In some cases, the specification of the arithmetic circuit is not given to the PIT solver, and the PIT solver can only input values into a "black box" that implements the circuit, and then analyze the output. Note that the solutions below assume that any operation (such as multiplication) in the given field takes constant time; further, all black-box algorithms below assume the size of the field is larger than the degree of the polynomial. The Schwartz–Zippel algorithm provides a practical probabilistic solution, by simply randomly testing inputs and checking whether the output is zero.


Then whatever A does (or fails to do) today will make no difference; similarly, whatever B does (or fails to do) today will make no difference: the outcome is already settled. Or again, suppose 'A wins' is today false. Then no matter what A does today (or fails to do), it will make no difference; similarly, no matter what B does (or fails to do), it will make no difference: the outcome is already settled.


The external cooling system was considerably larger than the machine itself. The electronic components were likewise improved over previous designs. The main CPU circuits moved to ECL-based logic, enabling a clock speed increase to 125 MHz (8 ns cycle time) from the 7600's 36.4 MHz (27.5 ns cycle time) an increase of about four times.


There are several approaches to platform virtualization. Examples of virtualization use cases: Running one or more applications that are not supported by the host OS: A virtual machine running the required guest OS could permit the desired applications to run, without altering the host OS. Evaluating an alternate operating system: The new OS could be run within a VM, without altering the host OS.


In mathematics, Gödel's speed-up theorem, proved by Gödel (1936), shows that there are theorems whose proofs can be drastically shortened by working in more powerful axiomatic systems. Kurt Gödel showed how to find explicit examples of statements in formal systems that are provable in that system but whose shortest proof is unimaginably long. For example, the statement: "This statement cannot be proved in Peano arithmetic in fewer than a googolplex symbols"is provable in Peano arithmetic (PA) but the shortest proof has at least a googolplex symbols, by an argument similar to the proof of Gödel's first incompleteness theorem: If PA is consistent, then it cannot prove the statement in fewer than a googolplex symbols, because the existence of such a proof would itself be a theorem of PA, a contradiction. But simply enumerating all strings of length up to a googolplex and checking that each such string is not a proof (in PA) of the statement, yields a proof of the statement (which is necessarily longer than a googolplex symbols).


In the branch of mathematics called graph theory, the strength of an undirected graph corresponds to the minimum ratio edges removed/components created in a decomposition of the graph in question. It is a method to compute partitions of the set of vertices and detect zones of high concentration of edges, and is analogous to graph toughness which is defined similarly for vertex removal.


However, the form of energy keeps changing. One may wonder if there is any such law for the conservation of information. In the classical world, information can be copied and deleted perfectly.


In the context of linear programming, one can think of any linear program as a covering problem if the coefficients in the constraint matrix, the objective function, and right-hand side are nonnegative. More precisely, consider the following general integer linear program: Such an integer linear program is called a covering problem if a j i , b j , c i ≥ 0 {\displaystyle a_{ji},b_{j},c_{i}\geq 0} for all i = 1 , … , n {\displaystyle i=1,\dots ,n} and j = 1 , … , m {\displaystyle j=1,\dots ,m} . Intuition: Assume having n {\displaystyle n} types of object and each object of type i {\displaystyle i} has an associated cost of c i {\displaystyle c_{i}} . The number x i {\displaystyle x_{i}} indicates how many objects of type i {\displaystyle i} we buy. If the constraints A x ≥ b {\displaystyle A\mathbf {x} \geq \mathbf {b} } are satisfied, it is said that x {\displaystyle \mathbf {x} } is a covering (the structures that are covered depend on the combinatorial context). Finally, an optimal solution to the above integer linear program is a covering of minimal cost.


Many developers feel that CW's are not efficient because of the amount of time they take and the time pressures that they are facing. A design team spends their time trying to resolve the problem, during the CW instead of after the results have been formulated. Evaluation time is spent re-designing, this inhibits the effectiveness of the walkthrough and leads to lengthy design discussions.


9. The utility and advantages of the patent property over the old modes or devices, if any, that had been used for working out similar results. 10.


Formal theories cannot dispense with primitive notions, under pain of infinite regress (per the regress problem). For example, in contemporary geometry, point, line, and contains are some primitive notions. Instead of attempting to define them, their interplay is ruled (in Hilbert's axiom system) by axioms like "For every two points there exists a line that contains them both".


In photography, reversal film or slide film is a type of photographic film that produces a positive image on a transparent base. Instead of negatives and prints, reversal film is processed to produce transparencies or diapositives (abbreviated as "diafilm" or "dia" in some languages like German or Hungarian). Reversal film is produced in various sizes, from 35 mm to roll film to 8×10 inch sheet film.


Although the recurrence is easily observed, it eventually became apparent that over much, much longer time periods, the system does eventually thermalize. Multiple competing theories have been proposed to explain the behavior of the system, and it remains a topic of active research. The original intent was to find a physics problem worthy of numerical simulation on the then-new MANIAC computer. Fermi felt that thermalization would pose such a challenge. As such, it represents one of the earliest uses of digital computers in mathematical research; simultaneously, the unexpected results launched the study of nonlinear systems.


This algorithm is implemented by RivEX, an ESRI ArcGIS 10.7 tool. The input to their algorithm is a network of the centre lines of the bodies of water, represented as arcs (or edges) joined at nodes. Lake boundaries and river banks should not be used as arcs, as these will generally form a non-tree network with an incorrect topology. Alternative stream ordering systems have been developed by Shreve and Hodgkinson et al. A statistical comparison of Strahler and Shreve systems, together with an analysis of stream/link lengths, is given by Smart.


In the MaxClique algorithm the approximate coloring algorithm is used to obtain set of color classes C. The ColorSort algorithm is an improved algorithm of the approximate coloring algorithm. In the approximate coloring algorithm vertices are colored one by one in the same order as they appear in a set of candidate vertices R so that if the next vertex p is non-adjacent to all vertices in the some color class it is added to this class and if p is adjacent to at least one vertex in every one of existing color classes it is put into a new color class. The MaxClique algorithm returns vertices R ordered by their colors. By looking at the MaxClique algorithm it is clear that vertices v ∈ R with colors C(v) < |Qmax| − |Q| + 1 will never be added to the current clique Q. Therefore, sorting those vertices by color is of no use to MaxClique algorithm.


Notice that if C o u t ( m ) = ( c 1 , ⋯ , c N ) {\displaystyle C_{out}(m)=(c_{1},\cdots ,c_{N})} , then C ∗ ( m ) = ( C i n 1 ( c 1 ) , ⋯ , C i n N ( c N ) ) {\displaystyle C^{*}(m)=(C_{in}^{1}(c_{1}),\cdots ,C_{in}^{N}(c_{N}))} . So for the lower bound Δ ( C ∗ ( m 1 ) , C ∗ ( m 2 ) ) {\displaystyle \Delta (C^{*}(m_{1}),C^{*}(m_{2}))} , we need to take into account the distance of C i n 1 , ⋯ , C i n N . {\displaystyle C_{in}^{1},\cdots ,C_{in}^{N}.}


In some programs, the required amount of memory depends on what the user may enter. In such cases the programmer needs to allocate memory dynamically. This is done by allocating memory at the heap rather than on the stack, where variables usually are stored (although variables can also be stored in the CPU registers). Dynamic memory allocation can only be made through pointers, and names – like with common variables – cannot be given.


In programming, the rule of least power is a design principle that "suggests choosing the least powerful  language suitable for a given purpose". Stated alternatively, given a choice among computer languages, classes of which range from descriptive (or declarative) to procedural, the less procedural, more descriptive the language one chooses, the more one can do with the data stored in that language. This rule is an application of the principle of least privilege to protocol design. The Rule of Least Power is an example in context of the centuries older principle known as Occam's razor in philosophy. In particular, arguments for and against the Rule of Least Power are subject to the same analysis as for Occam's razor.


R6RS specifies a more sophisticated transformation system, syntax-case, which has been available as a language extension to R5RS Scheme for some time. Invocations of macros and procedures bear a close resemblance—both are s-expressions—but they are treated differently. When the compiler encounters an s-expression in the program, it first checks to see if the symbol is defined as a syntactic keyword within the current lexical scope.


In set theory, a branch of mathematics, a set A {\displaystyle A} is called transitive if either of the following equivalent conditions hold: whenever x ∈ A {\displaystyle x\in A} , and y ∈ x {\displaystyle y\in x} , then y ∈ A {\displaystyle y\in A} . whenever x ∈ A {\displaystyle x\in A} , and x {\displaystyle x} is not an urelement, then x {\displaystyle x} is a subset of A {\displaystyle A} .Similarly, a class M {\displaystyle M} is transitive if every element of M {\displaystyle M} is a subset of M {\displaystyle M} .


{\displaystyle {\boldsymbol {\hat {\beta }}}=(X^{\mathrm {T} }X)^{-1}X^{\mathrm {T} }\mathbf {y} .} The matrix ( X T X ) − 1 X T {\displaystyle (X^{\mathrm {T} }X)^{-1}X^{\mathrm {T} }} is known as the Moore–Penrose pseudoinverse of X. The use of the matrix inverse in this formula requires that X is of full rank, i.e. there is not perfect multicollinearity among different explanatory variables (i.e. no explanatory variable can be perfectly predicted from the others). In such cases, the singular value decomposition can be used to compute the pseudoinverse.


In statistics, the theory of minimum norm quadratic unbiased estimation (MINQUE) was developed by C. R. Rao. Its application was originally to the problem of heteroscedasticity and the estimation of variance components in random effects models. The theory involves three stages: defining a general class of potential estimators as quadratic functions of the observed data, where the estimators relate to a vector of model parameters; specifying certain constraints on the desired properties of the estimators, such as unbiasedness; choosing the optimal estimator by minimising a "norm" which measures the size of the covariance matrix of the estimators. == References ==


In other cases, a word may be usable in multiple genders indifferently. For example, in Bulgarian the word пу̀стош, (pustosh, "wilderness") may be either masculine (definite form пу̀стоша, pustoshə) or feminine (definite form пустошта̀, pustoshta) without any change in meaning and no preference in usage. In Norwegian, many nouns can be either feminine or masculine according to the dialect, level of formality or whim of the speaker/writer. Even the two written forms of the language have many nouns whose gender is optional.


