In number theory, a congruum (plural congrua) is the difference between successive square numbers in an arithmetic progression of three squares. That is, if x 2 {\displaystyle x^{2}} , y 2 {\displaystyle y^{2}} , and z 2 {\displaystyle z^{2}} (for integers x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} ) are three square numbers that are equally spaced apart from each other, then the spacing between them, z 2 − y 2 = y 2 − x 2 {\displaystyle z^{2}-y^{2}=y^{2}-x^{2}} , is called a congruum. The congruum problem is the problem of finding squares in arithmetic progression and their associated congrua. It can be formalized as a Diophantine equation: find integers x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} such that When this equation is satisfied, both sides of the equation equal the congruum.


In mathematical optimization, the Rosenbrock function is a non-convex function, introduced by Howard H. Rosenbrock in 1960, which is used as a performance test problem for optimization algorithms. It is also known as Rosenbrock's valley or Rosenbrock's banana function. The global minimum is inside a long, narrow, parabolic shaped flat valley. To find the valley is trivial.


In mathematical optimization, the Ackley function is a non-convex function used as a performance test problem for optimization algorithms. It was proposed by David Ackley in his 1987 PhD dissertation.On a 2-dimensional domain it is defined by: f ( x , y ) = − 20 exp   − exp   + e + 20 {\displaystyle {\begin{aligned}f(x,y)=-20&{}\exp \left\\&{}-\exp \left+e+20\end{aligned}}} Its global optimum point is f ( 0 , 0 ) = 0. {\displaystyle f(0,0)=0.}


In number theory, the Ankeny–Artin–Chowla congruence is a result published in 1953 by N. C. Ankeny, Emil Artin and S. Chowla. It concerns the class number h of a real quadratic field of discriminant d > 0. If the fundamental unit of the field is ε = t + u d 2 {\displaystyle \varepsilon ={\frac {t+u{\sqrt {d}}}{2}}} with integers t and u, it expresses in another form h t u ( mod p ) {\displaystyle {\frac {ht}{u}}{\pmod {p}}\;} for any prime number p > 2 that divides d. In case p > 3 it states that − 2 m h t u ≡ ∑ 0 < k < d χ ( k ) k ⌊ k / p ⌋ ( mod p ) {\displaystyle -2{mht \over u}\equiv \sum _{0


In the following example, we use the keys JANEAUSTEN and AEROPLANES to encrypt the following plaintext: "Transposition ciphers scramble letters like puzzle pieces to create an indecipherable arrangement." The colors show how the letters are scrambled in each transposition step. While a single step only causes a minor rearrangement, the second step leads to a significant scrambling effect if the last row of the grid is incomplete.


The overlapping sequences are then aligned into contigs and merged. Usually, several samples are pooled in one run, and each sample is characterized by a short DNA fragment, the tag. In a demultiplexing step, sequences are sorted using these tags to reassemble the separate samples.


The problem of packet size distribution is fairly well-understood today. Existing models of packet sizes have proven to be valid and simple. Most packet size models do not consider the problem of order in packet sizes.


The denominator is the number of total such friendships, which is twice the total edges in the network (one from the u's perspective and the other from the v's). After this analysis, Feld goes on to make some more qualitative assumptions about the statistical correlation between the number of friends that two friends have, based on theories of social networks such as assortative mixing, and he analyzes what these assumptions imply about the number of people whose friends have more friends than they do. Based on this analysis, he concludes that in real social networks, most people are likely to have fewer friends than the average of their friends' numbers of friends.


A task as simple as counting backwards can change memory recall; however an empty delay interval has no effect. This is because the person can continue to rehearse the items in their working memory to be remembered without interference. Cohen (1989) found that there is better recall for an action in the presence of interference if that action is physically performed during the encoding phase. It has also been found that recalling some items can interfere and inhibit the recall of other items. Another stream of thought and evidence suggests that the effects of interference on recency and primacy are relative, determined by the ratio rule (retention interval to inter item presentation distractor rate) and they exhibit time-scale invariance.


In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept called mathematical fallacy. There is a distinction between a simple mistake and a mathematical fallacy in a proof, in that a mistake in a proof leads to an invalid proof while in the best-known examples of mathematical fallacies there is some element of concealment or deception in the presentation of the proof. For example, the reason why validity fails may be attributed to a division by zero that is hidden by algebraic notation. There is a certain quality of the mathematical fallacy: as typically presented, it leads not only to an absurd result, but does so in a crafty or clever way.


If the conditional distribution of Y {\displaystyle Y} given X {\displaystyle X} is a continuous distribution, then its probability density function is known as the conditional density function. The properties of a conditional distribution, such as the moments, are often referred to by corresponding names such as the conditional mean and conditional variance. More generally, one can refer to the conditional distribution of a subset of a set of more than two variables; this conditional distribution is contingent on the values of all the remaining variables, and if more than one variable is included in the subset then this conditional distribution is the conditional joint distribution of the included variables.


In the C programming language, Duff's device is a way of manually implementing loop unrolling by interleaving two syntactic constructs of C: the do-while loop and a switch statement. Its discovery is credited to Tom Duff in November 1983, when Duff was working for Lucasfilm and used it to speed up a real-time animation program. Loop unrolling attempts to reduce the overhead of conditional branching needed to check whether a loop is done, by executing a batch of loop bodies per iteration. To handle cases where the number of iterations is not divisible by the unrolled-loop increments, a common technique among assembly language programmers is to jump directly into the middle of the unrolled loop body to handle the remainder. Duff implemented this technique in C by using C's case label fall-through feature to jump into the unrolled body.


A notable series of analog calculating machines were developed by Leonardo Torres Quevedo since 1895, including one that was able to compute the roots of arbitrary polynomials of order eight, including the complex ones, with a precision down to thousandths. An important advance in analog computing was the development of the first fire-control systems for long range ship gunlaying. When gunnery ranges increased dramatically in the late 19th century it was no longer a simple matter of calculating the proper aim point, given the flight times of the shells.


The hierarchical softmax (introduced by Morin and Bengio in 2005) uses a binary tree structure where the outcomes (vocabulary words) are the leaves and the intermediate nodes are suitably selected "classes" of outcomes, forming latent variables. The desired probability (softmax value) of a leaf (outcome) can then be calculated as the product of the probabilities of all nodes on the path from the root to that leaf. Ideally, when the tree is balanced, this would reduce the computational complexity from O ( K ) {\displaystyle O(K)} to O ( log 2  K ) {\displaystyle O(\log _{2}K)} .


It gets stuck at a basic feasible solution (a corner of the feasible polytope) and changes bases in a cyclic way without decreasing the minimization target. Such cycles are avoided by Bland's rule for choosing a column to enter and a column to leave the basis. Bland's rule was developed by Robert G. Bland, now an Emeritus Professor of operations research at Cornell University, while he was a research fellow at the Center for Operations Research and Econometrics in Belgium.


In this way they also form two boxes, this time in consecutive corners, with areas a 2 {\displaystyle a^{2}} and b 2 {\displaystyle b^{2}} which will again lead to a second square of with the area 2 a b + a 2 + b 2 {\displaystyle 2ab+a^{2}+b^{2}} . English mathematician Sir Thomas Heath gives this proof in his commentary on Proposition I.47 in Euclid's Elements, and mentions the proposals of German mathematicians Carl Anton Bretschneider and Hermann Hankel that Pythagoras may have known this proof. Heath himself favors a different proposal for a Pythagorean proof, but acknowledges from the outset of his discussion "that the Greek literature which we possess belonging to the first five centuries after Pythagoras contains no statement specifying this or any other particular great geometric discovery to him." Recent scholarship has cast increasing doubt on any sort of role for Pythagoras as a creator of mathematics, although debate about this continues.


That is, a metric should score highly translations that humans score highly, and give low scores to those humans give low scores. Human judgment is the benchmark for assessing automatic metrics, as humans are the end-users of any translation output. The measure of evaluation for metrics is correlation with human judgment.


In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable solution, but which has not yet been solved (i.e., no solution for it is known). In the history of science, some of these supposed open problems were "solved" by means of showing that they were not well-defined. In mathematics, many open problems are concerned with the question of whether a certain definition is or is not consistent.


In the context of machine learning, D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} is often called the information gain achieved if P {\displaystyle P} would be used instead of Q {\displaystyle Q} which is currently used. By analogy with information theory, it is called the relative entropy of P {\displaystyle P} with respect to Q {\displaystyle Q} . Expressed in the language of Bayesian inference, D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} is a measure of the information gained by revising one's beliefs from the prior probability distribution Q {\displaystyle Q} to the posterior probability distribution P {\displaystyle P} . In other words, it is the amount of information lost when Q {\displaystyle Q} is used to approximate P {\displaystyle P} .


In the Discovery with Model method, a model is developed via prediction, clustering or by human reasoning knowledge engineering and then used as a component in another analysis, namely in prediction and relationship mining. In the prediction method use, the created model's predictions are used to predict a new variable. For the use of relationship mining, the created model enables the analysis between new predictions and additional variables in the study.


{\displaystyle S^{1}.} The proof is based on normal surface techniques originated by Hellmuth Kneser. Existence was proven by Kneser, but the exact formulation and proof of the uniqueness was done more than 30 years later by John Milnor.


This project, known as Hilbert's program, was seriously affected by Gödel's incompleteness theorems, which show that the consistency of formal theories of arithmetic cannot be established using methods formalizable in those theories. Gentzen showed that it is possible to produce a proof of the consistency of arithmetic in a finitary system augmented with axioms of transfinite induction, and the techniques he developed to do so were seminal in proof theory. A second thread in the history of foundations of mathematics involves nonclassical logics and constructive mathematics.


In mathematics, a pseudomanifold is a special type of topological space. It looks like a manifold at most of its points, but it may contain singularities. For example, the cone of solutions of z 2 = x 2 + y 2 {\displaystyle z^{2}=x^{2}+y^{2}} forms a pseudomanifold. A pseudomanifold can be regarded as a combinatorial realisation of the general idea of a manifold with singularities. The concepts of orientability, orientation and degree of a mapping make sense for pseudomanifolds and moreover, within the combinatorial approach, pseudomanifolds form the natural domain of definition for these concepts.


