In some cases, search engines covering specific regions may provide locally-specific extensions of microdata. For example, Yandex, a major search engine in Russia, supports microformats such as hCard (company contact information), hRecipe (food recipe), hReview (market reviews) and hProduct (product data) and provides its own format for definition of the terms and encyclopedic articles. This extension was made in order to solve transliteration problems between the Cyrillic and Latin alphabets. After the implementation of additional parameters from Schema's vocabulary, indexation of information in Russian-language web-pages became more successful.


In queueing theory, a discipline within the mathematical theory of probability, the arrival theorem (also referred to as the random observer property, ROP or job observer property) states that "upon arrival at a station, a job observes the system as if in steady state at an arbitrary instant for the system without that job. "The arrival theorem always holds in open product-form networks with unbounded queues at each node, but it also holds in more general networks. A necessary and sufficient condition for the arrival theorem to be satisfied in product-form networks is given in terms of Palm probabilities in Boucherie & Dijk, 1997.


Exploiting the behavior of a buffer overflow is a well-known security exploit. On many systems, the memory layout of a program, or the system as a whole, is well defined. By sending in data designed to cause a buffer overflow, it is possible to write into areas known to hold executable code and replace it with malicious code, or to selectively overwrite data pertaining to the program's state, therefore causing behavior that was not intended by the original programmer.


k ! ( n − k ) ! .


With the other uses of radio spectrum growing in the 1990s, the FCC made available some bands of spectrum as unlicensed channels. This included spectrum for cordless phones and Wi-Fi. As a result, some of these channels have been used for news gathering by websites and more informal news outlets. One major disadvantage of unlicensed use is that there is no frequency coordination, which can result in interference or blocking of signals.


In statistics, sequential estimation refers to estimation methods in sequential analysis where the sample size is not fixed in advance. Instead, data is evaluated as it is collected, and further sampling is stopped in accordance with a predefined stopping rule as soon as significant results are observed. The generic version is called the optimal Bayesian estimator, which is the theoretical underpinning for every sequential estimator (but cannot be instantiated directly). It includes a Markov process for the state propagation and measurement process for each state, which yields some typical statistical independence relations.


The so-called kommunalka became the most common type of accommodation for the residents of large cities. In each communal apartment one room belonged to one family, while bathroom, toilet and kitchen were shared. Such a scheme was widespread until the mid-1950s, and in some cities there are more communal apartments.


In physics and applied mathematics, analytical regularization is a technique used to convert boundary value problems which can be written as Fredholm integral equations of the first kind involving singular operators into equivalent Fredholm integral equations of the second kind. The latter may be easier to solve analytically and can be studied with discretization schemes like the finite element method or the finite difference method because they are pointwise convergent. In computational electromagnetics, it is known as the method of analytical regularization. It was first used in mathematics during the development of operator theory before acquiring a name.


In mathematics, statistics and elsewhere, sums of squares occur in a number of contexts:


In mathematics, a primefree sequence is a sequence of integers that does not contain any prime numbers. More specifically, it usually means a sequence defined by the same recurrence relation as the Fibonacci numbers, but with different initial conditions causing all members of the sequence to be composite numbers that do not all have a common divisor. To put it algebraically, a sequence of this type is defined by an appropriate choice of two composite numbers a1 and a2, such that the greatest common divisor g c d ( a 1 , a 2 ) {\displaystyle \mathrm {gcd} (a_{1},a_{2})} is equal to 1, and such that for n > 2 {\displaystyle n>2} there are no primes in the sequence of numbers calculated from the formula a n = a n − 1 + a n − 2 {\displaystyle a_{n}=a_{n-1}+a_{n-2}} .The first primefree sequence of this type was published by Ronald Graham in 1964.


