For this description, the following functions and operators are used: Head (data, a): returns the first a bits of the 'data' string. Tail (data, a): returns the last a bits of the 'data' string. Encrypt (K, data): use the underlying block cipher in encrypt mode on the 'data' string using the key K. Decrypt (K, data): use the underlying block cipher in decrypt mode on the 'data' string using the key K. XOR: Bitwise Exclusive-OR.


As with complete linkage and average distance, the difficulty of calculating cluster distances causes the nearest-neighbor chain algorithm to take time and space O(n2) to compute the single-linkage clustering. However, the single-linkage clustering can be found more efficiently by an alternative algorithm that computes the minimum spanning tree of the input distances using Prim's algorithm, and then sorts the minimum spanning tree edges and uses this sorted list to guide the merger of pairs of clusters. Within Prim's algorithm, each successive minimum spanning tree edge can be found by a sequential search through an unsorted list of the smallest edges connecting the partially constructed tree to each additional vertex. This choice saves the time that the algorithm would otherwise spend adjusting the weights of vertices in its priority queue. Using Prim's algorithm in this way would take time O(n2) and space O(n), matching the best bounds that could be achieved with the nearest-neighbor chain algorithm for distances with constant-time calculations.


In mathematics, a proof of impossibility is a proof that demonstrates that a particular problem cannot be solved as described in the claim, or that a particular set of problems cannot be solved in general. Such a case is also known as a negative proof, proof of an impossibility theorem, or negative result. Proofs of impossibility often are the resolutions to decades or centuries of work attempting to find a solution, eventually proving that there is no solution. Proving that something is impossible is usually much harder than the opposite task, as it is often necessary to develop a proof that works in general, rather than to just show a particular example.


In mathematics, the Edwards curves are a family of elliptic curves studied by Harold Edwards in 2007. The concept of elliptic curves over finite fields is widely used in elliptic curve cryptography. Applications of Edwards curves to cryptography were developed by Daniel J. Bernstein and Tanja Lange: they pointed out several advantages of the Edwards form in comparison to the more well known Weierstrass form.


The distance between two objects that are not points is usually defined to be the smallest distance among pairs of points from the two objects. Formulas are known for computing distances between different types of objects, such as the distance from a point to a line. In advanced mathematics, the concept of distance has been generalized to abstract metric spaces, and other distances than Euclidean have been studied. In some applications in statistics and optimization, the square of the Euclidean distance is used instead of the distance itself.


Gradient descent should not be confused with local search algorithms, although both are iterative methods for optimization. Gradient descent is generally attributed to Augustin-Louis Cauchy, who first suggested it in 1847. Jacques Hadamard independently proposed a similar method in 1907. Its convergence properties for non-linear optimization problems were first studied by Haskell Curry in 1944, with the method becoming increasingly well-studied and used in the following decades.A simple extension of gradient descent, stochastic gradient descent, serves as the most basic algorithm used for training most deep networks today.


The rows in the synopsis are ordered such that the coverage of all possible RB cases is easily comprehensible. The column rotation indicates whether a rotation contributes to the rebalancing. The column assignment shows an assignment of N before entering a subsequent iteration step.


All other groups, which are more complicated, are called "nontrivial". In graph theory, the trivial graph is a graph which has only 1 vertex and no edge. Database theory has a concept called functional dependency, written X → Y {\displaystyle X\to Y} .


In the case of a weighted network, this triplet is expanded to a quadruplet e = ( u , v , d , w ) {\displaystyle e=(u,v,d,w)} , where w {\displaystyle w} is the weight on the link between u {\displaystyle u} and v {\displaystyle v} in the dimension d {\displaystyle d} . Further, as is often useful in social network analysis, link weights may take on positive or negative values. Such signed networks can better reflect relations like amity and enmity in social networks.


In radio communication systems, information is transported across space using radio waves. At the sending end, the information to be sent, in the form of a time-varying electrical signal, is applied to a radio transmitter. The information, called the modulation signal, can be an audio signal representing sound from a microphone, a video signal representing moving images from a video camera, or a digital signal representing data from a computer. In the transmitter, an electronic oscillator generates an alternating current oscillating at a radio frequency, called the carrier wave because it creates the radio waves that "carry" the information through the air.


In planning, more precisely in classical planning, one is interested in knowing if one can attain a state from an initial state from a description of actions. The description of actions defines a graph of implicit states, which is of exponential size in the size of the description. In symbolic model checking, the model (the underlying graph) is described with the aid of a symbolic representation such as binary decision diagrams.


The royalty that is paid to the composer and publisher is determined by the method of assessment used by the PRO to gauge the use of the music, there being no external metrics as in mechanical royalties or the reporting system used in the UK. Very basically, a PRO aggregates the royalties that are due to all of the composers/songwriters "who are its members" and each composer and publisher is paid royalties based on the assessed frequency of the music's performance, post deductions of charges (which are many). The PROs are audited agencies.


A formal system is used to derive one expression from one or more other expressions. Although a formal language can be identified with its formulas, a formal system cannot be likewise identified by its theorems. Two formal systems F S {\displaystyle {\mathcal {FS}}} and F S ′ {\displaystyle {\mathcal {FS'}}} may have all the same theorems and yet differ in some significant proof-theoretic way (a formula A may be a syntactic consequence of a formula B in one but not another for instance).


The index signal is asserted when the shaft is in its reference orientation, which causes the encoder interface to jam the reference angle into its position counter. Some incremental encoder applications lack reference position detectors and therefore must implement homing by other means. For example a computer, when using a mouse or trackball pointing device, typically will home the device by assuming a central, initial screen position upon booting, and jam the corresponding counts into the X and Y position counters. In the case of panel encoders used as hand-operated controls (e.g., audio volume control), the initial position typically is retrieved from flash or other non-volatile memory upon power-up and jammed into the position counter, and upon power-down the current position count is saved to non-volatile memory to serve as the initial position for the next power-up.


Channels just above the AM broadcast band were selected manually by the user. Some of the frequencies used are now part of the expanded AM radio band, and can be heard by anyone with an AM radio. There are reports of people still using these phones, and using them as makeshift AM radio stations that can be heard for a couple of city blocks.


In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions. There are six convex and ten star regular 4-polytopes, giving a total of sixteen.


The "no"-answer version of this problem is stated as: "given a finite set of integers, does every non-empty subset have a nonzero sum?". The verifier-based definition of NP does not require an efficient verifier for the "no"-answers. The class of problems with such verifiers for the "no"-answers is called co-NP. In fact, it is an open question whether all problems in NP also have verifiers for the "no"-answers and thus are in co-NP. In some literature the verifier is called the "certifier", and the witness the "certificate".


λy. x, sometimes called the K combinator, is written as λ λ 2 with De Bruijn indices. The binder for the occurrence x is the second λ in scope.


In some cases, a person will be declared dead even without any remains or doctor's declaration. This is under one of two circumstances. First, if a person was known to be in mortal peril when last seen, they can often be declared dead shortly after. Examples would be the passengers of the Titanic that were not rescued after the ship sank. Second, if a person has not been seen for a certain period of time and there has been no evidence that they are alive. The amount of time that has passed varies by jurisdiction, from as little as four years in the US state of Georgia to twenty years in Italy.


In computer science the parity stripe or parity disk in a RAID provides error-correction. Parity bits are written at the rate of one parity bit per n bits, where n is the number of disks in the array. When a read error occurs, each bit in the error region is recalculated from its set of n bits.


The outputs of the replications are compared using a voting circuit. A machine with two replications of each element is termed dual modular redundant (DMR). The voting circuit can then only detect a mismatch and recovery relies on other methods.


In the fields of chemical graph theory, molecular topology, and mathematical chemistry, a topological index, also known as a connectivity index, is a type of a molecular descriptor that is calculated based on the molecular graph of a chemical compound. Topological indices are numerical parameters of a graph which characterize its topology and are usually graph invariant. Topological indices are used for example in the development of quantitative structure-activity relationships (QSARs) in which the biological activity or other properties of molecules are correlated with their chemical structure.


In non-Euclidean geometry, squares are more generally polygons with 4 equal sides and equal angles. In spherical geometry, a square is a polygon whose edges are great circle arcs of equal distance, which meet at equal angles. Unlike the square of plane geometry, the angles of such a square are larger than a right angle. Larger spherical squares have larger angles.


In the classic formalization of generative grammars first proposed by Noam Chomsky in the 1950s, a grammar G consists of the following components: A finite set N of nonterminal symbols, that is disjoint with the strings formed from G. A finite set Σ {\displaystyle \Sigma } of terminal symbols that is disjoint from N. A finite set P of production rules, each rule of the form ( Σ ∪ N ) ∗ N ( Σ ∪ N ) ∗ → ( Σ ∪ N ) ∗ {\displaystyle (\Sigma \cup N)^{*}N(\Sigma \cup N)^{*}\rightarrow (\Sigma \cup N)^{*}} where ∗ {\displaystyle {*}} is the Kleene star operator and ∪ {\displaystyle \cup } denotes set union. That is, each production rule maps from one string of symbols to another, where the first string (the "head") contains an arbitrary number of symbols provided at least one of them is a nonterminal. In the case that the second string (the "body") consists solely of the empty string—i.e., that it contains no symbols at all—it may be denoted with a special notation (often Λ {\displaystyle \Lambda } , e or ϵ {\displaystyle \epsilon } ) in order to avoid confusion.A distinguished symbol S ∈ N {\displaystyle S\in N} that is the start symbol, also called the sentence symbol.A grammar is formally defined as the tuple ( N , Σ , P , S ) {\displaystyle (N,\Sigma ,P,S)} . Such a formal grammar is often called a rewriting system or a phrase structure grammar in the literature.


In mathematics, especially in the field of commutative algebra, a connected ring is a commutative ring A that satisfies one of the following equivalent conditions: A possesses no non-trivial (that is, not equal to 1 or 0) idempotent elements; the spectrum of A with the Zariski topology is a connected space.


{\displaystyle f_{i-1}(r_{1}\dots r)=(1-r)f_{i}(r_{1},\dots ,r_{i-1},0)+rf_{i}(r_{1},\dots ,r_{i-1},1){\text{ if }}{\mathsf {S}}=\mathrm {R} .} If either fails then reject. V → P: V picks a random r in F and sends it to P.