Perform Cholesky decomposition (a variant of Gaussian elimination for symmetric matrices), ordering the elimination of the variables by the recursive structure of the partition: each of the two subgraphs formed by removing the separator is eliminated first, and then the separator vertices are eliminated.As a consequence of this algorithm, the fill-in (the set of nonzero matrix entries created in the Cholesky decomposition that are not part of the input matrix structure) is limited to at most the square of the separator size at each level of the recursive partition. In particular, for planar graphs (frequently arising in the solution of sparse linear systems derived from two-dimensional finite element method meshes) the resulting matrix has O(n log n) nonzeros, due to the planar separator theorem guaranteeing separators of size O(√n). For arbitrary graphs there is a nested dissection that guarantees fill-in within a O ( min { d log 4  n , m 1 / 4 log 3.5  n } ) {\displaystyle O(\min\{{\sqrt {d}}\log ^{4}n,m^{1/4}\log ^{3.5}n\})} factor of optimal, where d is the maximum degree and m is the number of non-zeros.


These conditions are very loose, and allow enormous flexibility in the choice of open sets. For example, every subset can be open (the discrete topology), or no subset can be open except the space itself and the empty set (the indiscrete topology).In practice, however, open sets are usually chosen to provide a notion of nearness that is similar to that of metric spaces, without having a notion of distance defined. In particular, a topology allows defining properties such as continuity, connectedness, and compactness, which were originally defined by means of a distance. The most common case of a topology without any distance is given by manifolds, which are topological spaces that, near each point, resemble an open set of a Euclidean space, but on which no distance is defined in general. Less intuitive topologies are used in other branches of mathematics; for example, the Zariski topology, which is fundamental in algebraic geometry and scheme theory.


In separate chaining, the process involves building a linked list with key–value pair for each search array index. The collided items are chained together through a single linked list, which can be traversed to access the item with a unique search key. : 464 Collision resolution through chaining with linked list is a common method of implementation of hash tables. Let T {\displaystyle T} and x {\displaystyle x} be the hash table and the node respectively, the operation involves as follows:: 258 Chained-Hash-Insert(T, k) insert x at the head of linked list T Chained-Hash-Search(T, k) search for an element with key k in linked list T Chained-Hash-Delete(T, k) delete x from the linked list T If the element is comparable either numerically or lexically, and inserted into the list by maintaining the total order, it results in faster termination of the unsuccessful searches. : 520–521


In statistics, dispersion (also called variability, scatter, or spread) is the extent to which a distribution is stretched or squeezed. Common examples of measures of statistical dispersion are the variance, standard deviation, and interquartile range. For instance, when the variance of data in a set is large, the data is widely scattered. On the other hand, when the variance is small, the data in the set is clustered. Dispersion is contrasted with location or central tendency, and together they are the most used properties of distributions.


The notion directly extends to the product of an arbitrary finite number of types (a n-ary product type), and in this case, it characterizes the expressions which behave as tuples of expressions of the corresponding types. A degenerated form of product type is the unit type: it is the product of no types. In call-by-value programming languages, a product type can be interpreted as a set of pairs whose first component is a value in the first type and whose second component is a value in the second type.


In the data warehouse process, data can be aggregated in data marts at different levels of abstraction. The user may start looking at the total sale units of a product in an entire region. Then the user looks at the states in that region. Finally, they may examine the individual stores in a certain state. Therefore, typically, the analysis starts at a higher level and drills down to lower levels of details.


12n(π(n+1)+1)where π(n) is the prime-counting function. This was subsequently improved by Hans Frederick Blichfeldt who replaced the 12 with a 6. Unpublished work on the finite case was also done by Boris Weisfeiler. Subsequently, Michael Collins, using the classification of finite simple groups, showed that in the finite case, one can take ƒ(n) = (n + 1)! when n is at least 71, and gave near complete descriptions of the behavior for smaller n.


Some studies suggest that these errors can be anywhere between 5 and 10 degrees.These goniometers come in different forms that some argue increase reliability. The universal standard goniometer is a plastic or metal tool with 1 degree increments.


In scalar languages such as C and Pascal, operations apply only to single values, so a+b expresses the addition of two numbers. In such languages, adding one array to another requires indexing and looping, the coding of which is tedious. In array-based languages, for example in Fortran, the nested for-loop above can be written in array-format in one line, or alternatively, to emphasize the array nature of the objects, While scalar languages like C do not have native array programming elements as part of the language proper, this does not mean programs written in these languages never take advantage of the underlying techniques of vectorization (i.e., utilizing a CPU's vector-based instructions if it has them or by using multiple CPU cores). Some C compilers like GCC at some optimization levels detect and vectorize sections of code that its heuristics determine would benefit from it. Another approach is given by the OpenMP API, which allows one to parallelize applicable sections of code by taking advantage of multiple CPU cores.


In the context of government documents, redaction (also called sanitization) generally refers more specifically to the process of removing sensitive or classified information from a document prior to its publication, during declassification.


Other authors call a set directed if and only if it is directed both upward and downward.Directed sets are a generalization of nonempty totally ordered sets. That is, all totally ordered sets are directed sets (contrast partially ordered sets, which need not be directed). Join-semilattices (which are partially ordered sets) are directed sets as well, but not conversely.


In mathematics, a matrix coefficient (or matrix element) is a function on a group of a special form, which depends on a linear representation of the group and additional data. Precisely, it is a function on a compact topological group G obtained by composing a representation of G on a vector space V with a linear map from the endomorphisms of V into V's underlying field. It is also called a representative function.


It is called the policy cycle as the final stage (evaluation) often leads back to the first stage (problem definition), thus restarting the cycle. Harold Lasswell's popular model of the policy cycle divided the process into seven distinct stages, asking questions of both how and why public policies should be made. With the stages ranging from (1) intelligence, (2) promotion, (3) prescription, (4) invocation, (5) application, (6) termination and (7) appraisal, this process inherently attempts to combine policy implementation to formulated policy goals.One version by James E. Anderson, in his Public Policy-Making (1974) has the following stages: Agenda setting (Problem identification) – The recognition of certain subject as a problem demanding further government attention.


In order to compete with Intel's Advanced Programmable Interrupt Controller (APIC), which had enabled the first Intel 486-based multiprocessor systems, in early 1995 AMD and Cyrix proposed as somewhat similar-in-purpose OpenPIC architecture supporting up to 32 processors. The OpenPIC architecture had at least declarative support from IBM and Compaq around 1995. No x86 motherboard was released with OpenPIC however. After the OpenPIC's failure in the x86 market, AMD licensed the Intel APIC Architecture for its AMD Athlon and later processors.


If we assume that a function f: R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } is globally continuous and separately differentiable on each variable (all partial derivatives exist everywhere), it is not true that f {\displaystyle f} will necessarily be differentiable. A counterexample in two dimensions is given by f ( x , y ) = 2 x 2 y + y 3 x 2 + y 2 . {\displaystyle f(x,y)={\dfrac {2x^{2}y+y^{3}}{x^{2}+y^{2}}}.} If in addition we define f ( 0 , 0 ) = 0 {\displaystyle f(0,0)=0} , this function is everywhere continuous and has well-defined partial derivatives in x {\displaystyle x} and y {\displaystyle y} everywhere (also at the origin), but is not differentiable at the origin.


In the field of bioinformatics, a sequence database is a type of biological database that is composed of a large collection of computerized ("digital") nucleic acid sequences, protein sequences, or other polymer sequences stored on a computer. The UniProt database is an example of a protein sequence database. As of 2013 it contained over 40 million sequences and is growing at an exponential rate. Historically, sequences were published in paper form, but as the number of sequences grew, this storage method became unsustainable.


Alternatively the two distributions can both be empirically estimated ones; this is called the two-sample case. The criterion is named after Harald Cramér and Richard Edler von Mises who first proposed it in 1928–1930. The generalization to two samples is due to Anderson.The Cramér–von Mises test is an alternative to the Kolmogorov–Smirnov test (1933).


In typical applications, e.g., when discretizing integral equations, preconditioning the resulting systems of linear equations, or solving elliptic partial differential equations, a rank proportional to log  ( 1 / ϵ ) γ {\displaystyle \log(1/\epsilon )^{\gamma }} with a small constant γ {\displaystyle \gamma } is sufficient to ensure an accuracy of ϵ {\displaystyle \epsilon } . Compared to many other data-sparse representations of non-sparse matrices, hierarchical matrices offer a major advantage: the results of matrix arithmetic operations like matrix multiplication, factorization or inversion can be approximated in O ( n k α log  ( n ) β ) {\displaystyle O(nk^{\alpha }\,\log(n)^{\beta })} operations, where α , β ∈ { 1 , 2 , 3 } . {\displaystyle \alpha ,\beta \in \{1,2,3\}.}


In mathematics, the signal magnitude area (abbreviated SMA or sma) is a statistical measure of the magnitude of a varying quantity.


If all coefficients in r {\displaystyle r} are negative, then z 0 {\displaystyle z_{0}} is an optimal solution, since all variables (including all non-basic variables) must be at least 0, so the second line implies z ≤ z 0 {\displaystyle z\leq z_{0}} . If some coefficients in r {\displaystyle r} are positive, then it may be possible to increase the maximization target.


In recent years, the speed of the CPU has grown many times in comparison to the access speed of the main memory. Care needs to be taken to reduce the number of times main memory is accessed in order to maintain performance. If, for instance, every instruction run in the CPU requires an access to memory, the computer gains nothing for increased CPU speed—a problem referred to as being memory bound. It is possible to make extremely fast memory, but this is only practical for small amounts of memory for cost, power and signal routing reasons.


In mathematical optimization, neighborhood search is a technique that tries to find good or near-optimal solutions to a combinatorial optimisation problem by repeatedly transforming a current solution into a different solution in the neighborhood of the current solution. The neighborhood of a solution is a set of similar solutions obtained by relatively simple modifications to the original solution. For a very large-scale neighborhood search, the neighborhood is large and possibly exponentially sized. The resulting algorithms can outperform algorithms using small neighborhoods because the local improvements are larger.


In statistics, linear regression is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of one explanatory variable is called simple linear regression; for more than one, the process is called multiple linear regression. This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable.In linear regression, the relationships are modeled using linear predictor functions whose unknown model parameters are estimated from the data. Such models are called linear models.


In some languages such as REXX, the precision of all calculations must be set before doing a calculation. Other languages, such as Python and Ruby, extend the precision automatically to prevent overflow.