{\displaystyle \forall x\,A.} And as a rule of inference it is from ⊢ ∀ x A {\displaystyle \vdash \forall xA} infer ⊢ A { x ↦ a } . {\displaystyle \vdash A\{x\mapsto a\}.} Irving Copi noted that universal instantiation "...follows from variants of rules for 'natural deduction', which were devised independently by Gerhard Gentzen and Stanisław Jaśkowski in 1934."


A source code repository is a place where large amounts of source code are kept, either publicly or privately. Source code repositories are used most basically for backups and versioning, and on multi-developer projects to handle various source code versions and to provide aid in resolving conflicts that arise from developers submitting overlapping modifications. Subversion, Git and Mercurial are examples of popular tools used to handle this workflow, which are common in open source projects. For smaller projects, its code may be kept as a non-managed set of files (even the Linux kernel was maintained as a set of files for many years).


In statistics, the concept of the shape of a probability distribution arises in questions of finding an appropriate distribution to use to model the statistical properties of a population, given a sample from that population. The shape of a distribution may be considered either descriptively, using terms such as "J-shaped", or numerically, using quantitative measures such as skewness and kurtosis. Considerations of the shape of a distribution arise in statistical data analysis, where simple quantitative descriptive statistics and plotting techniques such as histograms can lead on to the selection of a particular family of distributions for modelling purposes.


It was especially noted that it received a score of 45% from the online review aggregator Rotten Tomatoes—the worst score received by any Best Picture nominee in the site's history.Some critics think the term is overused. "If I were Oscar-blogging this year,< a long rant about the empty foolishness of the phrase 'Oscar bait' would be on the way," tweeted film historian Mark Harris in early December 2012, before some of that year's likely Oscar nominees had even been released. Four years later, he explained his objections in a conversation with fellow Vulture editor Kyle Buchanan.


The first figure of the syllogism (see term logic for an outline of syllogistic theory) is best adapted to demonstration, because it affords conclusions universally affirmative. This figure is commonly used by mathematicians. The demonstration of an affirmative proposition is preferable to that of a negative; the demonstration of a universal to that of a particular; and direct demonstration to a reductio ad absurdum.


In the rapid foundational changes of the 1950s Weil's approach became obsolete. In scheme theory, though, from 1957, generic points returned: this time à la Zariski. For example for R a discrete valuation ring, Spec(R) consists of two points, a generic point (coming from the prime ideal {0}) and a closed point or special point coming from the unique maximal ideal.


In the balanced ternary system the value of a digit n places left of the radix point is the product of the digit and 3n. This is useful when converting between decimal and balanced ternary. In the following the strings denoting balanced ternary carry the suffix, bal3. For instance, 10bal3 = 1 × 31 + 0 × 30 = 3dec 10𝖳bal3 = 1 × 32 + 0 × 31 + (−1) × 30 = 8dec −9dec = −1 × 32 + 0 × 31 + 0 × 30 = 𝖳00bal3 8dec = 1 × 32 + 0 × 31 + (−1) × 30 = 10𝖳bal3Similarly, the first place to the right of the radix point holds 3−1 = 1/3, the second place holds 3−2 = 1/9, and so on.


In science and information science, an ontology formally represents knowledge as a set of concepts within a domain, and the relationships between those concepts. It can be used to reason about the entities within that domain and may be used to describe the domain. More specifically, an ontology is a model for describing the world that consists of a set of types, properties, and relationship types. Exactly what is provided around these varies, but they are the essentials of an ontology.


In numerical mathematics, hierarchical matrices (H-matrices) are used as data-sparse approximations of non-sparse matrices. While a sparse matrix of dimension n {\displaystyle n} can be represented efficiently in O ( n ) {\displaystyle O(n)} units of storage by storing only its non-zero entries, a non-sparse matrix would require O ( n 2 ) {\displaystyle O(n^{2})} units of storage, and using this type of matrices for large problems would therefore be prohibitively expensive in terms of storage and computing time. Hierarchical matrices provide an approximation requiring only O ( n k log  ( n ) ) {\displaystyle O(nk\,\log(n))} units of storage, where k {\displaystyle k} is a parameter controlling the accuracy of the approximation.


In the domain of central processing unit (CPU) design, hazards are problems with the instruction pipeline in CPU microarchitectures when the next instruction cannot execute in the following clock cycle, and can potentially lead to incorrect computation results. Three common types of hazards are data hazards, structural hazards, and control hazards (branching hazards).There are several methods used to deal with hazards, including pipeline stalls/pipeline bubbling, operand forwarding, and in the case of out-of-order execution, the scoreboarding method and the Tomasulo algorithm.


In most jurisdictions the subdivision of the band into different operating modes is according to informal convention rather than legal requirement.


In mathematics, specifically group theory, a descendant tree is a hierarchical structure that visualizes parent-descendant relations between isomorphism classes of finite groups of prime power order p n {\displaystyle p^{n}} , for a fixed prime number p {\displaystyle p} and varying integer exponents n ≥ 0 {\displaystyle n\geq 0} . Such groups are briefly called finite p-groups. The vertices of a descendant tree are isomorphism classes of finite p-groups. Additionally to their order p n {\displaystyle p^{n}} , finite p-groups have two further related invariants, the nilpotency class c {\displaystyle c} and the coclass r = n − c {\displaystyle r=n-c} .


In reconstructions of the Proto-Indo-European language (PIE), the verb form that has traditionally been called "perfect" in fact signified stative aspect (a current state of being). The name was assigned based on similarity to the Greek or Latin perfect tense, before the stative nature of the form was fully recognized. For details of its formation, see Proto-Indo-European verbs.


In order to use Bluetooth, a device must be compatible with the subset of Bluetooth profiles (often called services or functions) necessary to use the desired services. A Bluetooth profile is a specification regarding an aspect of Bluetooth-based wireless communication between devices. It resides on top of the Bluetooth Core Specification and (optionally) additional protocols. While the profile may use certain features of the core specification, specific versions of profiles are rarely tied to specific versions of the core specification, making them independent of each other.


(It would also be possible to add clauses to the formula to ensure that the three color classes are disjoint, but this makes no difference to the result.) Thus, by Courcelle's theorem, 3-colorability of graphs of bounded treewidth may be tested in linear time. For this variation of graph logic, Courcelle's theorem can be extended from treewidth to clique-width: for every fixed MSO1 property Π {\displaystyle \Pi } , and every fixed bound b {\displaystyle b} on the clique-width of a graph, there is a linear-time algorithm for testing whether a graph of clique-width at most b {\displaystyle b} has property Π {\displaystyle \Pi } . The original formulation of this result required the input graph to be given together with a construction proving that it has bounded clique-width, but later approximation algorithms for clique-width removed this requirement.


These concepts are "marginal" because they can be found by summing values in a table along rows or columns, and writing the sum in the margins of the table. The distribution of the marginal variables (the marginal distribution) is obtained by marginalizing (that is, focusing on the sums in the margin) over the distribution of the variables being discarded, and the discarded variables are said to have been marginalized out.


The phone then identifies itself to the network through the control channel. Once this is successfully completed, the phone is said to be attached to the network. The key feature of a mobile phone is the ability to receive and make calls in any area where coverage is available.


In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order 246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 = 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 ≈ 8×1053. The finite simple groups have been completely classified. Every such group belongs to one of 18 countably infinite families, or is one of 26 sporadic groups that do not follow such a systematic pattern.


For example, an 8-bit byte can have values ranging from 00000000 to 11111111 (0 to 255 decimal) in binary form, which can be conveniently represented as 00 to FF in hexadecimal. In mathematics, a subscript is typically used to specify the base. For example, the decimal value 24,779 would be expressed in hexadecimal as 60CB16.


In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the Stirling numbers of the first kind count permutations according to their number of cycles (counting fixed points as cycles of length one). The Stirling numbers of the first and second kind can be understood as inverses of one another when viewed as triangular matrices. This article is devoted to specifics of Stirling numbers of the first kind. Identities linking the two kinds appear in the article on Stirling numbers.


In statistics, kernel density estimation (KDE) is the application of kernel smoothing for probability density estimation, i.e., a non-parametric method to estimate the probability density function of a random variable based on kernels as weights. KDE answers a fundamental data smoothing problem where inferences about the population are made, based on a finite data sample. In some fields such as signal processing and econometrics it is also termed the Parzen–Rosenblatt window method, after Emanuel Parzen and Murray Rosenblatt, who are usually credited with independently creating it in its current form. One of the famous applications of kernel density estimation is in estimating the class-conditional marginal densities of data when using a naive Bayes classifier, which can improve its prediction accuracy.


In other languages – including most Indo-European and Afro-Asiatic languages – third-person personal pronouns (at least those used to refer to people) intrinsically distinguish male from female. This feature commonly co-exists with a full system of grammatical gender, where all nouns are assigned to classes such as masculine, feminine and neuter. In languages with grammatical gender, even pronouns which are semantically gender-neutral may be required to take a gender for such purposes as grammatical agreement. Thus in French, for example, the first- and second-person personal pronouns may behave as either masculine or feminine depending on the sex of the referent; and indefinite pronouns such as quelqu'un ('someone') and personne ('no one') are treated conventionally as masculine, even though personne as a noun ('person') is only feminine regardless of the sex of the referent.


Some demonstrations prove only that the things are a certain way, rather than why they are so. The latter are the most perfect.


In replicated-write protocols, unlike the primary-based protocol, all updates are carried out to all replicas.


1,000 bits a second. Well, we'll surely be able to figure out something to do with that.'" — Saverah Warenstein, former programmer at Lincoln Laboratory, IBM


In the C programming language, data types constitute the semantics and characteristics of storage of data elements. They are expressed in the language syntax in form of declarations for memory locations or variables. Data types also determine the types of operations or methods of processing of data elements. The C language provides basic arithmetic types, such as integer and real number types, and syntax to build array and compound types. Headers for the C standard library, to be used via include directives, contain definitions of support types, that have additional properties, such as providing storage with an exact size, independent of the language implementation on specific hardware platforms.


