of computer design in the 1960s and 70s, the so-called orthogonal design. This style of memory access with wide variety of modes led to instruction sets with hundreds of different instructions, a style known today as CISC (Complex Instruction Set Computing).


In the UK "abandoned" calls must not be called back within 72 hours unless there is a dedicated agent available. GDPR (General Data Protection Regulation) also requires that the purpose of any call be related to the reason for collecting data in the first place.In the United States, if someone answers but no agent is available within 2 seconds of the person's greeting, Federal Communications Commission (FCC) regulations consider the call "abandoned" and require the dialer to play a recorded message. The FCC requires that predictive dialers abandon less than 3% of answered calls.In 1991 the Telephone Consumer Protection Act prohibited the use of an “automatic telephone dialing system” to contact “any telephone number assigned to a mobile telephone service” without “express prior consent” from the party being called.


In modular representation theory, while Maschke's theorem does not hold when the characteristic divides the group order, the group algebra may be decomposed as the direct sum of a maximal collection of two-sided ideals known as blocks. When the field F has characteristic 0, or characteristic coprime to the group order, there is still such a decomposition of the group algebra F as a sum of blocks (one for each isomorphism type of simple module), but the situation is relatively transparent when F is sufficiently large: each block is a full matrix algebra over F, the endomorphism ring of the vector space underlying the associated simple module. To obtain the blocks, the identity element of the group G is decomposed as a sum of primitive idempotents in Z(R), the center of the group algebra over the maximal order R of F. The block corresponding to the primitive idempotent e is the two-sided ideal e R. For each indecomposable R-module, there is only one such primitive idempotent that does not annihilate it, and the module is said to belong to (or to be in) the corresponding block (in which case, all its composition factors also belong to that block). In particular, each simple module belongs to a unique block. Each ordinary irreducible character may also be assigned to a unique block according to its decomposition as a sum of irreducible Brauer characters. The block containing the trivial module is known as the principal block.


For example, to the stem of bog (to move) is added -a giving as its subjunctive in the first person boga mé: First conjugation: Second conjugation: E.g. "go mbeannaí Dia thú" – May God bless you. There is also some irregularity in certain verbs in the subjunctive. The verb bí (to be) is the most irregular verb in Irish (as in most Indo-European languages): The Irish phrase for "thank you" – go raibh maith agat – uses the subjunctive of "bí" and literally means "may there be good at-you".


Technology companies have risen to meet the demand to help manage these complex systems. Cloud-based SCM technologies are at the forefront of next-generation supply chains due to their impact on optimization of time, resources, and inventory visibility. Cloud technologies facilitate work being processed offline from a mobile app which solves the common issue of inventory residing in areas with no online coverage or connectivity.


In number theory, a Wall–Sun–Sun prime or Fibonacci–Wieferich prime is a certain kind of prime number which is conjectured to exist, although none are known.


In the C++ programming language, static_cast is an operator that performs an explicit type conversion.


Hartman, Newman & Ziv (1991) and de Fraysseix, Ossona de Mendez & Pach (1991) proved that every bipartite planar graph can be represented as an intersection graph of horizontal and vertical line segments; for this result see also Czyzowicz, Kranakis & Urrutia (1998). De Castro et al. (2002) proved that every triangle-free planar graph can be represented as an intersection graph of line segments having only three directions; this result implies Grötzsch's theorem (Grötzsch 1959) that triangle-free planar graphs can be colored with three colors. de Fraysseix & Ossona de Mendez (2005) proved that if a planar graph G can be 4-colored in such a way that no separating cycle uses all four colors, then G has a representation as an intersection graph of segments.


Some early Soviet computer designers implemented systems based on ternary logic; that is, a bit could have three states: +1, 0, or -1, corresponding to positive, zero, or negative voltage. An early project for the U.S. Air Force, BINAC attempted to make a lightweight, simple computer by using binary arithmetic.


Other approaches are loosely based on the idea of distributional semantics applied to sentences. Skip-Thought trains an encoder-decoder structure for the task of neighboring sentences predictions. Though this has been shown to achieve worse performance than approaches such as InferSent or SBERT.


In protected mode, there are four privilege levels or rings, numbered from 0 to 3, with ring 0 being the most privileged and 3 being the least. The use of rings allows for system software to restrict tasks from accessing data, call gates or executing privileged instructions. In most environments, the operating system and some device drivers run in ring 0 and applications run in ring 3.


In the case of a continuously differentiable function F, a coordinate descent algorithm can be sketched as: The step size can be chosen in various ways, e.g., by solving for the exact minimizer of f(xi) = F(x) (i.e., F with all variables but xi fixed), or by traditional line search criteria.


In several scientific fields, "complexity" has a precise meaning: In computational complexity theory, the amounts of resources required for the execution of algorithms is studied. The most popular types of computational complexity are the time complexity of a problem equal to the number of steps that it takes to solve an instance of the problem as a function of the size of the input (usually measured in bits), using the most efficient algorithm, and the space complexity of a problem equal to the volume of the memory used by the algorithm (e.g., cells of the tape) that it takes to solve an instance of the problem as a function of the size of the input (usually measured in bits), using the most efficient algorithm. This allows classification of computational problems by complexity class (such as P, NP, etc.). An axiomatic approach to computational complexity was developed by Manuel Blum.


In some processor architectures, like that of the Intel 4004 4-bit processor, memory was divided into (256 byte) pages and special precautions had to be taken when the control flow crossed page boundaries, as some machine instructions exhibited different behaviour if located in the last few instructions of a page, so that only few instructions were recommended to jump between pages.


In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite set of real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the nth root of the product of n numbers, i.e., for a set of numbers a1, a2, ..., an, the geometric mean is defined as ( ∏ i = 1 n a i ) 1 n = a 1 a 2 ⋯ a n n {\displaystyle \left(\prod _{i=1}^{n}a_{i}\right)^{\frac {1}{n}}={\sqrt{a_{1}a_{2}\cdots a_{n}}}} or, equivalently, as the arithmetic mean in logscale: exp  ( 1 n ∑ i = 1 n ln  a i ) {\displaystyle \exp {\left({{\frac {1}{n}}\sum \limits _{i=1}^{n}\ln a_{i}}\right)}} Most commonly the numbers are restricted to being non-negative, to avoid complications related to negative numbers not having real roots, and frequently they are restricted to being positive, to enable the use of logarithms. For instance, the geometric mean of two numbers, say 2 and 8, is just the square root of their product, that is, 2 ⋅ 8 = 4 {\displaystyle {\sqrt {2\cdot 8}}=4} . As another example, the geometric mean of the three numbers 4, 1, and 1/32 is the cube root of their product (1/8), which is 1/2, that is, 4 ⋅ 1 ⋅ 1 / 32 3 = 1 / 2 {\displaystyle {\sqrt{4\cdot 1\cdot 1/32}}=1/2} .


Some points of clarification: first, the definitions above involve quantification over properties and hence higher-order logic. Second, in (1), expressions of the form ( ∀ X ( X x ↔ X y ) ) {\displaystyle (\forall X(Xx\leftrightarrow Xy))} capture the concept of sharing all properties, or being indiscernible with respect to a set of properties. Thus, (1) can be understood more intuitively as the claim that all objects that are indiscernible with respect to a base set of properties are indiscernible with respect to a supervenient set of properties, or, as it is also sometimes said, that B-twins are A-twins.


Formally, it is the sequence ( z i ) , i = 0 , 1 , 2 , … {\displaystyle (z_{i}),i=0,1,2,\ldots } given by z i := { x i / 2 if i is even, y ( i − 1 ) / 2 if i is odd. {\displaystyle z_{i}:={\begin{cases}x_{i/2}&{\text{ if }}i{\text{ is even,}}\\y_{(i-1)/2}&{\text{ if }}i{\text{ is odd. }}\end{cases}}}


Codes defined via a Hamming space necessarily have the same length for every codeword, so they are called block codes when it is necessary to distinguish them from variable-length codes that are defined by unique factorization on a monoid. The Hamming distance endows a Hamming space with a metric, which is essential in defining basic notions of coding theory such as error detecting and error correcting codes.Hamming spaces over non-field alphabets have also been considered, especially over finite rings (most notably over Z4) giving rise to modules instead of vector spaces and ring-linear codes (identified with submodules) instead of linear codes.


All arbitrary (i.e. zero-symmetric or zero-asymmetric) optimal sets of nodes in  when n = 2 have been determined by F. Schurer, and in an alternative fashion by H.-J. Rack and R. Vajda (2014). If we assume that we take −1 and 1 as nodes for interpolation, then as shown by H.-J.


In set theory, this multiplication principle is often taken to be the definition of the product of cardinal numbers. We have | S 1 | ⋅ | S 2 | ⋯ | S n | = | S 1 × S 2 × ⋯ × S n | {\displaystyle |S_{1}|\cdot |S_{2}|\cdots |S_{n}|=|S_{1}\times S_{2}\times \cdots \times S_{n}|} where × {\displaystyle \times } is the Cartesian product operator. These sets need not be finite, nor is it necessary to have only finitely many factors in the product; see cardinal number. An extension of the rule of product considers there are n different types of objects, say sweets, to be associated with k objects, say people. How many different ways can the people receive their sweets? Each person may receive any of the n sweets available, and there are k people, so there are n ⋯ ⋅ n ⏞ k = n k {\displaystyle \overbrace {n\cdots \cdot n} ^{k}=n^{k}} ways to do this.


In mathematics, the generalized minimal residual method (GMRES) is an iterative method for the numerical solution of an indefinite nonsymmetric system of linear equations. The method approximates the solution by the vector in a Krylov subspace with minimal residual. The Arnoldi iteration is used to find this vector. The GMRES method was developed by Yousef Saad and Martin H. Schultz in 1986.


Due to the nature of virtualization, the mapping of logical to physical requires some processing power and lookup tables. Therefore, every implementation will add some small amount of latency. In addition to response time concerns, throughput has to be considered.


