In mathematics, the power set (or powerset) of a set S is the set of all subsets of S, including the empty set and S itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. The powerset of S is variously denoted as P(S), 𝒫(S), P(S), P ( S ) {\displaystyle \mathbb {P} (S)} , ℘ ( S ) {\displaystyle \wp (S)} , or 2S. The notation 2S, meaning the set of all functions from S to a given set of two elements (e.g., {0, 1}), is used because the powerset of S can be identified with, equivalent to, or bijective to the set of all the functions from S to the given two elements set.Any subset of P(S) is called a family of sets over S.


In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers (numbers), the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in his Elements (c. 300 BC).


In the computer science fields of knowledge engineering and ontology, the Sigma knowledge engineering environment is an open source computer program for the development of formal ontologies. It is designed for use with the Suggested Upper Merged Ontology. It originally included only the Vampire theorem prover as its core deductive inference engine, but now allows use of many other provers that have participated in the CASC/CADE competitions.


In numerical analysis, hill climbing is a mathematical optimization technique which belongs to the family of local search. It is an iterative algorithm that starts with an arbitrary solution to a problem, then attempts to find a better solution by making an incremental change to the solution. If the change produces a better solution, another incremental change is made to the new solution, and so on until no further improvements can be found. For example, hill climbing can be applied to the travelling salesman problem.


Thus, the solution x 0 = A − 1 b {\displaystyle x_{0}=A^{-1}\mathbf {b} } only exists where an actual corner exists in the window N {\displaystyle N} . A methodology for performing automatic scale selection for this corner localization method has been presented by Lindeberg by minimizing the normalized residual d ~ min = c − b T A − 1 b trace  A {\displaystyle {\tilde {d}}_{\min }={\frac {c-b^{T}A^{-1}b}{\operatorname {trace} A}}} over scales.


p ) ⋅ β q ( σ ∘ ι p , p + 1 , . . .


It is guaranteed to be a vertex of the convex hull of the polygon. Alternatively, the vertex with the smallest Y-coordinate among the ones with the largest X-coordinates or the vertex with the smallest X-coordinate among the ones with the largest Y-coordinates (or any other of 8 "smallest, largest" X/Y combinations) will do as well. Once a vertex of the convex hull is chosen, one can then apply the formula using the previous and next vertices, even if those are not on the convex hull, as there can be no local concavity on this vertex.


In the case of broadcast packets over a switching loop, the situation may develop into a broadcast storm. In a very simple example, a switch with three ports A, B, and C has a normal node connected to port A while ports B and C are connected to each other in a loop. All ports have the same link speed and run in full duplex mode. Now, when a broadcast frame enters the switch through port A, this frame is forwarded to all ports but the source port, i.e. ports B and C. Both frames exiting ports B and C traverse the loop in opposite directions and reenter the switch through their counterpart port.


In most current use cases, the letters A–F or a–f represent the values 10–15, while the numerals 0–9 are used to represent their decimal values. There is no universal convention to use lowercase or uppercase, so each is prevalent or preferred in particular environments by community standards or convention; even mixed case is used. Seven-segment displays use mixed-case AbCdEF to make digits that can be distinguished from each other. There is some standardization of using spaces (rather than commas or another punctuation mark) to separate hex values in a long list. For instance, in the following hex dump, each 8-bit byte is a 2-digit hex number, with spaces between them, while the 32-bit offset at the start is an 8-digit hex number.


These algorithms are called auction algorithms, push-relabel algorithms, or preflow-push algorithms. Some of these algorithms were shown to be equivalent.Some of the local methods assume that the graph admits a perfect matching; if this is not the case, then some of these methods might run forever. : 3 A simple technical way to solve this problem is to extend the input graph to a complete bipartite graph, by adding artificial edges with very large weights.


The algebra is named for logicians Adolf Lindenbaum and Alfred Tarski. Starting in the academic year 1926-1927, Lindenbaum pioneered his method in Jan Łukasiewicz's mathematical logic seminar, and the method was popularized and generalized in subsequent decades through work by Tarski. The Lindenbaum–Tarski algebra is considered the origin of the modern algebraic logic.


This statistic is named after Major Percy Alexander MacMahon who showed in 1913 that the distribution of the major index on all permutations of a fixed length is the same as the distribution of inversions. That is, the number of permutations of length n with k inversions is the same as the number of permutations of length n with major index equal to k. (These numbers are known as Mahonian numbers, also in honor of MacMahon.)


In probability theory, the van den Berg–Kesten (BK) inequality or van den Berg–Kesten–Reimer (BKR) inequality states that the probability for two random events to both happen, and at the same time one can find "disjoint certificates" to show that they both happen, is at most the product of their individual probabilities. The special case for two monotone events (the notion as used in the FKG inequality) was first proved by van den Berg and Kesten in 1985, who also conjectured that the inequality holds in general, not requiring monotonicity. Reimer later proved this conjecture. : 159: 44 The inequality is applied to probability spaces with a product structure, such as in percolation problems. : 829


In the case of constituency tree, a subtree is defined as a node and all its children (e.g., ] ]] is a subtree of the two trees). Terminals are not considered subtree (e.g.  is not a subtree). The subtree kernel count the number of common subtrees between two given trees. In this example, there are seven common subtrees: ] ]], ] ]], ], ], ], ] (counted twice as it appears twice).


In mathematics, power iteration (also known as the power method) is an eigenvalue algorithm: given a diagonalizable matrix A {\displaystyle A} , the algorithm will produce a number λ {\displaystyle \lambda } , which is the greatest (in absolute value) eigenvalue of A {\displaystyle A} , and a nonzero vector v {\displaystyle v} , which is a corresponding eigenvector of λ {\displaystyle \lambda } , that is, A v = λ v {\displaystyle Av=\lambda v} . The algorithm is also known as the Von Mises iteration.Power iteration is a very simple algorithm, but it may converge slowly. The most time-consuming operation of the algorithm is the multiplication of matrix A {\displaystyle A} by a vector, so it is effective for a very large sparse matrix with appropriate implementation.


In the Japanese kantenji braille, the standard 8-dot Braille patterns 267, 1267, 2467, and 12467 are the patterns related to Braille pattern dots-135, since the two additional dots of kantenji patterns 0135, 1357, and 01357 are placed above the base 6-dot cell, instead of below, as in standard 8-dot braille.


From the definition of the square root, we have that b × b = b {\displaystyle {\sqrt {b}}\times {\sqrt {b}}=b} . Therefore, the exponent r {\displaystyle r} must be such that b r × b r = b {\displaystyle b^{r}\times b^{r}=b} . Using the fact that multiplying makes exponents add gives b r + r = b {\displaystyle b^{r+r}=b} .


In mathematics, anticommutativity is a specific property of some non-commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the inverse of the result with unswapped arguments. The notion inverse refers to a group structure on the operation's codomain, possibly with another operation. Subtraction is an anticommutative operation because commuting the operands of a − b gives b − a = −(a − b); for example, 2 − 10 = −(10 − 2) = −8. Another prominent example of an anticommutative operation is the Lie bracket. In mathematical physics, where symmetry is of central importance, these operations are mostly called antisymmetric operations, and are extended in an associative setting to cover more than two arguments.


We conclude that supp  ( δ p ) {\displaystyle \operatorname {supp} (\delta _{p})} is the closure of the singleton set { p } , {\displaystyle \{p\},} which is { p } {\displaystyle \{p\}} itself. In fact, a measure μ {\displaystyle \mu } on the real line is a Dirac measure δ p {\displaystyle \delta _{p}} for some point p {\displaystyle p} if and only if the support of μ {\displaystyle \mu } is the singleton set { p } . {\displaystyle \{p\}.} Consequently, Dirac measure on the real line is the unique measure with zero variance (provided that the measure has variance at all).


In some cases, a polygon has to be covered not with arbitrary rectangles but with rectangles from a finite family.


In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily the elementary arithmetic multiplication): x·y, or simply xy, denotes the result of applying the semigroup operation to the ordered pair (x, y). Associativity is formally expressed as that (x·y)·z = x·(y·z) for all x, y and z in the semigroup. Semigroups may be considered a special case of magmas, where the operation is associative, or as a generalization of groups, without requiring the existence of an identity element or inverses.


In telephony, a special information tone (SIT) is an in-band international standard call progress tone consisting of three rising tones indicating a call has failed. It usually precedes a recorded announcement describing the problem.Because the SIT is well known in many countries, callers can understand that their call has failed, even though they do not understand the language of the recorded announcement (e.g., when calling internationally) instead of assuming the recording is voicemail or some other intended function. Like a dial tone or busy signal, the SIT is an in-band signal intended both to be heard by the caller, and to be detected by automated dialing equipment to determine a call has failed. In North America, the AT&T/Bellcore SIT standard allows the frequency and duration of the tones to vary slightly - making eight distinct messages specifically for automated equipment; indicating not only a failed call, but also the specific reason for the failure (e.g., disconnected number, busy circuits, dialing error, etc.).


There exists a dedicated group of enthusiasts who use TVRO (TV receive-only) gear such as satellite dishes to peek in on backhaul signals that are available on any of the dozens of broadcast satellites that are visible from almost any point on Earth. In its early days, their hobby was strengthened by the fact that most backhaul was analog and in the clear (unscrambled or unencrypted) which made for a vast smorgasbord of free television available for the technically inclined amateur. In recent years, full-time content and cable channels have added encryption and conditional access, and occasional signals are steadily becoming digital, which has had a deleterious effect on the hobby.


In summer, the so-called “fast” search robot was launched, working in parallel with the actual pages intended for indexing. The base of the "fast robot" is updated every 1.5–2 hours.The ranking algorithm has been improved to increase search accuracy.Search capabilities have been expanded with the help of Yandex. Dictionaries ”and“ Yandex.


Kellerer and Woeginger study a variant in which there are at most 3*m jobs, and each machine must contain at most 3 jobs (this can be seen as a generalization of 3-partition problem). They show that MLPT attains at most ( 4 m − 1 ) / ( 3 m ) {\displaystyle (4m-1)/(3m)} of the minimum largest sum, which the same approximation ratio that LPT attains for the unconstrained problem. The bound is tight for MLPT.