In places, licensure may still be a lifelong privilege, but increasingly nowadays, it requires periodic review by peers and renewal. It is very common for license renewal to depend, at least in part, on academia. In the United Kingdom such regular upgrading of skills is often termed continuous professional development, or CPD. In many professions this is fast becoming a standard, mandatory and annual requirement.


In the Goidelic languages, dependent and independent verb forms are distinct verb forms; each tense of each verb exists in both forms. Verbs are often preceded by a particle which marks negation, or a question, or has some other force. The dependent verb forms are used after a particle, while independent forms are used when the verb is not subject to a particle. For example, in Irish, the past tense of the verb feic ("to see") has two forms: the independent form chonaic and the dependent form faca. The independent form is used when no particle precedes the verb, as in Chonaic mé Seán ("I saw John"). The dependent form is used when a particle such as ní ("not") precedes the verb, as in Ní fhaca mé Seán ("I did not see John").


Some mathematical operators have inherent associativity. For example, subtraction and division, as used in conventional math notation, are inherently left-associative. Addition and multiplication, by contrast, are both left and right associative.


In operating systems, a giant lock, also known as a big-lock or kernel-lock, is a lock that may be used in the kernel to provide concurrency control required by symmetric multiprocessing (SMP) systems. A giant lock is a solitary global lock that is held whenever a thread enters kernel space and released when the thread returns to user space; a system call is the archetypal example. In this model, threads in user space can run concurrently on any available processors or processor cores, but no more than one thread can run in kernel space; any other threads that try to enter kernel space are forced to wait. In other words, the giant lock eliminates all concurrency in kernel space.


In the Romance languages, prepositions combine with stressed pronominal forms that are distinct from the unstressed clitic pronouns used with verbs. In French, prepositions combine with disjunctive pronouns, which are also found in other syntactic contexts (see French disjunctive pronouns). In Portuguese, Spanish, Italian, and Romanian, prepositions generally combine with pronouns that are identical in form to nominative (subject) pronouns, but there are unique prepositional forms for the 1st and 2nd person singular (and 3rd person reflexive). This is also true in Catalan, but the 2nd person singular prepositional form is identical to the nominative.


Importantly, the law applies (as the name indicates) only when a large number of observations are considered. There is no principle that a small number of observations will coincide with the expected value or that a streak of one value will immediately be "balanced" by the others (see the gambler's fallacy). The LLN only applies to the average. Therefore, while other formulas that look similar are not verified, such as the raw deviation from "theoretical results": not only does it not converge toward zero as n increases, but it tends to increase in absolute value as n increases.


Squaring both sides yields 2b2 = a2. Since the expression on the left is an integer multiple of 2, the right expression is by definition divisible by 2. That is, a2 is even, which implies that a must also be even, as seen in the proposition above (in #Proof by contraposition).


The theory of Linear Programming dictates that under mild assumptions (if the linear program has an optimal solution, and if the feasible region does not contain a line), one can always find an extreme point or a corner point that is optimal. The obtained optimum is tested for being an integer solution. If it is not, there is guaranteed to exist a linear inequality that separates the optimum from the convex hull of the true feasible set.


In mathematical logic and computer science, the lambda-mu calculus is an extension of the lambda calculus introduced by M. Parigot. It introduces two new operators: the μ operator (which is completely different both from the μ operator found in computability theory and from the μ operator of modal μ-calculus) and the bracket operator. Proof-theoretically, it provides a well-behaved formulation of classical natural deduction. One of the main goals of this extended calculus is to be able to describe expressions corresponding to theorems in classical logic.


In most non-strict languages the non-strictness extends to data constructors. This allows conceptually infinite data structures (such as the list of all prime numbers) to be manipulated in the same way as ordinary finite data structures. It also allows for the use of very large but finite data structures such as the complete game tree of chess. Non-strictness has several disadvantages which have prevented widespread adoption: Because of the uncertainty regarding if and when expressions will be evaluated, non-strict languages generally must be purely functional to be useful.


Another constraint is adjacency. This constraint applies to the case when the "cake" is a disputed territory that has to be divided among neighboring countries. In this case, it may required that the piece allocated to each country is adjacent to its current territory; this constraint is handled by Hill's land division problem. In land division there are often two-dimensional geometric constraints, e.g., each piece should be a square or (more generally) a fat object.


Define the constant Φ {\displaystyle \Phi } as Φ = min S ⊂ X , π ( S ) ≤ 1 2 Q ( S × S c ) π ( S ) . {\displaystyle \Phi =\min _{S\subset X,\pi (S)\leq {\frac {1}{2}}}{\frac {Q(S\times S^{c})}{\pi (S)}}.} The operator K , {\displaystyle K,} acting on the space of functions from | X | {\displaystyle |X|} to | X | {\displaystyle |X|} , defined by ( K ϕ ) ( x ) = ∑ y K ( x , y ) ϕ ( y ) {\displaystyle (K\phi )(x)=\sum _{y}K(x,y)\phi (y)} has eigenvalues λ 1 ≥ λ 2 ≥ ⋯ ≥ λ n {\displaystyle \lambda _{1}\geq \lambda _{2}\geq \cdots \geq \lambda _{n}} .


In some sense the 0-1 indicator function is the most natural loss function for classification. It takes the value 0 if the predicted output is the same as the actual output, and it takes the value 1 if the predicted output is different from the actual output. For binary classification with Y = { − 1 , 1 } {\displaystyle Y=\{-1,1\}} , this is: V ( f ( x → ) , y ) = θ ( − y f ( x → ) ) {\displaystyle V(f({\vec {x}}),y)=\theta (-yf({\vec {x}}))} where θ {\displaystyle \theta } is the Heaviside step function.


In metadata, an identification scheme is used to identify unique records in a set. If a data element is used to identify a record within a data set, the data element uses the Identifier representation term. An identification scheme should be contrasted with a classification scheme. Classification schemes are used to classify individual records into categories. Many records in a data set may be in a single category.


In telecommunications, a remote digital terminal (RDT) typically accepts E1, T1 or OC-3 digital lines to communicate with a telephone Access network (AN) or telephone exchange (Local Digital Switch, LDS) on one side, and forms a local exchange (LE) on the other, which is connected to "plain old telephone service" (POTS) lines.


In statistics and machine learning, lasso (least absolute shrinkage and selection operator; also Lasso or LASSO) is a regression analysis method that performs both variable selection and regularization in order to enhance the prediction accuracy and interpretability of the resulting statistical model. It was originally introduced in geophysics, and later by Robert Tibshirani, who coined the term. Lasso was originally formulated for linear regression models. This simple case reveals a substantial amount about the estimator.


In the following table are the decimal values of the Kaktovik digits up to three places to the left and to the right of the units' place.


In relational algebra, a rename is a unary operation written as ρ a / b ( R ) {\displaystyle \rho _{a/b}(R)} where: R is a relation a and b are attribute names b is an attribute of RThe result is identical to R except that the b attribute in all tuples is renamed to a. For an example, consider the following invocation of ρ on an Employee relation and the result of that invocation: Formally, the semantics of the rename operator is defined as follows: ρ a / b ( R ) = { t : t ∈ R } , {\displaystyle \rho _{a/b}(R)=\{\ t:t\in R\ \},} where t  {\displaystyle t} is defined as the tuple t, with the b attribute renamed to a, so that: t  = { ( c , v ) | ( c , v ) ∈ t , c ≠ b } ∪ { ( a , t ( b ) ) } . {\displaystyle t=\{\ (c,v)\ |\ (c,v)\in t,\ c\neq b\ \}\cup \{\ (a,\ t(b))\ \}.} == References ==


Operation itself is triggered by writing data to a triggering operand of an operation. Thus, an operation is executed as a side effect of the triggering data transport. Therefore, executing an addition operation in TTA requires three data transport definitions, also called moves.


In probability theory, a joint distribution can be factorized into a product between conditional and marginal distributions Q ( X , Y ) = P ( X ∣ Y ) π ( Y ) {\displaystyle Q(X,Y)=P(X\mid Y)\pi (Y)} The analog of this rule in the kernel embedding framework states that C X Y π , {\displaystyle {\mathcal {C}}_{XY}^{\pi },} the joint embedding of Q ( X , Y ) , {\displaystyle Q(X,Y),} can be factorized as a composition of conditional embedding operator with the auto-covariance operator associated with π ( Y ) {\displaystyle \pi (Y)} C X Y π = C X ∣ Y C Y Y π {\displaystyle {\mathcal {C}}_{XY}^{\pi }={\mathcal {C}}_{X\mid Y}{\mathcal {C}}_{YY}^{\pi }} where C X Y π = E  , {\displaystyle {\mathcal {C}}_{XY}^{\pi }=\mathbb {E} ,} C Y Y π = E  . {\displaystyle {\mathcal {C}}_{YY}^{\pi }=\mathbb {E} .} In practical implementations, the kernel chain rule takes the following form C ^ X Y π = C ^ X ∣ Y C ^ Y Y π = Υ ( G + λ I ) − 1 G ~ diag  ( α ) Φ ~ T {\displaystyle {\widehat {\mathcal {C}}}_{XY}^{\pi }={\widehat {\mathcal {C}}}_{X\mid Y}{\widehat {\mathcal {C}}}_{YY}^{\pi }={\boldsymbol {\Upsilon }}(\mathbf {G} +\lambda \mathbf {I} )^{-1}{\widetilde {\mathbf {G} }}\operatorname {diag} ({\boldsymbol {\alpha }}){\boldsymbol {\widetilde {\Phi }}}^{T}}


to 4:00 p.m. Eastern time. Index ETFs are also sometimes weighted by revenue rather than market capitalization.


In proof theory and constructive mathematics, the principle of independence of premise states that if φ and ∃x θ are sentences in a formal theory and φ → ∃x θ is provable, then ∃x (φ → θ) is provable. Here x cannot be a free variable of φ, while θ can be a predicate depending on it. The main application of the principle is in the study of intuitionistic logic, where the principle is not generally valid. The principle is valid in classical logic.


In project management, communication management must address the following questions: What information needs to flow in and out of the project? Who needs what information? When is the information needed?


In mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra) is a mathematical structure consisting of two involutive rings R and A, where R is commutative and A has the structure of an associative algebra over R. Involutive algebras generalize the idea of a number system equipped with conjugation, for example the complex numbers and complex conjugation, matrices over the complex numbers and conjugate transpose, and linear operators over a Hilbert space and Hermitian adjoints. However, it may happen that an algebra admits no involution.


In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. A twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain. Well-known examples of convex functions of a single variable include a linear function f ( x ) = c x {\displaystyle f(x)=cx} (where c {\displaystyle c} is a real number), a quadratic function c x 2 {\displaystyle cx^{2}} ( c {\displaystyle c} as a nonnegative real number) and a exponential function c e x {\displaystyle ce^{x}} ( c {\displaystyle c} as a nonnegative real number).


In taxonomy, an undescribed taxon is a taxon (for example, a species) that has been discovered, but not yet formally described and named. The various Nomenclature Codes specify the requirements for a new taxon to be validly described and named. Until such a description has been published, the taxon has no formal or official name, although a temporary, informal name is often used. A published scientific name may not fulfil the requirements of the Codes for various reasons.


In probability theory, especially as that field is used in statistics, a group family of probability distributions is a family obtained by subjecting a random variable with a fixed distribution to a suitable family of transformations such as a location-scale family, or otherwise a family of probability distributions acted upon by a group.Consideration of a particular family of distributions as a group family can, in statistical theory, lead to the identification of an ancillary statistic.


In order for programs and interrupt handlers to work without interference and share the same hardware memory and access to the I/O system, in a multitasking operating systems running on a digital system with a single CPU/MCU it is required to have some sort of software and hardware facilities to keep track of an executing processes data (memory page addresses, registers etc.) and to save and recover them back to the state they were in before they were suspended. This is achieved by a context switching. : 3.3 The running programs are often assigned a Process Context IDentifiers (PCID). In Linux-based operating systems, a set of data stored in registers is usually saved into a process descriptor in memory to implement switching of context. PCIDs are also used.


However, if this would result in the same output level as the previous bit time, a 0 level is output instead. Initially, the encoder output present state is assumed at 0 level when the first bit arrives at the encoder input.The new line-coding scheme violates the encoding rule of NRZ-L when a sequence of 1s or 0s arrives and hence, it overcomes some of their deficiencies. During the violation period for a run of 1s or 0s, it operates on the same encoding rule of the polar RZ but with pulse occupancy of full period.


In model theory, a mathematical discipline, a β-model (from the French "bon ordre", well-ordering) is a model which is correct about statements of the form "X is well-ordered". The term was introduced by Mostowski (1959) as a strengthening of the notion of ω-model. In contrast to the notation for set-theoretic properties named by ordinals, such as ξ {\displaystyle \xi } -indescribability, the letter β here is only denotational.


In multi-channel systems, channels are used for separate purposes.


In the area of modern algebra known as group theory, a Tarski monster group, named for Alfred Tarski, is an infinite group G, such that every proper subgroup H of G, other than the identity subgroup, is a cyclic group of order a fixed prime number p. A Tarski monster group is necessarily simple. It was shown by Alexander Yu. Olshanskii in 1979 that Tarski groups exist, and that there is a Tarski p-group for every prime p > 1075. They are a source of counterexamples to conjectures in group theory, most importantly to Burnside's problem and the von Neumann conjecture.


In number theory, the sum of squares function is an arithmetic function that gives the number of representations for a given positive integer n as the sum of k squares, where representations that differ only in the order of the summands or in the signs of the numbers being squared are counted as different, and is denoted by rk(n).


The Tsallis distributions have been applied to problems in the fields of statistical mechanics, geology, anatomy, astronomy, economics, finance, and machine learning. The distributions are often used for their heavy tails. Note that Tsallis distributions are obtained as Box–Cox transformation over usual distributions, with deformation parameter λ = 1 − q {\displaystyle \lambda =1-q} . This deformation transforms exponentials into q-exponentials.


In October 2006, HTML inventor and W3C chair Tim Berners-Lee, introducing a major W3C effort to develop a new HTML specification, posted in his blog that, "The attempt to get the world to switch to XML ... all at once didn't work. The large HTML-generating public did not move ... Some large communities did shift and are enjoying the fruits of well-formed systems ... The plan is to charter a completely new HTML group." The current HTML5 working draft says "special attention has been given to defining clear conformance criteria for user agents in an effort to improve interoperability ... while at the same time updating the HTML specifications to address issues raised in the past few years." Ian Hickson, editor of the HTML5 specification criticizing the improper use of XHTML in 2002, is a member of the group developing this specification and is listed as one of the co-editors of the current working draft.Simon Pieters researched the XML-compliance of mobile browsers and concluded "the claim that XHTML would be needed for mobile devices is simply a myth".


Nor can there be an infinite number of middle terms between the first principle and the conclusion. In all demonstration, the first principles, the conclusion, and all the intermediate propositions, must be necessary, general and eternal truths. Of things that happen by chance, or contingently, or which can change, or of individual things, there is no demonstration.


We call it β 2 , {\displaystyle \beta _{2},} and define a new sequence ⟨ β 2 , i ⟩ {\displaystyle \langle \beta _{2,i}\rangle } similar to the previous sequence. We can repeat this process, getting a sequence of sequences ⟨ β j , i ⟩ {\displaystyle \langle \beta _{j,i}\rangle } where each element of a sequence is greater than every member of the previous sequences. Then for each i < α , {\displaystyle i<\alpha ,} ⟨ β j , i ⟩ {\displaystyle \langle \beta _{j,i}\rangle } is an increasing sequence contained in C i , {\displaystyle C_{i},} and all these sequences have the same limit (the limit of ⟨ β j , i ⟩ {\displaystyle \langle \beta _{j,i}\rangle } ).


In the Japanese kantenji braille, the standard 8-dot Braille patterns 356, 1356, 3456, and 13456 are the patterns related to Braille pattern dots-245, since the two additional dots of kantenji patterns 0245, 2457, and 02457 are placed above the base 6-dot cell, instead of below, as in standard 8-dot braille.


In the European Union, the proposed Artificial Intelligence Act includes requirements to disclose copyrighted material used to train generative AI systems, and to label any AI-generated output as such.In the United States, a group of companies including OpenAI, Alphabet, and Meta signed a voluntary agreement with the White House in July 2023 to watermark AI-generated content.In China, the Interim Measures for the Management of Generative AI Services introduced by the Cyberspace Administration of China regulates any public-facing generative AI. It includes requirements to watermark generated images or videos, regulations on training data and label quality, restrictions on personal data collection, and a guideline that generative AI must "adhere to socialist core values".


For detecting performance regressions, software performance tests are run on a regular basis, to monitor the response time and resource usage metrics of the software after subsequent changes. Unlike functional regression tests, the results of performance tests are subject to variance - that is, results can differ between tests due to variance in performance measurements; as a result, a decision must be made on whether a change in performance numbers constitutes a regression, based on experience and end-user demands. Approaches such as statistical significance testing and change point detection are sometimes used to aid in this decision.


In the Finnish language, there are three classes of vowels – front, back, and neutral, where each front vowel has a back vowel pairing. Grammatical endings such as case and derivational endings – but not enclitics – have only archiphonemic vowels U, O, A, which are realized as either back  or front  inside a single word. From vowel harmony it follows that the initial syllable of each single (non-compound) word controls the frontness or backness of the entire word. Non-initially, the neutral vowels are transparent to and unaffected by vowel harmony.


In mathematics, a noncototient is a positive integer n that cannot be expressed as the difference between a positive integer m and the number of coprime integers below it. That is, m − φ(m) = n, where φ stands for Euler's totient function, has no solution for m. The cototient of n is defined as n − φ(n), so a noncototient is a number that is never a cototient. It is conjectured that all noncototients are even. This follows from a modified form of the slightly stronger version of the Goldbach conjecture: if the even number n can be represented as a sum of two distinct primes p and q, then p q − φ ( p q ) = p q − ( p − 1 ) ( q − 1 ) = p + q − 1 = n − 1.


In number theory, Selberg's identity is an approximate identity involving logarithms of primes named after Atle Selberg. The identity, discovered jointly by Selberg and Paul Erdős, was used in the first elementary proof for the prime number theorem.


In probability theory, coupling is a proof technique that allows one to compare two unrelated random variables (distributions) X and Y by creating a random vector W whose marginal distributions correspond to X and Y respectively. The choice of W is generally not unique, and the whole idea of "coupling" is about making such a choice so that X and Y can be related in a particularly desirable way.


In the dynamics of the sandpile automaton defined above, some stable configurations ( 0 ≤ z ( v ) < 4 {\displaystyle 0\leq z(v)<4} for all v ∈ G ∖ { s } {\displaystyle v\in G\setminus \{s\}} ) appear infinitely often, while others can only appear a finite number of times (if at all). The former are referred to as recurrent configurations, while the latter are referred to as transient configurations. The recurrent configurations thereby consist of all stable non-negative configurations which can be reached from any other stable configuration by repeatedly adding grains of sand to vertices and toppling. It is easy to see that the minimally stable configuration z m {\displaystyle z_{m}} , where every vertex carries z m ( v ) = d e g ( v ) − 1 {\displaystyle z_{m}(v)=deg(v)-1} grains of sand, is reachable from any other stable configuration (add d e g ( v ) − z ( v ) − 1 ≥ 0 {\displaystyle deg(v)-z(v)-1\geq 0} grains to every vertex).


Kenkichi Iwasawa (1941) proved that a p-group G is an Iwasawa group if and only if one of the following cases happens: G is a Dedekind group, or G contains an abelian normal subgroup N such that the quotient group G/N is a cyclic group and if q denotes a generator of G/N, then for all n ∈ N, q−1nq = n1+ps where s ≥ 1 in general, but s ≥ 2 for p=2.In Berkovich & Janko (2008, p. 257), Iwasawa's proof was deemed to have essential gaps, which were filled by Franco Napolitani and Zvonimir Janko. Roland Schmidt (1994) has provided an alternative proof along different lines in his textbook.