The polytope described by the linear program upper bounding the sum of edges taken per vertex is integral in the case of bipartite graphs, that is, it exactly describes the matching polytope, while for general graphs it is non-integral. Hence, for bipartite graphs, it suffices to solve the corresponding linear program to obtain a valid matching. For general graphs, however, there are two other characterizations of the matching polytope one of which makes use of the blossom inequality for odd subsets of vertices and hence allows to relax the integer program to a linear program while still obtaining valid matchings. These characterizations are of further interest in Edmonds' famous blossom algorithm used for finding such matchings in general graphs.


In mathematics, especially homological algebra and other applications of abelian category theory, the short five lemma is a special case of the five lemma. It states that for the following commutative diagram (in any abelian category, or in the category of groups), if the rows are short exact sequences, and if g and h are isomorphisms, then f is an isomorphism as well. It follows immediately from the five lemma. The essence of the lemma can be summarized as follows: if you have a homomorphism f from an object B to an object B′, and this homomorphism induces an isomorphism from a subobject A of B to a subobject A′ of B′ and also an isomorphism from the factor object B/A to B′/A′, then f itself is an isomorphism. Note however that the existence of f (such that the diagram commutes) has to be assumed from the start; two objects B and B′ that simply have isomorphic sub- and factor objects need not themselves be isomorphic (for example, in the category of abelian groups, B could be the cyclic group of order four and B′ the Klein four-group).


In the case of a shared-secret pepper, a single compromised password (via password reuse or other attack) along with a user's salt can lead to an attack to discover the pepper, rendering it ineffective. If an attacker knows a plaintext password and a user's salt, as well as the algorithm used to hash the password, then discovering the pepper can be a matter of brute forcing the values of the pepper. This is why NIST recommends the secret value be at least 112 bits, so that discovering it by exhaustive search is intractable. The pepper must be generated anew for every application it is deployed in, otherwise a breach of one application would result in lowered security of another application.


In pseudocode, the algorithm works as follows:


In the context of linear programming and related problems in mathematical optimization, convex polytopes are often described by a system of linear inequalities that their points must obey. When a polytope is integral, linear programming can be used to solve integer programming problems for the given system of inequalities, a problem that can otherwise be more difficult. Some polyhedra arising from combinatorial optimization problems are automatically integral.


Regularisation is particularly important feature for non-linear models and also often used in linear adaptive filters to reduce statistical uncertainties. However because nonlinear filters typically have a much higher potential structural complexity (or higher dimensional feature space) compared to the subspace actually required, regularisation of some kind must deal with the under-determined model. Though some specific forms of parameter regularisation such as prescribed by Vapink's SRM & SVM address the dimensionality problem statistically to some extent, there remain further statistical and practical issues for truly adaptive non-linear filters.


Pal et al. designed features to measure a user's authority on a certain topic. For example, retweet impact refers to how many times a certain user has been retweeted on a certain topic. The impact is dampened by a factor measuring how many times the user had been retweeted by a unique author to avoid the cases when a user has fans who retweet regardless of the content. They first used a clustering approach to find the target cluster which has the highest average score across all features, and used a ranking algorithm to find the most authoritative users within the cluster. With these authority detection methods, social Q&A could be more effective in providing accurate answers to askers.


In most mathematics, the operations of addition and multiplication are associative. The associative law for addition, for example, states that ( a + b ) + c = a + ( b + c ) {\displaystyle (a+b)+c=a+(b+c)} . This means that once the associative law is stated, the parentheses are unnecessary and are usually omitted. More generally, any sum, of any number of terms, can be written without parentheses and any product, of any number of factors, can be written without parentheses.


In order to construct a shadow volume, project a ray from the light source through each vertex in the shadow casting object to some point (generally at infinity). These projections will together form a volume; any point inside that volume is in shadow, everything outside is lit by the light. For a polygonal model, the volume is usually formed by classifying each face in the model as either facing toward the light source or facing away from the light source. The set of all edges that connect a toward-face to an away-face form the silhouette with respect to the light source.


Written as a formula: r e l e v a n t _ r e t r i e v e d _ i n s t a n c e s a l l _ r e l e v a n t _ i n s t a n c e s {\displaystyle {\frac {relevant\_retrieved\_instances}{all\_{\mathbf {relevant}}\_instances}}} . Both precision and recall are therefore based on relevance.


In the context of group theory, a sequence G 0 → f 1 G 1 → f 2 G 2 → f 3 ⋯ → f n G n {\displaystyle G_{0}\;{\xrightarrow {f_{1}}}\;G_{1}\;{\xrightarrow {f_{2}}}\;G_{2}\;{\xrightarrow {f_{3}}}\;\cdots \;{\xrightarrow {f_{n}}}\;G_{n}} of groups and group homomorphisms is called exact, if the image (or range) of each homomorphism is equal to the kernel of the next: i m ( f k ) = k e r ( f k + 1 ) {\displaystyle \mathrm {im} (f_{k})=\mathrm {ker} (f_{k+1})} The sequence of groups and homomorphisms may be either finite or infinite. A similar definition can be made for certain other algebraic structures. For example, one could have an exact sequence of vector spaces and linear maps, or of modules and module homomorphisms.


The Sinclair ZX80 and ZX81 characters sets used x0C (ASCII: form feed). The ZX Spectrum and the BBC Micro used x60 (ASCII: `, grave).


First compute M p ≡ C d p mod p {\displaystyle M_{p}\equiv C^{d_{p}}{\bmod {p}}} and M q ≡ C d q mod q {\displaystyle M_{q}\equiv C^{d_{q}}{\bmod {q}}} . 2. Use the Chinese Remainder Theorem to compute the unique value of M ∈ Z N {\displaystyle M\in \mathbb {Z_{N}} } which satisfies M ≡ M p mod p {\displaystyle M\equiv M_{p}{\bmod {p}}} and M ≡ M q mod q {\displaystyle M\equiv M_{q}{\bmod {q}}} . The result of M {\displaystyle M} satisfies M ≡ C d mod N {\displaystyle M\equiv C^{d}{\bmod {N}}} as needed. The point is that Wiener’s attack does not apply here because the value of d mod λ ( N ) {\displaystyle d{\bmod {\lambda }}(N)} can be large.


In order to generate a single constraint from a pair of constraints applied to the same (pair of) variable, we formalize the notion of intersection of constraints and of order over constraints. Similarly, in order to define a new constraints from existing constraints, a notion of sum of constraint must also be defined.


When Euclidean space is represented by a Cartesian coordinate system in analytic geometry, Euclidean distance satisfies the Pythagorean relation: the squared distance between two points equals the sum of squares of the difference in each coordinate between the points. The theorem can be generalized in various ways: to higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and to objects that are not triangles at all but n-dimensional solids. The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power; popular references in literature, plays, musicals, songs, stamps, and cartoons abound.


In recent years a number of neural and deep-learning techniques have been proposed, some of which generalize traditional Matrix factorization algorithms via a non-linear neural architecture. While deep learning has been applied to many different scenarios: context-aware, sequence-aware, social tagging etc. its real effectiveness when used in a simple Collaborative filtering scenario has been put into question. Systematic analysis of publications applying deep learning or neural methods to the top-k recommendation problem, published in top conferences (SIGIR, KDD, WWW, RecSys, IJCAI), has shown that on average less than 40% of articles are reproducible, with as little as 14% in some conferences.


Newer Digital Spread Spectrum (DSS) variants spread their signal over a range of frequencies, providing more resistance to signal fade. This technology enabled the digital information to spread in pieces among several frequencies between the receiver and the base, thereby making it almost impossible to eavesdrop on the cordless conversation. The FCC only allows DSS model phones to transmit at the full power of 1 watt, which allows increased range over older analog and digital models.Virtually all new cordless phones sold in the US use DECT 6.0 on the 1.9 GHz band, though legacy phones can remain in use on the older 900 MHz, 2.4 GHz and 5.8 GHz bands.


In modern landscape design olive trees are frequently used as ornamental features for their distinctively gnarled trunks and "evergreen" silvery gray foliage.


The feasible set of the problem is separate from the objective function, which states the criterion to be optimized and which in the above example is x 2 + y 4 . {\displaystyle x^{2}+y^{4}.} In many problems, the feasible set reflects a constraint that one or more variables must be non-negative.


In the 1710s, Leibniz described Grandi's series in his correspondence with several other mathematicians. The letter with the most lasting impact was his first reply to Wolff, which he published in the Acta Eruditorum. In this letter, Leibniz attacked the problem from several angles. In general, Leibniz believed that the algorithms of calculus were a form of "blind reasoning" that ultimately had to be founded upon geometrical interpretations.


In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7, and 9 (summing to 25) is 5. Depending on the context, an average might be another statistic such as the median, or mode. For example, the average personal income is often given as the median—the number below which are 50% of personal incomes and above which are 50% of personal incomes—because the mean would be higher by including personal incomes from a few billionaires. For this reason, it is recommended to avoid using the word "average" when discussing measures of central tendency.


In number theory, Szpiro's conjecture relates to the conductor and the discriminant of an elliptic curve. In a slightly modified form, it is equivalent to the well-known abc conjecture. It is named for Lucien Szpiro, who formulated it in the 1980s. Szpiro's conjecture and its equivalent forms have been described as "the most important unsolved problem in Diophantine analysis" by Dorian Goldfeld, in part to its large number of consequences in number theory including Roth's theorem, the Mordell conjecture, the Fermat–Catalan conjecture, and Brocard's problem.


The general problem remains unsolved, apart from the best-known lower bound Ω ( ( log  n ) / n 2 ) {\displaystyle \Omega ((\log n)/n^{2})} (achievable; hence, Heilbronn's conjecture is not correct for general n {\displaystyle n} ) and upper bound exp  ( c log  n ) / n 8 / 7 {\displaystyle \exp(c{\sqrt {\log n}})/n^{8/7}} (proven by Komlos, Pintsz and Szemeredi in 1982 and 1981, respectively). Using the incompressibility method, the average case was studied. It was proven that if the area is too small (or large) it can be compressed below the Kolmogorov complexity of a uniformly-random arrangement (high Kolmogorov complexity). This proves that for the overwhelming majority of the arrangements (and the expectation), the area of the smallest triangle formed by three of n {\displaystyle n} points thrown uniformly at random in the unit square is Θ ( 1 / n 3 ) {\displaystyle \Theta (1/n^{3})} . In this case, the incompressibility method proves the lower and upper bounds of the property involved.