A field with a single derivation operator is called an ordinary differential field; if there is a finite set containing several commuting derivation operators the field is called a partial differential field. Here differential operators with derivatives ∂ x = ∂ ∂ x {\textstyle \partial _{x}={\frac {\partial }{\partial x}}} and ∂ y = ∂ ∂ y {\textstyle \partial _{y}={\frac {\partial }{\partial y}}} with coefficients from some differential field are considered. Its elements have the form ∑ i , j r i , j ( x , y ) ∂ x i ∂ y j {\textstyle \sum _{i,j}r_{i,j}(x,y)\partial _{x}^{i}\partial _{y}^{j}} ; almost all coefficients r i , j {\displaystyle r_{i,j}} are zero.


In mathematical logic, Rosser's trick is a method for proving Gödel's incompleteness theorems without the assumption that the theory being considered is ω-consistent (Smorynski 1977, p. 840; Mendelson 1977, p. 160). This method was introduced by J. Barkley Rosser in 1936, as an improvement of Gödel's original proof of the incompleteness theorems that was published in 1931. While Gödel's original proof uses a sentence that says (informally) "This sentence is not provable", Rosser's trick uses a formula that says "If this sentence is provable, there is a shorter proof of its negation".


In phonetics, a continuant is a speech sound produced without a complete closure in the oral cavity. By one defintion, continuant is a distinctive feature that refers to any sound produced with an incomplete closure of the vocal tract, thus encompassing all sounds (including vowels) except stops, affricates and nasals. By another definition, it refers exclusively to consonantal sounds produced with an incomplete closure of the oral cavity, prototypically approximants and fricatives, but sometimes also trills.Compare sonorants (resonants), a class of speech sounds which includes vowels, approximants and nasals (but not fricatives), and contrasts with obstruents.


Joan Clarke was also one such woman who made immense contributions to computing during World War II. She was a British mathematician and codebreaker who worked at Bletchley Park on breaking codes generated by Enigma machines, eventually developing Alan Turing's bombe technology to aid in deciphering complex Nazi messages.


In probability theory and information theory, adjusted mutual information, a variation of mutual information may be used for comparing clusterings. It corrects the effect of agreement solely due to chance between clusterings, similar to the way the adjusted rand index corrects the Rand index. It is closely related to variation of information: when a similar adjustment is made to the VI index, it becomes equivalent to the AMI. The adjusted measure however is no longer metrical.


In the external memory model, the number of memory transfers it needs to perform a sort of N {\displaystyle N} items on a machine with cache of size Z {\displaystyle Z} and cache lines of length L {\displaystyle L} is O ( N L log Z  N ) {\displaystyle O\left({\tfrac {N}{L}}\log _{Z}N\right)} , under the tall cache assumption that Z = Ω ( L 2 ) {\displaystyle Z=\Omega (L^{2})} . This number of memory transfers has been shown to be asymptotically optimal for comparison sorts. Funnelsort also achieves the asymptotically optimal runtime complexity of Θ ( N log  N ) {\displaystyle \Theta (N\log N)} .


Later authors such as Diana L. Paxson and Freya Aswynn follow Blum (1989) in drawing a direct correlation between runic divination and tarot divination. They may discuss runes in the context of "spreads" and advocate the usage of "rune cards". Modern authors like Ralph Blum sometimes include a "blank rune" in their sets. Some were to replace a lost rune, but according to Ralph Blum this was the god Odin's rune, the rune of the beginning and the end, representing "the divine in all human transactions".


In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts. The exponential distribution is not the same as the class of exponential families of distributions. This is a large class of probability distributions that includes the exponential distribution as one of its members, but also includes many other distributions, like the normal, binomial, gamma, and Poisson distributions.


In order to mount a brute-force or dictionary based WPA password cracking attack on a WiFi user with WPA or WPA2 enabled, a hacker must first sniff the WPA 4-way handshake. The user can be elicited to provide this sequence by first forcing them offline with the deauthentication attack.In a similar phishing style attack without password cracking, Wifiphisher starts with a deauthentication attack to disconnect the user from their legitimate base station, then mounts a man-in-the-middle attack to collect passwords supplied by an unwitting user.


Fuzzy logic is increasingly employed in diagnostic and medical equipment capable of measuring gradations of a condition. In information services, fuzzy concepts are frequently encountered because a customer or client asks a question about something which could be interpreted in different ways, or, a document is transmitted of a type or meaning which cannot be easily allocated to a known type or category, or to a known procedure. It might take considerable inquiry to "place" the information, or establish in what framework it should be understood.


These weights should exceed the weights of all existing matchings, to prevent appearance of artificial edges in the possible solution. As shown by Mulmuley, Vazirani and Vazirani, the problem of minimum weight perfect matching is converted to finding minors in the adjacency matrix of a graph. Using the isolation lemma, a minimum weight perfect matching in a graph can be found with probability at least 1⁄2. For a graph with n vertices, it requires O ( log 2  ( n ) ) {\displaystyle O(\log ^{2}(n))} time.


In terms of structure, a triglyph may be carved from a single block with a metope, or the triglyph block may have slots cut into it to allow a separately cut metope (in stone or wood) to be slid into place, as at the Temple of Aphaea. Of the two groups of 6th-century metopes from Foce del Sele, now in the museum at Paestum, the earlier uses the first method, the later the second. There may be some variation in design within a single structure to allow for corner contraction, an adjustment of the column spacing and arrangement of the Doric frieze in a temple to make the design appear more harmonious. In the evolution of the Doric order, the placing of the triglyphs evolved somewhat, especially at corners.


In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of n elements is called the alternating group of degree n, or the alternating group on n letters and denoted by An or Alt(n).


In telecommunication, signaling is the use of signals for controlling communications. This may constitute an information exchange concerning the establishment and control of a telecommunication circuit and the management of the network.


The technique is known as a "trivial hash function" or, when used specifically for branch tables, "double dispatch". For this to be feasible, the range of all possible values of the data needs to be small (e.g. an ASCII or EBCDIC character value which have a range of hexadecimal '00' – 'FF'. If the actual range is guaranteed to be smaller than this, the array can be truncated to less than 256 bytes).


There are several algorithms for subband tree structuring that find a set of optimal bases that provide the most desirable representation of the data relative to a particular cost function (entropy, energy compaction, etc.). There were relevant studies in signal processing and communications fields to address the selection of subband trees (orthogonal basis) of various kinds, e.g. regular, dyadic, irregular, with respect to performance metrics of interest including energy compaction (entropy), subband correlations and others. Discrete wavelet transform theory (continuous in the time variable) offers an approximation to transform discrete (sampled) signals. In contrast, the discrete-time subband transform theory enables a perfect representation of already sampled signals.


In mathematics, Legendre's three-square theorem states that a natural number can be represented as the sum of three squares of integers n = x 2 + y 2 + z 2 {\displaystyle n=x^{2}+y^{2}+z^{2}} if and only if n is not of the form n = 4 a ( 8 b + 7 ) {\displaystyle n=4^{a}(8b+7)} for nonnegative integers a and b. The first numbers that cannot be expressed as the sum of three squares (i.e. numbers that can be expressed as n = 4 a ( 8 b + 7 ) {\displaystyle n=4^{a}(8b+7)} ) are 7, 15, 23, 28, 31, 39, 47, 55, 60, 63, 71 ... (sequence A004215 in the OEIS).


In the Unified Modeling Language (UML), an object diagram focuses on some particular set of objects and attributes, and the links between these instances. A correlated set of object diagrams provides insight into how an arbitrary view of a system is expected to evolve over time. Early UML specifications described object diagrams as such: "An object diagram is a graph of instances, including objects and data values. A static object diagram is an instance of a class diagram; it shows a snapshot of the detailed state of a system at a point in time.


In mathematics, a uniform matroid is a matroid in which the independent sets are exactly the sets containing at most r elements, for some fixed integer r. An alternative definition is that every permutation of the elements is a symmetry.


Then, every process—including the X Server—should be able to command the kernel to perform mode-setting operations, and the kernel would ensure that concurrent operations don't result in an inconsistent state. The new kernel API and code added to the DRM module to perform these mode-setting operations was called Kernel Mode-Setting (KMS).Kernel Mode-Setting provides several benefits. The most immediate is of course the removal of duplicate mode-setting code, from both the kernel (Linux console, fbdev) and user space (X Server DDX drivers).


More generally, recall is simply the complement of the type II error rate (i.e., one minus the type II error rate). Precision is related to the type I error rate, but in a slightly more complicated way, as it also depends upon the prior distribution of seeing a relevant vs. an irrelevant item. The above cat and dog example contained 8 − 5 = 3 type I errors (false positives) out of 10 total cats (true negatives), for a type I error rate of 3/10, and 12 − 5 = 7 type II errors, for a type II error rate of 7/12. Precision can be seen as a measure of quality, and recall as a measure of quantity. Higher precision means that an algorithm returns more relevant results than irrelevant ones, and high recall means that an algorithm returns most of the relevant results (whether or not irrelevant ones are also returned).


The presence of an increasing amount of on-line data will necessitate greater use of automatic data processing in decision-making. Saving data on multiple memory locations will guarantee the safekeeping of the acts. Statistical data will be available in real time and under multiple profiles, with great benefits for top level decision-making.


In order to increase traffic capacity and safety, a route may have two or more separate roads for each direction of traffic. Alternatively, a given road might be declared one-way.


In probability and statistics, the class of exponential dispersion models (EDM) is a set of probability distributions that represents a generalisation of the natural exponential family. Exponential dispersion models play an important role in statistical theory, in particular in generalized linear models because they have a special structure which enables deductions to be made about appropriate statistical inference.


Implicit regularization is essentially ubiquitous in modern machine learning approaches, including stochastic gradient descent for training deep neural networks, and ensemble methods (such as random forests and gradient boosted trees).In explicit regularization, independent of the problem or model, there is always a data term, that corresponds to a likelihood of the measurement and a regularization term that corresponds to a prior. By combining both using Bayesian statistics, one can compute a posterior, that includes both information sources and therefore stabilizes the estimation process. By trading off both objectives, one chooses to be more addictive to the data or to enforce generalization (to prevent overfitting).