In mathematics, the Jacobsthal numbers are an integer sequence named after the German mathematician Ernst Jacobsthal. Like the related Fibonacci numbers, they are a specific type of Lucas sequence U n ( P , Q ) {\displaystyle U_{n}(P,Q)} for which P = 1, and Q = −2—and are defined by a similar recurrence relation: in simple terms, the sequence starts with 0 and 1, then each following number is found by adding the number before it to twice the number before that. The first Jacobsthal numbers are: 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, … (sequence A001045 in the OEIS)A Jacobsthal prime is a Jacobsthal number that is also prime. The first Jacobsthal primes are: 3, 5, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243, … (sequence A049883 in the OEIS)


For example, the statement "there is an integer n such that if there is a sequence of rooted trees T1, T2, ..., Tn such that Tk has at most k+10 vertices, then some tree can be homeomorphically embedded in a later one"is provable in Peano arithmetic, but the shortest proof has length at least A(1000), where A(0)=1 and A(n+1)=2A(n). The statement is a special case of Kruskal's theorem and has a short proof in second order arithmetic. If one takes Peano arithmetic together with the negation of the statement above, one obtains an inconsistent theory whose shortest known contradiction is equivalently long.


Most of the more successful systems use lexical statistics (that is, they consider the identities of the words involved, as well as their part of speech). However such systems are vulnerable to overfitting and require some kind of smoothing to be effective.Parsing algorithms for natural language cannot rely on the grammar having 'nice' properties as with manually designed grammars for programming languages. As mentioned earlier some grammar formalisms are very difficult to parse computationally; in general, even if the desired structure is not context-free, some kind of context-free approximation to the grammar is used to perform a first pass.


After period one, the market value changes to 120k:80k:600k. We now rebalance to increase exposure on the outperforming asset and reduce exposure to the worst-performing asset. Asset A is the best performer, so its rebalanced to be left at 120k, B is the worst performer, to its rebalanced to 60k, and C is the remaining, 800k-120k-60k=620k. We are now back to the original fixed weights of 120:60:620 or ratio-wise 2:1:remaining.


In mathematics, geology, and cartography, a surface map is a 2D perspective representation of a 3-dimensional surface. Surface maps usually represent real-world entities such as landforms or the surfaces of objects. They can, however, serve as an abstraction where the third, or even all of the dimensions correspond to non-spatial data. In this capacity they act more as graphs than maps.


In software engineering, a schema migration (also database migration, database change management) refers to the management of version-controlled, incremental and reversible changes to relational database schemas. A schema migration is performed on a database whenever it is necessary to update or revert that database's schema to some newer or older version. Migrations are performed programmatically by using a schema migration tool. When invoked with a specified desired schema version, the tool automates the successive application or reversal of an appropriate sequence of schema changes until it is brought to the desired state.


A simultaneous embedding in a grid of linear dimensions is also possible for any number of graphs that are all stars. Other pairs of graph types that always admit a simultaneous embedding, but that might need larger grid sizes, include a wheel graph and a cycle graph, a tree and a matching, or a pair of graphs both of which have maximum degree two. However, pairs of planar graphs and a matching, or of a Angelini, Geyer, Neuwirth and Kaufmann showed that a tree and a path exist, that have no simultaneous geometric embedding.Testing whether two graphs admit a simultaneous geometric embedding is NP-hard.


This is generally done at two levels, at the sentence level, where scores are calculated by the metric for a set of translated sentences, and then correlated against human judgment for the same sentences. And at the corpus level, where scores over the sentences are aggregated for both human judgments and metric judgments, and these aggregate scores are then correlated. Figures for correlation at the sentence level are rarely reported, although Banerjee et al. (2005) do give correlation figures that show that, at least for their metric, sentence-level correlation is substantially worse than corpus level correlation.


Making use of currying, n-ary type operators can be (re)written as a sequence of applications of unary type operators. Therefore, we can view the type operators as a simply typed lambda calculus, which has only one basic type, usually denoted ∗ {\displaystyle *} , and pronounced "type", which is the type of all types in the underlying language, which are now called proper types in order to distinguish them from the types of the type operators in their own calculus, which are called kinds. Type operators may bind type variables.


In mathematical terms, a vector optimization problem can be written as: C -  min x ∈ S f ( x ) {\displaystyle C\operatorname {-} \min _{x\in S}f(x)} where f: X → Z {\displaystyle f:X\to Z} for a partially ordered vector space Z {\displaystyle Z} . The partial ordering is induced by a cone C ⊆ Z {\displaystyle C\subseteq Z} . X {\displaystyle X} is an arbitrary set and S ⊆ X {\displaystyle S\subseteq X} is called the feasible set.


More generally, data depth, gives a center-outward ordering of data points, and thereby provides a mechanism for constructing rank statistics of various kinds of multidimensional data. For instance, functional data examples can be ordered using the method of band depth or a modified band depth. In contour data analysis, each observation is a feature-set (a subset of the domain), and therefore not a function.


Both Alice and Bob announce these bits publicly and run a check to see whether more than a certain number of them agree. If this check passes, Alice and Bob proceed to use information reconciliation and privacy amplification techniques to create some number of shared secret keys. Otherwise, they cancel and start over.


When substantiating a civil fraud claim, the plaintiff is generally required to prove that the defendant falsely represented a material fact, that this representation was made with intent to deceive, that the plaintiff reasonably relied on the representation, and the representation resulted in damages to the plaintiff. Some legal experts have recommended strengthening existing intellectual property laws to address the growing problem of art forgeries proliferating in the mass market. They argue that the existing legal regime is ineffective in combating this growing trend.


As an example, a 4-byte value consists of 8 nibbles, wherein the upper 7 nibbles store the digits of a 7-digit decimal value, and the lowest nibble indicates the sign of the decimal integer value. Standard sign values are 1100 (hex C) for positive (+) and 1101 (D) for negative (−). This convention comes from the zone field for EBCDIC characters and the signed overpunch representation.


In probability theory and statistics, the coefficient of variation (COV), also known as Normalized Root-Mean-Square Deviation (NRMSD), Percent RMS, and relative standard deviation (RSD), is a standardized measure of dispersion of a probability distribution or frequency distribution. It is defined as the ratio of the standard deviation σ {\displaystyle \sigma } to the mean μ {\displaystyle \mu } (or its absolute value, | μ | {\displaystyle |\mu |} ), and often expressed as a percentage ("%RSD"). The CV or RSD is widely used in analytical chemistry to express the precision and repeatability of an assay. It is also commonly used in fields such as engineering or physics when doing quality assurance studies and ANOVA gauge R&R, by economists and investors in economic models, and in neuroscience.


An important problem in computational logic is to determine fragments of well-known logics such as first-order logic that are as expressive as possible yet are decidable or more strongly have low computational complexity. The field of descriptive complexity theory aims at establishing a link between logics and computational complexity theory, by identifying logical fragments that exactly capture certain complexity classes. == References ==


All object-oriented programming (OOP) systems support encapsulation, but encapsulation is not unique to OOP. Implementations of abstract data types, modules, and libraries, among other systems, also offer encapsulation. The similarity has been explained by programming language theorists in terms of existential types.


The generalized version was popularized by Hoffmeister & Bäck and Mühlenbein et al. Finding the minimum of this function is a fairly difficult problem due to its large search space and its large number of local minima. On an n {\displaystyle n} -dimensional domain it is defined by: f ( x ) = A n + ∑ i = 1 n  {\displaystyle f(\mathbf {x} )=An+\sum _{i=1}^{n}\left} where A = 10 {\displaystyle A=10} and x i ∈  {\displaystyle x_{i}\in } . There are many extrema: The global minimum is at x = 0 {\displaystyle \mathbf {x} =\mathbf {0} } where f ( x ) = 0 {\displaystyle f(\mathbf {x} )=0} . The maximum function value for x i ∈  {\displaystyle x_{i}\in } is located around x i ∈  {\displaystyle x_{i}\in } :Here are all the values at 0.5 interval listed for the 2D Rastrigin function with x i ∈  {\displaystyle x_{i}\in }: The abundance of local minima underlines the necessity of a global optimization algorithm when needing to find the global minimum. Local optimization algorithms are likely to get stuck in a local minimum.


In the Japanese kantenji braille, the standard 8-dot Braille patterns 268, 1268, 2468, and 12468 are the patterns related to Braille pattern dots-156, since the two additional dots of kantenji patterns 0156, 1567, and 01567 are placed above the base 6-dot cell, instead of below, as in standard 8-dot braille.


Although von Neumann's name is popularly attached to the conjecture, its first written appearance seems to be due to Mahlon Marsh Day in 1957. The Tits alternative is a fundamental theorem which, in particular, establishes the conjecture within the class of linear groups. The historically first potential counterexample is Thompson group F. While its amenability is a wide open problem, the general conjecture was shown to be false in 1980 by Alexander Ol'shanskii; he demonstrated that Tarski monster groups, constructed by him, which are easily seen not to have free subgroups of rank 2, are not amenable.


The finer the granularity, the greater the potential for parallelism and hence speed-up, but the greater the overheads of synchronization and communication. Granularity disintegrators exist as well and are important to understand in order to determine the accurate level of granularity.In order to attain the best parallel performance, the best balance between load and communication overhead needs to be found. If the granularity is too fine, the performance can suffer from the increased communication overhead. On the other side, if the granularity is too coarse, the performance can suffer from load imbalance.


In mathematics, Krohn–Rhodes complexity is an important topic in the study of finite semigroups and automata. In Network theory complexity is the product of richness in the connections between components of a system, and defined by a very unequal distribution of certain measures (some elements being highly connected and some very few, see complex network). In software engineering, programming complexity is a measure of the interactions of the various elements of the software. This differs from the computational complexity described above in that it is a measure of the design of the software.Other fields introduce less precisely defined notions of complexity: A complex adaptive system has some or all of the following attributes:The number of parts (and types of parts) in the system and the number of relations between the parts is non-trivial – however, there is no general rule to separate "trivial" from "non-trivial"; The system has memory or includes feedback; The system can adapt itself according to its history or feedback; The relations between the system and its environment are non-trivial or non-linear; The system can be influenced by, or can adapt itself to, its environment; The system is highly sensitive to initial conditions.