Fortran did not have an equality operator (it was only possible to compare an expression to zero, using the arithmetic IF statement) until FORTRAN IV was released in 1962, since when it has used the four characters .EQ. to test for equality. The language B introduced the use of == with this meaning, which has been copied by its descendant C and most later languages where = means assignment. The equal sign is also used in defining attribute–value pairs, in which an attribute is assigned a value.


In mathematical physics and gauge theory, the ADHM construction or monad construction is the construction of all instantons using methods of linear algebra by Michael Atiyah, Vladimir Drinfeld, Nigel Hitchin, Yuri I. Manin in their paper "Construction of Instantons."


In mathematics, Krener's theorem is a result attributed to Arthur J. Krener in geometric control theory about the topological properties of attainable sets of finite-dimensional control systems. It states that any attainable set of a bracket-generating system has nonempty interior or, equivalently, that any attainable set has nonempty interior in the topology of the corresponding orbit. Heuristically, Krener's theorem prohibits attainable sets from being hairy.


In mathematics, Whitney's planarity criterion is a matroid-theoretic characterization of planar graphs, named after Hassler Whitney. It states that a graph G is planar if and only if its graphic matroid is also cographic (that is, it is the dual matroid of another graphic matroid). In purely graph-theoretic terms, this criterion can be stated as follows: There must be another (dual) graph G'=(V',E') and a bijective correspondence between the edges E' and the edges E of the original graph G, such that a subset T of E forms a spanning tree of G if and only if the edges corresponding to the complementary subset E-T form a spanning tree of G'.


Z is sometimes called the Boltzmann sum over states (or "Zustandssumme" in the original German). If we index the summation via the energy eigenvalues instead of all possible states, degeneracy must be taken into account. The probability of our system having energy ε i {\displaystyle \varepsilon _{i}} is simply the sum of the probabilities of all corresponding microstates: P ( ε i ) = 1 Z g i e − ε i / k T {\displaystyle P(\varepsilon _{i})={\frac {1}{Z}}g_{i}e^{-\varepsilon _{i}/kT}} where, with obvious modification, Z = ∑ j g j e − ε j / k T , {\displaystyle Z=\sum _{j}g_{j}e^{-\varepsilon _{j}/kT},} this is the same result as before.


In the case of time-dependent vector fields F: R n × R → R n {\displaystyle {\boldsymbol {F}}:\mathbb {R} ^{n}\times \mathbb {R} \to \mathbb {R} ^{n}} , one denotes φ t , t 0 ( x 0 ) = x ( t + t 0 ) , {\displaystyle \varphi ^{t,t_{0}}({\boldsymbol {x}}_{0})={\boldsymbol {x}}(t+t_{0}),} where x: R → R n {\displaystyle {\boldsymbol {x}}:\mathbb {R} \to \mathbb {R} ^{n}} is the solution of x ˙ ( t ) = F ( x ( t ) , t ) , x ( t 0 ) = x 0 . {\displaystyle {\dot {\boldsymbol {x}}}(t)={\boldsymbol {F}}({\boldsymbol {x}}(t),t),\qquad {\boldsymbol {x}}(t_{0})={\boldsymbol {x}}_{0}.} Then φ t , t 0 ( x 0 ) {\displaystyle \varphi ^{t,t_{0}}({\boldsymbol {x}}_{0})} is the time-dependent flow of F. It is not a "flow" by the definition above, but it can easily be seen as one by rearranging its arguments.


More generally, addition-chain exponentiation may also refer to exponentiation by non-minimal addition chains constructed by a variety of algorithms (since a shortest addition chain is very difficult to find). The shortest addition-chain algorithm requires no more multiplications than binary exponentiation and usually less. The first example of where it does better is for a15, where the binary method needs six multiplications but the shortest addition chain requires only five: a 15 = a × ( a ×  2 ) 2 {\displaystyle a^{15}=a\times (a\times ^{2})^{2}\!}


In the Unidirectional mode of operation, packets are only sent in one direction: from compressor to decompressor. This mode therefore makes ROHC usable over links where a return path from decompressor to compressor is unavailable or undesirable. In order to handle potential decompression errors, the compressor sends periodic refreshes of the stream context to the decompressor.


In mathematics, the pointwise product of two functions is another function, obtained by multiplying the images of the two functions at each value in the domain. If f and g are both functions with domain X and codomain Y, and elements of Y can be multiplied (for instance, Y could be some set of numbers), then the pointwise product of f and g is another function from X to Y which maps x in X to f (x)g(x) in Y.


count t max {\displaystyle t_{\max }}: max. countSince the number of indexed items per descriptor is usually distributed according to a power law, for larger ranges of values, a logarithmic representation makes sense.Implementations of tag clouds also include text parsing and filtering out unhelpful tags such as common words, numbers, and punctuation. There are also websites creating artificially or randomly weighted tag clouds, for advertising, or for humorous results.


n ) ( ∇ p 0 . n ) − π ( ∇ u 0 .


This approach allows the system to gather information about the frequency with which various constructions occur in specific contexts. (See machine learning.) Approaches which have been used include straightforward PCFGs (probabilistic context-free grammars), maximum entropy, and neural nets.


This concept can be extended to the multivariate context by an extension of the simple sum to a number of F {\displaystyle F} sums that cover all dimensions in the feature space: p w ( θ → ) = ∑ k 1 , . . . , k F = − ∞ ∞ p ( θ → + 2 π k 1 e 1 + ⋯ + 2 π k F e F ) {\displaystyle p_{w}({\vec {\theta }})=\sum _{k_{1},...,k_{F}=-\infty }^{\infty }{p({\vec {\theta }}+2\pi k_{1}\mathbf {e} _{1}+\dots +2\pi k_{F}\mathbf {e} _{F})}} where e k = ( 0 , … , 0 , 1 , 0 , … , 0 ) T {\displaystyle \mathbf {e} _{k}=(0,\dots ,0,1,0,\dots ,0)^{\mathsf {T}}} is the k {\displaystyle k} th Euclidean basis vector.


Usually, the price also includes a mark-up for profit over the cost of production. More generalized in the field of economics, cost is a metric that is totaling up as a result of a process or as a differential for the result of a decision.


In some developing countries, many people do not have access to banking facilities, especially in tier II and tier III cities. Taking the example of India, there are more mobile phone users than there are people with active bank accounts. Telecom operators, in such locations, have started offering mobile money wallets which allow adding funds easily through their existing mobile subscription number, by visiting physical recharge points close to their homes and offices and converting their cash into mobile wallet currency. This can be used for online transaction and eCommerce purchases.


In processing electronic audio signals, pre-emphasis refers to a system process designed to increase (within a frequency band) the magnitude of some (usually higher) frequencies with respect to the magnitude of other (usually lower) frequencies in order to improve the overall signal-to-noise ratio by minimizing the adverse effects of such phenomena as attenuation distortion or saturation of recording media in subsequent parts of the system. The mirror operation is called de-emphasis, and the system as a whole is called emphasis. Pre-emphasis is achieved with a pre-emphasis network which is essentially a calibrated filter. The frequency response is decided by special time constants.


An important aspect of this stage is verification. This effort verifies that the requirement has been implemented correctly. There are 4 methods of verification: analysis, inspection, testing, and demonstration.


The version without negations is sometimes called weakly injective. The existence of value collisions is a strong notion of non-injectivity. And regarding surjectivity, similar considerations exist for outlier-production in the codomain.


In the Middle East, Hasan Ibn al-Haytham, Latinized as Alhazen (c. 965 – c. 1040 CE) derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration, where the formulas for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid. Roshdi Rashed has argued that the 12th century mathematician Sharaf al-Dīn al-Tūsī must have used the derivative of cubic polynomials in his Treatise on Equations. Rashed's conclusion has been contested by other scholars, who argue that he could have obtained his results by other methods which do not require the derivative of the function to be known.


In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup.


"If the cause, assigned for any effect, be not sufficient to produce it, we must either reject that cause, or add to it such qualities as will give it a just proportion to the effect." Occam's razor: Explanations which require fewer unjustified assumptions are more likely to be correct; avoid unnecessary or improbable assumptions. Popper's falsifiability principle: For a theory to be considered scientific, it must be falsifiable. Sagan standard: Extraordinary claims require extraordinary evidence.


In the book written by Frank Rosenblatt, published in 1961, a three-layer Multilayer Perceptron (MLP) model with skip connections was presented (Chapter 15, p313 in ). The model was referred to as a "cross-coupled system", and the skip connections were forms of cross-coupled connections. In two books published in 1994 and 1996, "skip-layer" connections were presented in feed-forward MLP models: "The general definition  allows more than one hidden layer, and it also allows 'skip-layer' connections from input to output" (p261 in, p144 in ), "... which allows the non-linear units to perturb a linear functional form" (p262 in ). This description suggests that the non-linear MLP performs like a residual function (perturbation) added to a linear function.


In phonology, voicing (or sonorization) is a sound change where a voiceless consonant becomes voiced due to the influence of its phonological environment; shift in the opposite direction is referred to as devoicing or desonorization. Most commonly, the change is a result of sound assimilation with an adjacent sound of opposite voicing, but it can also occur word-finally or in contact with a specific vowel. For example, the English suffix -s is pronounced  when it follows a voiceless phoneme (cats), and  when it follows a voiced phoneme (dogs). This type of assimilation is called progressive, where the second consonant assimilates to the first; regressive assimilation goes in the opposite direction, as can be seen in have to .


It is impossible for an unprivileged application to access a page that has not been explicitly allocated to it, because every memory address either points to a page allocated to that application, or generates an interrupt called a page fault. Unallocated pages, and pages allocated to any other application, do not have any addresses from the application point of view. A page fault may not necessarily indicate an error.


In mathematics and theoretical computer science, a set constraint is an equation or an inequation between sets of terms. Similar to systems of (in)equations between numbers, methods are studied for solving systems of set constraints. Different approaches admit different operators (like "∪", "∩", "\", and function application) on sets and different (in)equation relations (like "=", "⊆", and "⊈") between set expressions. Systems of set constraints are useful to describe (in particular infinite) sets of ground terms. They arise in program analysis, abstract interpretation, and type inference.


In telecommunications, a long-distance call (U.S.) or trunk call (also known as a toll call in the U.K.) is a telephone call made to a location outside a defined local calling area.