The manuscripts available are of variable quality, and invariably incomplete. By careful analysis of the translations and originals, hypotheses have been made about the contents of the original text (copies of which are no longer available). Ancient texts which refer to the Elements itself, and to other mathematical theories that were current at the time it was written, are also important in this process.


Such procedures are often implemented in various convenience routines such as daemon(3) in Unix. Systems often start daemons at boot time that will respond to network requests, hardware activity, or other programs by performing some task. Daemons such as cron may also perform defined tasks at scheduled times.


In natural languages, the meaning of a complex spoken sentence can be understood by decomposing it into smaller lexical segments (roughly, the words of the language), associating a meaning to each segment, and combining those meanings according to the grammar rules of the language. Though lexical recognition is not thought to be used by infants in their first year, due to their highly limited vocabularies, it is one of the major processes involved in speech segmentation for adults. Three main models of lexical recognition exist in current research: first, whole-word access, which argues that words have a whole-word representation in the lexicon; second, decomposition, which argues that morphologically complex words are broken down into their morphemes (roots, stems, inflections, etc.) and then interpreted and; third, the view that whole-word and decomposition models are both used, but that the whole-word model provides some computational advantages and is therefore dominant in lexical recognition.To give an example, in a whole-word model, the word "cats" might be stored and searched for by letter, first "c", then "ca", "cat", and finally "cats". The same word, in a decompositional model, would likely be stored under the root word "cat" and could be searched for after removing the "s" suffix.


In software engineering, continuous integration (CI) is the practice of merging all developers' working copies to a shared mainline several times a day. Nowadays it is typically implemented in such a way that it triggers an automated build with testing. Grady Booch first proposed the term CI in his 1991 method, although he did not advocate integrating several times a day. Extreme programming (XP) adopted the concept of CI and did advocate integrating more than once per day – perhaps as many as tens of times per day.


In mathematical optimization, fractional programming is a generalization of linear-fractional programming. The objective function in a fractional program is a ratio of two functions that are in general nonlinear. The ratio to be optimized often describes some kind of efficiency of a system.


In order to find and communicate with clients in proximity of a device, the protocol makes use of both the server and client modes of Bluetooth LE, switching between the two frequently. In server mode the device advertises its EphID to be read by clients, with clients scanning for servers. When a client and server meet, the client reads the EphID and subsequently writes its own EphID to the server. The two devices then store the encounter in their respective contact logs in addition to a coarse timestamp and signal strength. The signal strength is later used as part of the infection reporting process to estimate the distance between an infected patient and the user.


Note that the polynomial is a q-analogue of the hook length formula. Furthermore, let λ be a rectangular partition of size n, and let X be the set of semi-standard Young tableaux of shape λ. Let C = Z/kZ act on X via k-promotion. Then ( X , C , q − κ ( λ ) s λ ( 1 , q , q 2 , … , q k − 1 ) ) {\displaystyle (X,C,q^{-\kappa (\lambda )}s_{\lambda }(1,q,q^{2},\dotsc ,q^{k-1}))} exhibit the cyclic sieving phenomenon.


In modern protocol design, protocols are layered to form a protocol stack. Layering is a design principle that divides the protocol design task into smaller steps, each of which accomplishes a specific part, interacting with the other parts of the protocol only in a small number of well-defined ways. Layering allows the parts of a protocol to be designed and tested without a combinatorial explosion of cases, keeping each design relatively simple. The communication protocols in use on the Internet are designed to function in diverse and complex settings.


In mathematics, especially in the area of abstract algebra dealing with ordered structures on abelian groups, the Hahn embedding theorem gives a simple description of all linearly ordered abelian groups. It is named after Hans Hahn.


In number theory, Büchi's problem, also known as the n squares' problem, is an open problem named after the Swiss mathematician Julius Richard Büchi. It asks whether there is a positive integer M such that every sequence of M or more integer squares, whose second difference is constant and equal to 2, is necessarily a sequence of squares of the form (x + i)2, i = 1, 2, ..., M,... for some integer x. In 1983, Douglas Hensley observed that Büchi's problem is equivalent to the following: Does there exist a positive integer M such that, for all integers x and a, the quantity (x + n)2 + a cannot be a square for more than M consecutive values of n, unless a = 0?


In practical applications, one would have several BDKs on record, possibly for different customers, or to contain the scope of key compromise. When processing transactions, it is important for the receiver to know which BDK was used to initialize the originating device. To achieve this, the 80-bit KSN is structured into three parts: as Key Set ID, a TRSM ID, and the transaction counter. The algorithm specifies that the transaction counter is 21-bits, but treats the remaining 59 bits opaquely (the algorithm only specifies that unused bits be 0-padded to a nibble boundary, and then 'f' padded to the 80-bit boundary).


In the above experiment, participants were presented with a list of 100 familiar words and were asked to read them aloud while simultaneously trying to remember each one. Subsequent to this, participants were asked to make a recognition decision based on the number of "yes" responses that were accompanied by some recollective experience. The results demonstrate the differing relationships between the "yes" and "no" conditions and "remember" and "know" memory performance. The outcome confirms that although familiarity and recollection may involve different processes, the remember/know exemplar does not probe them directly.


(This step is the same as in Factor analysis). Estimate the Discriminant Function Coefficients and determine the statistical significance and validity—Choose the appropriate discriminant analysis method. The direct method involves estimating the discriminant function so that all the predictors are assessed simultaneously.


First, they define a set of items that are called small. Let T be the total size of all small items. Then, they construct a matrix A representing all configurations with sum < 2.


In mathematics, a regular matroid is a matroid that can be represented over all fields.


In studies of the networks of citations between scientific papers, Derek de Solla Price showed in 1965 that the number of links to papers—i.e., the number of citations they receive—had a heavy-tailed distribution following a Pareto distribution or power law, and thus that the citation network is scale-free. He did not however use the term "scale-free network", which was not coined until some decades later. In a later paper in 1976, Price also proposed a mechanism to explain the occurrence of power laws in citation networks, which he called "cumulative advantage" but which is today more commonly known under the name preferential attachment. Recent interest in scale-free networks started in 1999 with work by Albert-László Barabási and Réka Albert at the University of Notre Dame who mapped the topology of a portion of the World Wide Web, finding that some nodes, which they called "hubs", had many more connections than others and that the network as a whole had a power-law distribution of the number of links connecting to a node.


In the early 1980s, NLP was advertised as an important advance in psychotherapy and counseling, and attracted some interest in counseling research and clinical psychology. However, as controlled trials failed to show any benefit from NLP and its advocates made increasingly dubious claims, scientific interest in NLP faded.Numerous literature reviews and meta-analyses have failed to show evidence for NLP's assumptions or effectiveness as a therapeutic method. While some NLP practitioners have argued that the lack of empirical support is due to insufficient research which tests NLP, the consensus scientific opinion is that NLP is pseudoscience and that attempts to dismiss the research findings based on these arguments "s an admission that NLP does not have an evidence base and that NLP practitioners are seeking a post-hoc credibility.


The equation a 2 = c + d 2 {\displaystyle a^{2}=c+d^{2}} shows that |a| > √c. Thus, if the nested radical is real, and if denesting is possible, then a > 0. Then the solution is


In mathematics, the Chevalley–Shephard–Todd theorem in invariant theory of finite groups states that the ring of invariants of a finite group acting on a complex vector space is a polynomial ring if and only if the group is generated by pseudoreflections. In the case of subgroups of the complex general linear group the theorem was first proved by G. C. Shephard and J. A. Todd (1954) who gave a case-by-case proof. Claude Chevalley (1955) soon afterwards gave a uniform proof. It has been extended to finite linear groups over an arbitrary field in the non-modular case by Jean-Pierre Serre.


Different kinds of Kolmogorov complexity are studied: the uniform complexity, prefix complexity, monotone complexity, time-bounded Kolmogorov complexity, and space-bounded Kolmogorov complexity. An axiomatic approach to Kolmogorov complexity based on Blum axioms (Blum 1967) was introduced by Mark Burgin in the paper presented for publication by Andrey Kolmogorov. The axiomatic approach encompasses other approaches to Kolmogorov complexity.


String metrics and edit distances. There are many ways of measuring distances between strings of characters, which may represent sentences in computational linguistics or code words in coding theory. Edit distances attempt to measure the number of changes necessary to get from one string to another.


This little program prints three “1”s to the right, reverses direction and moves left printing 0’s until it hits a blank. We will print all the symbols that our machine uses: Here at the end we find that a blank on the left has “come into play” so we leave it as part of the total configuration. Given that we have done our job correctly, we add the starting conditions and see “where the theorem goes”.


In naive set theory, a set is described as a well-defined collection of objects. These objects are called the elements or members of the set. Objects can be anything: numbers, people, other sets, etc. For instance, 4 is a member of the set of all even integers. Clearly, the set of even numbers is infinitely large; there is no requirement that a set be finite. The definition of sets goes back to Georg Cantor. He wrote in his 1915 article Beiträge zur Begründung der transfiniten Mengenlehre: “Unter einer 'Menge' verstehen wir jede Zusammenfassung M von bestimmten wohlunterschiedenen Objekten unserer Anschauung oder unseres Denkens (welche die 'Elemente' von M genannt werden) zu einem Ganzen.” – Georg Cantor “A set is a gathering together into a whole of definite, distinct objects of our perception or of our thought—which are called elements of the set.” – Georg Cantor


In mathematics, counterexamples are often used to prove the boundaries of possible theorems. By using counterexamples to show that certain conjectures are false, mathematical researchers can then avoid going down blind alleys and learn to modify conjectures to produce provable theorems. It is sometimes said that mathematical development consists primarily in finding (and proving) theorems and counterexamples.