In practice, there are a wide rarity of methods that are utilized, most of which are reconstruct 3-D information (volume) from 2-D signals (image). Typically used methods are CT, MRI, PET and SPECT. And the filtered back projection based on the principles introduced above are commonly applied.


In relational database management systems, a user-defined function provides a mechanism for extending the functionality of the database server by adding a function, that can be evaluated in standard query language (usually SQL) statements. The SQL standard distinguishes between scalar and table functions. A scalar function returns only a single value (or NULL), whereas a table function returns a (relational) table comprising zero or more rows, each row with one or more columns. User-defined functions in SQL are declared using the CREATE FUNCTION statement.


For example, for AES with a 256-bit key, K {\displaystyle K} is a 256-bit number and F {\displaystyle F} is a 128-bit number. Encrypting block P {\displaystyle P} with logical index (tweak) I {\displaystyle I} uses the following formula: X = F ⊗ I , C = E K ( P ⊕ X ) ⊕ X . {\displaystyle {\begin{aligned}X&=F\otimes I,\\C&=E_{K}(P\oplus X)\oplus X.\end{aligned}}} Here multiplication ⊗ {\displaystyle \otimes } and addition ⊕ {\displaystyle \oplus } are performed in the finite field ( GF ( 2 128 ) {\displaystyle {\text{GF}}\left(2^{128}\right)} for AES).


The solution corresponding to the original constrained optimization is always a saddle point of the Lagrangian function, which can be identified among the stationary points from the definiteness of the bordered Hessian matrix.The great advantage of this method is that it allows the optimization to be solved without explicit parameterization in terms of the constraints. As a result, the method of Lagrange multipliers is widely used to solve challenging constrained optimization problems. Further, the method of Lagrange multipliers is generalized by the Karush–Kuhn–Tucker conditions, which can also take into account inequality constraints of the form h ( x ) ≤ c {\displaystyle h(\mathbf {x} )\leq c} for a given constant c {\displaystyle c~} .


In synchronous duplex mode of operation hardware coupling is provided between two processors which execute same set of instructions and compare the results continuously. If mismatch occurs then the faulty processor is identified and taken out of service within a few milliseconds. When system is operating normally, the two processors have same data in memories at all times and simultaneously receive information from exchange environment.


A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connectedness, which allows distinguishing a circle from two non-intersecting circles.


This is especially useful when dealing with the semantics of different side effects, such as nontermination, mutable state or nondeterminism. Instead of giving two variants of the semantics, one for the call-by-name evaluation order and one for the call-by-value one, one can simply give a semantics for the CBPV term language; one gets two semantics for lambda-calculus by composing this CBPV semantics with the same CBV and CBN translations from lambda-calculus. == References ==


One way to construct a family of sets with these parameters, each two sharing an element, is to choose a single element to belong to all the subsets, and then form all of the subsets that contain the chosen element. The Erdős–Ko–Rado theorem states that when n {\displaystyle n} is large enough for the problem to be nontrivial ( n ≥ 2 r {\displaystyle n\geq 2r} ) this construction produces the largest possible intersecting families.


In number theory, many arithmetic functions are integer-valued.


In mathematics, a Lucas–Carmichael number is a positive composite integer n such that if p is a prime factor of n, then p + 1 is a factor of n + 1; n is odd and square-free.The first condition resembles the Korselt's criterion for Carmichael numbers, where -1 is replaced with +1. The second condition eliminates from consideration some trivial cases like cubes of prime numbers, such as 8 or 27, which otherwise would be Lucas–Carmichael numbers (since n3 + 1 = (n + 1)(n2 − n + 1) is always divisible by n + 1). They are named after Édouard Lucas and Robert Carmichael.


In psychology, data collection often relies on samples drawn from Western, Educated, Industrialized, Rich and Democratic (WEIRD) populations, which impedes the overall generalizability of findings. Crowdsourcing the data collection process by relying on multi-lab data collection and online crowdsourcing platforms (e.g., Amazon Mechanical Turk, Prolific) makes it easier to reach a wider audience of participants from different cultural backgrounds and non-WEIRD populations. When the research question makes it possible to rely on internet samples, it is also an easy way to recruit larger samples of participants with minimal financial input and within short amounts of time.Most of the time, members of the general public are recruited to undergo studies as research participants but they can also be recruited to collect data and observations.


In the character sets developed for computing, each upper- and lower-case letter is encoded as a separate character. In order to enable case folding and case conversion, the software needs to link together the two characters representing the case variants of a letter. (Some old character-encoding systems, such as the Baudot code, are restricted to one set of letters, usually represented by the upper-case variants.) Case-insensitive operations can be said to fold case, from the idea of folding the character code table so that upper- and lower-case letters coincide.


In the following example, there is a common Swedish surname Åström written in the two alternative methods, the first one with a precomposed Å (U+00C5) and ö (U+00F6), and the second one using a decomposed base letter A (U+0041) with a combining ring above (U+030A) and an o (U+006F) with a combining diaeresis (U+0308). Åström (U+00C5 U+0073 U+0074 U+0072 U+00F6 U+006D) Åström (U+0041 U+030A U+0073 U+0074 U+0072 U+006F U+0308 U+006D)Except for the different colors, the two solutions are equivalent and should render identically. In practice, however, some Unicode implementations still have difficulties with decomposed characters. In the worst case, combining diacritics may be disregarded or rendered as unrecognized characters after their base letters, as they are not included in all fonts.


Jackson and Mr C.B. Thompson. Proper time studies are based on repeated observation, so that motions performed on the same part differently by one or many workers can be recorded, to determine those values that are truly repetitive and measurable.


This discrete problem may be ill-conditioned, depending on the original problem and the chosen quadrature rule. Since the linear equations require O ( n 3 ) {\displaystyle O(n^{3})} operations to solve, high-order quadrature rules perform better because low-order quadrature rules require large n {\displaystyle n} for a given accuracy. Gaussian quadrature is normally a good choice for smooth, non-singular problems.


6. Evaluation and visualization Finally, the clustering models can be assessed by various metrics. And it is sometimes helpful to visualize the results by plotting the clusters into low (two) dimensional space. See multidimensional scaling as a possible approach.


In the field of artificial intelligence, the most difficult problems are informally known as AI-complete or AI-hard, implying that the difficulty of these computational problems, assuming intelligence is computational, is equivalent to that of solving the central artificial intelligence problem—making computers as intelligent as people, or strong AI. To call a problem AI-complete reflects an attitude that it would not be solved by a simple specific algorithm. AI-complete problems are hypothesised to include computer vision, natural language understanding, and dealing with unexpected circumstances while solving any real-world problem.Currently, AI-complete problems cannot be solved with modern computer technology alone, but would also require human computation. This property could be useful, for example, to test for the presence of humans as CAPTCHAs aim to do, and for computer security to circumvent brute-force attacks.


That is to say, when one or more values are missing for a case, most statistical packages default to discarding any case that has a missing value, which may introduce bias or affect the representativeness of the results. Imputation preserves all cases by replacing missing data with an estimated value based on other available information.


In the case where R X = 3 {\displaystyle R_{X}=3} and R Y = 7 {\displaystyle R_{Y}=7} , the length of an input variable y {\displaystyle \mathbf {y} } from source Y {\displaystyle Y} is 7 bits, therefore it can be sent lossless with 7 bits independent of any other bits. Based on the knowledge that x {\displaystyle \mathbf {x} } and y {\displaystyle \mathbf {y} } have Hamming distance at most one, for input x {\displaystyle \mathbf {x} } from source X {\displaystyle X} , since the receiver already has y {\displaystyle \mathbf {y} } , the only possible x {\displaystyle \mathbf {x} } are those with at most 1 distance from y {\displaystyle \mathbf {y} } . If we model the correlation between two sources as a virtual channel, which has input x {\displaystyle \mathbf {x} } and output y {\displaystyle \mathbf {y} } , as long as we get y {\displaystyle \mathbf {y} } , all we need to successfully "decode" x {\displaystyle \mathbf {x} } is "parity bits" with particular error correction ability, taking the difference between x {\displaystyle \mathbf {x} } and y {\displaystyle \mathbf {y} } as channel error. We can also model the problem with cosets partition.


In symmetric pairing-based cryptography the group G {\displaystyle G} is equipped with a pairing e: G × G → T {\displaystyle e:G\times G\to T} which is bilinear. This map gives an efficient algorithm to solve the decisional Diffie-Hellman problem. Given input ( g , g a , g b , h ) {\displaystyle (g,\,g^{a},\,g^{b},\,h)} , it is easy to check if h {\displaystyle h} is equal to g a b {\displaystyle g^{ab}} . This follows by using the pairing: note that e ( g a , g b ) = e ( g , g ) a b = e ( g , g a b ) .


The goal may be to visualize such data by points in Euclidean space whose distance matrix approximates a given dissimilarity matrix as well as possible — this is known as multidimensional scaling. Alternatively, given two sets of data already represented by points in Euclidean space, one may ask how similar they are in shape, that is, how closely can they be related by a distance-preserving transformation — this is Procrustes analysis. Some of the distances may also be missing or come unlabelled (as an unordered set or multiset instead of a matrix), leading to more complex algorithmic tasks, such as the graph realization problem or the turnpike problem (for points on a line).


In the field of linguistics, the term passive is applied to a wide range of grammatical structures. Linguists therefore find it difficult to define the term in a way that makes sense across all human languages. The canonical passive in European languages has the following properties: The subject is not an agent. There is a change in: word order; or in nominal morphology—the form of the nouns in the sentence.


In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix. The adjacency matrix of a simple undirected graph is a real symmetric matrix and is therefore orthogonally diagonalizable; its eigenvalues are real algebraic integers. While the adjacency matrix depends on the vertex labeling, its spectrum is a graph invariant, although not a complete one. Spectral graph theory is also concerned with graph parameters that are defined via multiplicities of eigenvalues of matrices associated to the graph, such as the Colin de Verdière number.