In telecommunications, remote call forwarding is a service feature that allows incoming calls to be forwarded to a remote call forwarding number, such as a cell phone or another office location and is designated by the call receiver.Customers may have a remote-forwarding telephone number in a central switching office without having any other local telephone service in that office. One common purpose for this service is to enable customers to retain their telephone number when they move to a location serviced by a different telephone exchange. The service is useful for business customers with widely advertised numbers which appear on headed paper, vehicles and various marketing literature. When customers ring, their calls are forwarded to the new location.


In order to maintain good programming practice, a number of compiler options and flags designed for safety have been enabled by default to help in spotting potential issues so they can be fixed earlier (-Wall, -Werror, -Wextra, -Wuninitialized). There have also been code readability updates which help future contributors in verifying program correctness (KNF, white-space, line-wrapping, etc.). Modification or removal of unneeded method wrappers and macros also help with code readability and auditing (Error and I/O abstraction library references). Changes were made to ensure that LibreSSL will be year 2038 compatible along with maintaining portability for other similar platforms. In addition, explicit_bzero and bn_clear calls were added to prevent the compiler from optimizing them out and prevent attackers from reading previously allocated memory.


Females only lay one test clutch and return shortly after laying it. Capable males prove their parental quality by defending the brood and not cannibalizing the eggs. Test eggs are energetically expensive to create, so this strategy is typically only used by large females at the beginning of the mating phase.


One special type of operand is the parenthesis group. An expression enclosed in parentheses is typically recursively evaluated to be treated as a single operand on the next evaluation level. Each operator is given a position, precedence, and an associativity.


In terms of information theory, entropy is considered to be a measure of the uncertainty in a message. To put it intuitively, suppose p = 0 {\displaystyle p=0} . At this probability, the event is certain never to occur, and so there is no uncertainty at all, leading to an entropy of 0. If p = 1 {\displaystyle p=1} , the result is again certain, so the entropy is 0 here as well.


In mathematics, primitive recursive set functions or primitive recursive ordinal functions are analogs of primitive recursive functions, defined for sets or ordinals rather than natural numbers. They were introduced by Jensen & Karp (1971).


Iteration length has changed from months to weeks or days with the rising popularity of Agile, DevOps, and Continuous Delivery. Traditional methods of testing, which rely heavily on manual testing and automated GUI tests that require frequent updating, cannot keep pace.


Therefore, there are O(V) saturating pushes on (u, v), and the total number of saturating pushes is at most 2| V || E |. This results in a time bound of O(VE) for the saturating push operations. Bounding the number of nonsaturating pushes can be achieved via a potential argument.


In the United States, since 2010 the Insurance Institute for Highway Safety uses a scheme it has developed that takes into account a combination of both vehicle footprint (length times width) and weight. The United States National Highway Traffic Safety Administration (NHTSA) separates vehicles into classes by the curb weight of the vehicle with standard equipment including the maximum capacity of fuel, oil, coolant, and air conditioning, if so equipped. The United States Federal Highway Administration has developed a classification scheme used for automatically calculating road use tolls. There are two broad categories depending on whether the vehicle carries passengers or commodities.


Suppose further that the researcher wants to estimate a bivariate linear model via least squares: Y i = β 0 + β 1 X 1 i + β 2 X 2 i + e i {\displaystyle Y_{i}=\beta _{0}+\beta _{1}X_{1i}+\beta _{2}X_{2i}+e_{i}} . If the researcher only has access to N = 2 {\displaystyle N=2} data points, then they could find infinitely many combinations ( β ^ 0 , β ^ 1 , β ^ 2 ) {\displaystyle ({\hat {\beta }}_{0},{\hat {\beta }}_{1},{\hat {\beta }}_{2})} that explain the data equally well: any combination can be chosen that satisfies Y ^ i = β ^ 0 + β ^ 1 X 1 i + β ^ 2 X 2 i {\displaystyle {\hat {Y}}_{i}={\hat {\beta }}_{0}+{\hat {\beta }}_{1}X_{1i}+{\hat {\beta }}_{2}X_{2i}} , all of which lead to ∑ i e ^ i 2 = ∑ i ( Y ^ i − ( β ^ 0 + β ^ 1 X 1 i + β ^ 2 X 2 i ) ) 2 = 0 {\displaystyle \sum _{i}{\hat {e}}_{i}^{2}=\sum _{i}({\hat {Y}}_{i}-({\hat {\beta }}_{0}+{\hat {\beta }}_{1}X_{1i}+{\hat {\beta }}_{2}X_{2i}))^{2}=0} and are therefore valid solutions that minimize the sum of squared residuals. To understand why there are infinitely many options, note that the system of N = 2 {\displaystyle N=2} equations is to be solved for 3 unknowns, which makes the system underdetermined.


In mathematics, a Euclidean plane is a Euclidean space of dimension two, denoted E2. It is a geometric space in which two real numbers are required to determine the position of each point. It is an affine space, which includes in particular the concept of parallel lines. It has also metrical properties induced by a distance, which allows to define circles, and angle measurement. A Euclidean plane with a chosen Cartesian coordinate system is called a Cartesian plane. The set R 2 {\displaystyle \mathbb {R} ^{2}} of the pairs of real numbers (the real coordinate plane), equipped with the dot product, is often called the Euclidean plane, since every Euclidean plane is isomorphic to it.


Some Boolean operations, in particular do not have inverses that may be defined as functions. In particular the disjunction "or" has inverses that allow two values. In natural language "or" represents alternate possibilities.


If all elements in each sampled cluster are sampled, then this is referred to as a "one-stage" cluster sampling plan. If a simple random subsample of elements is selected within each of these groups, this is referred to as a "two-stage" cluster sampling plan. A common motivation for cluster sampling is to reduce the total number of interviews and costs given the desired accuracy. For a fixed sample size, the expected random error is smaller when most of the variation in the population is present internally within the groups, and not between the groups.


In stochastic gradient descent, feature scaling can sometimes improve the convergence speed of the algorithm. In support vector machines, it can reduce the time to find support vectors. Note that feature scaling changes the SVM result.


In software testing, test automation is the use of software separate from the software being tested to control the execution of tests and the comparison of actual outcomes with predicted outcomes. Test automation can automate some repetitive but necessary tasks in a formalized testing process already in place, or perform additional testing that would be difficult to do manually. Test automation is critical for continuous delivery and continuous testing.


It also debuted atop the US Billboard Top Rock Albums chart and was later ranked at number 62 on the chart's year-end version. Elsewhere in North America, the album peaked at number 19 on the Billboard Canadian Albums Chart. In the Asia-Pacific region, Notes on a Conditional Form reached the top of the Australian Albums chart, number four on the New Zealand Albums chart, number 14 on the Billboard Japan Hot Albums chart and number 17 on the Japanese Albums chart.


In the case of a private VoIP system, the primary telephony system itself is located within the private infrastructure of the end-user organization. Usually, the system will be deployed on-premises at a site within the direct control of the organization. This can provide numerous benefits in terms of QoS control (see below), cost scalability, and ensuring privacy and security of communications traffic. However, the responsibility for ensuring that the VoIP system remains performant and resilient is predominantly vested in the end-user organization.


In principle, it should be possible to optimize x86 code to favor Code Morphing Software, or even for compilers to target the native VLIW architecture directly. However, writing in 2003, Linus Torvalds apparently dismissed these approaches as unrealistic: The native crusoe code – even if it was documented and available – is not very conducive to general-purpose OS stuff. It has no notion of memory protection, and there's no MMU for code accesses, so things like kernel modules simply wouldn't work. The translations are usually better than statically compiled native code (because the whole CPU is designed for speculation, and the static compilers don't know how to do that), and thus going to native mode is not necessarily a performance improvement.


For example, BOOK, CHOICE, or DECIDE. The lowercase letters l, o, s, x and z are rotationally symmetrical, while pairs such as b/q, d/p, m/w, n/u, and in some typefaces h/y and a/e, are rotations of each other. Thus, the words "sos", "pod", "suns", "yeah", "swims", "dollop", or "passed" form natural rotational ambigrams.


For high-dimensional problems, the exact computation of the Hessian is usually prohibitively expensive, and even its storage can be problematic, requiring O ( N 2 ) {\displaystyle O(N^{2})} memory (but see the limited-memory L-BFGS quasi-Newton method). The conjugate gradient method can also be derived using optimal control theory. In this accelerated optimization theory, the conjugate gradient method falls out as a nonlinear optimal feedback controller, u = k ( x , x ˙ ) := − γ a ∇ x f ( x ) − γ b x ˙ {\displaystyle u=k(x,{\dot {x}}):=-\gamma _{a}\nabla _{x}f(x)-\gamma _{b}{\dot {x}}} for the double integrator system, x ¨ = u {\displaystyle {\ddot {x}}=u} The quantities γ a > 0 {\displaystyle \gamma _{a}>0} and γ b > 0 {\displaystyle \gamma _{b}>0} are variable feedback gains.


Similarly, the trace of the matrix equals the sum of its eigenvalues. From this point of view, we can define the pseudo-determinant for a singular matrix to be the product of its nonzero eigenvalues (the density of multivariate normal distribution will need this quantity). In many applications, such as PageRank, one is interested in the dominant eigenvalue, i.e. that which is largest in absolute value. In other applications, the smallest eigenvalue is important, but in general, the whole spectrum provides valuable information about a matrix.


B1 generates a random bit b in a secure laboratory and gives it to A1 at a time t1. B0 and B1 communicate a {\displaystyle a} and b through a secure and authenticated channel. Similarly, A0 and A1 communicate a {\displaystyle a} and b through a secure and authenticated channel.


In probability theory, a random recursive tree is a rooted tree chosen uniformly at random from the recursive trees with a given number of vertices.


See  for more details on complex applications. Metaheuristics fall in two categories: trajectory-based metaheuristics and population-based metaheuristics. The main difference of these two kind of methods relies in the number of tentative solutions used in each step of the (iterative) algorithm.


In orbital mechanics, a frozen orbit is an orbit for an artificial satellite in which natural drifting due to the central body's shape has been minimized by careful selection of the orbital parameters. Typically, this is an orbit in which, over a long period of time, the satellite's altitude remains constant at the same point in each orbit. Changes in the inclination, position of the apsis of the orbit, and eccentricity have been minimized by choosing initial values so that their perturbations cancel out., which results in a long-term stable orbit that minimizes the use of station-keeping propellant.