In recent years many other matrix factorization models have been developed to exploit the ever increasing amount and variety of available interaction data and use cases. Hybrid matrix factorization algorithms are capable of merging explicit and implicit interactions or both content and collaborative data


Excessive use of whitespace, especially trailing whitespace at the end of lines, is considered a nuisance. However correct use of whitespace can make the code easier to read and help group related logic. Most languages only recognize ASCII characters as whitespace, or in some cases Unicode newlines as well, but not most of the characters listed above. The C language defines whitespace characters to be "space, horizontal tab, new-line, vertical tab, and form-feed". The HTTP network protocol requires different types of whitespace to be used in different parts of the protocol, such as: only the space character in the status line, CRLF at the end of a line, and "linear whitespace" in header values.


0&0&1&0&0\\0&0&0&0&1\\0&0&0&1&0\end{pmatrix}}.}


RANAP handles signaling for the Iu-PS - RNC and 3G SGSN and Iu-CS - RNC and 3G MSC . It also provides the signaling channel to transparently pass messages between the User Equipment (UE) and the CN. In LTE, RANAP has been replaced by S1AP. In SA (standalone) installations of 5G, S1AP will be replaced by NGAP.


Therefore, a single recursive call locates the desired element in the correct partition, and we build upon this for quickselect: // Returns the k-th smallest element of list within left..right inclusive // (i.e. left <= k <= right). function select(list, left, right, k) is if left = right then // If the list contains only one element, return list // return that element pivotIndex := ... // select a pivotIndex between left and right, // e.g., left + floor(rand() % (right − left + 1)) pivotIndex := partition(list, left, right, pivotIndex) // The pivot is in its final sorted position if k = pivotIndex then return list else if k < pivotIndex then return select(list, left, pivotIndex − 1, k) else return select(list, pivotIndex + 1, right, k) Note the resemblance to quicksort: just as the minimum-based selection algorithm is a partial selection sort, this is a partial quicksort, generating and partitioning only O ( log  n ) {\displaystyle O(\log n)} of its O ( n ) {\displaystyle O(n)} partitions. This simple procedure has expected linear performance, and, like quicksort, has quite good performance in practice. It is also an in-place algorithm, requiring only constant memory overhead if tail call optimization is available, or if eliminating the tail recursion with a loop: function select(list, left, right, k) is loop if left = right then return list pivotIndex := ... // select pivotIndex between left and right pivotIndex := partition(list, left, right, pivotIndex) if k = pivotIndex then return list else if k < pivotIndex then right := pivotIndex − 1 else left := pivotIndex + 1


, AN) with A0 = 0. Then the largest stopping time to exercise the American option in an optimal way is τ max := { N if A N = 0 , min { n ∈ { 0 , … , N − 1 } ∣ A n + 1 < 0 } if A N < 0. {\displaystyle \tau _{\text{max}}:={\begin{cases}N&{\text{if }}A_{N}=0,\\\min\{n\in \{0,\dots ,N-1\}\mid A_{n+1}<0\}&{\text{if }}A_{N}<0.\end{cases}}} Since A is predictable, the event {τmax = n} = {An = 0, An+1 < 0} is in Fn for every n ∈ {0, 1, .


In terms of the number of comparisons, the performance of binary search can be analyzed by viewing the run of the procedure on a binary tree. The root node of the tree is the middle element of the array. The middle element of the lower half is the left child node of the root, and the middle element of the upper half is the right child node of the root. The rest of the tree is built in a similar fashion.


{\displaystyle \mathbb {N} .} N {\displaystyle \mathbb {N} } is the intersection of all sets satisfying (1) and (2).There are many sets that satisfy (1) and (2) – for example, the set {1, 1.649, 2, 2.649, 3, 3.649, …} satisfies the definition. However, condition (3) specifies the set of natural numbers by removing the sets with extraneous members.


In the early 2000s, some Web developers began to question why Web authors ever made the leap into authoring in XHTML. Others countered that the problems ascribed to the use of XHTML could mostly be attributed to two main sources: the production of invalid XHTML documents by some Web authors and the lack of support for XHTML built into Internet Explorer 6. They went on to describe the benefits of XML-based Web documents (i.e. XHTML) regarding searching, indexing, and parsing as well as future-proofing the Web itself.


It is basically testing how well a system recovers from crashes, hardware failures, or other catastrophic problems Examples of recovery testing: While an application is running, suddenly restart the computer, and afterwards check the validness of the application's data integrity. While an application is receiving data from a network, unplug the connecting cable. After some time, plug the cable back in and analyze the application's ability to continue receiving data from the point at which the network connection disappeared.


It is the part of the observed score that would recur across different measurement occasions in the absence of error. Errors of measurement are composed of both random error and systematic error. It represents the discrepancies between scores obtained on tests and the corresponding true scores. This conceptual breakdown is typically represented by the simple equation: Observed test score = true score + errors of measurement


In probability theory, a stochastic process is said to have stationary increments if its change only depends on the time span of observation, but not on the time when the observation was started. Many large families of stochastic processes have stationary increments either by definition (e.g. Lévy processes) or by construction (e.g. random walks)


In the following, a short introduction to input-output analysis and its environmental extension for the calculation of material footprints or RME indicators is provided. The inter-industry flows within an economy form an n×n matrix Z and the total output of each industry forms an n×1 vector x. By dividing each flow into an industry (i.e., each element of Z) by the total output of that same industry, we obtain an n×n matrix of so-called technical coefficients A. In matrix algebra, this reads as follows: A = Z × x ^ − 1 {\displaystyle A=Z\times {\hat {x}}^{-1}} where: x ^ {\displaystyle {\hat {x}}} represents the vector x diagonlized into a matrix ( x ^ = I x → {\displaystyle {\hat {x}}=I{\vec {x}}} )Matrix A contains the multipliers for the inter-industry inputs required to supply one unit of industry output. A certain total economic output x is required to satisfy a given level of final demand y. This final demand may be domestic (for private households as well as the public sector) or foreign (exports) and can be written as an n×1 vector. When this vector of final demand y is multiplied by the Leontief inverse (I−A)−1, we obtain total output x. I is the identity matrix so that the following matrix equation is the result of equivalence operations in our previous equation: x → = ( I − A ) − 1 × y → {\displaystyle {\vec {x}}=\left(I-A\right)^{-1}\times {\vec {y}}} The Leontief inverse contains the multipliers for the direct and indirect inter-industry inputs required to provide 1 unit of output to final demand.


Although these examples offer alternate strategies for achieving the same abstraction, they do not fundamentally alter the need to support abstract nouns in code – all programming relies on an ability to abstract verbs as functions, nouns as data structures, and either as processes. Consider for example a sample Java fragment to represent some common farm "animals" to a level of abstraction suitable to model simple aspects of their hunger and feeding. It defines an Animal class to represent both the state of the animal and its functions: With the above definition, one could create objects of type Animal and call their methods like this: In the above example, the class Animal is an abstraction used in place of an actual animal, LivingThing is a further abstraction (in this case a generalisation) of Animal.


In software development or backward compatibility is a general notion of interoperation between software pieces that will not produce any errors when its functionality is invoked via API. The software is considered stable when its API that is used to invoke functions is stable across different versions. In operating systems upgraded to a newer versions are said to be backward compatible if executable and other files from previous versions will work as usual.In compilers backward compatibility may refer to the ability of a compiler of a newer version of the language to accept source code of programs or data that worked under the previous version.A data format is said to be backward compatible when a newer version of program that can open it opens it without errors just like its predecessor.


The Jordan–Hölder theorem is a more precise way of stating this fact about finite groups. However, a significant difference from integer factorization is that such "building blocks" do not necessarily determine a unique group, since there might be many non-isomorphic groups with the same composition series or, put in another way, the extension problem does not have a unique solution. Gorenstein (d.1992), Lyons, and Solomon are gradually publishing a simplified and revised version of the proof.


In phylogenetics, informative site is a term used when maximum parsimony is the optimality criterion for construction of a phylogenetic tree. It refers to a characteristic for which the number of character-state evolutionary changes of at this site depends on the topology of the tree. The charactetistics can take on multiple types of data, including morphological (such as the presence of wings, tentacles, etc.) or molecular information such as sequences of DNA or proteins. The informative site is a position in the relevant set of aligned sequences at which there are at least two different character states and each of those states occurs in at least two of the sequences.


In the encoded format a "1" bit indicates a flux transition, while a "0" indicates that the magnetic field on the disk does not change for that time interval.


In science and engineering, a system is the part of the universe that is being studied, while the environment is the remainder of the universe that lies outside the boundaries of the system. It is also known as the surroundings or neighborhood, and in thermodynamics, as the reservoir. Depending on the type of system, it may interact with the environment by exchanging mass, energy (including heat and work), linear momentum, angular momentum, electric charge, or other conserved properties. In some disciplines, such as information theory, information may also be exchanged. The environment is ignored in analysis of the system, except in regard to these interactions.


Advantages of this approach include the general increased quality of the data returned in searches and with proper tagging, ontologies finding entries that may not explicitly state the search term but are still relevant. One disadvantage of this approach is that the results that are returned come in the format of the database of their origin and as such, direct comparisons may be difficult. Another problem is that the terms used in tagging and searching can sometimes be ambiguous and may cause confusion among the results.


In telecommunications, multiple access schemes are orthogonal when an ideal receiver can completely reject arbitrarily strong unwanted signals from the desired signal using different basis functions. One such scheme is time-division multiple access (TDMA), where the orthogonal basis functions are nonoverlapping rectangular pulses ("time slots").


It was later argued that it has strong biological motivations and mathematical justifications. In 2011 it was found to enable better training of deeper networks, compared to the widely used activation functions prior to 2011, e.g., the logistic sigmoid (which is inspired by probability theory; see logistic regression) and its more practical counterpart, the hyperbolic tangent. The rectifier is, as of 2017, the most popular activation function for deep neural networks.Rectified linear units find applications in computer vision and speech recognition using deep neural nets and computational neuroscience.


In order to understand morphogenetic events, i.e. the growth and shaping of tissues and organs, it is necessary to analyze the packing of cells into tissues. In that context, an analysis of patterning processes can help to identify the underlying mechanisms that drive morphogenesis. == References ==


In the 1990s, there was a general confusion about the proper character encoding to use to write text in Latin Croatian on computers. An attempt was made to apply the 7-bit "YUSCII", later "CROSCII", which included the five letters with diacritics at the expense of five non-letter characters (, {, }, @), but it was ultimately unsuccessful. Because the ASCII character @ sorts before A, this led to jokes calling it žabeceda (žaba=frog, abeceda=alphabet). Other short-lived vendor-specific efforts were also undertaken.