In Dirac representation, the four contravariant gamma matrices are γ 0 = ( 1 0 0 0 0 1 0 0 0 0 − 1 0 0 0 0 − 1 ) , γ 1 = ( 0 0 0 1 0 0 1 0 0 − 1 0 0 − 1 0 0 0 ) , γ 2 = ( 0 0 0 − i 0 0 i 0 0 i 0 0 − i 0 0 0 ) , γ 3 = ( 0 0 1 0 0 0 0 − 1 − 1 0 0 0 0 1 0 0 ) . {\displaystyle {\begin{aligned}\gamma ^{0}&={\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}},&\gamma ^{1}&={\begin{pmatrix}0&0&0&1\\0&0&1&0\\0&-1&0&0\\-1&0&0&0\end{pmatrix}},\\\\\gamma ^{2}&={\begin{pmatrix}0&0&0&-i\\0&0&i&0\\0&i&0&0\\-i&0&0&0\end{pmatrix}},&\gamma ^{3}&={\begin{pmatrix}0&0&1&0\\0&0&0&-1\\-1&0&0&0\\0&1&0&0\end{pmatrix}}~.\end{aligned}}} γ 0 {\displaystyle \gamma ^{0}} is the time-like, Hermitian matrix. The other three are space-like, anti-Hermitian matrices.


In software engineering and mathematics, numerical error is the error in the numerical computations.


As a consequence, the cardinality of the real numbers, which is the same as that of the power set of the integers, is strictly larger than the cardinality of the integers; see Cardinality of the continuum for details. The theorem is named for German mathematician Georg Cantor, who first stated and proved it at the end of the 19th century.


In mathematics, the reflexive closure of a binary relation R {\displaystyle R} on a set X {\displaystyle X} is the smallest reflexive relation on X {\displaystyle X} that contains R . {\displaystyle R.} A relation is called reflexive if it relates every element of X {\displaystyle X} to itself. For example, if X {\displaystyle X} is a set of distinct numbers and x R y {\displaystyle xRy} means " x {\displaystyle x} is less than y {\displaystyle y} ", then the reflexive closure of R {\displaystyle R} is the relation " x {\displaystyle x} is less than or equal to y {\displaystyle y} ".


Note that the bit b i {\displaystyle b_{i}} is what decides which basis a i {\displaystyle a_{i}} is encoded in (either in the computational basis or the Hadamard basis). The qubits are now in states which are not mutually orthogonal, and thus it is impossible to distinguish all of them with certainty without knowing b {\displaystyle b} .


As the names of these two tests implies, destructive testing will destroy the part that is being tested while nondestructive testing enables the test piece to be used afterwards. There are several methods available in each of these types. This section outlines some requirements of testing plastic welds as well as the different types of destructive and non-destructive methods that are applicable to plastic welding and go over some of the advantages and disadvantages.


In addition: Rule A2 does not perform any reduction on its own. However, it is still useful, because of its "shuffling" effect that can create new opportunities for applying the other rules; Rule A1 is not used in practice, because it may increase proof size; Rules B1, B2, B2' and B3 are directly responsible for the reduction, as they produce a transformed root clause stronger than the original one; The application of a B rule may lead to an illegal proof (see the example below), as some literals missing in the transformed root clause may be involved in another resolution step along the path to the proof root. Therefore, the algorithm also has to "reconstruct" a legal proof when this happen.The following example shows a situation where the proof becomes illegal after the application of B2' rule: Applying rule B2' to the highlighted context: The proof is now illegal because the literal o {\displaystyle o} is missing from the transformed root clause.


In some early DSP and RISC processors, the documentation advises programmers to avoid such dependencies in adjacent and nearly adjacent instructions (called delay slots), or declares that the second instruction uses an old value rather than the desired value (in the example above, the processor might counter-intuitively copy the unincremented value), or declares that the value it uses is undefined. The programmer may have unrelated work that the processor can do in the meantime; or, to ensure correct results, the programmer may insert NOPs into the code, partly negating the advantages of pipelining.


Further, the single pass aspect slowed or prevented getting the compiler bootstrapped out of Z80 assembly language and onto Pascal itself. Since the compiler was monolithic, the conversion to Pascal could not be done one section at a time, but had to proceed as a wholesale replacement.


If one also assumes the Axiom of Choice, then all sets can be enumerated so that it coincides up to relabeling with the most general form of enumerations. Since set theorists work with infinite sets of arbitrarily large cardinalities, the default definition among this group of mathematicians of an enumeration of a set tends to be any arbitrary α-sequence exactly listing all of its elements. Indeed, in Jech's book, which is a common reference for set theorists, an enumeration is defined to be exactly this. Therefore, in order to avoid ambiguity, one may use the term finitely enumerable or denumerable to denote one of the corresponding types of distinguished countable enumerations.


In spring 2008 Nokia introduced a built in SIP VoIP client for the very first time to the mass market device (Nokia 6300i) running Series 40 operating system. Later that year (Nokia 6260 Slide was introduced introducing slightly updated SIP VoIP client. Nokia maintains a list of all phones that have an integrated VoIP client in Forum Nokia.Aircell's battle with some companies allowing VoIP calls on flights is another example of the growing conflict of interest between incumbent operators and new VoIP operators.


In queueing models, a discipline within the mathematical theory of probability, the quasi-birth–death process describes a generalisation of the birth–death process. : 118 As with the birth-death process it moves up and down between levels one at a time, but the time between these transitions can have a more complicated distribution encoded in the blocks.


In real-life data, particularly for large datasets, the notions of rough approximations were found to be excessively restrictive. Therefore, an extension of DRSA, based on stochastic model (Stochastic DRSA), which allows inconsistencies to some degree, has been introduced. Having stated the probabilistic model for ordinal classification problems with monotonicity constraints, the concepts of lower approximations are extended to the stochastic case. The method is based on estimating the conditional probabilities using the nonparametric maximum likelihood method which leads to the problem of isotonic regression. Stochastic dominance-based rough sets can also be regarded as a sort of variable-consistency model.


In statistics, a scan statistic or window statistic is a problem relating to the clustering of randomly positioned points. An example of a typical problem is the maximum size of a cluster of points on a line or the longest series of successes recorded by a moving window of fixed length.Joseph Naus first published on the problem in the 1960s, and has been called the "father of the scan statistic" in honour of his early contributions. The results can be applied in epidemiology, public health and astronomy to find unusual clusters of events.It was extended by Martin Kulldorff to multidimensional settings and varying window sizes in a 1997 paper, which is (as of 11 October 2015) the most cited article in its journal, Communications in Statistics – Theory and Methods. This work lead to the creation of the software SaTScan, a program trademarked by Martin Kulldorff that applies his methods to data.


In software development, formal methods are mathematical approaches to solving software (and hardware) problems at the requirements, specification, and design levels. Formal methods are most likely to be applied to safety-critical or security-critical software and systems, such as avionics software. Software safety assurance standards, such as DO-178C allows the usage of formal methods through supplementation, and Common Criteria mandates formal methods at the highest levels of categorization. For sequential software, examples of formal methods include the B-Method, the specification languages used in automated theorem proving, RAISE, and the Z notation.


In the case of coronals, the symbols ⟨t, d⟩ are normally used for the stop portion of the affricate regardless of place. For example,  is commonly seen for . The exemplar languages are ones that have been reported to have these sounds, but in several cases, they may need confirmation.


In mathematical logic, Löb's theorem states that in Peano arithmetic (PA) (or any formal system including PA), for any formula P, if it is provable in PA that "if P is provable in PA then P is true", then P is provable in PA. If Prov(P) means that the formula P is provable, we may express this more formally as If P A ⊢ P r o v ( P ) → P {\displaystyle {\mathit {PA}}\vdash {\mathrm {Prov} (P)\rightarrow P}} then P A ⊢ P {\displaystyle {\mathit {PA}}\vdash P} .An immediate corollary (the contrapositive) of Löb's theorem is that, if P is not provable in PA, then "if P is provable in PA, then P is true" is not provable in PA. For example, "If 1 + 1 = 3 {\displaystyle 1+1=3} is provable in PA, then 1 + 1 = 3 {\displaystyle 1+1=3} " is not provable in PA.Löb's theorem is named for Martin Hugo Löb, who formulated it in 1955. It is related to Curry's paradox.


Such groups are elementarily equivalent to the integers ( Z , + , < ) {\displaystyle (\mathbb {Z} ,+,<)} . Z-groups are an alternative presentation of Presburger arithmetic. occasionally, (Z)-group is used to mean a Zassenhaus group, a special type of permutation group.


The pixel X is predicted by the LOCO-I predictor according to the following guesses: X = { min ( A , B ) if C ≥ max ( A , B ) max ( A , B ) if C ≤ min ( A , B ) A + B − C otherwise . {\displaystyle X=\left\{{\begin{aligned}&\min(A,B)\quad \,{\mbox{if}}\,C\geq \max(A,B)\\&\max(A,B)\quad {\mbox{if}}\,C\leq \min(A,B)\\&A+B-C\quad \,{\mbox{otherwise}}.\\\end{aligned}}\right.} The three simple predictors are selected according to the following conditions: (1) it tends to pick B in cases where a vertical edge exists left of the X, (2) A in cases of an horizontal edge above X, or (3) A + B – C if no edge is detected.


In the era of rapid technology evolution, it is not the biggest that survives, but the fastest. The sooner the end product is delivered without major defects, the sooner feedback can be received, and incorporated into the next iteration. The shorter the iterations, the better the learning and communication within the team. With speed, decisions can be delayed.


This problem was solved, with an algorithm, in the same original paper by Gale and Shapley, in which the stable marriage problem was solved.The hospitals/residents problem with couples allows the set of residents to include couples who must be assigned together, either to the same hospital or to a specific pair of hospitals chosen by the couple (e.g., a married couple want to ensure that they will stay together and not be stuck in programs that are far away from each other). The addition of couples to the hospitals/residents problem renders the problem NP-complete.The assignment problem seeks to find a matching in a weighted bipartite graph that has maximum weight.


In relational databases, a table can be a (mathematical) set or a multiset, depending on the presence of unicity constraints on some columns (which turns it into a candidate key). SQL allows the selection of rows from a relational table: this operation will in general yield a multiset, unless the keyword DISTINCT is used to force the rows to be all different, or the selection includes the primary (or a candidate) key. In ANSI SQL the MULTISET keyword can be used to transform a subquery into a collection expression: is a general select that can be used as subquery expression of another more general query, while transforms the subquery into a collection expression that can be used in another query, or in assignment to a column of appropriate collection type.