In the early years of Windows Mobile devices were able to be managed and synced from a remote computer using ActiveSync; a data synchronization technology and protocol developed by Microsoft, originally released in 1996. This allowed servers running Microsoft Exchange Server, or other third party variants, to act as a personal information manager and share information such as email, calendar appointments, contacts or internet favorites. With the release of Windows Vista, ActiveSync was replaced with Windows Mobile Device Center. Device Center is included with Vista and Windows 7 and provides many front end enhancements, allowing a home user to sync PIM information with Microsoft Outlook 2003 and later, photos from Windows Photo Gallery, videos or music from Windows Media Player and favorites with Internet Explorer; without the need for a server back end. Devices at this time also included a base driver compatible with Mobile Device Center so a user can connect to a computer without a need for any configuration.


In statistics, a paired difference test is a type of location test that is used when comparing two sets of paired measurements to assess whether their population means differ. A paired difference test uses additional information about the sample that is not present in an ordinary unpaired testing situation, either to increase the statistical power, or to reduce the effects of confounders. Specific methods for carrying out paired difference tests are, for normally distributed difference t-test (where the population standard deviation of difference is not known) and the paired Z-test (where the population standard deviation of the difference is known), and for differences that may not be normally distributed the Wilcoxon signed-rank test as well as the paired permutation test.


In the erasure channel model a symbol gets either transmitted correctly with probability 1 − p e {\displaystyle 1-p_{e}} or is erased with probability p e {\displaystyle p_{e}} (see Figure \ref{fig:Sudoku3x3channel}). The channel introduces no errors, i.e. no channel input is changed to another symbol. The example in Figure \ref{fig:Sudoku3x3BSC} shows the transmission of a 3 × 3 {\displaystyle 3\times 3} Sudoku code.


Write operations invalidate other copies, but often don't wait for their acknowledgements. Read operations typically don't check every redundant copy prior to answering, potentially missing the preceding write operation. The large amount of metadata signal traffic would require specialized hardware and short distances to be handled with acceptable performance (i.e., act like a non-clustered storage device or database).Whenever strong data consistency is expected, look for these indicators: the use of InfiniBand, Fibrechannel or similar low-latency networks to avoid performance degradation with increasing cluster size and number of redundant copies.


In mathematics, the matrix ( 1 − 1 1 1 ) {\displaystyle {\begin{pmatrix}1&-1\\1&1\end{pmatrix}}} is sometimes called the quincunx matrix. It is a 2×2 Hadamard matrix, and its rows form the basis of a diagonal square lattice consisting of the integer points whose coordinates both have the same parity; this lattice is a two-dimensional analogue of the three-dimensional body-centered cubic lattice.


But, in balanced ternary, we can't omit the rightmost trailing infinite −1s after the radix point in order to gain a representations of integer or terminating fraction. Donald Knuth has pointed out that truncation and rounding are the same operation in balanced ternary—they produce exactly the same result (a property shared with other balanced numeral systems). The number 1/2 is not exceptional; it has two equally valid representations, and two equally valid truncations: 0.1 (round to 0, and truncate to 0) and 1.𝖳 (round to 1, and truncate to 1).


In the UMTS cellular communication system, received signal code power (RSCP) denotes the power measured by a receiver on a particular physical communication channel. It is used as an indication of signal strength, as a handover criterion, in downlink power control, and to calculate path loss. In CDMA systems, a physical channel corresponds to a particular spreading code, hence the name (Received signal code power). RSCP is also called Receiver Side Call Power.


With an odd radix, double rounding is also equivalent to directly rounding to the final precision, unlike with an even radix. The basic operations—addition, subtraction, multiplication, and division—are done as in regular ternary. Multiplication by two can be done by adding a number to itself, or subtracting itself after a-trit-left-shifting. An arithmetic shift left of a balanced ternary number is the equivalent of multiplication by a (positive, integral) power of 3; and an arithmetic shift right of a balanced ternary number is the equivalent of division by a (positive, integral) power of 3.