Then given a query in natural language, the embedding for the query can be generated. A top k similarity search algorithm is then used between the query embedding and the document chunk embeddings to retrieve the most relevant document chunks as context information for question answering tasks. This approach is also known formally as retrieval augmented generationThough not as predominant as BERTScore, sentence embeddings are commonly used for sentence similarity evaluation which sees common use for the task of optimizing a Large language model's generation parameters is often performed via comparing candidate sentences against reference sentences. By using the cosine-similarity of the sentence embeddings of candidate and reference sentences as the evaluation function, a grid-search algorithm can be utilized to automate hyperparameter optimization. Sentence embedding is used by the deep learning software libraries PyTorch and TensorFlow.


To properly evaluate the truth (or falsehood) of a sentence, one must make reference to an interpretation of the theory. For first-order theories, interpretations are commonly called structures. Given a structure or interpretation, a sentence will have a fixed truth value. A theory is satisfiable when it is possible to present an interpretation in which all of its sentences are true. The study of algorithms to automatically discover interpretations of theories that render all sentences as being true is known as the satisfiability modulo theories problem.


In mathematics, the Burr–Erdős conjecture was a problem concerning the Ramsey number of sparse graphs. The conjecture is named after Stefan Burr and Paul Erdős, and is one of many conjectures named after Erdős; it states that the Ramsey number of graphs in any sparse family of graphs should grow linearly in the number of vertices of the graph. The conjecture was proven by Choongbum Lee. Thus it is now a theorem.


In plane geometry, constructing the diagonal of a square results in a triangle whose three angles are in the ratio 1: 1: 2, adding up to 180° or π radians. Hence, the angles respectively measure 45° (π/4), 45° (π/4), and 90° (π/2). The sides in this triangle are in the ratio 1: 1: √2, which follows immediately from the Pythagorean theorem. Of all right triangles, the 45° - 45° - 90° degree triangle has the smallest ratio of the hypotenuse to the sum of the legs, namely √2/2.


In optimization theory, the duality principle states that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. In general given two dual pairs separated locally convex spaces ( X , X ∗ ) {\displaystyle \left(X,X^{*}\right)} and ( Y , Y ∗ ) . {\displaystyle \left(Y,Y^{*}\right).} Then given the function f: X →  , {\displaystyle f:X\to ,} we can define the primal problem as finding x {\displaystyle x} such that inf x ∈ X f ( x ) .


In military terminology, vulnerability is a subset of survivability, the others being susceptibility and recoverability. Vulnerability is defined in various ways depending on the nation and service arm concerned, but in general it refers to the near-instantaneous effects of a weapon attack. In aviation it is defined as the inability of an aircraft to withstand the damage caused by the man-made hostile environment. In some definitions, recoverability (damage control, firefighting, restoration of capability) is included in vulnerability. Some military services develop their own concept of vulnerability.


In pure Prolog, normal DCG rules with no extra arguments on the functors, such as the previous example, can only express context-free grammars; there is only one argument on the left side of the production. However, context-sensitive grammars can also be expressed with DCGs, by providing extra arguments, such as in the following example: This set of DCG rules describes the grammar which generates the language that consists of strings of the form a n b n c n {\displaystyle a^{n}b^{n}c^{n}} . This set of DCG rules describes the grammar which generates the language that consists of strings of the form a n b n c n {\displaystyle a^{n}b^{n}c^{n}} , by structurally representing n


In statistics, k-medians clustering is a cluster analysis algorithm. It is a variation of k-means clustering where instead of calculating the mean for each cluster to determine its centroid, one instead calculates the median. This has the effect of minimizing error over all clusters with respect to the 1-norm distance metric, as opposed to the squared 2-norm distance metric (which k-means does.) This relates directly to the k-median problem with respect to the 1-norm, which is the problem of finding k centers such that the clusters formed by them are the most compact.


In scientific visualization the asymptotic decider is an algorithm developed by Nielson and Hamann in 1991 that creates isosurfaces from a given scalar field. It was proposed as an improvement to the marching cubes algorithm, which can produce some "bad" topology, but can also be considered an algorithm in its own right.


Definition. The constraint equations (6) define an n {\displaystyle \displaystyle n} -dimensional manifold M {\displaystyle \displaystyle M} within the configuration space of the Newtonian dynamical system (3). This manifold M {\displaystyle \displaystyle M} is called the configuration space of the constrained system.


In remote areas, an additional stage of clustering is used, in order to reduce travel requirements.Although cluster sampling and stratified sampling bear some superficial similarities, they are substantially different. In stratified sampling, a random sample is drawn from all the strata, where in cluster sampling only the selected clusters are studied, either in single- or multi-stage. Advantages Cost and speed that the survey can be done in Convenience of finding the survey sample Normally more accurate than cluster sampling for the same size sampleDisadvantages Not as accurate as Simple Random Sample if the sample is the same size More testing is difficult to do


In order to bound the competitive ratio for tango trees, we must find a lower bound on the performance of the optimal offline tree that we use as a benchmark. Once we find an upper bound on the performance of the tango tree, we can divide them to bound the competitive ratio.


Additivity and sigma-additivity are particularly important properties of measures. They are abstractions of how intuitive properties of size (length, area, volume) of a set sum when considering multiple objects. Additivity is a weaker condition than σ-additivity; that is, σ-additivity implies additivity. The term modular set function is equivalent to additive set function; see modularity below.


In mathematics, the absolute value or modulus of a real number x {\displaystyle x} , denoted | x | {\displaystyle |x|} , is the non-negative value of x {\displaystyle x} without regard to its sign. Namely, | x | = x {\displaystyle |x|=x} if x {\displaystyle x} is a positive number, and | x | = − x {\displaystyle |x|=-x} if x {\displaystyle x} is negative (in which case negating x {\displaystyle x} makes − x {\displaystyle -x} positive), and | 0 | = 0 {\displaystyle |0|=0} . For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3.


Narrowing reduces those worlds by eliminating combinations of different worlds from the same world set. Narrowing rules also detect situations where some combinations of worlds are shown to be impossible. No back tracking is required in the use of narrowing. By packaging the possible values in a value set all combinations of values may be considered at the same time. Evaluation proceeds as for a functional language, combining combinations of values in value sets, with narrowing rules eliminating impossible values from the sets.


Two companies have emerged that specialize in building multi-core processor devices using the MIPS architecture. Raza Microelectronics, Inc. bought the product line from failing SandCraft and later produced devices that contained eight cores for the telecommunication and networking markets. Cavium, originally a security processor vendor also produced devices with eight CPU cores, and later up to 32 cores, for the same markets. Both of these firms designed their cores in-house, only licensing the architecture instead of buying cores from MIPS.


In mathematics, Bézout's identity (also called Bézout's lemma), named after Étienne Bézout, is the following theorem: Here the greatest common divisor of 0 and 0 is taken to be 0. The integers x and y are called Bézout coefficients for (a, b); they are not unique. A pair of Bézout coefficients can be computed by the extended Euclidean algorithm, and this pair is, in the case of integers one of the two pairs such that | x | ≤ | b / d | {\displaystyle |x|\leq |b/d|} and | y | ≤ | a / d | ; {\displaystyle |y|\leq |a/d|;} equality occurs only if one of a and b is a multiple of the other.


In telecommunications there may be different reasons why a handover might be conducted: when the phone is moving away from the area covered by one cell and entering the area covered by another cell the call is transferred to the second cell in order to avoid call termination when the phone gets outside the range of the first cell; when the capacity for connecting new calls of a given cell is used up and an existing or new call from a phone, which is located in an area overlapped by another cell, is transferred to that cell in order to free-up some capacity in the first cell for other users, who can only be connected to that cell; in non-CDMA networks when the channel used by the phone becomes interfered by another phone using the same channel in a different cell, the call is transferred to a different channel in the same cell or to a different channel in another cell in order to avoid the interference; again in non-CDMA networks when the user behaviour changes, e.g. when a fast-travelling user, connected to a large, umbrella-type of cell, stops then the call may be transferred to a smaller macro cell or even to a micro cell in order to free capacity on the umbrella cell for other fast-traveling users and to reduce the potential interference to other cells or users (this works in reverse too, when a user is detected to be moving faster than a certain threshold, the call can be transferred to a larger umbrella-type of cell in order to minimize the frequency of the handovers due to this movement); in CDMA networks a handover (see further down) may be induced in order to reduce the interference to a smaller neighboring cell due to the "near–far" effect even when the phone still has an excellent connection to its current cell.The most basic form of handover is when a phone call in progress is redirected from its current cell (called source) to a new cell (called target). In terrestrial networks the source and the target cells may be served from two different cell sites or from one and the same cell site (in the latter case the two cells are usually referred to as two sectors on that cell site). Such a handover, in which the source and the target are different cells (even if they are on the same cell site) is called inter-cell handover. The purpose of inter-cell handover is to maintain the call as the subscriber is moving out of the area covered by the source cell and entering the area of the target cell.


In pure Prolog, NAF literals of the form n o t p {\displaystyle \mathrm {not} ~p} can occur in the body of clauses and can be used to derive other NAF literals. For example, given only the four clauses p ← q ∧ n o t r {\displaystyle p\leftarrow q\land \mathrm {not} ~r} q ← s {\displaystyle q\leftarrow s} q ← t {\displaystyle q\leftarrow t} t ← {\displaystyle t\leftarrow } NAF derives n o t s {\displaystyle \mathrm {not} ~s} , n o t r {\displaystyle \mathrm {not} ~r} and p {\displaystyle ~p} as well as t {\displaystyle ~t} and q {\displaystyle ~q} .


A function is most often denoted by letters such as f, g and h, and the value of a function f at an element x of its domain is denoted by f(x); the numerical value resulting from the function evaluation at a particular input value is denoted by replacing x with this value; for example, the value of f at x = 4 is denoted by f(4). When the function is not named and is represented by an expression E, the value of the function at, say, x = 4 may be denoted by E|x=4. For example, the value at 4 of the function that maps x to ( x + 1 ) 2 {\displaystyle (x+1)^{2}} may be denoted by ( x + 1 ) 2 | x = 4 {\displaystyle \left.