{\displaystyle |S|.} Obviously | S | = Δ ( C o u t ( m 1 ) , C o u t ( m 2 ) ) ⩾ ( 1 − R ) N {\displaystyle |S|=\Delta (C_{out}(m_{1}),C_{out}(m_{2}))\geqslant (1-R)N} . Due to the Wozencraft Ensemble Theorem, there are at most ε N {\displaystyle \varepsilon N} linear codes having distance less than H q − 1 ( 1 2 − ε ) ⋅ 2 k , {\displaystyle H_{q}^{-1}({\tfrac {1}{2}}-\varepsilon )\cdot 2k,} so T ⩾ | S | − ε N ⩾ ( 1 − R ) N − ε N = ( 1 − R − ε ) N .


In regular grammars, the left hand side is again only a single nonterminal symbol, but now the right-hand side is also restricted. The right side may be the empty string, or a single terminal symbol, or a single terminal symbol followed by a nonterminal symbol, but nothing else. (Sometimes a broader definition is used: one can allow longer strings of terminals or single nonterminals without anything else, making languages easier to denote while still defining the same class of languages.) The language { a n b n ∣ n ≥ 1 } {\displaystyle \left\{a^{n}b^{n}\mid n\geq 1\right\}} defined above is not regular, but the language { a n b m ∣ m , n ≥ 1 } {\displaystyle \left\{a^{n}b^{m}\mid m,n\geq 1\right\}} (at least 1 a {\displaystyle a} followed by at least 1 b {\displaystyle b} , where the numbers may be different) is, as it can be defined by the grammar G 3 {\displaystyle G_{3}} with N = { S , A , B } {\displaystyle N=\left\{S,A,B\right\}} , Σ = { a , b } {\displaystyle \Sigma =\left\{a,b\right\}} , S {\displaystyle S} the start symbol, and the following production rules: S → a A {\displaystyle S\rightarrow aA} A → a A {\displaystyle A\rightarrow aA} A → b B {\displaystyle A\rightarrow bB} B → b B {\displaystyle B\rightarrow bB} B → ϵ {\displaystyle B\rightarrow \epsilon } All languages generated by a regular grammar can be recognized in O ( n ) {\displaystyle O(n)} time by a finite-state machine. Although in practice, regular grammars are commonly expressed using regular expressions, some forms of regular expression used in practice do not strictly generate the regular languages and do not show linear recognitional performance due to those deviations.


In mathematics, Dickson's lemma states that every set of n {\displaystyle n} -tuples of natural numbers has finitely many minimal elements. This simple fact from combinatorics has become attributed to the American algebraist L. E. Dickson, who used it to prove a result in number theory about perfect numbers. However, the lemma was certainly known earlier, for example to Paul Gordan in his research on invariant theory.


Furthermore, rectangles can exist (a statement equivalent to the parallel postulate) only in Euclidean geometry. A Saccheri quadrilateral is a quadrilateral which has two sides of equal length, both perpendicular to a side called the base. The other two angles of a Saccheri quadrilateral are called the summit angles and they have equal measure.


In number theory, Tijdeman's theorem states that there are at most a finite number of consecutive powers. Stated another way, the set of solutions in integers x, y, n, m of the exponential diophantine equation y m = x n + 1 , {\displaystyle y^{m}=x^{n}+1,} for exponents n and m greater than one, is finite.


In probability theory, a Pitman–Yor process denoted PY(d, θ, G0), is a stochastic process whose sample path is a probability distribution. A random sample from this process is an infinite discrete probability distribution, consisting of an infinite set of atoms drawn from G0, with weights drawn from a two-parameter Poisson-Dirichlet distribution. The process is named after Jim Pitman and Marc Yor.


Sequential search models are used in the field of finance to examine how investors look for information on stocks and other financial assets. The assumption that consumers know what they are looking for and what the standard of the product or service should be is one of the limitations of sequential search models. This presumption might not always be accurate in practical circumstances. Another drawback is that sequential search models don't account for the possibility that customers could find out more about the calibre of a good or service as they search further.


Forwarding (described below) helps correct such errors by depending on the fact that the output of i1 (which is 3) can be used by subsequent instructions before the value 3 is committed to/stored in Register 1. Forwarding applied to the example means that there is no wait to commit/store the output of i1 in Register 1 (in this example, the output is 3) before making that output available to the subsequent instruction (in this case, i2).


He believed that scientific analysis would lead to the discovery of the "one best way" to do things or carrying out an operation. This, according to him could help save cost and time.


However, Veblen's theorem applies also to disconnected graphs, and can be generalized to infinite graphs in which every vertex has finite degree (Sabidussi 1964). If a countably infinite graph G has no odd-degree vertices, then it may be written as a union of disjoint (finite) simple cycles if and only if every finite subgraph of G can be extended (by including more edges and vertices from G) to a finite Eulerian graph. In particular, every countably infinite graph with only one end and with no odd vertices can be written as a union of disjoint cycles (Sabidussi 1964).


n j = ∑ i = 1 N ( q i = j ) {\displaystyle n_{j}=\sum _{i=1}^{N}(q_{i}=j)} The total number of permutation of q → {\displaystyle {\vec {q}}} can be represented by a multinomial as N ! ∏ j = 0 k n j ! {\displaystyle {\frac {N!


IBM however developed their Multiprocessor Interrupt Controller (MPIC) based on the OpenPIC register specification. In the reference IBM design, the processors share the MPIC over a DCR bus, with their access to the bus controlled by a DCR Arbiter. MPIC supports up to four processors and up to 128 interrupt sources.


In pharmacology and biochemistry, mode of action (MoA) describes a functional or anatomical change, resulting from the exposure of a living organism to a substance. In comparison, a mechanism of action (MOA) describes such changes at the molecular level.A mode of action is important in classifying chemicals, as it represents an intermediate level of complexity in between molecular mechanisms and physiological outcomes, especially when the exact molecular target has not yet been elucidated or is subject to debate. A mechanism of action of a chemical could be "binding to DNA" while its broader mode of action would be "transcriptional regulation". However, there is no clear consensus and the term mode of action is also often used, especially in the study of pesticides, to describe molecular mechanisms such as action on specific nuclear receptors or enzymes.


In the academic field of films and cinema, several studies involving research have discovered a positive connection between film critics evaluating films and how well the films perform with audience members. Also, studies involving research in the fields of films and cinema have discovered a connection between film critics evaluating films and audience members having interests or no interests in viewing those films. Based in the perspective of an audience member, a review serves as more than an object that is useful for making decisions. Listening to a review from a critic, watching a critic's review, and reading a critic's review are all ways in which the review is useful to an audience member. The critic's review is able to be referenced in conversations where audience members communicate with other individuals, and audience members can communicate messages about the artistic film that was critically examined or connect the criticism to problems that occur in society.


In real mode, the interrupt table is called IVT (interrupt vector table). Up to the 80286, the IVT always resided at the same location in memory, ranging from 0x0000 to 0x03ff, and consisted of 256 far pointers. Hardware interrupts may be mapped to any of the vectors by way of a programmable interrupt controller. On the 80286 and later, the size and locations of the IVT can be changed in the same way as it is done with the IDT (Interrupt descriptor table) in protected mode (i.e., via the LIDT (Load Interrupt Descriptor Table Register) instruction) though it does not change the format of it.


Deterministic context-free grammars were particularly useful because they could be parsed sequentially by a deterministic pushdown automaton, which was a requirement due to computer memory constraints. In 1965, Donald Knuth invented the LR(k) parser and proved that there exists an LR(k) grammar for every deterministic context-free language. This parser still required a lot of memory.


In the Japanese kantenji braille, the standard 8-dot Braille patterns 3568, 13568, 34568, and 134568 are the patterns related to Braille pattern dots-2456, since the two additional dots of kantenji patterns 02456, 24567, and 024567 are placed above the base 6-dot cell, instead of below, as in standard 8-dot braille.


In the branch of linguistics known as pragmatics, a presupposition (or PSP) is an implicit assumption about the world or background belief relating to an utterance whose truth is taken for granted in discourse. Examples of presuppositions include: Jane no longer writes fiction. Presupposition: Jane once wrote fiction. Have you stopped eating meat?


In mathematics, the capacitated arc routing problem (CARP) is that of finding the shortest tour with a minimum graph/travel distance of a mixed graph with undirected edges and directed arcs given capacity constraints for objects that move along the graph that represent snow-plowers, street sweeping machines, or winter gritters, or other real-world objects with capacity constraints. The constraint can be imposed for the length of time the vehicle is away from the central depot, or a total distance traveled, or a combination of the two with different weighting factors. There are many different variations of the CARP described in the book Arc Routing:Problems, Methods, and Applications by Ángel Corberán and Gilbert Laporte.Solving the CARP involves the study of graph theory, arc routing, operations research, and geographical routing algorithms to find the shortest path efficiently. The CARP is NP-hard arc routing problem.


In the early 1990s, when desktop mapping and presentation graphics became accessible to the average office user, data2map hired software engineers and GIS-specialists to develop several vector-map series for easy customization by the end user. The prime objective was to enable office users and professional graphics artists to visualize geo-referenced information on pre-designed country- and world-maps within their favorite standard off-the-shelf software.


These also may include full copies of the system images from which machines have their operating systems initially installed, or available for repair of any system files that may get corrupted during a machine's lifecycle. Some software may require quite substantial storage space or might be undergoing rapid (perhaps internal) development. In those cases the software may be installed on, and configured to be run directly from, the file servers.


In mathematics, Pythagorean addition is a binary operation on the real numbers that computes the length of the hypotenuse of a right triangle, given its two sides. According to the Pythagorean theorem, for a triangle with sides a {\displaystyle a} and b {\displaystyle b} , this length can be calculated as where ⊕ {\displaystyle \oplus } denotes the Pythagorean addition operation.This operation can be used in the conversion of Cartesian coordinates to polar coordinates. It also provides a simple notation and terminology for some formulas when its summands are complicated; for example, the energy-momentum relation in physics becomes It is implemented in many programming libraries as the hypot function, in a way designed to avoid errors arising due to limited-precision calculations performed on computers. In its applications to signal processing and propagation of measurement uncertainty, the same operation is also called addition in quadrature.


Over time, less key travel was accepted in the market, finally landing on 0.110 inches (2.79 mm). Coincident with this, Key Tronic was the first company to introduce a keyboard that was only about one inch thick.