Previous work includes the work of D. Coppersmith about the DLP in fields of characteristic two. The discrete logarithm problem in a finite field consists of solving the equation a x = b {\displaystyle a^{x}=b} for a , b ∈ F p n {\displaystyle a,b\in \mathbb {F} _{p^{n}}} , p {\displaystyle p} a prime number and n {\displaystyle n} an integer. The function f: F p n → F p n , x ↦ a x {\displaystyle f:\mathbb {F} _{p^{n}}\to \mathbb {F} _{p^{n}},x\mapsto a^{x}} for a fixed a ∈ F p n {\displaystyle a\in \mathbb {F} _{p^{n}}} is a one-way function used in cryptography. Several cryptographic methods are based on the DLP such as the Diffie-Hellman key exchange, the El Gamal cryptosystem and the Digital Signature Algorithm.


In signal processing this process is known as discrete convolution.


In particular, the value of the coefficient of determination will shrink relative to the original data. To lessen the chance or amount of overfitting, several techniques are available (e.g., model comparison, cross-validation, regularization, early stopping, pruning, Bayesian priors, or dropout). The basis of some techniques is either (1) to explicitly penalize overly complex models or (2) to test the model's ability to generalize by evaluating its performance on a set of data not used for training, which is assumed to approximate the typical unseen data that a model will encounter.


In mathematics, a Q-matrix is a square matrix whose associated linear complementarity problem LCP(M,q) has a solution for every vector q.


Some consider it to be an anti-pattern. There are valid forms of the pattern, including the use of the volatile keyword in Java and explicit memory barriers in C++.The pattern is typically used to reduce locking overhead when implementing "lazy initialization" in a multi-threaded environment, especially as part of the Singleton pattern. Lazy initialization avoids initializing a value until the first time it is accessed.


In the context of a jump process that takes states in a state space S {\displaystyle \mathrm {S} } , consider the set of random variables ( X n , T n ) {\displaystyle (X_{n},T_{n})} , where T n {\displaystyle T_{n}} represents the jump times and X n {\displaystyle X_{n}} represents the associated states in the sequence of states (see Figure). Let the sequence of inter-arrival times τ n = T n − T n − 1 {\displaystyle \tau _{n}=T_{n}-T_{n-1}} . In order for the sequence ( X n , T n ) {\displaystyle (X_{n},T_{n})} to be considered a Markov renewal process the following condition should hold: Pr ( τ n + 1 ≤ t , X n + 1 = j ∣ ( X 0 , T 0 ) , ( X 1 , T 1 ) , … , ( X n = i , T n ) ) = Pr ( τ n + 1 ≤ t , X n + 1 = j ∣ X n = i ) ∀ n ≥ 1 , t ≥ 0 , i , j ∈ S {\displaystyle {\begin{aligned}&\Pr(\tau _{n+1}\leq t,X_{n+1}=j\mid (X_{0},T_{0}),(X_{1},T_{1}),\ldots ,(X_{n}=i,T_{n}))\\={}&\Pr(\tau _{n+1}\leq t,X_{n+1}=j\mid X_{n}=i)\,\forall n\geq 1,t\geq 0,i,j\in \mathrm {S} \end{aligned}}}


In the fields of machine learning, the theory of computation, and random matrix theory, a probability distribution over vectors is said to be in isotropic position if its covariance matrix is equal to the identity matrix.


There are two running instances of each role. In this example, the cache is distributed across all instances of the dedicated Cache1 role. A dedicated topology has the advantage of scaling the caching tier independently of any other role in the cloud service. For the best Caching performance, a dedicated topology is recommended because the role instances do not share their resources with other application code and services.


However, the values are frequently represented as 1 or 0, which corresponds to counting the number of successes in a single trial: 1 (success) or 0 (failure); see § Counting. Often, binary data is used to represent one of two conceptually opposed values, e.g.: the outcome of an experiment ("success" or "failure") the response to a yes–no question ("yes" or "no") presence or absence of some feature ("is present" or "is not present") the truth or falsehood of a proposition ("true" or "false", "correct" or "incorrect")However, it can also be used for data that is assumed to have only two possible values, even if they are not conceptually opposed or conceptually represent all possible values in the space. For example, binary data is often used to represent the party choices of voters in elections in the United States, i.e. Republican or Democratic.


In order to explain column-level encryption it is important to outline basic database structure. A typical relational database is divided into tables that are divided into columns that each have rows of data. Whilst TDE usually encrypts an entire database, column-level encryption allows for individual columns within a database to be encrypted. It is important to establish that the granularity of column-level encryption causes specific strengths and weaknesses to arise when compared to encrypting an entire database.


Formally, a sequence of numbers s 0 , s 1 , s 2 , s 3 , … {\displaystyle s_{0},s_{1},s_{2},s_{3},\ldots } is constant-recursive if it satisfies a recurrence relation where c i {\displaystyle c_{i}} are constants. For example, the Fibonacci sequence satisfies the recurrence relation F n = F n − 1 + F n − 2 , {\displaystyle F_{n}=F_{n-1}+F_{n-2},} where F n {\displaystyle F_{n}} is the n {\displaystyle n} th Fibonacci number. Constant-recursive sequences are studied in combinatorics and the theory of finite differences.


In his formal published treatises, Archimedes solved the same problem using the method of exhaustion. The 15th century saw the work of Nicholas of Cusa, further developed in the 17th century by Johannes Kepler, in particular, the calculation of the area of a circle by representing the latter as an infinite-sided polygon. Simon Stevin's work on the decimal representation of all numbers in the 16th century prepared the ground for the real continuum.


In order to evaluate the outcomes of a program, the evaluator first needs to monitor the process in order to assess the implementation of the intervention. The reason for this is that many program failures are due to failures in the implementation of the program. Therefore, in order to determine whether or not the planned outcomes have been reached, the evaluator needs to assess how the intervention was implemented.


Computer manufacturers who had a significant business in the defense, aerospace, or related industries, also offered Ada compilers and tools on their platforms; these included Concurrent Computer Corporation, Cray Research, Inc., Digital Equipment Corporation, Harris Computer Systems, and Siemens Nixdorf Informationssysteme AG.In 1991, the US Department of Defense began to require the use of Ada (the Ada mandate) for all software, though exceptions to this rule were often granted. The Department of Defense Ada mandate was effectively removed in 1997, as the DoD began to embrace commercial off-the-shelf (COTS) technology. Similar requirements existed in other NATO countries: Ada was required for NATO systems involving command and control and other functions, and Ada was the mandated or preferred language for defense-related applications in countries such as Sweden, Germany, and Canada.By the late 1980s and early 1990s, Ada compilers had improved in performance, but there were still barriers to fully exploiting Ada's abilities, including a tasking model that was different from what most real-time programmers were used to.Because of Ada's safety-critical support features, it is now used not only for military applications, but also in commercial projects where a software bug can have severe consequences, e.g., avionics and air traffic control, commercial rockets such as the Ariane 4 and 5, satellites and other space systems, railway transport and banking.


In the following examples, computed values are in bold, while Register numbers are not.For example, to write the value 3 to register 1, (which already contains a 6), and then add 7 to register 1 and store the result in register 2, i.e.: i0: R1 = 6 i1: R1 = 3 i2: R2 = R1 + 7 = 10 Following execution, register 2 should contain the value 10. However, if i1 (write 3 to register 1) does not fully exit the pipeline before i2 starts executing, it means that R1 does not contain the value 3 when i2 performs its addition. In such an event, i2 adds 7 to the old value of register 1 (6), and so register 2 contains 13 instead, i.e.: i0: R1 = 6 i2: R2 = R1 + 7 = 13 i1: R1 = 3 This error occurs because i2 reads Register 1 before i1 has committed/stored the result of its write operation to Register 1. So when i2 is reading the contents of Register 1, register 1 still contains 6, not 3.


By taking the determiner, a function word, to be head over the noun, a structure is established that is analogous to the structure of the finite clause, with a complementizer. Apart from the minimalist program, however, the DP hypothesis is rejected by most other modern theories of syntax and grammar, in part because these theories lack the relevant functional categories. Dependency grammars, for instance, almost all assume the traditional NP analysis of noun phrases. For illustrations of different analyses of noun phrases depending on whether the DP hypothesis is rejected or accepted, see the next section.


In mathematics, a residuated Boolean algebra is a residuated lattice whose lattice structure is that of a Boolean algebra. Examples include Boolean algebras with the monoid taken to be conjunction, the set of all formal languages over a given alphabet Σ under concatenation, the set of all binary relations on a given set X under relational composition, and more generally the power set of any equivalence relation, again under relational composition. The original application was to relation algebras as a finitely axiomatized generalization of the binary relation example, but there exist interesting examples of residuated Boolean algebras that are not relation algebras, such as the language example.


In mathematics, a signature matrix is a diagonal matrix whose diagonal elements are plus or minus 1, that is, any matrix of the form: A = ( ± 1 0 ⋯ 0 0 0 ± 1 ⋯ 0 0 ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 ⋯ ± 1 0 0 0 ⋯ 0 ± 1 ) {\displaystyle A={\begin{pmatrix}\pm 1&0&\cdots &0&0\\0&\pm 1&\cdots &0&0\\\vdots &\vdots &\ddots &\vdots &\vdots \\0&0&\cdots &\pm 1&0\\0&0&\cdots &0&\pm 1\end{pmatrix}}} Any such matrix is its own inverse, hence is an involutory matrix. It is consequently a square root of the identity matrix. Note however that not all square roots of the identity are signature matrices.


The distinction between data and derived value is illustrated by the information ladder. However, data has staged a comeback with the popularisation of the term big data, which refers to the collection and analyses of massive sets of data. Several organisations have established data management centers (DMC) for their operations.