In telecommunications, long-term evolution (LTE) is a standard for wireless broadband communication for mobile devices and data terminals, based on the GSM/EDGE and UMTS/HSPA standards. It improves on those standards' capacity and speed by using a different radio interface and core network improvements. LTE is the upgrade path for carriers with both GSM/UMTS networks and CDMA2000 networks. Because LTE frequencies and bands differ from country to country, only multi-band phones can use LTE in all countries where it is supported.


Additionally, DBA functions are often automated using restructuring tools that are tightly coupled to an active data dictionary. Software frameworks aimed at rapid application development sometimes include high-level data dictionary facilities, which can substantially reduce the amount of programming required to build menus, forms, reports, and other components of a database application, including the database itself. For example, PHPLens includes a PHP class library to automate the creation of tables, indexes, and foreign key constraints portably for multiple databases.


In real network problems, people are interested in determining the likelihood of occurring double links (with opposite directions) between vertex pairs. This problem is fundamental for several reasons. First, in the networks that transport information or material (such as email networks, World Wide Web (WWW), World Trade Web, or Wikipedia ), mutual links facilitate the transportation process. Second, when analyzing directed networks, people often treat them as undirected ones for simplicity; therefore, the information obtained from reciprocity studies helps to estimate the error introduced when a directed network is treated as undirected (for example, when measuring the clustering coefficient). Finally, detecting nontrivial patterns of reciprocity can reveal possible mechanisms and organizing principles that shape the observed network's topology.


In mathematical morphology, the closing of a set (binary image) A by a structuring element B is the erosion of the dilation of that set, A ∙ B = ( A ⊕ B ) ⊖ B , {\displaystyle A\bullet B=(A\oplus B)\ominus B,\,} where ⊕ {\displaystyle \oplus } and ⊖ {\displaystyle \ominus } denote the dilation and erosion, respectively. In image processing, closing is, together with opening, the basic workhorse of morphological noise removal. Opening removes small objects, while closing removes small holes.


In statistics and research design, an index is a composite statistic – a measure of changes in a representative group of individual data points, or in other words, a compound measure that aggregates multiple indicators. Indexes – also known as composite indicators – summarize and rank specific observations.Much data in the field of social sciences and sustainability are represented in various indices such as Gender Gap Index, Human Development Index or the Dow Jones Industrial Average. The ‘Report by the Commission on the Measurement of Economic Performance and Social Progress’, written by Joseph Stiglitz, Amartya Sen, and Jean-Paul Fitoussi in 2009 suggests that these measures have experienced a dramatic growth in recent years due to three concurring factors: improvements in the level of literacy (including statistical) increased complexity of modern societies and economies, and widespread availability of information technology.According to Earl Babbie, items in indexes are usually weighted equally, unless there are some reasons against it (for example, if two items reflect essentially the same aspect of a variable, they could have a weight of 0.5 each).According to the same author, constructing the items involves four steps. First, items should be selected based on their content validity, unidimensionality, the degree of specificity in which a dimension is to be measured, and their amount of variance.


In statistics, regression validation is the process of deciding whether the numerical results quantifying hypothesized relationships between variables, obtained from regression analysis, are acceptable as descriptions of the data. The validation process can involve analyzing the goodness of fit of the regression, analyzing whether the regression residuals are random, and checking whether the model's predictive performance deteriorates substantially when applied to data that were not used in model estimation.


In the equations below, the lowercase variables represent vectors. Matrices W q {\displaystyle W_{q}} and U q {\displaystyle U_{q}} contain, respectively, the weights of the input and recurrent connections, where the subscript q {\displaystyle _{q}} can either be the input gate i {\displaystyle i} , output gate o {\displaystyle o} , the forget gate f {\displaystyle f} or the memory cell c {\displaystyle c} , depending on the activation being calculated. In this section, we are thus using a "vector notation". So, for example, c t ∈ R h {\displaystyle c_{t}\in \mathbb {R} ^{h}} is not just one unit of one LSTM cell, but contains h {\displaystyle h} LSTM cell's units.


If there is a dependence of each state on an overall entity (e.g. a map or simply an overall state variable), one typically uses SLAM (simultaneous localization and mapping) techniques, which include the sequential estimator as a special case (when the overall state variable has just one state). It will estimate the state sequence and the overall entity. There are also none-causal variants, that have all measurements at the same time, batches of measurements or revert the state evolution to go backwards again.


For each node on the phylogeny, the nodal support is the percentage of pseudoreplicates containing that node.The statistical rigor of the bootstrap test has been empirically evaluated using viral populations with known evolutionary histories, finding that 70% bootstrap support corresponds to a 95% probability that the clade exists. However, this was tested under ideal conditions (e.g. no change in evolutionary rates, symmetric phylogenies). In practice, values above 70% are generally supported and left to the researcher or reader to evaluate confidence.


In mathematics, a summability kernel is a family or sequence of periodic integrable functions satisfying a certain set of properties, listed below. Certain kernels, such as the Fejér kernel, are particularly useful in Fourier analysis. Summability kernels are related to approximation of the identity; definitions of an approximation of identity vary, but sometimes the definition of an approximation of the identity is taken to be the same as for a summability kernel.


In sociolinguistics, a variety, also called an isolect or lect, is a specific form of a language or language cluster. This may include languages, dialects, registers, styles, or other forms of language, as well as a standard variety. The use of the word "variety" to refer to the different forms avoids the use of the term language, which many people associate only with the standard language, and the term dialect, which is often associated with non-standard varieties thought of as less prestigious or "correct" than the standard. Linguists speak of both standard and non-standard (vernacular) varieties. "Lect" avoids the problem in ambiguous cases of deciding whether two varieties are distinct languages or dialects of a single language. Variation at the level of the lexicon, such as slang and argot, is often considered in relation to particular styles or levels of formality (also called registers), but such uses are sometimes discussed as varieties as well.


In order to add first order protection against phase-flips within a single degree of freedom, a higher dimension manifold is required. The 4-component cat code uses the even-parity submanifold of the superposition of 4 coherent states to encode information. The odd-parity submanifold is also 2-dimensional and serves as an error space since a single photon loss switches the parity of the state. Hence, monitoring the parity is sufficient to detect errors caused by single photon loss.


In statistics and machine learning, ensemble methods use multiple learning algorithms to obtain better predictive performance than could be obtained from any of the constituent learning algorithms alone. Unlike a statistical ensemble in statistical mechanics, which is usually infinite, a machine learning ensemble consists of only a concrete finite set of alternative models, but typically allows for much more flexible structure to exist among those alternatives.


A slide is a specially mounted individual transparency intended for projection onto a screen using a slide projector. This allows the photograph to be viewed by a large audience at once. The most common form is the 35 mm slide, with the image framed in a 2×2 inch cardboard or plastic mount. Some specialized labs produce photographic slides from digital camera images in formats such as JPEG, from computer-generated presentation graphics, and from a wide variety of physical source material such as fingerprints, microscopic sections, paper documents, astronomical images, etc. Reversal film is sometimes used as motion picture film, mostly in the 16 mm, Super 8 and 8 mm "cine" formats, to yield a positive image on the camera original. This avoids the expense of using negative film, which requires additional film and processing to create a positive film print for projection.


The three-dimensional angular momentum for a point particle is classically represented as a pseudovector r × p, the cross product of the particle's position vector r (relative to some origin) and its momentum vector; the latter is p = mv in Newtonian mechanics. Unlike linear momentum, angular momentum depends on where this origin is chosen, since the particle's position is measured from it. Angular momentum is an extensive quantity; that is, the total angular momentum of any composite system is the sum of the angular momenta of its constituent parts.


In the classical untyped lambda calculus, every function has a fixed point. A particular implementation of fix is Curry's paradoxical combinator Y, represented by Y = λ f . ( λ x . f ( x x ) ) ( λ x .


In the Dominican Republic, number portability in both mobile and local telephony was launched September 30, 2009. In March, 2009, the Dominican Telecommunications Institute (INDOTEL) selected Informática El Corte Inglés to administer the number portability.


In the above UML class diagram, the Client class refers (1) to the Aggregate interface for creating an Iterator object (createIterator()) and (2) to the Iterator interface for traversing an Aggregate object (next(),hasNext()). The Iterator1 class implements the Iterator interface by accessing the Aggregate1 class. The UML sequence diagram shows the run-time interactions: The Client object calls createIterator() on an Aggregate1 object, which creates an Iterator1 object and returns it to the Client. The Client uses then Iterator1 to traverse the elements of the Aggregate1 object.


However, the only way the process can use or set '-' values in its VAS is to ask the OS to map them to bytes from a file. A common way to use VAS memory in this way is to map it to the page file. The page file is a single file, but multiple distinct sets of contiguous bytes can be mapped into a VAS: 0 4 GiB VAS |---vvv--------vvvvvv---vvvv----vv---v----vvv--| mapping ||| |||||| |||| || | ||| file bytes app kernel user system_page_file And different parts of the page file can map into the VAS of different processes: 0 4 GiB VAS 1 |---vvvv-------vvvvvv---vvvv----vv---v----vvv--| mapping |||| |||||| |||| || | ||| file bytes app1 app2 kernel user system_page_file mapping |||| |||||| |||| || | VAS 2 |--------vvvv--vvvvvv---vvvv-------vv---v------| On Microsoft Windows 32-bit, by default, only 2 GiB are made available to processes for their own use.


In real-time computing, priority inheritance is a method for eliminating unbounded priority inversion. Using this programming method, a process scheduling algorithm increases the priority of a process (A) to the maximum priority of any other process waiting for any resource on which A has a resource lock (if it is higher than the original priority of A). The basic idea of the priority inheritance protocol is that when a job blocks one or more high-priority jobs, it ignores its original priority assignment and executes its critical section at an elevated priority level. After executing its critical section and releasing its locks, the process returns to its original priority level.