Examples of their use are: In quantum teleportation, a classical channel together with a previously prepared entangled quantum state are used to transmit quantum information between two parties. Neither the classical channel nor the previously prepared quantum state alone can do this task. In quantum cryptography, a classical channel is used along with a quantum channel in protocols for quantum key exchange. In quantum communication, the information rate of a noisy quantum channel can be increased when used together with a noise-free classical channel. In particular, some very noisy quantum channels cannot transmit quantum information if used alone but can transmit it if used together with a classical channel (which is not capable of transmitting quantum information by itself).


In number theory, an arithmetic number is an integer for which the average of its positive divisors is also an integer. For instance, 6 is an arithmetic number because the average of its divisors is 1 + 2 + 3 + 6 4 = 3 , {\displaystyle {\frac {1+2+3+6}{4}}=3,} which is also an integer. However, 2 is not an arithmetic number because its only divisors are 1 and 2, and their average 3/2 is not an integer. The first numbers in the sequence of arithmetic numbers are 1, 3, 5, 6, 7, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 27, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, ... (sequence A003601 in the OEIS).


In older distributors, adjusting the ignition timing is usually achieved through both mechanical advance and vacuum advance. Mechanical advance adjusts the timing based on the engine speed (rpm), using a set of hinged weights attached to the distributor shaft. These weights cause the breaker points mounting plate to slightly rotate, thereby advancing the ignition timing. Vacuum advance typically uses manifold vacuum to adjust the ignition timing, for example to improve fuel economy and driveability when minimal power is required from the engine. Most distributors used on electronic fuel injection engines use electronics to adjust the ignition timing, instead of vacuum and centrifugal systems. This allows the ignition timing to be optimised based on factors other than engine speed and manifold vacuum.


Current views vary on whether all languages have a verb phrase; some schools of generative grammar (such as principles and parameters) hold that all languages have a verb phrase, while others (such as lexical functional grammar) take the view that at least some languages lack a verb phrase constituent, including those languages with a very free word order (the so-called non-configurational languages, such as Japanese, Hungarian, or Australian aboriginal languages), and some languages with a default VSO order (several Celtic and Oceanic languages). Phrase structure grammars view both finite and nonfinite verb phrases as constituent phrases and, consequently, do not draw any key distinction between them. Dependency grammars (described below) are much different in this regard.


In metropolitan areas, FOMA uses the UMTS band I around 2100 MHz, which has been originally assigned to IMT-2000 services worldwide, except in the Americas. In order to improve coverage in rural and mountainous areas, NTT DoCoMo also offers FOMA services in the 800 MHz band originally assigned to the 2G PDC mova service, which corresponds to UMTS band VI and is similar to band V used in the United States. These extended service areas are branded FOMA Plus-Area (FOMAプラスエリア) and require multiband terminals. == References ==


To emphasize that there are an infinite number of terms, a series may be called an infinite series. Such a series is represented (or denoted) by an expression like or, using the summation sign, The infinite sequence of additions implied by a series cannot be effectively carried on (at least in a finite amount of time). However, if the set to which the terms and their finite sums belong has a notion of limit, it is sometimes possible to assign a value to a series, called the sum of the series.


In Kent's coin tossing protocol, Alice has two agents A0 and A1, and Bob has two agents B0 and B1. Ai and Bi are at location Li, for i ∈ { 0 , 1 } {\displaystyle i\in \{0,1\}} .


Network theory has applications in many disciplines, including statistical physics, particle physics, computer science, electrical engineering, biology, archaeology, linguistics, economics, finance, operations research, climatology, ecology, public health, sociology, psychology, and neuroscience. Applications of network theory include logistical networks, the World Wide Web, Internet, gene regulatory networks, metabolic networks, social networks, epistemological networks, etc.; see List of network theory topics for more examples. Euler's solution of the Seven Bridges of Königsberg problem is considered to be the first true proof in the theory of networks.


Starting from the basic fact stated above that, for any positive integer n {\displaystyle n} , b n {\displaystyle b^{n}} is n {\displaystyle n} occurrences of b {\displaystyle b} all multiplied by each other, several other properties of exponentiation directly follow. In particular: In other words, when multiplying a base raised to one exponent by the same base raised to another exponent, the exponents add. From this basic rule that exponents add, we can derive that b 0 {\displaystyle b^{0}} must be equal to 1 for any b ≠ 0 {\displaystyle b\neq 0} , as follows.


In star topology, every peripheral node (computer workstation or any other peripheral) is connected to a central node called a hub or switch. The hub is the server and the peripherals are the clients. The network does not necessarily have to resemble a star to be classified as a star network, but all of the peripheral nodes on the network must be connected to one central hub. All traffic that traverses the network passes through the central hub, which acts as a signal repeater.


Under one definition, an undirected hypergraph ( X , E ) {\displaystyle (X,E)} is a directed hypergraph which has a symmetric edge set: If ( D , C ) ∈ E {\displaystyle (D,C)\in E} then ( C , D ) ∈ E {\displaystyle (C,D)\in E} . For notational simplicity one can remove the "duplicate" hyperedges since the modifier "undirected" is precisely informing us that they exist: If ( D , C ) ∈ E {\displaystyle (D,C)\in E} then ( C , D ) ∈ → E {\displaystyle (C,D){\vec {\in }}E} where ∈ → {\displaystyle {\vec {\in }}} means implicitly in. While graph edges connect only 2 nodes, hyperedges connect an arbitrary number of nodes.


In real-time computing, the priority ceiling protocol is a synchronization protocol for shared resources to avoid unbounded priority inversion and mutual deadlock due to wrong nesting of critical sections. In this protocol each resource is assigned a priority ceiling, which is a priority equal to the highest priority of any task which may lock the resource. The protocol works by temporarily raising the priorities of tasks in certain situations, thus it requires a scheduler that supports dynamic priority scheduling.


In multi-agent system research, distributed knowledge is all the knowledge that a community of agents possesses and might apply in solving a problem. Distributed knowledge is approximately what "a wise man knows", or what someone who has complete knowledge of what each member of the community knows knows. Distributed knowledge might also be called the aggregate knowledge of a community, as it represents all the knowledge that a community might bring to bear to solve a problem.


In mathematical logic, various sublanguages of set theory are decidable. These include: Sets with Monotone, Additive, and Multiplicative Functions. Sets with restricted quantifiers. == References ==


In order to define a specific proof of knowledge, one need not only define the language, but also the witnesses the verifier should know. In some cases proving membership in a language may be easy, while computing a specific witness may be hard. This is best explained using an example: Let ⟨ g ⟩ {\displaystyle \langle g\rangle } be a cyclic group with generator g {\displaystyle g} in which solving the discrete logarithm problem is believed to be hard. Deciding membership of the language L = { x ∣ g w = x } {\displaystyle L=\{x\mid g^{w}=x\}} is trivial, as every x {\displaystyle x} is in ⟨ g ⟩ {\displaystyle \langle g\rangle } . However, finding the witness w {\displaystyle w} such that g w = x {\displaystyle g^{w}=x} holds corresponds to solving the discrete logarithm problem.


In mathematics, specifically in order theory and functional analysis, the order dual of an ordered vector space X {\displaystyle X} is the set Pos  ( X ∗ ) − Pos  ( X ∗ ) {\displaystyle \operatorname {Pos} \left(X^{*}\right)-\operatorname {Pos} \left(X^{*}\right)} where Pos  ( X ∗ ) {\displaystyle \operatorname {Pos} \left(X^{*}\right)} denotes the set of all positive linear functionals on X {\displaystyle X} , where a linear function f {\displaystyle f} on X {\displaystyle X} is called positive if for all x ∈ X , {\displaystyle x\in X,} x ≥ 0 {\displaystyle x\geq 0} implies f ( x ) ≥ 0. {\displaystyle f(x)\geq 0.} The order dual of X {\displaystyle X} is denoted by X + {\displaystyle X^{+}} . Along with the related concept of the order bound dual, this space plays an important role in the theory of ordered topological vector spaces.


Typically both transactions will be cancelled and rolled back, and then they will be started again in a different order, automatically, so that the deadlock doesn't occur again. Or sometimes, just one of the deadlocked transactions will be cancelled, rolled back, and automatically restarted after a short delay. Deadlocks can also occur among three or more transactions. The more transactions involved, the more difficult they are to detect, to the point that transaction processing systems find there is a practical limit to the deadlocks they can detect.


In statistics, the Nemenyi test is a post-hoc test intended to find the groups of data that differ after a global statistical test (such as the Friedman test) has rejected the null hypothesis that the performance of the comparisons on the groups of data is similar. The test makes pair-wise tests of performance. The test is named after Peter Nemenyi.The test is sometimes referred to as the "Nemenyi–Damico–Wolfe test", when regarding one-sided multiple comparisons of "treatments" versus "control", but it can also be referred to as the "Wilcoxon–Nemenyi–McDonald–Thompson test", when regarding two-sided multiple comparisons of "treatments" versus "treatments".


To make the perfect and pluperfect tenses, the first consonant of the verb's root is usually repeated with the vowel ε (e), for example: γράφω, γέγραφα (gégrapha) "I write, I have written", λῡ́ω, λέλυκα (lū́ō, léluka) "I free, I have freed", διδάσκω, δεδίδαχα (didáskō, dedídakha) "I teach, I have taught" (all present, perfect). This is called "reduplication". Some verbs, however, where reduplication is not convenient, use an augment instead, e.g. ἔσχον, ἔσχηκα (éskhon, éskhēka) "I had, I have had" (aorist, perfect), εὑρίσκω, ηὕρηκα (heurískō, hēúrēka) "I find, I have found" (present, perfect). This reduplication or perfect-tense augment appears in every part of the verb, not in the indicative only.


Knuth himself attributes the first study of multisets to the Indian mathematician Bhāskarāchārya, who described permutations of multisets around 1150. Other names have been proposed or used for this concept, including list, bunch, bag, heap, sample, weighted set, collection, and suite. : 694


In the Picture Concepts task, children have presented with a series of pictures on two or three rows and asked which pictures (one from each row) belong together based on some common characteristic. This task assesses the child's ability to discover the underlying characteristic (e.g., rule, concept, trend, class membership) that governs a set of materials.