The associativity of this product follows from that of the group product. The product of group subsets therefore defines a natural monoid structure on the power set of G. A lot more can be said in the case where S and T are subgroups. The product of two subgroups S and T of a group G is itself a subgroup of G if and only if ST = TS.


In the Scrum framework, which claims to be consistent with agile values and principles, the scrum master role is accountable for ensuring the scrum process is followed and for coaching the scrum team through that process. A common pitfall is for a scrum master to act as a contributor. While not prohibited by the Scrum framework, the scrum master needs to ensure they have the capacity to act in the role of scrum master first and not work on development tasks. A scrum master's role is to facilitate the process rather than create the product.Having the scrum master also multitasking may result in too many context switches to be productive. Additionally, as a scrum master is responsible for ensuring roadblocks are removed so that the team can make forward progress, the benefit gained by individual tasks moving forward may not outweigh roadblocks that are deferred due to lack of capacity.


For algorithms operating on this representation, this requires an All-to-All communication step as well as O ( m ) {\displaystyle {\mathcal {O}}(m)} message buffer sizes, as each PE potentially has outgoing edges to every other PE.2D partitioning: Every processor gets a submatrix of the adjacency matrix. Assume the processors are aligned in a rectangle p = p r × p c {\displaystyle p=p_{r}\times p_{c}} , where p r {\displaystyle p_{r}} and p c {\displaystyle p_{c}} are the amount of processing elements in each row and column, respectively. Then each processor gets a submatrix of the adjacency matrix of dimension ( n / p r ) × ( n / p c ) {\displaystyle (n/p_{r})\times (n/p_{c})} .


In the early days of telephony, companies used manual telephone switchboards, and switchboard operators connected calls by inserting a pair of phone plugs into the appropriate jacks. They were gradually phased out and replaced by automated systems, first those allowing direct dialing within a local area, then for long-distance and international direct dialing.


(Some authors write it as fg. )such that the following axiom holds: (associativity) if f: A → B, g: B → C and h: C → D then h ∘ (g ∘ f) = (h ∘ g) ∘ f. == References ==


In the Japanese kantenji braille, the standard 8-dot Braille patterns 235678, 1235678, 2345678, and 12345678 are the patterns related to Braille pattern dots-123456, since the two additional dots of kantenji patterns 0123456, 1234567, and 01234567 are placed above the base 6-dot cell, instead of below, as in standard 8-dot braille.


A Poisson compounded with Log(p)-distributed random variables has a negative binomial distribution. In other words, if N is a random variable with a Poisson distribution, and Xi, i = 1, 2, 3, ... is an infinite sequence of independent identically distributed random variables each having a Log(p) distribution, then ∑ i = 1 N X i {\displaystyle \sum _{i=1}^{N}X_{i}} has a negative binomial distribution. In this way, the negative binomial distribution is seen to be a compound Poisson distribution. R. A. Fisher described the logarithmic distribution in a paper that used it to model relative species abundance.


A near-bijection from the unit square to the unit interval can be obtained by interleaving the digits of the decimal representations of the Cartesian coordinates of points in the square. The ambiguities of decimal, exemplified by the two decimal representations of 1 = 0.999..., cause this to be an injection rather than a bijection, but this issue can be repaired by using the Schröder–Bernstein theorem. == References ==


In some applications, we want to find a small cut in a graph G = ( V , E ) {\displaystyle G=(V,E)} that partitions the graph into nearly equal-size pieces. We usually call a cut b-balanced or a (b,1 − b)-separator (for b ≤ 1/2) if b π ( V ) ≤ π ( U ) ≤ ( 1 − b ) π ( V ) {\displaystyle b\pi (V)\leq \pi (U)\leq (1-b)\pi (V)} where π ( U ) {\displaystyle \pi (U)} is the sum of the node weights in U. This is also an NP-hard problem. An approximation algorithm has been designed for this problem, and the core idea is that G has a b-balanced cut of size S, then we find a b′-balanced cut of size O ( S log  n b − b ′ ) {\displaystyle O\left(S\log {\frac {n}{b}}-b'\right)} for any b' where b′ < b and b′ ≤ 1/3. Then we repeat the process then finally obtain the result that total weight of the edges in the cut is at most O ( S log  n b − b ′ ) {\displaystyle O\left({\frac {S\log n}{b-b'}}\right)} .


The mixed product is an example of an operation of arity 3, also called ternary operation. Generally, the arity is taken to be finite. However, infinitary operations are sometimes considered, in which case the "usual" operations of finite arity are called finitary operations. A partial operation is defined similarly to an operation, but with a partial function in place of a function.


In some contexts a regularized version of the least squares solution may be preferable. Tikhonov regularization (or ridge regression) adds a constraint that ‖ β ‖ 2 2 {\displaystyle \|\beta \|_{2}^{2}} , the squared ℓ 2 {\displaystyle \ell _{2}} -norm of the parameter vector, is not greater than a given value to the least squares formulation, leading to a constrained minimization problem. This is equivalent to the unconstrained minimization problem where the objective function is the residual sum of squares plus a penalty term α ‖ β ‖ 2 2 {\displaystyle \alpha \|\beta \|_{2}^{2}} and α {\displaystyle \alpha } is a tuning parameter (this is the Lagrangian form of the constrained minimization problem).In a Bayesian context, this is equivalent to placing a zero-mean normally distributed prior on the parameter vector.


In the context of cryptography, encryption serves as a mechanism to ensure confidentiality. Since data may be visible on the Internet, sensitive information such as passwords and personal communication may be exposed to potential interceptors. The process of encrypting and decrypting messages involves keys. The two main types of keys in cryptographic systems are symmetric-key and public-key (also known as asymmetric-key).Many complex cryptographic algorithms often use simple modular arithmetic in their implementations.


In software testing, recovery testing is the activity of testing how well an application is able to recover from crashes, hardware failures and other similar problems. Recovery testing is the forced failure of the software in a variety of ways to verify that recovery is properly performed. Recovery testing should not be confused with reliability testing, which tries to discover the specific point at which failure occurs. Recovery testing is basically done in order to check how fast and better the application can recover against any type of crash or hardware failure etc. Recovery testing is simulating failure modes or actually causing failures in a controlled environment.


Stewart in 1971, it was the first numerically stable method that could be systematically applied to solve such equations. The algorithm works by using the real Schur decompositions of A {\displaystyle A} and B {\displaystyle B} to transform A X − X B = C {\displaystyle AX-XB=C} into a triangular system that can then be solved using forward or backward substitution. In 1979, G. Golub, C. Van Loan and S. Nash introduced an improved version of the algorithm, known as the Hessenberg–Schur algorithm. It remains a standard approach for solving Sylvester equations when X {\displaystyle X} is of small to moderate size.


In other words, it is similar to VIEWS in databases modelling. Like VIEW in Oracle or MySQL. For each page there is One abstract Table of data. But it is merged from other tables. Uses WebML-OQL (WebML-Object Query Language)


The middle compartment contains the attributes of the class. They are left-aligned and the first letter is lowercase. The bottom compartment contains the operations the class can execute. They are also left-aligned and the first letter is lowercase.In the design of a system, a number of classes are identified and grouped together in a class diagram that helps to determine the static relations between them. In detailed modeling, the classes of the conceptual design are often split into subclasses.In order to further describe the behavior of systems, these class diagrams can be complemented by a state diagram or UML state machine.


In probability, and statistics, a multivariate random variable or random vector is a list or vector of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value. The individual variables in a random vector are grouped together because they are all part of a single mathematical system — often they represent different properties of an individual statistical unit. For example, while a given person has a specific age, height and weight, the representation of these features of an unspecified person from within a group would be a random vector. Normally each element of a random vector is a real number. Random vectors are often used as the underlying implementation of various types of aggregate random variables, e.g. a random matrix, random tree, random sequence, stochastic process, etc. More formally, a multivariate random variable is a column vector X = ( X 1 , … , X n ) T {\displaystyle \mathbf {X} =(X_{1},\dots ,X_{n})^{\mathsf {T}}} (or its transpose, which is a row vector) whose components are scalar-valued random variables on the same probability space as each other, ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},P)} , where Ω {\displaystyle \Omega } is the sample space, F {\displaystyle {\mathcal {F}}} is the sigma-algebra (the collection of all events), and P {\displaystyle P} is the probability measure (a function returning each event's probability).


In practice, it is more efficient to use an extended variant of Euclid's algorithm to calculate the Jacobi symbol ( a n ) {\displaystyle \left({\frac {a}{n}}\right)} . If n {\displaystyle n} is an odd prime, this is equal to the Legendre symbol, and decides whether a {\displaystyle a} is a quadratic residue modulo n {\displaystyle n} . On the other hand, since the equivalence of a n − 1 2 {\displaystyle a^{\frac {n-1}{2}}} to the Jacobi symbol holds for all odd primes, but not necessarily for composite numbers, calculating both and comparing them can be used as a primality test, specifically the Solovay–Strassen primality test. Composite numbers for which the congruence holds for a given a {\displaystyle a} are called Euler–Jacobi pseudoprimes to base a {\displaystyle a} .


In mathematical logic, the theory of infinite sets was first developed by Georg Cantor. Although this work has become a thoroughly standard fixture of classical set theory, it has been criticized in several areas by mathematicians and philosophers. Cantor's theorem implies that there are sets having cardinality greater than the infinite cardinality of the set of natural numbers. Cantor's argument for this theorem is presented with one small change.


At any point in a trip the camel may leave any amount of bananas that it is carrying at a camp post, or may collect any amount of bananas that was left at a camp post on a previous trip, as long as its banana load never exceeds 1 unit. The problem asks for the maximum units of bananas that can be transported to the market. In the travelers across the desert problem, the starting base has unlimited units of supplies.


In psychological coercion, the threatened injury regards the victim's relationships with other people. The most obvious example is blackmail, where the threat consists of the dissemination of damaging information. However, many other types are possible e.g. "emotional blackmail", which typically involves threats of rejection from or disapproval by a peer-group, or creating feelings of guilt/obligation via a display of anger or hurt by someone whom the victim loves or respects.