One start code, one house code, and one function code is known as an X10 frame and represent the minimum components of a valid X10 data packet. Each frame is sent twice in succession to make sure the receivers understand it over any power line noise for purposes of redundancy, reliability, and to accommodate line repeaters. After allowing for retransmission, line control, etc., data rates are around 20 bit/s, making X10 data transmission so slow that the technology is confined to turning devices on and off or other very simple operations.


We also add a team node for each team and connect each game node {i, j} with i < j to V, and connects each of them from s by an edge with capacity rij – which represents the number of plays between these two teams. We also add a team node for each team and connect each game node {i, j} with two team nodes i and j to ensure one of them wins. One does not need to restrict the flow value on these edges. Finally, edges are made from team node i to the sink node t and the capacity of wk + rk – wi is set to prevent team i from winning more than wk + rk. Let S be the set of all teams participating in the league and let r ( S − { k } ) = ∑ i , j ∈ { S − { k } } i < j r i j {\displaystyle r(S-\{k\})=\sum _{i,j\in \{S-\{k\}\} \atop i


In probability theory, a branch of mathematics Poisson-Dirichlet distributions are probability distributions on the set of nonnegative, non-decreasing sequences with sum 1, depending on two parameters α ∈ [ 0 , 1 ) {\displaystyle \alpha \in [0,1)} and θ ∈ ( − α , ∞ ) {\displaystyle \theta \in (-\alpha ,\infty )} . It can be defined as follows. One considers independent random variables ( Y n ) n ≥ 1 {\displaystyle (Y_{n})_{n\geq 1}} such that Y n {\displaystyle Y_{n}} follows the beta distribution of parameters 1 − α {\displaystyle 1-\alpha } and θ + n α {\displaystyle \theta +n\alpha } . Then, the Poisson-Dirichlet distribution P D ( α , θ ) {\displaystyle PD(\alpha ,\theta )} of parameters α {\displaystyle \alpha } and θ {\displaystyle \theta } is the law of the random decreasing sequence containing Y 1 {\displaystyle Y_{1}} and the products Y n ∏ k = 1 n − 1 ( 1 − Y k ) {\displaystyle Y_{n}\prod _{k=1}^{n-1}(1-Y_{k})} . This definition is due to Jim Pitman and Marc Yor. It generalizes Kingman's law, which corresponds to the particular case α = 0 {\displaystyle \alpha =0} .


(Why do we have all these abbreviations anyway? Saving bandwidth?


In mathematics, a Kleene algebra ( KLAY-nee; named after Stephen Cole Kleene) is an idempotent (and thus partially ordered) semiring endowed with a closure operator. It generalizes the operations known from regular expressions.


(1) This is a bank. This bank accepted my identification. (2) She is a bank teller.


Created in 1978, it is still used today for applications involving digital signatures. Using number theory, the RSA algorithm selects two prime numbers, which help generate both the encryption and decryption keys.A publicly available public-key encryption application called Pretty Good Privacy (PGP) was written in 1991 by Phil Zimmermann, and distributed free of charge with source code. PGP was purchased by Symantec in 2010 and is regularly updated.


In number theory, Lucas's theorem expresses the remainder of division of the binomial coefficient ( m n ) {\displaystyle {\tbinom {m}{n}}} by a prime number p in terms of the base p expansions of the integers m and n. Lucas's theorem first appeared in 1878 in papers by Édouard Lucas.


In the classical setting, RG can be viewed as the following problem. Alice, Bob, and the referee is given some statement. Alice is trying to convince the referee that the statement is true while Bob is trying to convince the referee that the statement is false. The referee, who has limited computing power, will look at the proofs provided by Alice and Bob, ask them questions, and at the end of the day decide which player is correct (wins).


In set theory and order theory, the Cantor–Bernstein theorem states that the cardinality of the second type class, the class of countable order types, equals the cardinality of the continuum. It was used by Felix Hausdorff and named by him after Georg Cantor and Felix Bernstein. Cantor constructed a family of countable order types with the cardinality of the continuum, and in his 1901 inaugural dissertation Bernstein proved that such a family can have no higher cardinality. == References ==


In partition calculus, part of combinatorial set theory, a branch of mathematics, the Erdős–Rado theorem is a basic result extending Ramsey's theorem to uncountable sets. It is named after Paul Erdős and Richard Rado. It is sometimes also attributed to Đuro Kurepa who proved it under the additional assumption of the generalised continuum hypothesis, and hence the result is sometimes also referred to as the Erdős–Rado–Kurepa theorem.


One picks a set ι to be the type of individuals. For example, ι might be the set of natural numbers, or the set of atoms (in a set theory with atoms) or any other set one is interested in. Then if τ1,...,τm are types, the type (τ1,...,τm) is the power set of the product τ1×...×τm, which can also be thought of informally as the set of (propositional predicative) functions from this product to a 2-element set {true,false}. The ramified type (τ1,...,τm|σ1,...,σn) can be modeled as the product of the type (τ1,...,τm,σ1,...,σn) with the set of sequences of n quantifiers (∀ or ∃) indicating which quantifier should be applied to each variable σi. (One can vary this slightly by allowing the σs to be quantified in any order, or allowing them to occur before some of the τs, but this makes little difference except to the bookkeeping.)


In summary, each perspective focuses attention on the same fundamental questions, then answers those questions from that viewpoint, creating different descriptive representations (i.e., models), which translate from higher to lower perspectives. The basic model for the focus (or product abstraction) remains constant. The basic model of each column is uniquely defined, yet related across and down the matrix. In addition, the six categories of enterprise architecture components, and the underlying interrogatives that they answer, form the columns of the Zachman Framework and these are: Inventory Sets — What Process Flows — How Distribution Networks — Where Responsibility Assignments — Who Timing Cycles — When Motivation Intentions — WhyIn Zachman's opinion, the single factor that makes his framework unique is that each element on either axis of the matrix is explicitly distinguishable from all the other elements on that axis. The representations in each cell of the matrix are not merely successive levels of increasing detail, but actually are different representations — different in context, meaning, motivation, and use. Because each of the elements on either axis is explicitly different from the others, it is possible to define precisely what belongs in each cell.


Stack overflow may be difficult to avoid when using recursive procedures since many compilers assume that the recursion stack is a contiguous area of memory, and some allocate a fixed amount of space for it. Compilers may also save more information in the recursion stack than is strictly necessary, such as return address, unchanging parameters, and the internal variables of the procedure. Thus, the risk of stack overflow can be reduced by minimizing the parameters and internal variables of the recursive procedure or by using an explicit stack structure.


In telecommunications, a block check character (BCC) is a character added to a transmission block to facilitate error detection. In longitudinal redundancy checking and cyclic redundancy checking, block check characters are computed for, and added to, each message block transmitted. This block check character is compared with a second block check character computed by the receiver to determine whether the transmission is error free.


In the above UML class diagram, the Client class refers to the common AbstractExpression interface for interpreting an expression interpret(context). The TerminalExpression class has no children and interprets an expression directly. The NonTerminalExpression class maintains a container of child expressions (expressions) and forwards interpret requests to these expressions. The object collaboration diagram shows the run-time interactions: The Client object sends an interpret request to the abstract syntax tree.


However, if all voter factions have the same proportion of strategic and honest voters, simulations show that any significant proportion of honest voters will lead to results which tend to be more satisfying to voters than approval voting, and indeed, more satisfying than any other method with the same unbiased proportion of strategic voters.Strategic voters are faced with the initial tactic as to how highly to score their second-choice candidate. The voter may want to retain expression of a high preference of their favorite candidate over their second choice. But that does not allow the same voter to express a high preference of their second choice over any others. In a simulation study using polling data collected under a majority judgment method, that method's designers found that score voting was more vulnerable to strategy than any other method they studied, including plurality.


In reverse accumulation AD, the dependent variable to be differentiated is fixed and the derivative is computed with respect to each sub-expression recursively. In a pen-and-paper calculation, the derivative of the outer functions is repeatedly substituted in the chain rule: In reverse accumulation, the quantity of interest is the adjoint, denoted with a bar w ¯ i {\displaystyle {\bar {w}}_{i}} ; it is a derivative of a chosen dependent variable with respect to a subexpression w i {\displaystyle w_{i}}: Using the chain rule, if w i {\displaystyle w_{i}} has successors in the computational graph: w ¯ i = ∑ j ∈ { successors of i } w ¯ j ∂ w j ∂ w i {\displaystyle {\bar {w}}_{i}=\sum _{j\in \{{\text{successors of i}}\}}{\bar {w}}_{j}{\frac {\partial w_{j}}{\partial w_{i}}}} Reverse accumulation traverses the chain rule from outside to inside, or in the case of the computational graph in Figure 3, from top to bottom. The example function is scalar-valued, and thus there is only one seed for the derivative computation, and only one sweep of the computational graph is needed to calculate the (two-component) gradient. This is only half the work when compared to forward accumulation, but reverse accumulation requires the storage of the intermediate variables wi as well as the instructions that produced them in a data structure known as a "tape" or a Wengert list (however, Wengert published forward accumulation, not reverse accumulation), which may consume significant memory if the computational graph is large.


See Linearization§Transformation, below, for more details. In general, there is no closed-form expression for the best-fitting parameters, as there is in linear regression. Usually numerical optimization algorithms are applied to determine the best-fitting parameters.


Eliminating the systematic error improves accuracy but does not change precision. A measurement system is considered valid if it is both accurate and precise. Related terms include bias (non-random or directed effects caused by a factor or factors unrelated to the independent variable) and error (random variability).


For instance, two-dimensional convex hulls can be computed using predicates that test the sign of quadratic polynomials, and therefore may require twice as many bits of precision within these calculations as the input numbers. When integer arithmetic cannot be used (for instance, when the result of a calculation is an algebraic number rather than an integer or rational number), a second method is to use symbolic algebra to perform all computations with exactly represented algebraic numbers rather than numerical approximations to them. A third method, sometimes called a "floating point filter", is to compute numerical predicates first using an inexact method based on floating-point arithmetic, but to maintain bounds on how accurate the result is, and repeat the calculation using slower symbolic algebra methods or numerically with additional precision when these bounds do not separate the calculated value from zero.