In statistics, cluster sampling is a sampling plan used when mutually homogeneous yet internally heterogeneous groupings are evident in a statistical population. It is often used in marketing research. In this sampling plan, the total population is divided into these groups (known as clusters) and a simple random sample of the groups is selected. The elements in each cluster are then sampled.


The symbol "≈" is also used for this purpose. In physics and astronomy, a tilde can be used between two expressions (e.g. h ~ 10−34 J s) to state that the two are of the same order of magnitude.In statistics and probability theory, the tilde means "is distributed as"; see random variable(e.g. X ~ B(n,p) for a binomial distribution). A tilde can also be used to represent geometric similarity (e.g. ∆ABC ~ ∆DEF, meaning triangle ABC is similar to DEF).


In mathematics, the surgery structure set S ( X ) {\displaystyle {\mathcal {S}}(X)} is the basic object in the study of manifolds which are homotopy equivalent to a closed manifold X. It is a concept which helps to answer the question whether two homotopy equivalent manifolds are diffeomorphic (or PL-homeomorphic or homeomorphic). There are different versions of the structure set depending on the category (DIFF, PL or TOP) and whether Whitehead torsion is taken into account or not.


If two of these symbols are used, one on the left and the mirror image of it on the right, it almost always indicates a set, as in { a , b , c } {\displaystyle \{a,b,c\}} , the set containing three members, a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} . But if it is used only on the left, it groups two or more simultaneous equations. There are other symbols of grouping.


In the 80386 microprocessor and later, virtual 8086 mode (also called virtual real mode, V86-mode, or VM86) allows the execution of real mode applications that are incapable of running directly in protected mode while the processor is running a protected mode operating system. It is a hardware virtualization technique that allowed multiple 8086 processors to be emulated by the 386 chip. It emerged from the painful experiences with the 80286 protected mode, which by itself was not suitable to run concurrent real-mode applications well. John Crawford developed the Virtual Mode bit at the register set, paving the way to this environment.VM86 mode uses a segmentation scheme identical to that of real mode (for compatibility reasons), which creates 20-bit linear addresses in the same manner as 20-bit physical addresses are created in real mode, but are subject to protected mode's memory paging mechanism.


In order to prevent Intel x86-specific code from slipping into the operating system, due to developers being used to developing on x86 chips, Windows NT 3.1 was initially developed using non-x86 development systems and then ported to the x86 architecture. This work was initially based on the Intel i860-based Dazzle system and, later, the MIPS R4000-based Jazz platform. Both systems were designed internally at Microsoft.Windows NT 3.1 was released for Intel x86 PC compatible, PC-98, DEC Alpha, and ARC-compliant MIPS platforms.


In n-dimensional Euclidean space over the real numbers, R n {\displaystyle \mathbb {R} ^{n}} , the standard basis is denoted e1, e2, e3, ... en. Each basis vector ei points along the positive xi axis, with the basis being orthonormal. Component j of ei is given by the Kronecker delta: A vector in R n {\displaystyle \mathbb {R} ^{n}} takes the form: Similarly for the order-2 tensor above, for each vector a and b in R n {\displaystyle \mathbb {R} ^{n}}: or more generally:


Mathematical theorems are known about combinations of relation properties, such as "A transitive relation is irreflexive if, and only if, it is asymmetric". Of particular importance are relations that satisfy certain combinations of properties. A partial order is a relation that is reflexive, antisymmetric, and transitive, an equivalence relation is a relation that is reflexive, symmetric, and transitive, a function is a relation that is right-unique and left-total (see below).Since relations are sets, they can be manipulated using set operations, including union, intersection, and complementation, and satisfying the laws of an algebra of sets. Beyond that, operations like the converse of a relation and the composition of relations are available, satisfying the laws of a calculus of relations.The above concept of relation has been generalized to admit relations between members of two different sets (heterogeneous relation, like "lies on" between the set of all points and that of all lines in geometry), relations between three or more sets (Finitary relation, like "person x lives in town y at time z"), and relations between classes (like "is an element of" on the class of all sets, see Binary relation § Sets versus classes).


In the early 1960s Peter Swinnerton-Dyer used the EDSAC-2 computer at the University of Cambridge Computer Laboratory to calculate the number of points modulo p (denoted by Np) for a large number of primes p on elliptic curves whose rank was known. From these numerical results Birch & Swinnerton-Dyer (1965) conjectured that Np for a curve E with rank r obeys an asymptotic law ∏ p ≤ x N p p ≈ C log  ( x ) r as x → ∞ {\displaystyle \prod _{p\leq x}{\frac {N_{p}}{p}}\approx C\log(x)^{r}{\mbox{ as }}x\rightarrow \infty } where C is a constant. Initially this was based on somewhat tenuous trends in graphical plots; this induced a measure of skepticism in J. W. S. Cassels (Birch's Ph.D.


It is easy to find an initial solution that visits all the cities but will likely be very poor compared to the optimal solution. The algorithm starts with such a solution and makes small improvements to it, such as switching the order in which two cities are visited. Eventually, a much shorter route is likely to be obtained.


In mid-1822 Hamilton began a systematic study of Laplace's Mécanique Céleste. In November and December 1822 he completed his first three original mathematical papers. On his first visit to Dunsink Observatory, he showed two of them to Brinkley, who asked for a more developed form.


In relation to the 2016 United States presidential election, individuals associated with online message boards, such as 4chan, noted a similarity between Kek and the character Pepe the Frog. This was later paired with images of Pepe, resulting in a resurgence of interest in the ancient deity.Believers in Kek say that repeating integers, often called “dubs”, are the prima materia of reality, and that their occurrence invoke the deity. Elon Musk has made numerous references to Pepe and even to kek, among others within the perceived Illuminati such as Donald Trump, who tweeted himself as a version of the frog. Believers have cited this as evidence of memetic synchronicity.


The Darmon–Granville theorem uses Faltings's theorem to show that for every specific choice of exponents (x, y, z), there are at most finitely many coprime solutions for (A, B, C). : p.


Together with K.-M. Chung, D. Schuch, W. Ulmer and B. Zeiger a unified understanding of molecular interactions was developed based on a nonlinear Schrödinger equation. Hartmann thereby pioneered the discovery of one self-interacting field as the foundation of chemistry. H. Hartman emerited in 1982 and died two years later. H. Hartmann was honoured as member of the Deutsche Akademie der Naturforscher Leopoldina, the Gesellschaft Österreichischer Chemiker, the Accademia Nazionale die Lincei, the Royal Danish Academy of Sciences and Letters, the Comitato Premio of Fondazione Balzan, the International Academy of Quantum Molecular Science, and the Akademie der Wissenschaften und der Literatur zu Mainz.


In most theories up to the eighteenth century, light was pictured as being made up of particles. Since particle models cannot easily account for the refraction, diffraction and birefringence of light, wave theories of light were proposed by René Descartes (1637), Robert Hooke (1665), and Christiaan Huygens (1678); however, particle models remained dominant, chiefly due to the influence of Isaac Newton. In the early 19th century, Thomas Young and August Fresnel clearly demonstrated the interference and diffraction of light, and by 1850 wave models were generally accepted. James Clerk Maxwell's 1865 prediction that light was an electromagnetic wave – which was confirmed experimentally in 1888 by Heinrich Hertz's detection of radio waves – seemed to be the final blow to particle models of light.


In a linear normalization, the weight t i {\displaystyle t_{i}} of a descriptor is mapped to a size scale of 1 through f, where t min {\displaystyle t_{\min }} and t max {\displaystyle t_{\max }} are specifying the range of available weights. s i = ⌈ f max ⋅ ( t i − t min ) t max − t min ⌉ {\displaystyle s_{i}=\left\lceil {\frac {f_{\max }\cdot (t_{i}-t_{\min })}{t_{\max }-t_{\min }}}\right\rceil } for t i > t min {\displaystyle t_{i}>t_{\min }} ; else s i = 1 {\displaystyle s_{i}=1} s i {\displaystyle s_{i}}: display fontsize f max {\displaystyle f_{\max }}: max. fontsize t i {\displaystyle t_{i}}: count t min {\displaystyle t_{\min }}: min.


In number theory, a factorion in a given number base b {\displaystyle b} is a natural number that equals the sum of the factorials of its digits. The name factorion was coined by the author Clifford A. Pickover.


Consider a class that contains the following procedure to print a string on standard output after a new line: The following snippet, assumed to be in the same class, uses print_on_new_line to demonstrate the mixing of open arguments and open targets in agents used as arguments to the same routine. This example uses the procedure do_all for linear structures, which executes the routine modeled by an agent for each item in the structure. The sequence of three instructions prints the strings in my_list, converts the strings to lowercase, and then prints them again. Procedure do_all iterates across the structure executing the routine substituting the current item for either the open argument (in the case of the agents based on print_on_new_line), or the open target (in the case of the agent based on to_lower). Open and closed arguments and targets also allow the use of routines that call for more arguments than are required by closing all but the necessary number of arguments: The Eiffel agent mechanism is detailed in the Eiffel ISO/ECMA standard document.


It was introduced by Gottlob Frege, developed by Moses Schönfinkel, and further developed by Haskell Curry.Uncurrying is the dual transformation to currying, and can be seen as a form of defunctionalization. It takes a function f {\displaystyle f} whose return value is another function g {\displaystyle g} , and yields a new function f ′ {\displaystyle f'} that takes as parameters the arguments for both f {\displaystyle f} and g {\displaystyle g} , and returns, as a result, the application of f {\displaystyle f} and subsequently, g {\displaystyle g} , to those arguments. The process can be iterated.


In queueing theory, an adversarial queueing network is a model where the traffic to the network is supplied by an opponent rather than as the result of a stochastic process. The model has seen use in describing the impact of packet injections on the performance of communication networks. The model was first introduced in 1996.The stability of an adversarial queueing network can be determined by considering a fluid limit. == References ==


Each base station is adapted to communicate over a radio link with at least one user's mobile station located within the radio range of said base station. The advantage is that the equipment for WiFi, 5G and other protocols can be centralized in one place, with remote antennas attached via fiber optic serving all protocols. It greatly reduces the equipment and maintenance cost of the network.RoF technology enables convergence of fixed and mobile networks.