In the TCP/IP network model, the transport layer of the network (Layer 4) is responsible for the reliable transport of data. The TCP protocol is the principal means of ensuring reliable unicast (point-to-point) transport. TCP does this through an ACK mechanism. With the ACK mechanism, data packets are sequentially numbered, and the sender does not send a packet in a sequence until it receives an acknowledgement (ACK) from the receiver that the previous packet has been successfully received.


The most common method is the Gram–Schmidt process. Which creates a set of orthogonal basis vectors, which can then easily be normalized. This method begins by first selecting any standard basis β={v1, v2,...,vn}.


In statistical hypothesis testing, the alternative hypothesis is one of the proposed proposition in the hypothesis test. In general the goal of hypothesis test is to demonstrate that in the given condition, there is sufficient evidence supporting the credibility of alternative hypothesis instead of the exclusive proposition in the test (null hypothesis). It is usually consistent with the research hypothesis because it is constructed from literature review, previous studies, etc. However, the research hypothesis is sometimes consistent with the null hypothesis.


David Hilbert argued in favor of the study of the infinite, saying "No one shall expel us from the Paradise that Cantor has created." Mathematicians began to search for axiom systems that could be used to formalize large parts of mathematics. In addition to removing ambiguity from previously naive terms such as function, it was hoped that this axiomatization would allow for consistency proofs.


In matrix theory, the Rule of Sarrus is a mnemonic device for computing the determinant of a 3 × 3 {\displaystyle 3\times 3} matrix named after the French mathematician Pierre Frédéric Sarrus.Consider a 3 × 3 {\displaystyle 3\times 3} matrix M =  , {\displaystyle M={\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}},} then its determinant can be computed by the following scheme. Write out the first two columns of the matrix to the right of the third column, giving five columns in a row. Then add the products of the diagonals going from top to bottom (solid) and subtract the products of the diagonals going from bottom to top (dashed). This yields det ( M ) = det  = a 11 a 22 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 − a 31 a 22 a 13 − a 32 a 23 a 11 − a 33 a 21 a 12 .


In probability theory and statistics, the Jensen–Shannon divergence is a method of measuring the similarity between two probability distributions. It is also known as information radius (IRad) or total divergence to the average. It is based on the Kullback–Leibler divergence, with some notable (and useful) differences, including that it is symmetric and it always has a finite value. The square root of the Jensen–Shannon divergence is a metric often referred to as Jensen–Shannon distance.


In some applications, the processing of an item Y by a stage A may depend on the results or effect of processing a previous item X by some later stage B of the pipeline. In that case, stage A cannot correctly process item Y until item X has cleared stage B. This situation occurs very often in instruction pipelines. For example, suppose that Y is an arithmetic instruction that reads the contents of a register that was supposed to have been modified by an earlier instruction X. Let A be the stage that fetches the instruction operands, and B be the stage that writes the result to the specified register. If stage A tries to process instruction Y before instruction X reaches stage B, the register may still contain the old value, and the effect of Y would be incorrect.


It is algebraically simpler, though in practice less robust, than the average absolute deviation. A useful property of the standard deviation is that, unlike the variance, it is expressed in the same unit as the data. The standard deviation of a population or sample and the standard error of a statistic (e.g., of the sample mean) are quite different, but related.


{\displaystyle s(G)=\sum _{(u,v)\in E}\deg(u)\cdot \deg(v).} This is maximized when high-degree nodes are connected to other high-degree nodes. Now define S ( G ) = s ( G ) s max , {\displaystyle S(G)={\frac {s(G)}{s_{\max }}},} where smax is the maximum value of s(H) for H in the set of all graphs with degree distribution identical to that of G. This gives a metric between 0 and 1, where a graph G with small S(G) is "scale-rich", and a graph G with S(G) close to 1 is "scale-free". This definition captures the notion of self-similarity implied in the name "scale-free".


In most command-line interfaces or text editors, the text cursor, also known as a caret, is an underscore, a solid rectangle, or a vertical line, which may be flashing or steady, indicating where text will be placed when entered (the insertion point). In text mode displays, it was not possible to show a vertical bar between characters to show where the new text would be inserted, so an underscore or block cursor was used instead. In situations where a block was used, the block was usually created by inverting the pixels of the character using the boolean math exclusive or function. On text editors and word processors of modern design on bitmapped displays, the vertical bar is typically used instead.


In statistics, the precision matrix or concentration matrix is the matrix inverse of the covariance matrix or dispersion matrix, P = Σ − 1 {\displaystyle P=\Sigma ^{-1}} . For univariate distributions, the precision matrix degenerates into a scalar precision, defined as the reciprocal of the variance, p = 1 σ 2 {\displaystyle p={\frac {1}{\sigma ^{2}}}} .Other summary statistics of statistical dispersion also called precision (or imprecision) include the reciprocal of the standard deviation, p = 1 σ {\displaystyle p={\frac {1}{\sigma }}} ; the standard deviation itself and the relative standard deviation; as well as the standard error and the confidence interval (or its half-width, the margin of error).


In telecommunication, a public land mobile network (PLMN) is a combination of wireless communication services offered by a specific operator in a specific country. A PLMN typically consists of several cellular technologies like GSM/2G, UMTS/3G, LTE/4G, offered by a single operator within a given country, often referred to as a cellular network.


In model checking, a field of computer science, a region is a convex polytope in R d {\displaystyle \mathbb {R} ^{d}} for some dimension d {\displaystyle d} , and more precisely a zone, satisfying some minimality property. The regions partition R d {\displaystyle \mathbb {R} ^{d}} . The set of zones depends on a set K {\displaystyle K} of constraints of the form x ≤ c {\displaystyle x\leq c} , x ≥ c {\displaystyle x\geq c} , x 1 ≤ x 2 + c {\displaystyle x_{1}\leq x_{2}+c} and x 1 ≥ x 2 + c {\displaystyle x_{1}\geq x_{2}+c} , with x 1 {\displaystyle x_{1}} and x 2 {\displaystyle x_{2}} some variables, and c {\displaystyle c} a constant.


In object-oriented programming, the command pattern is a behavioral design pattern in which an object is used to encapsulate all information needed to perform an action or trigger an event at a later time. This information includes the method name, the object that owns the method and values for the method parameters. Four terms always associated with the command pattern are command, receiver, invoker and client. A command object knows about receiver and invokes a method of the receiver.


Often the functions to be minimized are not f i {\displaystyle f_{i}} but | f i − z i ∗ | {\displaystyle |f_{i}-z_{i}^{*}|} for some scalars z i ∗ {\displaystyle z_{i}^{*}} . Then f T c h b ( x , w ) = max i w i | f i ( x ) − z i ∗ | . {\displaystyle f_{Tchb}(x,w)=\max _{i}w_{i}|f_{i}(x)-z_{i}^{*}|.} All three functions are named in honour of Pafnuty Chebyshev.


In the first normal form each field contains a single value. A field may not contain a set of values or a nested record. Subject contains a set of subject values, meaning it does not comply. To solve the problem, the subjects are extracted into a separate Subject table: In Subject, ISBN is a foreign key: It refers to the primary key in Book, and makes the relationship between these two tables explicit. Instead of one table in unnormalized form, there are now two tables conforming to the 1NF.


Due to the large game trees of complex games such as chess, algorithms that are designed to play this class of games will use partial game trees, which makes computation feasible on modern computers. Various methods exist to solve game trees. If a complete game tree can be generated, a deterministic algorithm, such as backward induction or retrograde analysis can be used. Randomized algorithms and minimax algorithms such as MCTS can be used in cases where a complete game tree is not feasible.


Lord Denning, first of the High Court of Justice, later of the Court of Appeal, provided a famous example of this evolutionary process in his development of the concept of estoppel starting in the High Trees case: Central London Property Trust Ltd v. High Trees House Ltd  K.B. 130.


In science, a prediction is a rigorous, often quantitative, statement, forecasting what would be observed under specific conditions; for example, according to theories of gravity, if an apple fell from a tree it would be seen to move towards the center of the earth with a specified and constant acceleration. The scientific method is built on testing statements that are logical consequences of scientific theories. This is done through repeatable experiments or observational studies. A scientific theory whose predictions are contradicted by observations and evidence will be rejected.


The resulting control-flow graph for the sequences of "if"s thus has many more nodes and almost twice as many edges, with these not adding any useful information. However, the simple branches in the if statements are individually conceptually easier than the complex branch of a switch statement. In terms of cyclomatic complexity, both of these options increase it by k−1 if given k cases.


These give a sense in which the functions x• and x\ are pseudoinverses or adjoints of each other, and likewise for •x and /x. This last definition is purely in terms of inequalities, noting that monotonicity can be axiomatized as x•y ≤ (x∨z)•y and similarly for the other operations and their arguments. Moreover, any inequality x ≤ y can be expressed equivalently as an equation, either x∧y = x or x∨y = y. This along with the equations axiomatizing lattices and monoids then yields a purely equational definition of residuated lattices, provided the requisite operations are adjoined to the signature (L, ≤, •, I) thereby expanding it to (L, ∧, ∨, •, I, /, \).


In mathematics, a Ditkin set, introduced by (Ditkin 1939), is a closed subset of the circle such that a function f vanishing on the set can be approximated by functions φnf with φ vanishing in a neighborhood of the set.


Two likelihood functions are equivalent if one is a scalar multiple of the other. The likelihood principle is this: All information from the data that is relevant to inferences about the value of the model parameters is in the equivalence class to which the likelihood function belongs. The strong likelihood principle applies this same criterion to cases such as sequential experiments where the sample of data that is available results from applying a stopping rule to the observations earlier in the experiment.


However, it is argued that chunking, rather than moving straight to short division, gives a better introduction to division, in part because the focus is always holistic, focusing throughout on the whole calculation and its meaning, rather than just rules for generating successive digits. The more freeform nature of chunking also means that it requires more genuine understanding – rather than just the ability to follow a ritualised procedure – to be successful.An alternative way of performing chunking involves the use of the standard long division tableau – except that the partial quotients are stacked up on the top of each other above the long division sign, and that all numbers are spelled out in full. By allowing one to subtract more chunks than what one currently has, it is also possible to expand chunking into a fully bidirectional method as well.