In pseudocode, the pairwise summation algorithm for an array x of length n ≥ 0 can be written: s = pairwise(x) if n ≤ N base case: naive summation for a sufficiently small array s = 0 for i = 1 to n s = s + x else divide and conquer: recursively sum two halves of the array m = floor(n / 2) s = pairwise(x) + pairwise(x) end if For some sufficiently small N, this algorithm switches to a naive loop-based summation as a base case, whose error bound is O(Nε). The entire sum has a worst-case error that grows asymptotically as O(ε log n) for large n, for a given condition number (see below). In an algorithm of this sort (as for divide and conquer algorithms in general), it is desirable to use a larger base case in order to amortize the overhead of the recursion.


If all weights are integers, then the run-time can be improved to O ( m n + n 2 log  log  n ) {\displaystyle O(mn+n^{2}\log \log n)} , but the resulting algorithm is only weakly-polynomial. If the weights are integers, and all weights are at most C (where C>1 is some integer), then the problem can be solved in O ( m n log  ( n ⋅ C ) ) {\displaystyle O(m{\sqrt {n}}\log(n\cdot C))} weakly-polynomial time in a method called weight scaling.In addition to the global methods, there are local methods which are based on finding local updates (rather than full augmenting paths). These methods have worse asymptotic runtime guarantees, but they often work better in practice.


If we compute the cross products, we have M = | e x e y e z x A − x 0 0 0 − F 0 | + | e x e y e z − x 0 0 0 R 0 0 | = F ( x − x A ) e z − R 0 x e z = − F x A L ( L − x ) e z . {\displaystyle \mathbf {M} =\left|{\begin{matrix}\mathbf {e} _{x}&\mathbf {e} _{y}&\mathbf {e} _{z}\\x_{A}-x&0&0\\0&-F&0\end{matrix}}\right|+\left|{\begin{matrix}\mathbf {e} _{x}&\mathbf {e} _{y}&\mathbf {e} _{z}\\-x&0&0\\0&R_{0}&0\end{matrix}}\right|=F(x-x_{A})\,\mathbf {e} _{z}-R_{0}x\,\mathbf {e} _{z}=-{\frac {Fx_{A}}{L}}(L-x)\,\mathbf {e} _{z}\,.} Thanks to the equilibrium, the internal bending moment due to external forces to the left of X must be exactly balanced by the internal turning force obtained by considering the part of the beam to the right of X M + M x z = 0 . {\displaystyle \mathbf {M} +\mathbf {M} _{xz}=\mathbf {0} \,.} which is clearly the case.


This combinator may be used in implementing Curry's paradox. The heart of Curry's paradox is that untyped lambda calculus is unsound as a deductive system, and the Y combinator demonstrates this by allowing an anonymous expression to represent zero, or even many values. This is inconsistent in mathematical logic.


In number theory, Lemoine's conjecture, named after Émile Lemoine, also known as Levy's conjecture, after Hyman Levy, states that all odd integers greater than 5 can be represented as the sum of an odd prime number and an even semiprime.


In the IMS it is possible for a user to have multiple terminals (e.g. a mobile phone, a computer) or application instances (e.g. video telephony, instant messaging, voice mail) that are identified with the same public identity (i.e. SIP URI). Therefore, a mechanism is needed in order to route requests to the desired device or application. That is what a Globally Routable User Agent URI (GRU) is: a URI that identifies a specific user agent instance (i.e. terminal or application instance) and it does it globally (i.e. it is valid to route messages to that user agent from any other user agent on the Internet). These URIs are constructed by adding the gr parameter to a SIP URI, either to the public SIP URI with a value that identifies the user agent instance, or to a specially created URI that does not reveal the relationship between the GRUU and the user's identity, for privacy purposes. They are commonly obtained during the registration process: the registering user agent sends a Uniform Resource Name (URN) that uniquely identifies that SIP instance, and the registrar (i.e. S-CSCF) builds the GRUU, associates it to the registered identity and SIP instance and sends it back to the user agent in the response. When the S-CSCF receives a request for that GRUU, it will be able to route the request to the registered SIP instance.


If this presupposition is satisfied, ∼ {\displaystyle \sim } passes along its overt argument's ordinary denotation while "resetting" its focus denotation. In other words, when the presupposition is satisfied,  ] o =  ] o {\displaystyle \!]_{o}=\! ]_{o}} and  ] f = {  ] o } {\displaystyle \!]_{f}=\{\! ]_{o}\}} .


In mathematics, a topological group G is called a discrete group if there is no limit point in it (i.e., for each element in G, there is a neighborhood which only contains that element). Equivalently, the group G is discrete if and only if its identity is isolated.A subgroup H of a topological group G is a discrete subgroup if H is discrete when endowed with the subspace topology from G. In other words there is a neighbourhood of the identity in G containing no other element of H. For example, the integers, Z, form a discrete subgroup of the reals, R (with the standard metric topology), but the rational numbers, Q, do not. Any group can be endowed with the discrete topology, making it a discrete topological group. Since every map from a discrete space is continuous, the topological homomorphisms between discrete groups are exactly the group homomorphisms between the underlying groups.


In set theory, an extender is a system of ultrafilters which represents an elementary embedding witnessing large cardinal properties. A nonprincipal ultrafilter is the most basic case of an extender. A (κ, λ)-extender can be defined as an elementary embedding of some model M {\displaystyle M} of ZFC− (ZFC minus the power set axiom) having critical point κ ε M, and which maps κ to an ordinal at least equal to λ. It can also be defined as a collection of ultrafilters, one for each n {\displaystyle n} -tuple drawn from λ.


The alternative hypothesis corresponds to the position against the defendant. Specifically, the null hypothesis also involves the absence of a difference or the absence of an association. Thus, the null hypothesis can never be that there is a difference or an association.


In a paper published in 1982, Benioff further developed his original model of quantum mechanical Turing machines. This work put quantum computers on a solid theoretical foundation. Richard Feynman then produced a universal quantum simulator. Building on the work of Benioff and Feynman, Deutsch proposed that quantum mechanics can be used to solve computational problems faster than classical computers, and in 1994, Shor described a factoring algorithm that is considered to have an exponential speedup over classical computers.After Benioff and his peers in the field published several more papers on quantum computers, the idea began to gain traction with industry, banking, and government agencies. The field is now a fast-growing area of research that could have applications in cybersecurity, cryptography, quantum system modeling and more.


The algorithm was first published in 1944 by Kenneth Levenberg, while working at the Frankford Army Arsenal. It was rediscovered in 1963 by Donald Marquardt, who worked as a statistician at DuPont, and independently by Girard, Wynne and Morrison.The LMA is used in many software applications for solving generic curve-fitting problems. By using the Gauss–Newton algorithm it often converges faster than first-order methods. However, like other iterative optimization algorithms, the LMA finds only a local minimum, which is not necessarily the global minimum.


Proving that such a system is thermodynamically stable is called the stability of matter problem and it is very difficult due to the long range of the Coulomb potential. Stability should be a consequence of screening effects, but those are hard to quantify. Let us denote by H N , K = − ∑ i = 1 N Δ x i 2 − ∑ k = 1 K Δ R k 2 M k − ∑ i = 1 N ∑ k = 1 K z k | x i − R k | + ∑ 1 ≤ i < j ≤ N 1 | x i − x j | + ∑ 1 ≤ k < m ≤ K z k z m | R k − R m | {\displaystyle H_{N,K}=-\sum _{i=1}^{N}{\frac {\Delta _{x_{i}}}{2}}-\sum _{k=1}^{K}{\frac {\Delta _{R_{k}}}{2M_{k}}}-\sum _{i=1}^{N}\sum _{k=1}^{K}{\frac {z_{k}}{|x_{i}-R_{k}|}}+\sum _{1\leq i


Rp is defined for sets S1 and S2 of cardinal number n and S3 of cardinal number n-1 as: Rp(S1,S2) iff (S3 ⊂ S1, S3 ⊂ S2)Meaning that S1 and S2 each have all the pitch-classes of S3 (transposed or inverted), plus one. he pc similarity relation Rp is not especially significant when taken alone, since by that measure a given set may be similar to many others. When Rp is combined with the  similarity measures, however, a considerable reduction is effected.


In mobile telephony a bearer service is a link between two points, which is defined by a certain set of characteristics. Whenever user equipment (UE) is being provided with any service (CS/PS service), the service has to be associated with a Radio Bearer specifying the configuration for layer 2 and physical layer in order to have its QoS clearly defined. Radio bearers are channels offered by Layer 2 to higher layers for the transfer of either user or control data. In other words, Layer 2 offers to the upper layers the service of information transmission between the UE and the UTRAN by means of the Radio Bearers (RBs) and Signaling Radio Bearers (SRBs). Therefore, the service access points between Layer 2 and upper layers are RBs.


In software development, test effort refers to the expenses for (still to come) tests. There is a relation with test costs and failure costs (direct, indirect, costs for fault correction). Some factors which influence test effort are: maturity of the software development process, quality and testability of the testobject, test infrastructure, skills of staff members, quality goals and test strategy.


Some functional elements, described using a software language, may actually be implemented as hardware (FPGA, ASIC) Hardware views – Describes the hardware engineering aspects of the system, hardware design, selection and implementation of all of the physical components to be assembled into the system. There may be many of these views, each specific to a different engineering discipline. Communications Protocol view – Describes the end to end design of the communications protocols and related data transport and data management services, shows the protocol stacks as they are implemented on each of the physical components of the system.


In present-day vocabulary, the term cast steel is almost always used in its sense referring to steel castings. Between the late 19th and mid 20th centuries, this was not always true, which is worth understanding if one is reading historical documents; see cast steel for details.


In the following, label is an optional identifier terminated by a colon, and block is a sequence of one of more Perl statements surrounded by braces. All looping constructs except for the C-style for-loop can have a continue block that is executed after each iteration of the loop body, before the loop condition is evaluated again. label for ( expr1 ; expr2 ; expr3 ) block This is the so-called C-style for loop. The first expression is evaluated prior to the first loop iteration.