"World's first GSM call was actually made by me. I called Marjo Jousinen, in Salo. ", Lonka informed. The following year saw the sending of the first short messaging service (SMS or "text message") message, MS and Vodafone UK and Telecom Finland signed the first international roaming agreement.


Many digital modulation methods such as COFDM use the electromagnetic spectrum very efficiently, allowing for a very tight spectral mask. This allows placement of broadcast stations or other transmissions on channels right next to each other without interference, allowing for an increase in a band's total capacity. Conversely, it is allowing the U.S. to eliminate TV channels 52 to 69, freeing up 108 MHz (from approximately 700 to 800 MHz) for emergency services and to be auctioned off to the highest bidder, while still retaining (although moving) all existing TV stations.


As a result, students inherit all the attributes common to all persons. Additionally, students have unique attributes that other persons don't have. Object-oriented languages model subset/superset relationships using inheritance.


The driving force for the cathodic protection current is the difference in electrode potential between the anode and the cathode. During the initial phase of high current, the potential of the steel surface is polarized (pushed) more negative protecting the steel which hydroxide ion generation at the steel surface and ionic migration restore the concrete environment. Over time the galvanic anode continues to corrode, consuming the anode material until eventually it must be replaced.Galvanic or sacrificial anodes are made in various shapes and sizes using alloys of zinc, magnesium, and aluminum.


Then [0, 1⁄2) is open in S but not in R {\displaystyle \mathbb {R} } . Likewise [1⁄2, 1) is closed in S but not in R {\displaystyle \mathbb {R} } . S is both open and closed as a subset of itself but not as a subset of R {\displaystyle \mathbb {R} } .


Therefore, there is varying degree of accuracy when it comes to locating a person using cell phone data. These datasets are anonymized by the phone companies so as to hide and protect the identity of actual users. As example of its usage, researchers used the trajectory of 100,000 cell phone users within a period of six months, while in much larger scale trajectories of three million cell phone users were analyzed.


In optimization theory, semi-infinite programming (SIP) is an optimization problem with a finite number of variables and an infinite number of constraints, or an infinite number of variables and a finite number of constraints. In the former case the constraints are typically parameterized.


In mathematics, a statistical manifold is a Riemannian manifold, each of whose points is a probability distribution. Statistical manifolds provide a setting for the field of information geometry. The Fisher information metric provides a metric on these manifolds. Following this definition, the log-likelihood function is a differentiable map and the score is an inclusion.


In mathematics, the Herzog–Schönheim conjecture is a combinatorial problem in the area of group theory, posed by Marcel Herzog and Jochanan Schönheim in 1974.Let G {\displaystyle G} be a group, and let A = { a 1 G 1 , … , a k G k } {\displaystyle A=\{a_{1}G_{1},\ \ldots ,\ a_{k}G_{k}\}} be a finite system of left cosets of subgroups G 1 , … , G k {\displaystyle G_{1},\ldots ,G_{k}} of G {\displaystyle G} . Herzog and Schönheim conjectured that if A {\displaystyle A} forms a partition of G {\displaystyle G} with k > 1 {\displaystyle k>1} , then the (finite) indices  , … ,  {\displaystyle ,\ldots ,} cannot be distinct. In contrast, if repeated indices are allowed, then partitioning a group into cosets is easy: if H {\displaystyle H} is any subgroup of G {\displaystyle G} with index k =  < ∞ {\displaystyle k=<\infty } then G {\displaystyle G} can be partitioned into k {\displaystyle k} left cosets of H {\displaystyle H} .


On the other hand, if the running time of the algorithm is much smaller than D communication rounds, then the nodes in the network must produce their output without having the possibility to obtain information about distant parts of the network. In other words, the nodes must make globally consistent decisions based on information that is available in their local D-neighbourhood. Many distributed algorithms are known with the running time much smaller than D rounds, and understanding which problems can be solved by such algorithms is one of the central research questions of the field.


In order to implement a successful PAT project, a combination of three main PAT tools is essential: Multivariate data acquisition and data analysis tools: usually advanced software packages which aid in design of experiments, collection of raw data and statistically analyzing this data in order to determine what parameters are CPP. Process analytical chemistry (PAC) tools: in-line and on-line analytical instruments used to measure those parameters that have been defined as CPP. These include mainly near infrared spectroscopy (NIRS); but also include biosensors, Raman spectroscopy, fiber optics and others. Continuous improvement and/or knowledge management tools: paper systems or software packages which accumulate Quality Control data acquired over time for specific processes with the aim of defining process weaknesses and implementing and monitoring process improvement initiatives. These products may be the same or separated from the statistical analysis tools above.


In recent years, a number of commentators have called for legislative reforms to the state secrets privilege. These reforms center around several ideas: Requiring judges to review each piece of evidence that the executive claims is subject to the privilege. Requiring the executive to craft alternative evidence that is not subject to the privilege, for the opposing party to use in place of the original, privileged evidence. Such substitute evidence should only be required when it is possible to do so without harming national security.


In the Call Center space, the following type of call transfers can be undertaken and take on a slightly different meaning: Warm transfer: (also known as a live or hot transfer) the call center operator dials a number and talks to the person who has picked up the call before transferring the caller over to them. This could also be a three-way conference before the call center operator drops-off. One common example of a warm transfer is when a receptionist or virtual receptionist takes a call for the business and notifies the party attempting to be reached who the person is and the nature of their call. Tepid transfer: This involves the call center operator dialing a number and transferring the caller on to the called number without conferencing or speaking to the third party.


Each language had its own format for passing parameters into procedure calls, the file formats that they generated were often quite different. In general terms, it was not always possible to write different portions of a program in different languages, although doing so often has real utility.


In probability theory, the Azuma–Hoeffding inequality (named after Kazuoki Azuma and Wassily Hoeffding) gives a concentration result for the values of martingales that have bounded differences. Suppose { X k: k = 0 , 1 , 2 , 3 , … } {\displaystyle \{X_{k}:k=0,1,2,3,\dots \}} is a martingale (or super-martingale) and | X k − X k − 1 | ≤ c k , {\displaystyle |X_{k}-X_{k-1}|\leq c_{k},\,} almost surely. Then for all positive integers N and all positive reals ϵ {\displaystyle \epsilon } , P ( X N − X 0 ≥ ϵ ) ≤ exp  ( − ϵ 2 2 ∑ k = 1 N c k 2 ) . {\displaystyle {\text{P}}(X_{N}-X_{0}\geq \epsilon )\leq \exp \left({-\epsilon ^{2} \over 2\sum _{k=1}^{N}c_{k}^{2}}\right).}


While there is ongoing debate on whether checkpointing is the dominating I/O workload on distributed computing systems, there is general consensus that checkpointing is one of the major I/O workloads.There are two main approaches for checkpointing in the distributed computing systems: coordinated checkpointing and uncoordinated checkpointing. In the coordinated checkpointing approach, processes must ensure that their checkpoints are consistent. This is usually achieved by some kind of two-phase commit protocol algorithm.


The driven exam can be completed in the Class B bus the driver wishes to operate. It is common for restrictions to be issued on the driver's license disallowing them to operate higher classes of vehicles than they tested on. A common restriction is a driver with a Class B CDL that performs their P endorsement test on a class B bus, resulting in their license bearing the "no class A passenger vehicle" restriction, which is notated with an 'M', in addition to their P endorsement.


In mathematics, economics, and computer science, the stable marriage problem (also stable matching problem or SMP) is the problem of finding a stable matching between two equally sized sets of elements given an ordering of preferences for each element. A matching is a bijection from the elements of one set to the elements of the other set. A matching is not stable if: In other words, a matching is stable when there does not exist any pair (A, B) which both prefer each other to their current partner under the matching.


In quantum computing, the threshold theorem (or quantum fault-tolerance theorem) states that a quantum computer with a physical error rate below a certain threshold can, through application of quantum error correction schemes, suppress the logical error rate to arbitrarily low levels. This shows that quantum computers can be made fault-tolerant, as an analogue to von Neumann's threshold theorem for classical computation. This result was proven (for various error models) by the groups of Dorit Aharanov and Michael Ben-Or; Emanuel Knill, Raymond Laflamme, and Wojciech Zurek; and Alexei Kitaev independently. These results built off a paper of Peter Shor, which proved a weaker version of the threshold theorem.


In statistics, a symmetric probability distribution is a probability distribution—an assignment of probabilities to possible occurrences—which is unchanged when its probability density function (for continuous probability distribution) or probability mass function (for discrete random variables) is reflected around a vertical line at some value of the random variable represented by the distribution. This vertical line is the line of symmetry of the distribution. Thus the probability of being any given distance on one side of the value about which symmetry occurs is the same as the probability of being the same distance on the other side of that value.


In project management; Risks that are external to the project and the project manager cannot control. Good examples of external risks are changes in government legislation, changes in strategy from senior managers, and the economy.


In terms of representation scenarios, design knowledge can also be categorized into off-line and on-line knowledge. Design process knowledge can be categorized into ontologies.


In mathematics, the Drazin inverse, named after Michael P. Drazin, is a kind of generalized inverse of a matrix. Let A be a square matrix. The index of A is the least nonnegative integer k such that rank(Ak+1) = rank(Ak).


This led MIT Technology Review to name it 'the machine that saved Moore's law'. The first prototype in 2006 produced one wafer in 23 hours.


Coin errors that occur on the die are generally more desirable than errors made at the time of the strike. For example, a doubled die, where a date or another device appears twice slightly offset, is often a highly desired error. Strike errors are generally unique, whereas all coins struck with an error die will have the same characteristic.


In programming language design, there are a wide variety of factors to consider. Some factors may be mutually exclusive (e.g. security versus speed). It may be necessary to consider whether a programming language will perform better interpreted, or compiled, if a language should be dynamically or statically typed, if inheritance will be in a language, and the general syntax of the language. Many factors involved with the design of a language can be decided on by the goals behind the language.


For example, a client may receive an event stating that a given key has been pressed while the Shift modifier was down. If this key would normally generate the character "a", the client (and not the server) associates this event to the character "A". While the translation from keycodes to keysyms is done by the client, the table that represents this association is maintained by the server.