In the Plan 9 operating system from Bell Labs (mid-1980s onward), union mounting is a central concept, replacing several older Unix conventions with union directories; for example, several directories containing executables, unioned together at a single /bin directory, replace the PATH variable for command lookup in the shell.Plan 9 union semantics are greatly simplified compared to the implementations for POSIX-style operating systems: the union of two directories is simply the concatenation of their contents, so a directory listing of the union may display duplicate names. Also, no effort is made to recursively merge subdirectories, leading to an extremely simple implementation. Directories are unioned in a controllable order; u/name, where u is a union directory, denotes the file called name in the first constituent directory that contains such a file.


In mathematics, a monogenic semigroup is a semigroup generated by a single element. Monogenic semigroups are also called cyclic semigroups.


In recent years, sentence embedding has seen a growing level of interest due to its applications in natural language queryable knowledge bases through the usage of vector indexing for semantic search. LangChain for instance utilizes sentence transformers for purposes of indexing documents. In particular, an indexing is generated by generating embeddings for chunks of documents and storing (document chunk, embedding) tuples.


In mathematics, a fact is a statement (called a theorem) that can be proven by logical argument from certain axioms and definitions.


Unlike stochastic independence and uncorrelatedness, mean independence is not symmetric: it is possible for Y {\displaystyle Y} to be mean-independent of X {\displaystyle X} even though X {\displaystyle X} is mean-dependent on Y {\displaystyle Y} . The concept of mean independence is often used in econometrics to have a middle ground between the strong assumption of independent random variables ( X 1 ⊥ X 2 {\displaystyle X_{1}\perp X_{2}} ) and the weak assumption of uncorrelated random variables ( Cov  ( X 1 , X 2 ) = 0 ) . {\displaystyle (\operatorname {Cov} (X_{1},X_{2})=0).}


In mathematics, many sequences of numbers or of polynomials are indexed by nonnegative integers, for example, the Bernoulli numbers and the Bell numbers. In both mechanics and statistics, the zeroth moment is defined, representing total mass in the case of physical density, or total probability, i.e. one, for a probability distribution. The zeroth law of thermodynamics was formulated after the first, second, and third laws, but considered more fundamental, thus its name. In biology, an organism is said to have zero-order intentionality if it shows "no intention of anything at all". This would include a situation where the organism's genetically predetermined phenotype results in a fitness benefit to itself, because it did not "intend" to express its genes. In the similar sense, a computer may be considered from this perspective a zero-order intentional entity, as it does not "intend" to express the code of the programs it runs.In biological or medical experiments, initial measurements made before any experimental time has passed are said to be on the 0 day of the experiment.In genomics, both 0-based and 1-based systems are used for genome coordinates.Patient zero (or index case) is the initial patient in the population sample of an epidemiological investigation.


In statistical mechanics, the softargmax function is known as the Boltzmann distribution (or Gibbs distribution):: 7 the index set 1 , … , k {\displaystyle {1,\,\dots ,\,k}} are the microstates of the system; the inputs z i {\displaystyle z_{i}} are the energies of that state; the denominator is known as the partition function, often denoted by Z; and the factor β is called the coldness (or thermodynamic beta, or inverse temperature).


In mathematics, particularly in abstract algebra, a semigroup with involution or a *-semigroup is a semigroup equipped with an involutive anti-automorphism, which—roughly speaking—brings it closer to a group because this involution, considered as unary operator, exhibits certain fundamental properties of the operation of taking the inverse in a group: uniqueness, double application "cancelling itself out", and the same interaction law with the binary operation as in the case of the group inverse. It is thus not a surprise that any group is a semigroup with involution. However, there are significant natural examples of semigroups with involution that are not groups.


In mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomial P in one variable, they allow expressing the sums of the k-th powers of all roots of P (counted with their multiplicity) in terms of the coefficients of P, without actually finding those roots. These identities were found by Isaac Newton around 1666, apparently in ignorance of earlier work (1629) by Albert Girard. They have applications in many areas of mathematics, including Galois theory, invariant theory, group theory, combinatorics, as well as further applications outside mathematics, including general relativity.


In order to speak to larger groups of people, a need arose to increase the volume of the human voice. The earliest devices used to achieve this were acoustic megaphones. Some of the first examples, from fifth-century-BC Greece, were theater masks with horn-shaped mouth openings that acoustically amplified the voice of actors in amphitheaters. In 1665, the English physicist Robert Hooke was the first to experiment with a medium other than air with the invention of the "lovers' telephone" made of stretched wire with a cup attached at each end.In 1856, Italian inventor Antonio Meucci developed a dynamic microphone based on the generation of electric current by moving a coil of wire to various depths in a magnetic field.


On x86-64 processors in native long mode, the address translation scheme uses PAE but adds a fourth table, the 512-entry page-map level 4 table, and extends the page directory pointer table to 512 entries instead of the original 4 entries it has in protected mode. This means that 48 bits of virtual page number are translated, giving a virtual address space of up to 256 TB. For some processors, a mode can be enabled with a fifth table, the 512-entry page-map level 5 table; this means that 57 bits of virtual page number are translated, giving a virtual address space of up to 128 PB. : 141–153 In the page table entries, in the original specification, 40 bits of physical page number are implemented. Page table structures


In mathematics, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. Today, these dualities are usually collected under the label Stone duality, since they form a natural generalization of Stone's representation theorem for Boolean algebras. These concepts are named in honor of Marshall Stone. Stone-type dualities also provide the foundation for pointless topology and are exploited in theoretical computer science for the study of formal semantics. This article gives pointers to special cases of Stone duality and explains a very general instance thereof in detail.


Except for certain logical issues, in this case CbD specializes to traditional treatments of contextuality in quantum physics. In particular, for consistently connected cyclic systems the noncontextuality criterion above reduces to D ( C n ) ≤ n − 2 , {\displaystyle D\left({\mathcal {C}}_{n}\right)\leq n-2,} which includes the Bell/CHSH inequality ( n = 4 {\displaystyle n=4} ), KCBS inequality ( n = 5 {\displaystyle n=5} ), and other famous inequalities. That nonlocality is a special case of contextuality follows in CbD from the fact that being jointly distributed for random variables is equivalent to being measurable functions of one and the same random variable (this generalizes Arthur Fine's analysis of Bell's theorem).


In number theory, all powers of three are perfect totient numbers. The sums of distinct powers of three form a Stanley sequence, the lexicographically smallest sequence that does not contain an arithmetic progression of three elements. A conjecture of Paul Erdős states that this sequence contains no powers of two other than 1, 4, and 256.


The Multicast Dissemination Protocol (MDP) was an initial attempt to ensure reliability and to address the problem of ACK implosions through the use of NACKs. MDP used selective negative acknowledgement (NACK) to support reliability. Additionally, MDP implemented probabilistic techniques to suppress redundant NACKs in the multicast group (and thereby avoid NACK implosions).


The SCP responds with a geographic number, e.g. 0121 XXX XXXX, and the call is actually routed to a phone. By this architecture: 08XXX (non-geographic numbers) can be set up in a few SCP nodes rather than having to be set up in every telephone exchange in the country. geographic numbers can be hidden revenue can be generated by non-telecoms companies from people making telephone calls to services – e.g. telephone voting


One of the first system calls made by the C standard I/O library is in an isatty() call used to determine if the program is being run interactively by a human (in which case isatty() will succeed and the library will write its output a line at a time so the user sees a regular flow of text) or as part of a pipeline (in which case it writes a block at a time for efficiency). If a library routine fails for some reason unrelated to a system call (for example, because a user name wasn't found in the password file) and a naïve programmer blindly calls the normal error reporting routine perror() on every failure, the leftover ENOTTY will result in an utterly inappropriate "Not a typewriter" (or "Not a teletype", or "Inappropriate ioctl for device") being delivered to the user. For many years the UNIX mail program sendmail contained this bug: when mail was delivered from another system, the mail program was being run non-interactively. If the destination address was local, but referred to a user name not found in the local password file, the message sent back to the originator of the email was the announcement that the person they were attempting to communicate with was not a typewriter.


In mathematics, finite field arithmetic is arithmetic in a finite field (a field containing a finite number of elements) contrary to arithmetic in a field with an infinite number of elements, like the field of rational numbers. There are infinitely many different finite fields. Their number of elements is necessarily of the form pn where p is a prime number and n is a positive integer, and two finite fields of the same size are isomorphic. The prime p is called the characteristic of the field, and the positive integer n is called the dimension of the field over its prime field. Finite fields are used in a variety of applications, including in classical coding theory in linear block codes such as BCH codes and Reed–Solomon error correction, in cryptography algorithms such as the Rijndael (AES) encryption algorithm, in tournament scheduling, and in the design of experiments.


In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variables). It is named after the mathematician Joseph-Louis Lagrange. The basic idea is to convert a constrained problem into a form such that the derivative test of an unconstrained problem can still be applied. The relationship between the gradient of the function and gradients of the constraints rather naturally leads to a reformulation of the original problem, known as the Lagrangian function.The method can be summarized as follows: In order to find the maximum or minimum of a function f ( x ) {\displaystyle f(x)} subjected to the equality constraint g ( x ) = 0 {\displaystyle g(x)=0} , form the Lagrangian function, L ( x , λ ) ≡ f ( x ) + λ ⋅ g ( x ) {\displaystyle {\mathcal {L}}(x,\lambda )\equiv f(x)+\lambda \cdot g(x)} and find the stationary points of L {\displaystyle {\mathcal {L}}} considered as a function of x {\displaystyle x} and the Lagrange multiplier λ {\displaystyle \lambda ~} .


In order to allow the system to work even with the high inter-unit latencies, each processor used an 8-deep instruction pipeline. Branches used a variable delay slot, the end of which was signaled by a bit in the next instruction. The bit indicated that the results of the branch had to be re-merged at this point, stalling the processor until this took place.