In multilinear algebra, the tensor rank decomposition or the r a n k − R {\displaystyle rank-R} decomposition of a tensor is the decomposition of a tensor in terms of a sum of minimum R {\displaystyle R} r a n k − 1 {\displaystyle rank-1} tensors. This is an open problem. Canonical polyadic decomposition (CPD) is a variant of the rank decomposition which computes the best fitting K {\displaystyle K} r a n k − 1 {\displaystyle rank-1} terms for a user specified K {\displaystyle K} .


In papers published since 2008, ASJP has employed a similarity judgment program based on Levenshtein distance (LD). This approach was found to produce better classificatory results measured against expert opinion than the method used initially. LD is defined as the minimum number of successive changes necessary to convert one word into another, where each change is the insertion, deletion, or substitution of a symbol. Within the Levenshtein approach, differences in word length can be corrected for by dividing LD by the number of symbols of the longer of the two compared words.


That is, a node is given a high hub score by linking to nodes that are considered to be authorities on the subject.The Hub score and Authority score for a node is calculated with the following algorithm: Start with each node having a hub score and authority score of 1. Run the authority update rule Run the hub update rule Normalize the values by dividing each Hub score by square root of the sum of the squares of all Hub scores, and dividing each Authority score by square root of the sum of the squares of all Authority scores. Repeat from the second step as necessary.


Two octads intersect (have 1's in common) in 0, 2, or 4 coordinates in the binary vector representation (these are the possible intersection sizes in the subset representation). An octad and a dodecad intersect at 2, 4, or 6 coordinates. Up to relabeling coordinates, W is unique.The binary Golay code, G23 is a perfect code.


In mathematical morphology, a structuring element is a shape, used to probe or interact with a given image, with the purpose of drawing conclusions on how this shape fits or misses the shapes in the image. It is typically used in morphological operations, such as dilation, erosion, opening, and closing, as well as the hit-or-miss transform. According to Georges Matheron, knowledge about an object (e.g., an image) depends on the manner in which we probe (observe) it. In particular, the choice of a certain structuring element for a particular morphological operation influences the information one can obtain.


Since we continue to use a g - 1 coding scheme, it is in fact the −1 coded group that will not produce data, hence the fact that we are least interested in that group. A code of 0 is assigned to all other groups. The b values should be interpreted such that the experimental group is being compared against the mean of all groups combined (or weighted grand mean in the case of weighted effects coding).


In number theory, a prime triplet is a set of three prime numbers in which the smallest and largest of the three differ by 6. In particular, the sets must have the form (p, p + 2, p + 6) or (p, p + 4, p + 6). With the exceptions of (2, 3, 5) and (3, 5, 7), this is the closest possible grouping of three prime numbers, since one of every three sequential odd numbers is a multiple of three, and hence not prime (except for 3 itself).


In mathematics, a (B, N) pair is a structure on groups of Lie type that allows one to give uniform proofs of many results, instead of giving a large number of case-by-case proofs. Roughly speaking, it shows that all such groups are similar to the general linear group over a field. They were introduced by the mathematician Jacques Tits, and are also sometimes known as Tits systems.


In networks with unit capacities, a much stronger time bound holds. Each blocking flow can be found in O ( E ) {\displaystyle O(E)} time, and it can be shown that the number of phases does not exceed O ( E ) {\displaystyle O({\sqrt {E}})} and O ( V 2 / 3 ) {\displaystyle O(V^{2/3})} . Thus the algorithm runs in O ( min { V 2 / 3 , E 1 / 2 } E ) {\displaystyle O(\min\{V^{2/3},E^{1/2}\}E)} time.In networks that arise from the bipartite matching problem, the number of phases is bounded by O ( V ) {\displaystyle O({\sqrt {V}})} , therefore leading to the O ( V E ) {\displaystyle O({\sqrt {V}}E)} time bound. The resulting algorithm is also known as Hopcroft–Karp algorithm. More generally, this bound holds for any unit network — a network in which each vertex, except for source and sink, either has a single entering edge of capacity one, or a single outgoing edge of capacity one, and all other capacities are arbitrary integers.


In spite of the negative theoretical results on the joint spectral radius computability, methods have been proposed that perform well in practice. Algorithms are even known, which can reach an arbitrary accuracy in an a priori computable amount of time. These algorithms can be seen as trying to approximate the unit ball of a particular vector norm, called the extremal norm. One generally distinguishes between two families of such algorithms: the first family, called polytope norm methods, construct the extremal norm by computing long trajectories of points.


In mathematics, any vector space V {\displaystyle V} has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on V , {\displaystyle V,} together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the algebraic dual space. When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the continuous dual space. Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite-dimensional vector spaces.


If two dice are thrown and their values added, the resulting distribution is no longer uniform because not all sums have equal probability. Although it is convenient to describe discrete uniform distributions over integers, such as this, one can also consider discrete uniform distributions over any finite set.


Another example is lay, which may be the past tense of lie, but is also an independent verb (regular in pronunciation, but with irregular spelling: lay–laid–laid). In fact the past tense verb lay derives from a causative of the verb from which lie derives. The two verbs are sometimes confused, with lay used in the intransitive senses prescriptively reserved for lie.


Only O ( N l o g ( N ) ) {\displaystyle O(Nlog(N))} operations are needed to detect edges, where N {\displaystyle N} is the number of pixels. During the following years, other problems have been considered: classification, segmentation, inpainting and super-resolution. This approach can be applied to gray-level or color images.


However, context-free rules cannot express agreement or reference (anaphora), where two different parts of the sentence must agree with each other in some way. These can be readily expressed in W-grammars.


A system or application will hit a bottleneck if the work arrives at a comparatively faster pace relative to other processing components. According to the theory of constraints, improving on the occurrences of hot-spot point of the bottleneck constraint improves the overall processing speed of the software. A thought-provoking stipulation of the theory reveals that improving the efficiency of a particular process stage rather than the constraint can generate even more delay and decrease overall processing capabilities of a software. The process of tracking down bottlenecks (also referred as "hot spots" - sections of the code that execute most frequently - i.e. have the highest execution count) is called performance analysis. Reduction is achieved with the utilization of specialized tools such as performance analyzers or profilers, the objective being to make particular sections of code perform as effectively as possible to improve overall algorithmic efficiency.


In mathematics and computer programming, exponentiating by squaring is a general method for fast computation of large positive integer powers of a number, or more generally of an element of a semigroup, like a polynomial or a square matrix. Some variants are commonly referred to as square-and-multiply algorithms or binary exponentiation. These can be of quite general use, for example in modular arithmetic or powering of matrices. For semigroups for which additive notation is commonly used, like elliptic curves used in cryptography, this method is also referred to as double-and-add.


Reidl "in 1998 extends and proposes a new way to taxonomize information visualization techniques by using the Data State Model. Many of the techniques share similar operating steps that can easily be reused. The Data State Model not only helps researchers understand the space of design, but also helps implementers understand how information visualization techniques can be applied more broadly".In 1999 Stuart Card, Jock D. Mackinlay, and Ben Shneiderman present their own interpretation of this pattern, dubbing it the information visualization reference model.


In the case of neighborhood interchangeable algorithm, if we assign the worst case bound to each loop. Then for n variables, which have at most d values for a variable, then we have a bound of: O ( n d ( n − l ) ∗ d ) = O ( n 2 d 2 ) {\displaystyle O(nd(n-l)*d)=O(n^{2}d^{2})} . Similarly, the complexity analysis of the k-interchangeability algorithm for a worst case O ( n k − 1 ) {\displaystyle O(n^{k-1})} , with ( k − 1 ) {\displaystyle (k-1)} -tuples of variables and d k − 1 {\displaystyle d^{k-1}} , for ( k − 1 ) {\displaystyle (k-1)} -tuples of values, then the bound is: O ( n d n k − l d k − 1 ) = O ( n k d k ) {\displaystyle O(ndn^{k-l}d^{k-1})=O(n^{k}d^{k})} .


To oversimplify, the fundamental lemma of the project posits a direct connection between the generalized fundamental representation of a finite field with its group extension to the automorphic forms under which it is invariant. This is accomplished through abstraction to higher dimensional integration, by an equivalence to a certain analytical group as an absolute extension of its algebra. Consequently, this allows an analytical functional construction of powerful invariance transformations for a number field to its own algebraic structure.


In the bin packing problem, there are n items with different sizes. The goal is to pack the items into a minimum number of bins, where each bin can contain at most B. A feasible configuration is a set of sizes with a sum of at most B. Example: suppose the item sizes are 3,3,3,3,3,4,4,4,4,4, and B=12. Then the possible configurations are: 3333; 333; 33, 334; 3, 34, 344; 4, 44, 444. If we had only three items of size 3, then we could not use the 3333 configuration.Denote by S the set of different sizes (and their number).


In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. Formally, a ring is an abelian group whose operation is called addition, with a second binary operation called multiplication that is associative, is distributive over the addition operation, and has a multiplicative identity element.


In sorting n objects, merge sort has an average and worst-case performance of O(n log n). If the running time of merge sort for a list of length n is T(n), then the recurrence relation T(n) = 2T(n/2) + n follows from the definition of the algorithm (apply the algorithm to two lists of half the size of the original list, and add the n steps taken to merge the resulting two lists). The closed form follows from the master theorem for divide-and-conquer recurrences. The number of comparisons made by merge sort in the worst case is given by the sorting numbers.


In mathematics, Minkowski's theorem is the statement that every convex set in R n {\displaystyle \mathbb {R} ^{n}} which is symmetric with respect to the origin and which has volume greater than 2 n {\displaystyle 2^{n}} contains a non-zero integer point (meaning a point in Z n {\displaystyle \mathbb {Z} ^{n}} that is not the origin). The theorem was proved by Hermann Minkowski in 1889 and became the foundation of the branch of number theory called the geometry of numbers. It can be extended from the integers to any lattice L {\displaystyle L} and to any symmetric convex set with volume greater than 2 n d ( L ) {\displaystyle 2^{n}\,d(L)} , where d ( L ) {\displaystyle d(L)} denotes the covolume of the lattice (the absolute value of the determinant of any of its bases).