In philosophy, a core ontology is a basic and minimal ontology consisting only of the minimal concepts required to understand the other concepts. It must be based on a core glossary in some human language so humans can comprehend the concepts and distinctions made. Each natural language tends to rely on its own conceptual metaphor structure, and so tends to have its own core ontology (according to W. V. Quine). It could be said also to represent the moral core of a human linguistic culture, and to self-correct so as to better represent core cultural ideas.


The scale normalized Laplacian (in L 1 {\displaystyle L_{1}} -norm) is frequently used as a blob detector and for automatic scale selection in computer vision applications; see Laplacian of Gaussian and scale space. The relation between this Laplacian of the Gaussian operator and the difference-of-Gaussians operator is explained in appendix A in Lindeberg (2015). The Mexican hat wavelet can also be approximated by derivatives of cardinal B-splines.


In software engineering, a domain model is a conceptual model of the domain that incorporates both behavior and data. In ontology engineering, a domain model is a formal representation of a knowledge domain with concepts, roles, datatypes, individuals, and rules, typically grounded in a description logic.


In order to reduce failures, a precise knowledge of bond strength quality measurement during product design and subsequent manufacture is of vital importance. The best place to start is with the failure mode. This is based on the assumption that there is a particular failure mode, or range of modes, that may occur within a product. It is therefore reasonable to assume that the bond test should replicate the mode, or modes of interest.


In the 2003 version, the codes were as follows: PL1 CENTRALNY PL11 Lodzkie PL111 Lodzki PL112 Piotrkowsko-skierniewicki PL113 Miasto Lodz PL12 Mazowieckie PL121 Ciechanowsko-plocki PL122 Ostrolecko-siedlecki PL124 Radomski PL126 Warszawski PL127 Miasto Warszawa PL2 POLUDNIOWY PL21 Malopolskie PL211 Krakowsko-tarnowski PL212 Nowosadecki PL213 Miasto Krakow PL22 Slaskie PL224 Czestochowski PL225 Bielsko-bialski PL226 Centralny slaski PL227 Rybnicko-jastrzebski PL3 WSCHODNI PL31 Lubelskie PL311 Bialskopodlaski PL312 Chelmsko-zamojski PL313 Lubelski PL32 Podkarpackie PL321 Rzeszowsko-tarnobrzeski PL322 Krosniensko-przemyski PL33 Swietokrzyskie PL330 Swietokrzyski PL34 Podlaskie PL341 Bialostocko-suwalski PL342 Lomzynski PL4 POLNOCNO-ZACHODNI PL41 Wielkopolskie PL411 Pilski PL412 Poznanski PL413 Kaliski PL414 Koninski PL415 Miasto Poznan PL42 Zachodniopomorskie PL421 Szczecinski PL422 Koszalinski PL43 Lubuskie PL431 Gorzowski PL432 Zielonogorski PL5 POLUDNIOWO-ZACHODNI PL51 Dolnoslaskie PL511 Jeleniogorsko-walbrzyski PL512 Legnicki PL513 Wroclawski PL514 Miasto Wroclaw PL52 Opolskie PL520 Opolski PL6 POLNOCNY PL61 Kujawsk


In mathematics, a filter on a set X {\displaystyle X} is a family B {\displaystyle {\mathcal {B}}} of subsets such that: X ∈ B {\displaystyle X\in {\mathcal {B}}} and ∅ ∉ B {\displaystyle \emptyset \notin {\mathcal {B}}} if A ∈ B {\displaystyle A\in {\mathcal {B}}} and B ∈ B {\displaystyle B\in {\mathcal {B}}} , then A ∩ B ∈ B {\displaystyle A\cap B\in {\mathcal {B}}} If A , B ⊂ X , A ∈ B {\displaystyle A,B\subset X,A\in {\mathcal {B}}} , and A ⊂ B {\displaystyle A\subset B} , then B ∈ B {\displaystyle B\in {\mathcal {B}}} A filter on a set may be thought of as representing a "collection of large subsets", one intuitive example being the neighborhood filter. Filters appear in order theory, model theory, and set theory, but can also be found in topology, from which they originate. The dual notion of a filter is an ideal. Filters were introduced by Henri Cartan in 1937 and as described in the article dedicated to filters in topology, they were subsequently used by Nicolas Bourbaki in their book Topologie Générale as an alternative to the related notion of a net developed in 1922 by E. H. Moore and Herman L. Smith. Order filters are generalizations of filters from sets to arbitrary partially ordered sets. Specifically, a filter on a set is just a proper order filter in the special case where the partially ordered set consists of the power set ordered by set inclusion.


Alternatives for I include e and 1'. Alternative notations for the residuals are x → y for x\y and y ← x for y/x, suggested by the similarity between residuation and implication in logic, with the multiplication of the monoid understood as a form of conjunction that need not be commutative. When the monoid is commutative the two residuals coincide. When not commutative, the intuitive meaning of the monoid as conjunction and the residuals as implications can be understood as having a temporal quality: x•y means x and then y, x → y means had x (in the past) then y (now), and y ← x means if-ever x (in the future) then y (at that time), as illustrated by the natural language example at the end of the examples.


In telecommunications, direct-sequence spread spectrum (DSSS) is a spread-spectrum modulation technique primarily used to reduce overall signal interference. The direct-sequence modulation makes the transmitted signal wider in bandwidth than the information bandwidth. After the despreading or removal of the direct-sequence modulation in the receiver, the information bandwidth is restored, while the unintentional and intentional interference is substantially reduced.Swiss inventor, Gustav Guanella proposed a "means for and method of secret signals". With DSSS, the message symbols are modulated by a sequence of complex values known as spreading sequence.


In number theory, it is often important to find factors of an integer number N. Any number N has four obvious factors: ±1 and ±N. These are called "trivial factors". Any other factor, if it exists, would be called "nontrivial". The homogeneous matrix equation A x = 0 {\displaystyle A\mathbf {x} =\mathbf {0} } , where A {\displaystyle A} is a fixed matrix, x {\displaystyle \mathbf {x} } is an unknown vector, and 0 {\displaystyle \mathbf {0} } is the zero vector, has an obvious solution x = 0 {\displaystyle \mathbf {x} =\mathbf {0} } .


In statistical mechanics, configuration entropy is the portion of a system's entropy that is related to discrete representative positions of its constituent particles. For example, it may refer to the number of ways that atoms or molecules pack together in a mixture, alloy or glass, the number of conformations of a molecule, or the number of spin configurations in a magnet. The name might suggest that it relates to all possible configurations or particle positions of a system, excluding the entropy of their velocity or momentum, but that usage rarely occurs.


In the Intel 80386 and later, protected mode retains the segmentation mechanism of 80286 protected mode, but a paging unit has been added as a second layer of address translation between the segmentation unit and the physical bus. Also, importantly, address offsets are 32-bit (instead of 16-bit), and the segment base in each segment descriptor is also 32-bit (instead of 24-bit). The general operation of the segmentation unit is otherwise unchanged. The paging unit may be enabled or disabled; if disabled, operation is the same as on the 80286.


In ASCII and Unicode, the character is encoded at U+0004 . It can be referred to as Ctrl+D, ^D in caret notation. Unicode provides the character U+2404 ␄ SYMBOL FOR END OF TRANSMISSION for when EOT needs to be displayed graphically. In addition, U+2301 ⌁ ELECTRIC ARROW can also be used as a graphic representation of EOT; it is defined in Unicode as "symbol for End of Transmission".


In metalogic, 'syntax' has to do with formal languages or formal systems without regard to any interpretation of them, whereas, 'semantics' has to do with interpretations of formal languages. The term 'syntactic' has a slightly wider scope than 'proof-theoretic', since it may be applied to properties of formal languages without any deductive systems, as well as to formal systems. 'Semantic' is synonymous with 'model-theoretic'.


In mathematics, a right group is an algebraic structure consisting of a set together with a binary operation that combines two elements into a third element while obeying the right group axioms. The right group axioms are similar to the group axioms, but while groups can have only one identity and any element can have only one inverse, right groups allow for multiple one-sided identity elements and multiple one-sided inverse elements. It can be proven (theorem 1.27 in ) that a right group is isomorphic to the direct product of a right zero semigroup and a group, while a right abelian group is the direct product of a right zero semigroup and an abelian group. Left group and left abelian group are defined in analogous way, by substituting right for left in the definitions. The rest of this article will be mostly concerned about right groups, but everything applies to left groups by doing the appropriate right/left substitutions.


Then instead of storing only a single node in each entry of prev we would store all nodes satisfying the relaxation condition. For example, if both r and source connect to target and both of them lie on different shortest paths through target (because the edge cost is the same in both cases), then we would add both r and source to prev. When the algorithm completes, prev data structure will actually describe a graph that is a subset of the original graph with some edges removed. Its key property will be that if the algorithm was run with some starting node, then every path from that node to any other node in the new graph will be the shortest path between those nodes in the original graph, and all paths of that length from the original graph will be present in the new graph. Then to actually find all these shortest paths between two given nodes we would use a path finding algorithm on the new graph, such as depth-first search.


Hence the inputs to the neural network are contributions, future needs and wealth, and the output the percentage split predicted. On each arc there is a statistical weight. Using back propagation the neural network learns the necessary pattern to recognize the prediction.


In probability theory and statistics, the continuous uniform distributions or rectangular distributions are a family of symmetric probability distributions. Such a distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds. The bounds are defined by the parameters, a {\displaystyle a} and b , {\displaystyle b,} which are the minimum and maximum values. The interval can either be closed (i.e.  {\displaystyle } ) or open (i.e. ( a , b ) {\displaystyle (a,b)} ).


Some researchers identify various steps in the process of problem solving. These steps include recognizing the problem, trying to understand its nature, identifying general criteria the solution should meet, deciding how these criteria should be prioritized, monitoring the progress, and evaluating the results.An important distinction concerns the type of problem that is faced. For well-structured problems, it is easy to determine which steps need to be taken to solve them, but executing these steps may still be difficult.