{\displaystyle {\sigma ^{2}+\left({\frac {\text{Third central moment}}{2\sigma ^{2}}}\right)^{2}}\leq {\frac {1}{4}}(M-m)^{2}.} If one additionally assumes knowledge of the expectation, then the stronger Bhatia–Davis inequality holds σ 2 ≤ ( M − μ ) ( μ − m ) {\displaystyle \sigma ^{2}\leq (M-\mu )(\mu -m)} where μ is the expectation of the random variable.In the case of an independent sample of n observations from a bounded probability distribution, the von Szokefalvi Nagy inequality gives a lower bound to the variance of the sample mean: σ 2 ≥ ( M − m ) 2 2 n . {\displaystyle \sigma ^{2}\geq {\frac {(M-m)^{2}}{2n}}.}


In multi-processor systems (more than one Opteron on a single motherboard), the CPUs communicate using the Direct Connect Architecture over high-speed HyperTransport links. Each CPU can access the main memory of another processor, transparent to the programmer. The Opteron approach to multi-processing is not the same as standard symmetric multiprocessing; instead of having one bank of memory for all CPUs, each CPU has its own memory. Thus the Opteron is a Non-Uniform Memory Access (NUMA) architecture.


In order to extract the desired distortion information, at any given location in the spherical coordinate system, the values of K {\displaystyle K} can be computed directly. The Jacobian, J {\displaystyle J} , can be computed analytically from the mapping function itself, but it is often simpler to numerically approximate the values at any location on the map using central differences. Once these values are computed, SVD can be applied to each transformation matrix to extract the local distortion information. Remember that, because distortion is local, every location on the map will have its own transformation.


These probabilities are used to determine what the target is using a maximum likelihood decision. This method has been shown to be able to distinguish between vehicle types (wheeled vs tracked vehicles for example), and even decide how many people are present up to three people with a high probability of success.CNN-Based Target Recognition Convolutional neural network (CNN)-based target recognition is able to outperform the conventional methods.


For these reasons, constant interfaces may be considered an anti-pattern. Use of this pattern has a few other downsides: It pollutes the class namespace with read-only variables that may not be of use. Contrary to the compile-time tactical utility of implementing a constant interface, the incidental run-time artifacts have little practical purpose (cf. marker interfaces which also have no methods but are useful at run-time).


As x and y must be rational, the square of ± 2 x y {\displaystyle \pm 2{\sqrt {xy}}} must be rational. This implies that α = 0 {\displaystyle \alpha =0} in the expression of ± 2 x y {\displaystyle \pm 2{\sqrt {xy}}} as α + β c . {\displaystyle \alpha +\beta {\sqrt {c}}.}


In phrase structure grammars such as generative grammar, the verb phrase is one headed by a verb. It may be composed of only a single verb, but typically it consists of combinations of main and auxiliary verbs, plus optional specifiers, complements (not including subject complements), and adjuncts. For example: Yankee batters hit the ball well enough to win their first World Series since 2000. Mary saw the man through the window.


However, to improve the usability and reliability of keys, many single-access keys incorporate reticulation, changing the tree structure into a directed acyclic graph. Single-access keys have been in use for several hundred years. They may be printed in various styles (e. g., linked, nested, indented, graphically branching) or used as interactive, computer-aided keys.


The first two of these were to resolve the continuum hypothesis and prove the consistency of elementary arithmetic, respectively; the tenth was to produce a method that could decide whether a multivariate polynomial equation over the integers has a solution. Subsequent work to resolve these problems shaped the direction of mathematical logic, as did the effort to resolve Hilbert's Entscheidungsproblem, posed in 1928. This problem asked for a procedure that would decide, given a formalized mathematical statement, whether the statement is true or false.


In mathematics, the three classical Pythagorean means are the arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM). These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians because of their importance in geometry and music.


The cosets of Inn(G) with respect to outer automorphisms are then the elements of Out(G); this is an instance of the fact that quotients of groups are not, in general, (isomorphic to) subgroups. If the inner automorphism group is trivial (when a group is abelian), the automorphism group and outer automorphism group are naturally identified; that is, the outer automorphism group does act on the group. For example, for the alternating group, An, the outer automorphism group is usually the group of order 2, with exceptions noted below. Considering An as a subgroup of the symmetric group, Sn, conjugation by any odd permutation is an outer automorphism of An or more precisely "represents the class of the (non-trivial) outer automorphism of An", but the outer automorphism does not correspond to conjugation by any particular odd element, and all conjugations by odd elements are equivalent up to conjugation by an even element.


A second publication from 1968 described the theory and the technical details of the instrument and had Hadravský and Robert Galambos, the head of the group at Yale, as additional authors. In 1970 the US patent was granted. It was filed in 1967.


As an example, given a PostBQP algorithm A with success probability 2/3, we can construct another algorithm which runs three independent copies of A, outputs a postselection bit equal to the conjunction of the three "inner" ones, and outputs an output bit equal to the majority of the three "inner" ones; the new algorithm will be correct with conditional probability ( 2 / 3 ) 3 + 3 ( 1 / 3 ) ( 2 / 3 ) 2 = 20 / 27 {\displaystyle (2/3)^{3}+3(1/3)(2/3)^{2}=20/27} , greater than the original 2/3. PostBQP is closed under intersection. Suppose we have PostBQP circuit families for two languages L 1 {\displaystyle L_{1}} and L 2 {\displaystyle L_{2}} , with respective postselection qubits and output qubits P 1 , P 2 , Q 1 , Q 2 {\displaystyle P_{1},P_{2},Q_{1},Q_{2}} .


In number theory, the von Staudt–Clausen theorem is a result determining the fractional part of Bernoulli numbers, found independently by Karl von Staudt (1840) and Thomas Clausen (1840). Specifically, if n is a positive integer and we add 1/p to the Bernoulli number B2n for every prime p such that p − 1 divides 2n, we obtain an integer, i.e., B 2 n + ∑ ( p − 1 ) | 2 n 1 p ∈ Z . {\displaystyle B_{2n}+\sum _{(p-1)|2n}{\frac {1}{p}}\in \mathbb {Z} .} This fact immediately allows us to characterize the denominators of the non-zero Bernoulli numbers B2n as the product of all primes p such that p − 1 divides 2n; consequently the denominators are square-free and divisible by 6. These denominators are 6, 30, 42, 30, 66, 2730, 6, 510, 798, 330, 138, 2730, 6, 870, 14322, 510, 6, 1919190, 6, 13530, ... (sequence A002445 in the OEIS).The sequence of integers B 2 n + ∑ ( p − 1 ) | 2 n 1 p {\displaystyle B_{2n}+\sum _{(p-1)|2n}{\frac {1}{p}}} is 1, 1, 1, 1, 1, 1, 2, -6, 56, -528, 6193, -86579, 1425518, -27298230, ... (sequence A000146 in the OEIS).


Let π ( x ) {\displaystyle \pi (x)} , the prime-counting function, denote the number of primes less than or equal to x {\displaystyle x} . If q {\displaystyle q} is a positive integer and a {\displaystyle a} is coprime to q {\displaystyle q} , we let π ( x ; q , a ) {\displaystyle \pi (x;q,a)} denote the number of primes less than or equal to x {\displaystyle x} which are equal to a {\displaystyle a} modulo q {\displaystyle q} . Dirichlet's theorem on primes in arithmetic progressions then tells us that π ( x ; q , a ) ≈ π ( x ) φ ( q ) {\displaystyle \pi (x;q,a)\approx {\frac {\pi (x)}{\varphi (q)}}} where φ {\displaystyle \varphi } is Euler's totient function.


Barzilai and Borwein proved their method converges R-superlinearly for quadratic minimization in two dimensions. Raydan demonstrates convergence in general for quadratic problems.


In mathematics, the unknotting problem is the problem of algorithmically recognizing the unknot, given some representation of a knot, e.g., a knot diagram. There are several types of unknotting algorithms. A major unresolved challenge is to determine if the problem admits a polynomial time algorithm; that is, whether the problem lies in the complexity class P.


If portfolio weights are largely a function of estimation errors, then Ex-post performance of a growth-optimal portfolio may differ fantastically from the ex-ante prediction. Parameter uncertainty and estimation errors are a large topic in portfolio theory. An approach to counteract the unknown risk is to invest less than the Kelly criterion (e.g., half).


Every finite distributive lattice can be represented as a lattice of stable matchings. The number of elements in the lattice can vary from an average case of e − 1 n ln  n {\displaystyle e^{-1}n\ln n} to a worst-case of exponential. Computing the number of elements is #P-complete.


Atlantis, is continued. Ais and her crew face many other challenges, most notably internal problems, caused first by the rebellion of one of the deputies, and then by open warfare led by the non-Delosiani Dr. Satham, a scrupulous scientist and his henchmen.While the uplift experiment seems to progress, eventually the Great Brain, the highest authority on Des (Delos), decides to stop and destroy the work of Ais and her colleagues. However, Ais and some of her subordinates questioned (→ reflection) the strange order and refuse to abandon the uplift (whether the command actually came directly from the Great Brain is not known.


In the Heilbronn triangle problem, throw n {\displaystyle n} points in the unit square and determine the maximum of the minimal area of a triangle formed by three of the points over all possible arrangements. This problem was solved for small arrangements, and much work was done on asymptotic expression as a function of n {\displaystyle n} . The original conjecture of Heilbronn was O ( 1 / n 2 ) {\displaystyle O(1/n^{2})} during the early 1950s. Paul Erdős proved that this bound is correct for n {\displaystyle n} , a prime number.


He invented the Adams operations in K-theory, which are derived from the exterior powers; they are now also widely used in purely algebraic contexts. Adams introduced them in a 1962 paper to solve the famous vector fields on spheres problem. Subsequently he used them to investigate the Adams conjecture, which is concerned (in one instance) with the image of the J-homomorphism in the stable homotopy groups of spheres.


In other projects the need to move into Prolog was considered unnecessary because the printed logico-linguistic models provided an easy to use guide to decision making. For example, a system for mortgage loan approval


In the context of data race detection for programs using lock based synchronization, lockset-based techniques provide an unsound, yet lightweight mechanism for detecting data races. These techniques primarily detect violations of the lockset principle. which says that all accesses of a given memory location must be protected by a common lock. Such techniques are also used to filter out candidate race reports in more expensive analyses.