The Kolmogorov–Smirnov statistic quantifies a distance between the empirical distribution function of the sample and the cumulative distribution function of the reference distribution, or between the empirical distribution functions of two samples. The null distribution of this statistic is calculated under the null hypothesis that the sample is drawn from the reference distribution (in the one-sample case) or that the samples are drawn from the same distribution (in the two-sample case). In the one-sample case, the distribution considered under the null hypothesis may be continuous (see Section 2), purely discrete or mixed (see Section 2.2).


In relational database management systems, a unique key is a candidate key. All the candidate keys of a relation can uniquely identify the records of the relation, but only one of them is used as the primary key of the relation. The remaining candidate keys are called unique keys because they can uniquely identify a record in a relation. Unique keys can consist of multiple columns.


}}\end{cases}}} Euler's criterion can be concisely reformulated using the Legendre symbol: ( a p ) ≡ a p − 1 2 ( mod p ) . {\displaystyle \left({\frac {a}{p}}\right)\equiv a^{\tfrac {p-1}{2}}{\pmod {p}}.} The criterion first appeared in a 1748 paper by Leonhard Euler.


In straight line fitting, the model is y i = α + β x i + ε i {\displaystyle y_{i}=\alpha +\beta x_{i}+\varepsilon _{i}\,} where y i {\displaystyle y_{i}} is the response variable, x i {\displaystyle x_{i}} is the explanatory variable, εi is the random error, and α {\displaystyle \alpha } and β {\displaystyle \beta } are parameters. The mean, and predicted, response value for a given explanatory value, xd, is given by y ^ d = α ^ + β ^ x d , {\displaystyle {\hat {y}}_{d}={\hat {\alpha }}+{\hat {\beta }}x_{d},} while the actual response would be y d = α + β x d + ε d {\displaystyle y_{d}=\alpha +\beta x_{d}+\varepsilon _{d}\,} Expressions for the values and variances of α ^ {\displaystyle {\hat {\alpha }}} and β ^ {\displaystyle {\hat {\beta }}} are given in linear regression.


Within distributed morphology (DM), this is where morphological structure is constructed, where the hierarchical syntactic structure is transformed into a linearized structure, and syntactic features are replaced with vocabulary items, among other things.According to some theories of prosody, the prosodic representation is derived with direct reference to the hierarchical syntactic structure. For example, Selkirk (2011, and others) proposes that prosodic structure is constructed by a process of matching, although imperfectly, prosodic constituents to syntactic constituents. Kahnemuyipour (2009) demonstrates, using evidence from several languages, how information structure can be represented in the transfer from syntax to phonology, arguing that transfer can only be uni-directional, from syntax to phonology.


{\displaystyle p_{w}(s)={\frac {\pi s}{2}}e^{-\pi s^{2}/4}.} Here, s = S D {\displaystyle s={\frac {S}{D}}} where S is a particular spacing and D is the mean distance between neighboring intervals.In a mixed sequence (spin and parity are different), the probability density function can be obtained by randomly superimposing simple sequences.The above result is exact for 2 × 2 {\displaystyle 2\times 2} real symmetric matrices M {\displaystyle M} , with elements that are independent standard gaussian random variables, with joint distribution proportional to e − 1 2 T r ( M 2 ) = e − 1 2 T r ( a b b c ) 2 = e − 1 2 a 2 − 1 2 c 2 − b 2 . {\displaystyle e^{-{\frac {1}{2}}{\rm {Tr}}(M^{2})}=e^{-{\frac {1}{2}}{\rm {Tr}}\left({\begin{array}{cc}a&b\\b&c\\\end{array}}\right)^{2}}=e^{-{\frac {1}{2}}a^{2}-{\frac {1}{2}}c^{2}-b^{2}}.}


In modified policy iteration (van Nunen 1976; Puterman & Shin 1978), step one is performed once, and then step two is repeated several times. Then step one is again performed once and so on.


In the years 1994 to 2004 the elder 5-ton van 'VaMoRs' was used to develop the capabilities needed for driving on networks of minor (also unsealed) roads and for cross-country driving including avoidance of negative obstacles like ditches. Turning off onto crossroads of unknown width and intersection angles required a big effort, but has been achieved with "Expectation-based, Multi-focal, Saccadic vision" (EMS-vision). This vertebrate-type vision uses animation capabilities based on knowledge about subject classes (including the autonomous vehicle itself) and their potential behaviour in certain situations.


In quantitative linguistics, linguistic laws are statistical regularities emerging across different linguistic scales (i.e. phonemes, syllables, words or sentences) that can be formulated mathematically and that have been deduced from certain theoretical assumptions. They are also required to have been successfully tested through the use of data, that is, not to have been refuted by empirical evidence. Among the main linguistic laws proposed by various authors, the following can be highlighted: Zipf's law: The frequency of words is inversely proportional to their rank in frequency lists. Similar distribution between rank and frequency of sounds, phonemes, and letters can be observed.


The numbers 3 {\displaystyle {\sqrt {3}}} and 2 3 {\displaystyle 2{\sqrt {3}}} are also commensurable because their ratio, 3 2 3 = 1 2 {\textstyle {\frac {\sqrt {3}}{2{\sqrt {3}}}}={\frac {1}{2}}} , is a rational number. However, the numbers 3 {\textstyle {\sqrt {3}}} and 2 are incommensurable because their ratio, 3 2 {\textstyle {\frac {\sqrt {3}}{2}}} , is an irrational number. More generally, it is immediate from the definition that if a and b are any two non-zero rational numbers, then a and b are commensurable; it is also immediate that if a is any irrational number and b is any non-zero rational number, then a and b are incommensurable. On the other hand, if both a and b are irrational numbers, then a and b may or may not be commensurable.


Information can be transmitted either during the given time interval, or encoded as the presence or absence of a change in the received signal.Significant conditions are recognized by an appropriate device called a receiver, demodulator, or decoder. The decoder translates the actual signal received into its intended logical value such as a binary digit (0 or 1), an alphabetic character, a mark, or a space. Each significant instant is determined when the appropriate device assumes a condition or state usable for performing a specific function, such as recording, processing, or gating.


In mathematics, in the branch of combinatorics, a graded poset is a partially-ordered set (poset) P equipped with a rank function ρ from P to the set N of all natural numbers. ρ must satisfy the following two properties: The rank function is compatible with the ordering, meaning that for all x and y in the order, if x < y then ρ(x) < ρ(y), and The rank is consistent with the covering relation of the ordering, meaning that for all x and y, if y covers x then ρ(y) = ρ(x) + 1.The value of the rank function for an element of the poset is called its rank. Sometimes a graded poset is called a ranked poset but that phrase has other meanings; see Ranked poset. A rank or rank level of a graded poset is the subset of all the elements of the poset that have a given rank value.Graded posets play an important role in combinatorics and can be visualized by means of a Hasse diagram.


Most variants of realizability begin with a theorem that any statement that is provable in the formal system being studied is realizable. The realizer, however, usually gives more information about the formula than a formal proof would directly provide. Beyond giving insight into intuitionistic provability, realizability can be applied to prove the disjunction and existence properties for intuitionistic theories and to extract programs from proofs, as in proof mining. It is also related to topos theory via realizability topoi.


In mathematics and computing, the method of complements is a technique to encode a symmetric range of positive and negative integers in a way that they can use the same algorithm (or mechanism) for addition throughout the whole range. For a given number of places half of the possible representations of numbers encode the positive numbers, the other half represents their respective additive inverses. The pairs of mutually additive inverse numbers are called complements. Thus subtraction of any number is implemented by adding its complement.


This means that the justification ensures that the belief is true. For example, Richard Kirkham argues that the justification required for knowledge must be based on self-evident premises that deductively entail the held belief.


In telecommunication, trellis modulation (also known as trellis coded modulation, or simply TCM) is a modulation scheme that transmits information with high efficiency over band-limited channels such as telephone lines. Gottfried Ungerboeck invented trellis modulation while working for IBM in the 1970s, and first described it in a conference paper in 1976. It went largely unnoticed, however, until he published a new, detailed exposition in 1982 that achieved sudden and widespread recognition. In the late 1980s, modems operating over plain old telephone service (POTS) typically achieved 9.6 kbit/s by employing four bits per symbol QAM modulation at 2,400 baud (symbols/second). This bit rate ceiling existed despite the best efforts of many researchers, and some engineers predicted that without a major upgrade of the public phone infrastructure, the maximum achievable rate for a POTS modem might be 14 kbit/s for two-way communication (3,429 baud × 4 bits/symbol, using QAM).14 kbit/s is only 40% of the theoretical maximum bit rate predicted by Shannon's theorem for POTS lines (approximately 35 kbit/s). Ungerboeck's theories demonstrated that there was considerable untapped potential in the system, and by applying the concept to new modem standards, speed rapidly increased to 14.4, 28.8 and ultimately 33.6 kbit/s.


Indeed, a greedy algorithm finds the k-combination corresponding to N: take ck maximal with ( c k k ) ≤ N {\displaystyle {\tbinom {c_{k}}{k}}\leq N} , then take ck−1 maximal with ( c k − 1 k − 1 ) ≤ N − ( c k k ) {\displaystyle {\tbinom {c_{k-1}}{k-1}}\leq N-{\tbinom {c_{k}}{k}}} , and so forth. Finding the number N, using the formula above, from the k-combination (ck, ..., c2, c1) is also known as "ranking", and the opposite operation (given by the greedy algorithm) as "unranking"; the operations are known by these names in most computer algebra systems, and in computational mathematics.The originally used term "combinatorial representation of integers" was shortened to "combinatorial number system" by Knuth, who also gives a much older reference; the term "combinadic" is introduced by James McCaffrey (without reference to previous terminology or work). Unlike the factorial number system, the combinatorial number system of degree k is not a mixed radix system: the part ( c i i ) {\displaystyle {\tbinom {c_{i}}{i}}} of the number N represented by a "digit" ci is not obtained from it by simply multiplying by a place value. The main application of the combinatorial number system is that it allows rapid computation of the k-combination that is at a given position in the lexicographic ordering, without having to explicitly list the k-combinations preceding it; this allows for instance random generation of k-combinations of a given set. Enumeration of k-combinations has many applications, among which are software testing, sampling, quality control, and the analysis of lottery games.


It waits for the child that slept 10 seconds first; all the others have long since exited, so all of the messages (except the first) appear in quick succession. There is no possibility of random ordering here, since it is driven by a loop in a single process. The first parent message actually appeared before any of the children messages - the parent was able to continue into the second loop before any of the child processes were able to start.