The other 2 GiB are used by the operating system. On later 32-bit editions of Microsoft Windows, it is possible to extend the user-mode virtual address space to 3 GiB while only 1 GiB is left for kernel-mode virtual address space by marking the programs as IMAGE_FILE_LARGE_ADDRESS_AWARE and enabling the /3GB switch in the boot.ini file.On Microsoft Windows 64-bit, in a process running an executable that was linked with /LARGEADDRESSAWARE:NO, the operating system artificially limits the user mode portion of the process's virtual address space to 2 GiB. This applies to both 32- and 64-bit executables. Processes running executables that were linked with the /LARGEADDRESSAWARE:YES option, which is the default for 64-bit Visual Studio 2010 and later, have access to more than 2 GiB of virtual address space: up to 4 GiB for 32-bit executables, up to 8 TiB for 64-bit executables in Windows through Windows 8, and up to 128 TiB for 64-bit executables in Windows 8.1 and later.Allocating memory via C's malloc establishes the page file as the backing store for any new virtual address space. However, a process can also explicitly map file bytes.


This is how Minsky solves the problem, but the Gödel numbering he uses represents a great inconvenience to the model, and the result is nothing at all like our intuitive notion of a "stored program computer". Elgot and Robinson (1964) come to a similar conclusion with respect to a RASP that is "finitely determined". Indeed it can access an unbounded number of registers (e.g. to fetch instructions from them) but only if the RASP allows "self modification" of its program instructions, and has encoded its "data" in a Gödel number (Fig. 2 p.


In practical applications, the following considerations are commonly taken into account. One does not need to construct the convex hull of a polygon to find a suitable vertex. A common choice is the vertex of the polygon with the smallest X-coordinate. If there are several of them, the one with the smallest Y-coordinate is picked.


In neurophysiology, commutation is the process by which the brain's neural circuits exhibit non-commutativity. Physiologist Douglas B. Tweed and coworkers have considered whether certain neural circuits in the brain exhibit noncommutativity and state: In noncommutative algebra, order makes a difference to multiplication, so that a × b ≠ b × a {\displaystyle a\times b\neq b\times a} . This feature is necessary for computing rotary motion, because order makes a difference to the combined effect of two rotations. It has therefore been proposed that there are non-commutative operators in the brain circuits that deal with rotations, including motor system circuits that steer the eyes, head and limbs, and sensory system circuits that handle spatial information.


In applications that do not need to run older binary software, compressed RISCs are growing to dominate sales. Another approach to RISCs was the minimal instruction set computer (MISC), niladic, or zero-operand instruction set. This approach realized that most space in an instruction was used to identify the operands of the instruction.


In the United States, Great Britain, and Germany, the concept of a national centralized server model of healthcare data has been poorly received. Issues of privacy and security in such a model have been of concern.In the European Union (EU), a new directly binding instrument, a regulation of the European Parliament and of the council, was passed in 2016 to go into effect in 2018 to protect the processing of personal data, including that for purposes of health care, the General Data Protection Regulation. Threats to health care information can be categorized under three headings: Human threats, such as employees or hackers Natural and environmental threats, such as earthquakes, hurricanes and fires. Technology failures, such as a system crashingThese threats can either be internal, external, intentional and unintentional.


In real analysis, the symbol ∞ {\displaystyle \infty } , called "infinity", is used to denote an unbounded limit. The notation x → ∞ {\displaystyle x\rightarrow \infty } means that x {\displaystyle x} increases without bound, and x → − ∞ {\displaystyle x\to -\infty } means that x {\displaystyle x} decreases without bound. For example, if f ( t ) ≥ 0 {\displaystyle f(t)\geq 0} for every t {\displaystyle t} , then ∫ a b f ( t ) d t = ∞ {\displaystyle \int _{a}^{b}f(t)\,dt=\infty } means that f ( t ) {\displaystyle f(t)} does not bound a finite area from a {\displaystyle a} to b . {\displaystyle b.}


In terms of length, word count is typically anywhere from 1,000 to 4,000 for short stories; however, some have 15,000 words and are still classed as short stories. Stories of fewer than 1,000 words are sometimes referred to as "short short stories", or "flash fiction".Short stories have no set length. In terms of word count, there is no official demarcation between an anecdote, a short story, and a novel. Rather, the form's parameters are given by the rhetorical and practical context in which a given story is produced and considered so that what constitutes a short story may differ between genres, countries, eras, and commentators. Like the novel, the short story's predominant shape reflects the demands of the available markets for publication, and the evolution of the form seems closely tied to the evolution of the publishing industry and the submission guidelines of its constituent houses.As a point of reference for the genre writer, the Science Fiction and Fantasy Writers of America define short story length in the Nebula Awards for science fiction submission guidelines as having fewer than 7,500 words.


In settings with deadlines, it is possible that, if the job is completed by the deadline, there is a profit pj. Otherwise, there is no profit. The goal is to maximize the profit. Single-machine scheduling with deadlines is NP-hard; Sahni presents both exact exponential-time algorithms and a polynomial-time approximation algorithm.


Pr  > 2 3 , if f ( x , y ) = 0 {\displaystyle \Pr>{\frac {2}{3}},{\textrm {if}}\,f(x,y)=0} Pr  > 2 3 , if f ( x , y ) = 1 {\displaystyle \Pr>{\frac {2}{3}},{\textrm {if}}\,f(x,y)=1} A randomized protocol is a deterministic protocol that uses an extra random string in addition to its normal input. There are two models for this: a public string is a random string that is known by both parties beforehand, while a private string is generated by one party and must be communicated to the other party. A theorem presented below shows that any public string protocol can be simulated by a private string protocol that uses O(log n) additional bits compared to the original.


Using the relation between ψ 2 m {\displaystyle \psi _{2m}} and ψ 2 m + 1 {\displaystyle \psi _{2m+1}} , along with the equation of the curve, the functions ψ n 2 {\displaystyle \psi _{n}^{2}} , ψ 2 n y , ψ 2 n + 1 {\displaystyle {\frac {\psi _{2n}}{y}},\psi _{2n+1}} , ϕ n {\displaystyle \phi _{n}} are all in K  {\displaystyle K} . Let p > 3 {\displaystyle p>3} be prime and let E: y 2 = x 3 + A x + B {\displaystyle E:y^{2}=x^{3}+Ax+B} be an elliptic curve over the finite field F p {\displaystyle \mathbb {F} _{p}} , i.e., A , B ∈ F p {\displaystyle A,B\in \mathbb {F} _{p}} . The ℓ {\displaystyle \ell } -torsion group of E {\displaystyle E} over F ¯ p {\displaystyle {\bar {\mathbb {F} }}_{p}} is isomorphic to Z / ℓ × Z / ℓ {\displaystyle \mathbb {Z} /\ell \times \mathbb {Z} /\ell } if ℓ ≠ p {\displaystyle \ell \neq p} , and to Z / ℓ {\displaystyle \mathbb {Z} /\ell } or { 0 } {\displaystyle \{0\}} if ℓ = p {\displaystyle \ell =p} .


In the context of IBM mainframe computers in the S/360 line, a data set (IBM preferred) or dataset is a computer file having a record organization. Use of this term began with, e.g., DOS/360, OS/360, and is still used by their successors, including the current z/OS. Documentation for these systems historically preferred this term rather than file. A data set is typically stored on a direct access storage device (DASD) or magnetic tape, however unit record devices, such as punch card readers, card punches, line printers and page printers can provide input/output (I/O) for a data set (file).Data sets are not unstructured streams of bytes, but rather are organized in various logical record and block structures determined by the DSORG (data set organization), RECFM (record format), and other parameters.


Tasks such as network access and file access are often heavily intertwined with the distinctive implementations of each platform. The java.net and java.io libraries implement an abstraction layer in native OS code, then provide a standard interface for the Java applications to perform those tasks. Finally, when some underlying platform does not support all of the features a Java application expects, the class libraries work to gracefully handle the absent components, either by emulation to provide a substitute, or at least by providing a consistent way to check for the presence of a specific feature.


* file exists within a given directory, the web server may be configured to provide an automatically generated listing of the files within the directory instead. With the Apache web server, for example, this behavior is provided by the mod_autoindex module and controlled by the Options +Indexes directive in the web server configuration files. These automated directory listings are sometimes a security risk because they enumerate sensitive files which may not be intended for public access, in a process known as a directory indexing attack. Such a security misconfiguration may also assist in other attacks, such as a path or directory traversal attack.


As such, it is important for people and organizations to need know that the information and data they are storing, using, or sending over computer networks or storing on computer systems is secure. However, developers of software and hardware are faced with many challenges in developing a system that can be both user friendly, accessible 24/7 on almost any device and be truly secure. Security leaks happen, even to individuals and organizations that have security measures in place to protect their data and information (e.g., firewalls, encryption, strong passwords).


Therefore, a deaf individual may misinterpret or become confused about the meaning of written text that is based on a spoken language.Researchers Zhao, et al. (2000), developed a prototype called TEAM (translation from English to ASL by machine) that completed English to American Sign Language (ASL) translations. The program would first analyze the syntactic, grammatical, and morphological aspects of the English text. Following this step, the program accessed a sign synthesizer, which acted as a dictionary for ASL. This synthesizer housed the process one must follow to complete ASL signs, as well as the meanings of these signs. Once the entire text is analyzed and the signs necessary to complete the translation are located in the synthesizer, a computer generated human appeared and would use ASL to sign the English text to the user.


In some cases, however, optimization relies on using more elaborate algorithms, making use of "special cases" and special "tricks" and performing complex trade-offs. A "fully optimized" program might be more difficult to comprehend and hence may contain more faults than unoptimized versions. Beyond eliminating obvious antipatterns, some code level optimizations decrease maintainability.


The CPU excels in serial processing and accurate mathematical computation. The overarching goal of the requirements management effort for a software project would thus be to make sure the work being automated gets assigned to the proper processor. For instance, “Don’t make the human remember where she is in the interface. Make the interface report the human’s location in the system at all times.” Or “Don’t make the human enter the same data in two screens. Make the system store the data and fill in the second screen as needed.” The deliverable from the Feasibility stage is the budget and schedule for the project.