Seven dogs were missed (false negatives), and seven cats were correctly excluded (true negatives). The program's precision is then 5/8 (true positives / selected elements) while its recall is 5/12 (true positives / relevant elements). Adopting a hypothesis-testing approach from statistics, in which, in this case, the null hypothesis is that a given item is irrelevant (i.e., not a dog), absence of type I and type II errors (i.e., perfect specificity and sensitivity of 100% each) corresponds respectively to perfect precision (no false positive) and perfect recall (no false negative).


And if you cannot do that, simplify your design until you can.”At a SIGPLAN symposium in 1973, Tony Hoare discussed various language aspects in some detail. He also identifies a number of shortcomings in (then) current programming languages.“a programming language is a tool which should assist the programmer in the most difficult aspects of his art, namely program design, documentation, and debugging.”“objective criteria for good language design may be summarized in five catch phrases: simplicity, security, fast translation, efficient object code, and readability.” "It is absurd to make elaborate security checks on debugging runs, when no trust is put in the results, and then remove them in production runs, when an erroneous result could be expensive or disastrous.


In probability theory and mathematical statistics, the law of total cumulance is a generalization to cumulants of the law of total probability, the law of total expectation, and the law of total variance. It has applications in the analysis of time series. It was introduced by David Brillinger.It is most transparent when stated in its most general form, for joint cumulants, rather than for cumulants of a specified order for just one random variable. In general, we have κ ( X 1 , … , X n ) = ∑ π κ ( κ ( X i: i ∈ B ∣ Y ): B ∈ π ) , {\displaystyle \kappa (X_{1},\dots ,X_{n})=\sum _{\pi }\kappa (\kappa (X_{i}:i\in B\mid Y):B\in \pi ),} where κ(X1, ..., Xn) is the joint cumulant of n random variables X1, ..., Xn, and the sum is over all partitions π {\displaystyle \pi } of the set { 1, ..., n } of indices, and "B ∈ π;" means B runs through the whole list of "blocks" of the partition π, and κ(Xi: i ∈ B | Y) is a conditional cumulant given the value of the random variable Y. It is therefore a random variable in its own right—a function of the random variable Y.


Using a car as an analogy, if the user steps on the gas without first starting the engine, the car does not crash, fail, or throw an exception - it simply fails to accelerate. Sequential coupling can be refactored with the template method pattern to overcome the problems posed by the usage of this anti-pattern. == References ==


In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming.The name of the algorithm is derived from the concept of a simplex and was suggested by T. S. Motzkin. Simplices are not actually used in the method, but one interpretation of it is that it operates on simplicial cones, and these become proper simplices with an additional constraint. The simplicial cones in question are the corners (i.e., the neighborhoods of the vertices) of a geometric object called a polytope. The shape of this polytope is defined by the constraints applied to the objective function.


In all page table formats supported by x86 and x86-64, the 12 least significant bits of the page table entry are either interpreted by the memory management unit or are reserved for operating system use. In processors that implement the "no-execute" or "execution disable" feature, the most significant bit (bit 63) is the NX bit. The next eleven most significant bits (bits 52 through 62) are reserved for operating system use by both Intel and AMD's architecture specifications.


In the United States, commercial truck classification is determined based on the vehicle's gross vehicle weight rating (GVWR). The classes are numbered 1 through 8. Trucks are also classified more broadly by the Federal Highway Administration (FHWA), which groups classes 1 and 2 as light duty, 3 through 6 as medium duty, and 7 and 8 as heavy duty. The Environmental Protection Agency (EPA) has a separate system of emissions classifications for trucks.


In any case, such definitions (also called bridge laws or correspondence rules) were held to serve three important purposes. In the first place, by connecting the uninterpreted formalism with the observation language, they permit the assignment of synthetic content to theories. In the second, according to whether they express a factual or a purely conventional content, they allow for the subdivision of science into two parts: one factual and independent of human conventions, the other non-empirical and conventional.


In contrast, a program where both Stack and Queue are subclasses of a type Bag, whose specification for get is merely that it removes some element, does satisfy behavioral subtyping and allows clients to safely reason about correctness based on the presumed types of the objects they interact with. Indeed, any object that satisfies the Stack or Queue specification also satisfies the Bag specification. It is important to stress that whether a type S is a behavioral subtype of a type T depends only on the specification (i.e. the documentation) of type T; the implementation of type T, if it has any, is completely irrelevant to this question.


In model checking, a subfield of computer science, a timed word is an extension of the notion of words, in a formal language, in which each letter is associated with a positive time tag. The sequence of time tags must be non-decreasing, which intuitively means that letters are received. For example, a system receiving a word over a network may associate to each letter the time at which the letter is received. The non-decreasing condition here means that the letters are received in the correct order. A timed language is a set of timed words.


Fido is a dog. Therefore Fido is a mammal." Formally, the rule as an axiom schema is given as ∀ x A ⇒ A { x ↦ a } , {\displaystyle \forall x\,A\Rightarrow A\{x\mapsto a\},} for every formula A and every term a, where A { x ↦ a } {\displaystyle A\{x\mapsto a\}} is the result of substituting a for each free occurrence of x in A. A { x ↦ a } {\displaystyle \,A\{x\mapsto a\}} is an instance of ∀ x A .


In mathematical logic, focused proofs are a family of analytic proofs that arise through goal-directed proof-search, and are a topic of study in structural proof theory and reductive logic. They form the most general definition of goal-directed proof-search—in which someone chooses a formula and performs hereditary reductions until the result meets some condition. The extremal case where reduction only terminates when axioms are reached forms the sub-family of uniform proofs.A sequent calculus is said to have the focusing property when focused proofs are complete for some terminating condition. For System LK, System LJ, and System LL, uniform proofs are focused proofs where all the atoms are assigned negative polarity. Many other sequent calculi has been shown to have the focusing property, notably the nested sequent calculi of both the classical and intuitionistic variants of the modal logics in the S5 cube.


This is generally called roaming from a customer perspective, but also called visiting when describing the underlying technical process. Each geographic area has a database called the Visitor Location Register (VLR), which contains details of all the mobiles currently in that area. Whenever a phone attaches, or visits, a new area, the Visitor Location Register must contact the Home Location Register to obtain the details for that phone.


In mathematical finite group theory, the Brauer–Fowler theorem, proved by Brauer & Fowler (1955), states that if a group G has even order g > 2 then it has a proper subgroup of order greater than g1/3. The technique of the proof is to count involutions (elements of order 2) in G. Perhaps more important is another result that the authors derive from the same count of involutions, namely that up to isomorphism there are only a finite number of finite simple groups with a given centralizer of an involution. This suggested that finite simple groups could be classified by studying their centralizers of involutions, and it led to the discovery of several sporadic groups. Later it motivated a part of the classification of finite simple groups.


In statistics the mean squared prediction error (MSPE), also known as mean squared error of the predictions, of a smoothing, curve fitting, or regression procedure is the expected value of the squared prediction errors (PE), the square difference between the fitted values implied by the predictive function g ^ {\displaystyle {\widehat {g}}} and the values of the (unobservable) true value g. It is an inverse measure of the explanatory power of g ^ , {\displaystyle {\widehat {g}},} and can be used in the process of cross-validation of an estimated model. Knowledge of g would be required in order to calculate the MSPE exactly; in practice, MSPE is estimated.


For example, under this assumption, a poset may be defined as a small posetal category, a distributive lattice as a small posetal distributive category, a Heyting algebra as a small posetal finitely cocomplete cartesian closed category, and a Boolean algebra as a small posetal finitely cocomplete *-autonomous category. Conversely, categories, distributive categories, finitely cocomplete cartesian closed categories, and finitely cocomplete *-autonomous categories can be considered the respective categorifications of posets, distributive lattices, Heyting algebras, and Boolean algebras. == References ==


In mathematics, the Hardy–Ramanujan–Littlewood circle method is a technique of analytic number theory. It is named for G. H. Hardy, S. Ramanujan, and J. E. Littlewood, who developed it in a series of papers on Waring's problem.


Unary relations can be viewed as a collection of members (such as the collection of Nobel laureates) having some property (such as that of having been awarded the Nobel prize). Binary relations are the most commonly studied form of finitary relations. When X1 = X2 it is called a homogeneous relation, for example: Equality and inequality, denoted by signs such as = and < in statements such as "5 < 12", or Divisibility, denoted by the sign | in statements such as "13|143".Otherwise it is a heterogeneous relation, for example: Set membership, denoted by the sign ∈ in statements such as "1 ∈ N".


In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum (least upper bound). Complete Boolean algebras are used to construct Boolean-valued models of set theory in the theory of forcing. Every Boolean algebra A has an essentially unique completion, which is a complete Boolean algebra containing A such that every element is the supremum of some subset of A. As a partially ordered set, this completion of A is the Dedekind–MacNeille completion. More generally, if κ is a cardinal then a Boolean algebra is called κ-complete if every subset of cardinality less than κ has a supremum.


In QCA's next step, inferential logic or Boolean algebra is used to simplify or reduce the number of inferences to the minimum set of inferences supported by the data. This reduced set of inferences is termed the "prime implicates" by QCA adherents. For instance, if the presence of conditions A and B is always associated with the presence of a particular value of D, regardless of the observed value of C, then the value that C takes is irrelevant.


In some languages, aspect and time are very clearly separated, making them much more distinct to their speakers. There are a number of languages that mark aspect much more saliently than time. Prominent in this category are Chinese and American Sign Language, which both differentiate many aspects but rely exclusively on optional time-indicating terms to pinpoint an action with respect to time. In other language groups, for example in most modern Indo-European languages (except Slavic languages and some Indo-Aryan languages like Hindi), aspect has become almost entirely conflated, in the verbal morphological system, with time.