In order to route calls to a phone, cell towers listen for a signal sent from the phone and negotiate which tower is best able to communicate with the phone. As the phone changes location, the antenna towers monitor the signal, and the phone is "roamed" to an adjacent tower as appropriate. By comparing the relative signal strength from multiple antenna towers, a general location of a phone can be roughly determined.


Here α can be large. The following form of this inequality is often seen in more analytic contexts: with the same assumptions on f, for every p ∈ U there is a possibly smaller open neighborhood W of p and constants θ ∈ (0,1) and c > 0 such that | f ( x ) − f ( p ) | θ ≤ c | ∇ f ( x ) | . {\displaystyle |f(x)-f(p)|^{\theta }\leq c|\nabla f(x)|.} A special case of the Łojasiewicz inequality, due to Polyak, is commonly used to prove linear convergence of gradient descent algorithms.


These ran UTX-32, Gould's version of Unix based on a BSD 4.2 kernel developed by Purdue University to support multiprocessor systems. The Powernode 9080 was a symmetrical dual processor system, with both processors having full access to memory and the I/O bus, and capable of being booted up from either processor. It was the first such commercially available system to run any version of Unix.The CPU for these system ballooned to about a dozen boards because of the low-density ECL chip footprint.


In the category of sets, injections, surjections, and bijections correspond precisely to monomorphisms, epimorphisms, and isomorphisms, respectively.


Failure modes analysis is used to identify and order the impediments to reuse in a given organization. Reusability metrics indicate the likelihood that an artifact is reusable. Reuse library metrics are used to manage and track usage of a reuse repository.


In probability theory, the theory of large deviations concerns the asymptotic behaviour of remote tails of sequences of probability distributions. While some basic ideas of the theory can be traced to Laplace, the formalization started with insurance mathematics, namely ruin theory with Cramér and Lundberg. A unified formalization of large deviation theory was developed in 1966, in a paper by Varadhan. Large deviations theory formalizes the heuristic ideas of concentration of measures and widely generalizes the notion of convergence of probability measures. Roughly speaking, large deviations theory concerns itself with the exponential decline of the probability measures of certain kinds of extreme or tail events.


In the United States, the term "standard of review" has several different meanings in different contexts and thus there are several standards of review on appeal used in federal courts depending on the nature of the question being appealed and the body that made the decision.


In the area of mathematics called combinatorial group theory, the Schreier coset graph is a graph associated with a group G, a generating set {xi: i in I } of G, and a subgroup H ≤ G. The Schreier graph encodes the abstract structure of a group modulo an equivalence relation formed by the coset. The graph is named after Otto Schreier, who used the term “Nebengruppenbild”. An equivalent definition was made in an early paper of Todd and Coxeter.


The maximum number of edges for an undirected graph is ( | V | 2 ) = | V | ( | V | − 1 ) 2 {\displaystyle {\binom {|V|}{2}}={\frac {|V|(|V|-1)}{2}}} , so the maximal density is 1 (for complete graphs) and the minimal density is 0 (Coleman & Moré 1983). For families of graphs of increasing size, one often calls them sparse if D → 0 {\displaystyle D\rightarrow 0} as | V | → ∞ {\displaystyle |V|\rightarrow \infty } . Sometimes, in computer science, a more restrictive definition of sparse is used like | E | = O ( | V | log  | V | ) {\displaystyle |E|=O(|V|\log |V|)} or even | E | = O ( | V | ) {\displaystyle |E|=O(|V|)} .


( k − 1 ) ! ( k + ρ ) ! .


One may also regard the unit type as the type of 0-tuples, i.e. the product of no types. The unit type is the terminal object in the category of types and typed functions. It should not be confused with the zero or bottom type, which allows no values and is the initial object in this category.


The UNIVAC 1004 was a plug-board programmed punched card data processing system, introduced in 1962 by UNIVAC. Total memory was 961 characters (6 bits) of core memory.


Common notions such as injectivity and surjectivity can be expressed in a bounded fashion as well, and thus so is bijectivity. Both of these tie in to notions of size. Importantly, injection existence between any two sets provides a preorder.


It is a generalization and improvement of the MINRES method due to Paige and Saunders in 1975. The MINRES method requires that the matrix is symmetric, but has the advantage that it only requires handling of three vectors. GMRES is a special case of the DIIS method developed by Peter Pulay in 1980. DIIS is applicable to non-linear systems.


{\displaystyle \forall y\in Y,\exists x\in X{\text{ such that }}y=f(x).} The function is bijective (one-to-one and onto, one-to-one correspondence, or invertible) if each element of the codomain is mapped to by exactly one element of the domain. That is, the function is both injective and surjective.


This resulted in the move to use microkernels which used a minimal set of the operating system functions. Systems such as Mach at Carnegie Mellon University and ChorusOS at INRIA were examples of early microkernels.The separation of the operating system into separate components became necessary as supercomputers developed different types of nodes, e.g., compute nodes versus I/O nodes. Thus modern supercomputers usually run different operating systems on different nodes, e.g., using a small and efficient lightweight kernel such as CNK or CNL on compute nodes, but a larger system such as a Linux-derivative on server and I/O nodes.


The value of ψ ( n ) {\displaystyle \psi (n)} for the first few integers n {\displaystyle n} is: 1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24, ... (sequence A001615 in the OEIS).The function ψ ( n ) {\displaystyle \psi (n)} is greater than n {\displaystyle n} for all n {\displaystyle n} greater than 1, and is even for all n {\displaystyle n} greater than 2. If n {\displaystyle n} is a square-free number then ψ ( n ) = σ ( n ) {\displaystyle \psi (n)=\sigma (n)} , where σ ( n ) {\displaystyle \sigma (n)} is the divisor function. The ψ {\displaystyle \psi } function can also be defined by setting ψ ( p n ) = ( p + 1 ) p n − 1 {\displaystyle \psi (p^{n})=(p+1)p^{n-1}} for powers of any prime p {\displaystyle p} , and then extending the definition to all integers by multiplicativity.


In the following two examples, we always assume it is difficult to factorize a large composite number (see Integer factorization).


Researchers have begun to use deep learning techniques for language modeling as well. In the long history of speech recognition, both shallow form and deep form (e.g. recurrent nets) of artificial neural networks had been explored for many years during 1980s, 1990s and a few years into the 2000s. But these methods never won over the non-uniform internal-handcrafting Gaussian mixture model/Hidden Markov model (GMM-HMM) technology based on generative models of speech trained discriminatively.


Eve first captures the photon sent by Alice and then generates another photon to send to Bob. Eve manipulates the phase and timing of the "faked" photon in a way that prevents Bob from detecting the presence of an eavesdropper. The only way to eliminate this vulnerability is to eliminate differences in photodetector efficiency, which is difficult to do given finite manufacturing tolerances that cause optical path length differences, wire length differences, and other defects.


In the United States there is a rule that wrong-side failures are to be reported to the Federal Railroad Administration.


Double negation elimination occurs in classical logics but not in intuitionistic logic. In the context of a formula in the conjunctive normal form, a literal is pure if the literal's complement does not appear in the formula. In Boolean functions, each separate occurrence of a variable, either in inverse or uncomplemented form, is a literal. For example, if A {\displaystyle A} , B {\displaystyle B} and C {\displaystyle C} are variables then the expression A ¯ B C {\displaystyle {\bar {A}}BC} contains three literals and the expression A ¯ C + B ¯ C ¯ {\displaystyle {\bar {A}}C+{\bar {B}}{\bar {C}}} contains four literals. However, the expression A ¯ C + B ¯ C {\displaystyle {\bar {A}}C+{\bar {B}}C} would also be said to contain four literals, because although two of the literals are identical ( C {\displaystyle C} appears twice) these qualify as two separate occurrences.


This value indicates the mapping of the two items. Zero values indicate that no relationship exists. It must be determined if a relationship must be made.


In more general sense, each column is treated as a polynomial over GF  ( 2 8 ) {\displaystyle \operatorname {GF} (2^{8})} and is then multiplied modulo 01 16 ⋅ z 4 + 01 16 {\displaystyle {01}_{16}\cdot z^{4}+{01}_{16}} with a fixed polynomial c ( z ) = 03 16 ⋅ z 3 + 01 16 ⋅ z 2 + 01 16 ⋅ z + 02 16 {\displaystyle c(z)={03}_{16}\cdot z^{3}+{01}_{16}\cdot z^{2}+{01}_{16}\cdot z+{02}_{16}} . The coefficients are displayed in their hexadecimal equivalent of the binary representation of bit polynomials from GF  ( 2 )  {\displaystyle \operatorname {GF} (2)} . The MixColumns step can also be viewed as a multiplication by the shown particular MDS matrix in the finite field GF  ( 2 8 ) {\displaystyle \operatorname {GF} (2^{8})} . This process is described further in the article Rijndael MixColumns.


Two mappings are homotopic if one can be continuously deformed into the other. These homotopy classes form a group, called the n-th homotopy group, π n ( X ) , {\displaystyle \pi _{n}(X),} of the given space X with base point. Topological spaces with differing homotopy groups are never equivalent (homeomorphic), but topological spaces that are not homeomorphic can have the same homotopy groups. The notion of homotopy of paths was introduced by Camille Jordan.


This is typically subject to constraints such as: A variable must be initialized before its value is used. In strongly typed languages, each variable is assigned a type, and all uses of the variable must respect its type. Often, its type must be declared explicitly, before use.W-grammars are based on the idea of providing the nonterminal symbols of context-free grammars with attributes (or affixes) that pass information between the nodes of the parse tree, used to constrain the syntax and to specify the semantics.


In statistics a minimum-variance unbiased estimator (MVUE) or uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter. For practical statistics problems, it is important to determine the MVUE if one exists, since less-than-optimal procedures would naturally be avoided, other things being equal. This has led to substantial development of statistical theory related to the problem of optimal estimation. While combining the constraint of unbiasedness with the desirability metric of least variance leads to good results in most practical settings—making MVUE a natural starting point for a broad range of analyses—a targeted specification may perform better for a given problem; thus, MVUE is not always the best stopping point.


In pure integer programming problems, the feasible set is the set of integers (or some subset thereof). In linear programming problems, the feasible set is a convex polytope: a region in multidimensional space whose boundaries are formed by hyperplanes and whose corners are vertices. Constraint satisfaction is the process of finding a point in the feasible region.