In mathematics, specifically in the theory of generalized functions, the limit of a sequence of distributions is the distribution that sequence approaches. The distance, suitably quantified, to the limiting distribution can be made arbitrarily small by selecting a distribution sufficiently far along the sequence. This notion generalizes a limit of a sequence of functions; a limit as a distribution may exist when a limit of functions does not. The notion is a part of distributional calculus, a generalized form of calculus that is based on the notion of distributions, as opposed to classical calculus, which is based on the narrower concept of functions.


Let us suppose we have a function s i z e ( f ) {\displaystyle size(f)} that can measure the complexity of f {\displaystyle f} . Let Oracle ( x ) {\displaystyle {\text{Oracle}}(x)} be an oracle that, whenever called, returns an example x {\displaystyle x} and its correct label f ( x ) {\displaystyle f(x)} . When no noise corrupts the data, we can define learning in the Valiant setting:Definition: We say that f {\displaystyle f} is efficiently learnable using H {\displaystyle {\mathcal {H}}} in the Valiant setting if there exists a learning algorithm A {\displaystyle {\mathcal {A}}} that has access to Oracle ( x ) {\displaystyle {\text{Oracle}}(x)} and a polynomial p ( ⋅ , ⋅ , ⋅ , ⋅ ) {\displaystyle p(\cdot ,\cdot ,\cdot ,\cdot )} such that for any 0 < ε ≤ 1 {\displaystyle 0<\varepsilon \leq 1} and 0 < δ ≤ 1 {\displaystyle 0<\delta \leq 1} it outputs, in a number of calls to the oracle bounded by p ( 1 ε , 1 δ , n , size ( f ) ) {\displaystyle p\left({\frac {1}{\varepsilon }},{\frac {1}{\delta }},n,{\text{size}}(f)\right)} , a function h ∈ H {\displaystyle h\in {\mathcal {H}}} that satisfies with probability at least 1 − δ {\displaystyle 1-\delta } the condition error ( h ) ≤ ε {\displaystyle {\text{error}}(h)\leq \varepsilon } . In the following we will define learnability of f {\displaystyle f} when data have suffered some modification.


In the United States, flood openings are used to provide for the automatic equalization of hydrostatic pressure on either side of a wall. Building codes usually require the installation of flood openings in the walls of structures located in A-type flood zones recognized by the Federal Emergency Management Agency. Various agencies in the United States define necessary characteristics for flood openings.


Though the most straightforward application is to finite sets (such as pigeons and boxes), it is also used with infinite sets that cannot be put into one-to-one correspondence. To do so requires the formal statement of the pigeonhole principle, which is "there does not exist an injective function whose codomain is smaller than its domain". Advanced mathematical proofs like Siegel's lemma build upon this more general concept.


In number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, states that Every odd number greater than 5 can be expressed as the sum of three primes. (A prime may be used more than once in the same sum. )This conjecture is called "weak" because if Goldbach's strong conjecture (concerning sums of two primes) is proven, then this would also be true.


label for var ( list ) block label for var ( list ) block continue block label foreach var ( list ) block label foreach var ( list ) block continue block In foreach, var is a scalar variable that defaults to $_ if omitted. For each element of list, var is aliased to the element, and the loop body is executed once. The keywords for and foreach are synonyms and are always interchangeable.


In the above situation, one can prove that the value of any flow through a network is less than or equal to the capacity of any s-t cut, and that furthermore a flow with maximal value and a cut with minimal capacity exist. The main theorem links the maximum flow value with the minimum cut capacity of the network. Max-flow min-cut theorem. The maximum value of an s-t flow is equal to the minimum capacity over all s-t cuts.


For the case of a system of particles, the space V consists of functions called wave functions or state vectors. In the case of transformation laws in quantum mechanics, the requisite automorphisms are unitary (or antiunitary) linear transformations of the Hilbert space V. Under Galilean relativity or special relativity, the mathematics of frames of reference is particularly simple, considerably restricting the set of physically meaningful observables. In quantum mechanics, measurement of observables exhibits some seemingly unintuitive properties.


In overlapping domain decomposition methods, the subdomains overlap by more than the interface. Overlapping domain decomposition methods include the Schwarz alternating method and the additive Schwarz method. Many domain decomposition methods can be written and analyzed as a special case of the abstract additive Schwarz method.


It is therefore up to an opponent to prove that they don't exist. Such hypotheses can be used to expose the pretensions of dogmatism. Kant explicitly praises Hume on his critique of religion for being beyond the field of natural science.


If not, the function is evaluated, and another entry is added to the lookup table for reuse. Lazy evaluation is difficult to combine with imperative features such as exception handling and input/output, because the order of operations becomes indeterminate. The opposite of lazy evaluation is eager evaluation, sometimes known as strict evaluation. Eager evaluation is the evaluation strategy employed in most programming languages.


Leveraging the ANDF format, the DWARF standardized debugging format, and the OMI protocol for communicating with target board debug monitors, SCORE was able to provide a common building and debugging environment for real-time application developers. Support for Embedded C++ was added to SCORE in 2003, by which time it could integrate with a variety of target board scenarios on Intel x86 and Power PC processors. The C and Embedded C++ compilers for ANDF came from a licensing arrangement for the TenDRA Compiler (later DDC-I became the maintainer of those compilers). Subsequently, Ada 95 support for the older 1750A and TMS320C4x processors was added to SCORE.


Scaleform provides GFx for high performance Flash UI and high-quality video playback, and an Input Method Editor (IME) add-on for in-game Asian chat support. Other middleware is used for performance optimisation—for example 'Simplygon' helps to optimise and generate level of detail meshes, and 'Umbra' adds occlusion culling optimisations to 3d graphics. Some middleware contains full source code, others just provide an API reference for a compiled binary library. Some middleware programs can be licensed either way, usually for a higher fee for full source code.


However, this conclusion is not a mathematical certainty; there exist undirected graphs (such as the graph formed by removing a single edge from a large complete graph) that are unlikely to arise as social networks but in which most vertices have higher degree than the average of their neighbors' degrees. The Friendship Paradox may be restated in graph theory terms as "the average degree of a randomly selected node in a network is less than the average degree of neighbors of a randomly selected node", but this leaves unspecified the exact mechanism of averaging (i.e., macro vs micro averaging). Let G = ( V , E ) {\displaystyle G=(V,E)} be an undirected graph with | V | = N {\displaystyle |V|=N} and | E | = M {\displaystyle |E|=M} , having no isolated nodes.


A natural example is strings with concatenation as the binary operation, and the empty string as the identity element. Restricting to non-empty strings gives an example of a semigroup that is not a monoid. Positive integers with addition form a commutative semigroup that is not a monoid, whereas the non-negative integers do form a monoid.


In the 1980s IBM researcher Harlan Mills oversaw the development of the COBOL Structuring Facility, which applied a structuring algorithm to COBOL code. Mills's transformation involved the following steps for each procedure. Identify the basic blocks in the procedure. Assign a unique label to each block's entry path, and label each block's exit paths with the labels of the entry paths they connect to.


Fine-grained multithreading—such as in a barrel processor—issues instructions for different threads after every cycle, while coarse-grained multithreading only switches to issue instructions from another thread when the current executing thread causes some long latency events (like page fault etc.). Coarse-grain multithreading is more common for less context switch between threads. For example, Intel's Montecito processor uses coarse-grained multithreading, while Sun's UltraSPARC T1 uses fine-grained multithreading.


In number theory, Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers Q {\displaystyle \mathbb {Q} } is equivalent to either the usual real absolute value or a p-adic absolute value.


In mathematics, the plus sign "+" almost invariably indicates an operation that satisfies the axioms assigned to addition in the type of algebraic structure that is known as a field. For boolean algebra, this means that the logical operation signified by "+" is not the same as the inclusive disjunction signified by "∨" but is actually equivalent to the logical inequality operator signified by "≠", or what amounts to the same thing, the exclusive disjunction signified by "XOR" or "⊕". Naturally, these variations in usage have caused some failures to communicate between mathematicians and switching engineers over the years. At any rate, one has the following array of corresponding forms for the symbols associated with logical inequality: x + y x ≢ y J x y x X O R y x ≠ y {\displaystyle {\begin{aligned}x&+y&x&\not \equiv y&Jxy\\x&\mathrm {~XOR~} y&x&\neq y\end{aligned}}} This explains why "EQ" is often called "XNOR" in the combinational logic of circuit engineers, since it is the negation of the XOR operation; "NXOR" is a less commonly used alternative. Another rationalization of the admittedly circuitous name "XNOR" is that one begins with the "both false" operator NOR and then adds the eXception "or both true".


In the decision version of problems, the problem of producing an integer flow satisfying all demands is NP-complete, even for only two commodities and unit capacities (making the problem strongly NP-complete in this case). If fractional flows are allowed, the problem can be solved in polynomial time through linear programming, or through (typically much faster) fully polynomial time approximation schemes.


The notion of counting may be extended to them in the sense of establishing (the existence of) a bijection with some well-understood set. For instance, if a set can be brought into bijection with the set of all natural numbers, then it is called "countably infinite." This kind of counting differs in a fundamental way from counting of finite sets, in that adding new elements to a set does not necessarily increase its size, because the possibility of a bijection with the original set is not excluded.


In mathematical logic and computer science, the calculus of constructions (CoC) is a type theory created by Thierry Coquand. It can serve as both a typed programming language and as constructive foundation for mathematics. For this second reason, the CoC and its variants have been the basis for Coq and other proof assistants. Some of its variants include the calculus of inductive constructions (which adds inductive types), the calculus of (co)inductive constructions (which adds coinduction), and the predicative calculus of inductive constructions (which removes some impredicativity).


In mathematics graph theory a process graph or P-graph is a directed bipartite graph used in workflow modeling.


An attacker would then be able to run system commands using the application's privileges. Texas Instruments calculators (particularly the TI-85 and TI-82) were originally designed to use only interpreted programs written in dialects of TI-BASIC; however, after users discovered bugs that could be exploited to allow native Z-80 code to run on the calculator hardware, TI released programming data to support third-party development. (This did not carry on to the ARM-based TI-Nspire, for which jailbreaks using Ndless have been found but are still actively fought against by Texas Instruments.) Some versions of the iPhone allow an unauthorised user to access the phone while it is locked.