In telecommunications networks, a node (Latin: nodus, ‘knot’) is either a redistribution point or a communication endpoint. The definition of a node depends on the network and protocol layer referred to. A physical network node is an electronic device that is attached to a network, and is capable of creating, receiving, or transmitting information over a communication channel. A passive distribution point such as a distribution frame or patch panel is consequently not a node.


Rete II gives about a 100 to 1 order of magnitude performance improvement in more complex problems as shown by KnowledgeBased Systems Corporation benchmarks. Rete II can be characterized by two areas of improvement; specific optimizations relating to the general performance of the Rete network (including the use of hashed memories in order to increase performance with larger sets of data), and the inclusion of a backward chaining algorithm tailored to run on top of the Rete network. Backward chaining alone can account for the most extreme changes in benchmarks relating to Rete vs. Rete II. Rete II is implemented in the commercial product Advisor from FICO, formerly called Fair Isaac Jess (at least versions 5.0 and later) also adds a commercial backward chaining algorithm on top of the Rete network, but it cannot be said to fully implement Rete II, in part due to the fact that no full specification is publicly available.


In statistics and decision theory, a frequently used loss function is the 0-1 loss function L ( y ^ , y ) =  {\displaystyle L({\hat {y}},y)=\left} using Iverson bracket notation, i.e. it evaluates to 1 when y ^ ≠ y {\displaystyle {\hat {y}}\neq y} , and 0 otherwise.


In the following, fn is any of jn, yn, h(1)n, h(2)n for n = 0, ±1, ±2, ...


{\displaystyle y^{2}=x(x-a^{n})(x+b^{n}).\,} Such an elliptic curve would enjoy very special properties, which are due to the appearance of high powers of integers in its equation and the fact that an + bn = cn is an nth power as well. In the summer of 1986, Ken Ribet demonstrated that, just as Gerhard Frey had anticipated, a special case of the Taniyama–Shimura conjecture (still not proved at the time), together with the now proved epsilon conjecture, implies Fermat's Last Theorem. Thus, if the Taniyama–Shimura conjecture is true for semistable elliptic curves, then Fermat's Last Theorem would be true.


In most Forth systems, the body of a code definition consists of either machine language, or some form of threaded code. The original Forth which follows the informal FIG standard (Forth Interest Group), is a TIL (Threaded Interpretive Language). This is also called indirect-threaded code, but direct-threaded and subroutine threaded Forths have also become popular in modern times. The fastest modern Forths, such as SwiftForth, VFX Forth, and iForth, compile Forth to native machine code.


In mathematics, the Schwartz kernel theorem is a foundational result in the theory of generalized functions, published by Laurent Schwartz in 1952. It states, in broad terms, that the generalized functions introduced by Schwartz (Schwartz distributions) have a two-variable theory that includes all reasonable bilinear forms on the space D {\displaystyle {\mathcal {D}}} of test functions. The space D {\displaystyle {\mathcal {D}}} itself consists of smooth functions of compact support.


Different classification methods may have different percentages of error for a given classification project. It is important that the remote sensor chooses a classification method that works best with the number of classifications used while providing the least amount of error.


In prior versions of C++, only functions, classes or type aliases could be templated. C++14 allows the creation of variables that are templated. An example given in the proposal is a variable pi that can be read to get the value of pi for various types (e.g., 3 when read as an integral type; the closest value possible with float, double or long double precision when read as float, double or long double, respectively; etc.). The usual rules of templates apply to such declarations and definitions, including specialization.


For almost all tournaments, the size is at least ( n 2 ) / 2 − 1.73 n 3 / 2 {\displaystyle {\tbinom {n}{2}}/2-1.73n^{3/2}} . Every directed acyclic graph D {\displaystyle D} can be embedded as a subgraph of a larger tournament graph, in such a way that D {\displaystyle D} is the unique minimum feedback arc set of the tournament. The size of this tournament has been defined as the "reversing number" of D {\displaystyle D} , and among directed acyclic graphs with the same number of vertices it is largest when D {\displaystyle D} is itself an (acyclic) tournament.A directed graph has an Euler tour whenever it is strongly connected and each vertex has equal numbers of incoming and outgoing edges.


In mechanics, the eigenvectors of the moment of inertia tensor define the principal axes of a rigid body. The tensor of moment of inertia is a key quantity required to determine the rotation of a rigid body around its center of mass.


In some versions of Ring-LWE there is a security reduction to the shortest-vector problem (SVP) in a lattice as a lower bound on the security. The SVP is known to be NP-hard. Specific ring-LWE systems that have provable security reductions include a variant of Lyubashevsky's ring-LWE signatures defined in a paper by Güneysu, Lyubashevsky, and Pöppelmann.


In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n is also a positive integer. In other words, there are infinitely many primes that are congruent to a modulo d. The numbers of the form a + nd form an arithmetic progression a , a + d , a + 2 d , a + 3 d , … , {\displaystyle a,\ a+d,\ a+2d,\ a+3d,\ \dots ,\ } and Dirichlet's theorem states that this sequence contains infinitely many prime numbers. The theorem, named after Peter Gustav Lejeune Dirichlet, extends Euclid's theorem that there are infinitely many prime numbers. Stronger forms of Dirichlet's theorem state that for any such arithmetic progression, the sum of the reciprocals of the prime numbers in the progression diverges and that different such arithmetic progressions with the same modulus have approximately the same proportions of primes. Equivalently, the primes are evenly distributed (asymptotically) among the congruence classes modulo d containing a's coprime to d.


When one processor is taken out then the other processor operates independently. When the faulty processor is repaired and brought in service then memory contents of the active processor are copied into its memory and the two are synchronized and comparator is enabled. It is possible that a comparator fault occurs only due to transient failure which is not shown even when check out program is run.


In mathematics, the Erdős–Ko–Rado theorem limits the number of sets in a family of sets for which every two sets have at least one element in common. Paul Erdős, Chao Ko, and Richard Rado proved the theorem in 1938, but did not publish it until 1961. It is part of the field of combinatorics, and one of the central results of extremal set theory. The theorem applies to families of sets that all have the same size, r {\displaystyle r} , and are all subsets of some larger set of size n {\displaystyle n} .


In most modern operating systems (OSs), a large body of reusable code is provided to simplify the programmer's job. This code is typically provided as a set of dynamically loadable libraries that applications can call at runtime. Because the Java platform is not dependent on any specific operating system, applications cannot rely on any of the pre-existing OS libraries.


Difficulties with such dependencies can be easily avoided if we reinsert the upper unit node η ′ {\displaystyle \eta ^{\prime }} after reinserting the unit node η {\displaystyle \eta } (i.e. after reinsertion, η ′ {\displaystyle \eta ^{\prime }} must appear below η {\displaystyle \eta } , to cancel the extra literal ℓ ¯ {\displaystyle {\overline {\ell }}} from η {\displaystyle \eta } ’s clause). This can be ensured by collecting the unit nodes in a queue during a bottom-up traversal of the proof and reinserting them in the order they were queued. The algorithm for fixing a proof containing many roots performs a top-down traversal of the proof, recomputing the resolvents and replacing broken nodes (e.g. nodes having deletedNodeMarker as one of their parents) by their surviving parents (e.g. the other parent, in case one parent was deletedNodeMarker). When unit nodes are collected and removed from a proof of a clause κ {\displaystyle \kappa } and the proof is fixed, the clause κ ′ {\displaystyle \kappa ^{\prime }} in the root node of the new proof is not equal to κ {\displaystyle \kappa } anymore, but contains (some of) the duals of the literals of the unit clauses that have been removed from the proof. The reinsertion of unit nodes at the bottom of the proof resolves κ ′ {\displaystyle \kappa ^{\prime }} with the clauses of (some of) the collected unit nodes, in order to obtain a proof of κ {\displaystyle \kappa } again.


Then the new state is ( a 1 , a 2 , … , a r ; z ′ ) {\displaystyle (a_{1},a_{2},\dots ,a_{r};z')} . By iterating the state change an FCSR generates an infinite, eventually periodic sequence of numbers in S {\displaystyle S} . FCSRs have been used in the design of stream ciphers (such as the F-FCSR generator), in the cryptanalysis of the summation combiner stream cipher (the reason Goresky and Klapper invented them), and in generating pseudorandom numbers for quasi-Monte Carlo (under the name Multiply With Carry (MWC) generator - invented by Couture and L'Ecuyer,) generalizing work of Marsaglia and Zaman.FCSRs are analyzed using number theory.


In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities. This greatly simplifies operations such as finding the maximum or minimum of a multivariate function and solving systems of differential equations. The notation used here is commonly used in statistics and engineering, while the tensor index notation is preferred in physics.


In most traditional PCRs the resulting products are analyzed after the PCR has been completed. This is called end-point analysis and is normally qualitative of nature rather than being quantitative. For this sort of analysis, products are mostly analyzed on an agarose gel and visualized using ethidium bromide as a fluorescent dye. Direct correlation between signal strength and initial sample concentration is not possible using end-point analysis since PCR efficiency decreases as the reaction nears the plateau phase.


In the branch of abstract algebra known as ring theory, a left primitive ring is a ring which has a faithful simple left module. Well known examples include endomorphism rings of vector spaces and Weyl algebras over fields of characteristic zero.


+ 1 {\displaystyle (p-1)!+1} . Both are named for 18th-century English mathematician John Wilson; in 1770, Edward Waring credited the theorem to Wilson, although it had been stated centuries earlier by Ibn al-Haytham.The only known Wilson primes are 5, 13, and 563 (sequence A007540 in the OEIS).


As a result, the vast majority of a program's time was being spent in only five instructions. Further, even when the program called one of those five instructions, the microcode required a finite time to decode it, even if it was just to call the internal hardware. On faster machines, this overhead was considerable.


This becomes important, for example, when the Levene's test fails to satisfy the rather generous conditions for normality associated with that test and is a default alternative under those conditions for certain statistical software programs like the VarianceEquivalenceTest routine in Mathematica. In addition to Levene's test, other parametric tests for equality of variance include the Bartlett, Brown-Forsythe, and Fisher Ratio tests. == References ==


This permits the existence of directory hierarchies, i.e., directories containing sub-directories. A name that refers to a file within a directory must be typically unique. In other words, there must be no identical names within a directory.