In physics, especially quantum field theory, regularization is a method of modifying observables which have singularities in order to make them finite by the introduction of a suitable parameter called the regulator. The regulator, also known as a "cutoff", models our lack of knowledge about physics at unobserved scales (e.g. scales of small size or large energy levels). It compensates for (and requires) the possibility that "new physics" may be discovered at those scales which the present theory is unable to model, while enabling the current theory to give accurate predictions as an "effective theory" within its intended scale of use.


The main categories for non-iid data can be summarized as follows: Covariate shift: local nodes may store examples that have different statistical distributions compared to other nodes. An example occurs in natural language processing datasets where people typically write the same digits/letters with different stroke widths or slants. Prior probability shift: local nodes may store labels that have different statistical distributions compared to other nodes.


Non-redundant information is most often obtained through contacts in different clusters. When two separate clusters possess non-redundant information, there is said to be a structural hole between them. Thus, a network that bridges structural holes will provide network benefits that are in some degree additive, rather than overlapping.


In real analysis, the following result is called Young's convolution inequality:Suppose f {\displaystyle f} is in the Lebesgue space L p ( R d ) {\displaystyle L^{p}(\mathbb {R} ^{d})} and g {\displaystyle g} is in L q ( R d ) {\displaystyle L^{q}(\mathbb {R} ^{d})} and with 1 ≤ p , q , r ≤ ∞ . {\displaystyle 1\leq p,q,r\leq \infty .} Then Here the star denotes convolution, L p {\displaystyle L^{p}} is Lebesgue space, and denotes the usual L p {\displaystyle L^{p}} norm. Equivalently, if p , q , r ≥ 1 {\displaystyle p,q,r\geq 1} and 1 p + 1 q + 1 r = 2 {\textstyle {\frac {1}{p}}+{\frac {1}{q}}+{\frac {1}{r}}=2} then


In mathematics, Coxeter matroids are generalization of matroids depending on a choice of a Coxeter group W and a parabolic subgroup P. Ordinary matroids correspond to the case when P is a maximal parabolic subgroup of a symmetric group W. They were introduced by Gelfand and Serganova (1987, 1987b), who named them after H. S. M. Coxeter. Borovik, Gelfand & White (2003) give a detailed account of Coxeter matroids.


In the context of telecommunications, a terminal is a device which ends a telecommunications link and is the point at which a signal enters or leaves a network. Examples of terminal equipment include telephones, fax machines, computer terminals, printers and workstations. An end instrument is a piece of equipment connected to the wires at the end of a telecommunications link. In telephony, this is usually a telephone connected to a local loop. End instruments that relate to data terminal equipment include printers, computers, barcode readers, automated teller machines (ATMs) and the console ports of routers.


In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset. The rank of A determines the size of the largest free abelian group contained in A. If A is torsion-free then it embeds into a vector space over the rational numbers of dimension rank A. For finitely generated abelian groups, rank is a strong invariant and every such group is determined up to isomorphism by its rank and torsion subgroup. Torsion-free abelian groups of rank 1 have been completely classified. However, the theory of abelian groups of higher rank is more involved. The term rank has a different meaning in the context of elementary abelian groups.


Rather than adopting an updated set of primary colors, proponents of split-primary theory explain this lack of chroma by the purported presence of chemical impurities, small amounts of other colors, in the paints, or biases away from the ideal primary toward one or the other of the adjacent colors. Every red paint, for example, is said to be tainted with, or biased toward, either blue or yellow, every blue paint toward either red or green, and every yellow toward either green or orange. These biases are said to result in mixtures that contain sets of complementary colors, darkening the resulting color.


Languages such as Japanese and Chinese form possessive constructions with nouns using possessive particles, in the same way as described for pronouns above. An example from Japanese is neko no iro ("the cat's color"), where neko means "cat", no is the particle, and iro means "color".


In the (n2 − 1) -dimensional adjoint representation, the generators are represented by (n2 − 1) × (n2 − 1) matrices, whose elements are defined by the structure constants themselves: ( T a ) j k = − i f a j k . {\displaystyle \left(T_{a}\right)_{jk}=-if_{ajk}.}


8. Specifying relevant conditions to which the concept applies, as a procedure (computer programming, formal concept analysis). 9.


Output: prefix sum ⨁ 0 ≤ j ≤ i m j {\displaystyle \bigoplus _{0\leq j\leq i}m_{j}} of processor i {\displaystyle i} . x := m i {\displaystyle x:=m_{i}} σ := m i {\displaystyle \sigma :=m_{i}} for 0 ≤ k ≤ d − 1 {\displaystyle 0\leq k\leq d-1} do y := i XOR 2 k {\displaystyle y:=i{\text{ XOR }}2^{k}} Send σ {\displaystyle \sigma } to y {\displaystyle y} Receive m {\displaystyle m} from y {\displaystyle y} σ := σ ⊕ m {\displaystyle \sigma :=\sigma \oplus m} if bit k {\displaystyle k} in i {\displaystyle i} is set then x := x ⊕ m {\displaystyle x:=x\oplus m} endfor The algorithm works as follows.


In mathematics a group is a set together with a binary operation on the set called multiplication that obeys the group axioms. The axiom of choice is an axiom of ZFC set theory which in one form states that every set can be wellordered. In ZF set theory, i.e. ZFC without the axiom of choice, the following statements are equivalent: For every nonempty set X there exists a binary operation • such that (X, •) is a group. The axiom of choice is true.


The distance of products to each other indicate either how different they are. The dimensions must be labelled by the researcher. This requires subjective judgement and is often very challenging. See perceptual mapping.


In telecommunications, a terminal adapter or TA acts as an interface between a terminal device, such as a computer or telephone, and a communications network (typically an integrated services digital network).


In some applications it may be acceptable to retrieve a "good guess" of the nearest neighbor. In those cases, we can use an algorithm which doesn't guarantee to return the actual nearest neighbor in every case, in return for improved speed or memory savings. Often such an algorithm will find the nearest neighbor in a majority of cases, but this depends strongly on the dataset being queried. Algorithms that support the approximate nearest neighbor search include locality-sensitive hashing, best bin first and balanced box-decomposition tree based search.


In mathematics, a Rosati involution, named after Carlo Rosati, is an involution of the rational endomorphism ring of an abelian variety induced by a polarization. Let A {\displaystyle A} be an abelian variety, let A ^ = P i c 0 ( A ) {\displaystyle {\hat {A}}=\mathrm {Pic} ^{0}(A)} be the dual abelian variety, and for a ∈ A {\displaystyle a\in A} , let T a: A → A {\displaystyle T_{a}:A\to A} be the translation-by- a {\displaystyle a} map, T a ( x ) = x + a {\displaystyle T_{a}(x)=x+a} . Then each divisor D {\displaystyle D} on A {\displaystyle A} defines a map ϕ D: A → A ^ {\displaystyle \phi _{D}:A\to {\hat {A}}} via ϕ D ( a ) =  {\displaystyle \phi _{D}(a)=} . The map ϕ D {\displaystyle \phi _{D}} is a polarization if D {\displaystyle D} is ample.


In software engineering, software system safety optimizes system safety in the design, development, use, and maintenance of software systems and their integration with safety-critical hardware systems in an operational environment.


In mathematics, a graph partition is the reduction of a graph to a smaller graph by partitioning its set of nodes into mutually exclusive groups. Edges of the original graph that cross between the groups will produce edges in the partitioned graph. If the number of resulting edges is small compared to the original graph, then the partitioned graph may be better suited for analysis and problem-solving than the original. Finding a partition that simplifies graph analysis is a hard problem, but one that has applications to scientific computing, VLSI circuit design, and task scheduling in multiprocessor computers, among others.


As with Racter, the question is how much the programmer filtered the output of the program, keeping only the occasional interesting output. Also, mathematics being a very specialized domain, it is doubtful whether the techniques used can be abstracted to general cognition. Another mathematical program, called Geometry, was celebrated for making an insightful discovery of an original proof that an isosceles triangle has equal base angles.


In simple terms, a convex function refers to a function whose graph is shaped like a cup ∪ {\displaystyle \cup } (or a straight line like a linear function), while a concave function's graph is shaped like a cap ∩ {\displaystyle \cap } . Convex functions play an important role in many areas of mathematics.


In philately, an invert error occurs when part of a stamp is printed upside-down. Inverts are perhaps the most spectacular of postage stamp errors, not only because of their striking visual appearance, but because some are quite rare, and highly valued by stamp collectors.


In mathematics and computer programming, the order of operations is a collection of rules that reflect conventions about which operations to perform first in order to evaluate a given mathematical expression. These rules are formalized with a ranking of the operators. The rank of an operator is called its precedence, and an operation with a higher precedence is performed before operations with lower precedence. Calculators generally perform operations with the same precedence from left to right, but some programming languages and calculators adopt different conventions.


Which is succeeded by an account of the fellowship examination, in 1823 (G. and W. B. Whittaker, London, 1823)Question 1249 in the Gentleman's Diary; or Mathematical Repository for 1829 (so appearing in late 1828) takes up the theme, with solutions appearing in the issue for the following year. One of the solvers, T. S. Davies then generalized the result in Question 1265 that year, presenting his own solution the following year, drawing on a paper he had already contributed to the Philosophical Magazine in 1826. There are no cross-references in this material to that described above.


In software engineering, the use of models is an alternative to more common code-based development techniques. A model always conforms to a unique metamodel. One of the currently most active branch of Model Driven Engineering is the approach named model-driven architecture proposed by OMG.


In mathematics, a harmonic divisor number, or Ore number (named after Øystein Ore who defined it in 1948), is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are: 1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190 (sequence A001599 in the OEIS).


The null hypothesis may be true, whereas we reject H0. On the other hand, the alternative hypothesis H1 may be true, whereas we do not reject H0. Two types of error are distinguished: type I error and type II error.


In numerical analysis, interpolative decomposition (ID) factors a matrix as the product of two matrices, one of which contains selected columns from the original matrix, and the other of which has a subset of columns consisting of the identity matrix and all its values are no greater than 2 in absolute value.