In strategic planning, resource allocation is a plan for using available resources, for example human resources, especially in the near term, to achieve goals for the future. It is the process of allocating scarce resources among the various projects or business units.There are a number of approaches to solving resource allocation problems e.g. resources can be allocated using a manual approach, an algorithmic approach (see below), or a combination of both.There may be contingency mechanisms such as a priority ranking of items excluded from the plan, showing which items to fund if more resources should become available and a priority ranking of some items included in the plan, showing which items should be sacrificed if total funding must be reduced.


In order to match certain signatures, an IDS is required to keep state related to the connections it is monitoring. For example, an IDS must maintain "TCP control blocks" (TCBs), chunks of memory which track information such as sequence numbers, window sizes, and connection states (ESTABLISHED, RELATED, CLOSED, etc.), for each TCP connection monitored by the IDS. Once all of the IDS's random-access memory (RAM) is consumed, it is forced to utilize virtual memory on the hard disk which is much slower than RAM, leading to performance problems and dropped packets similar to the effects of CPU exhaustion.If the IDS doesn't garbage collect TCBs correctly and efficiently, an attacker can exhaust the IDS's memory by starting a large number of TCP connections very quickly. Similar attacks can be made by fragmenting a large number of packets into a larger number of smaller packets, or send a large number of out-of-order TCP segments.


In mathematics and computer science, the middle-square method is a method of generating pseudorandom numbers. In practice it is a highly flawed method for many practical purposes, since its period is usually very short and it has some severe weaknesses; repeated enough times, the middle-square method will either begin repeatedly generating the same number or cycle to a previous number in the sequence and loop indefinitely.


In probability theory and statistics, Fisher's noncentral hypergeometric distribution is a generalization of the hypergeometric distribution where sampling probabilities are modified by weight factors. It can also be defined as the conditional distribution of two or more binomially distributed variables dependent upon their fixed sum. The distribution may be illustrated by the following urn model.


Instead, they bear a legend statement "Authorized by Drawer". This type of instrument is usually used by credit card companies, utility companies, or telemarketers. Remotely created checks are vulnerable to fraud.


In the United States, privacy concerns have been addressed by the US Congress via the passage of regulatory controls such as the Health Insurance Portability and Accountability Act (HIPAA). The HIPAA requires individuals to give their "informed consent" regarding information they provide and its intended present and future uses. According to an article in Biotech Business Week, "'n practice, HIPAA may not offer any greater protection than the longstanding regulations in the research arena,' says the AAHC. More importantly, the rule's goal of protection through informed consent is approach a level of incomprehensibility to average individuals."


In response to a series of earthquakes near Comrie in Scotland in 1839, a committee was formed in the United Kingdom in order to produce better detection devices for earthquakes. The outcome of this was an inverted pendulum seismometer constructed by James David Forbes, first presented in a report by David Milne-Home in 1842, which recorded the measurements of seismic activity through the use of a pencil placed on paper above the pendulum. The designs provided did not prove effective, according to Milne's reports. It was Milne who coined the word seismometer in 1841, to describe this instrument.


Merchant codes and product codes are used at the point of sale (required by law by certain merchants by certain states in the U.S.) to restrict sales if they do not qualify. Because of the extra checking and documenting that goes on, later, the statement can be used to substantiate these purchases for tax deductions.


At the same time, the availability of published material online opens more doors for plagiarism, unauthorized use, or re-use of the material. Some publishers are trying to address these concerns. For example, in 2011, HarperCollins limited the number of times that one of its e-books could be lent in a public library. Other publishers, such as Penguin, are attempting to incorporate e-book elements into their regular paper publications.


In simplest terms it is a measure of the probability of finding a particle at a distance of r {\displaystyle r} away from a given reference particle, relative to that for an ideal gas. The general algorithm involves determining how many particles are within a distance of r {\displaystyle r} and r + d r {\displaystyle r+dr} away from a particle. This general theme is depicted to the right, where the red particle is our reference particle, and blue particles are those whose centers are within the circular shell, dotted in orange.


In mathematics, the Carathéodory kernel theorem is a result in complex analysis and geometric function theory established by the Greek mathematician Constantin Carathéodory in 1912. The uniform convergence on compact sets of a sequence of holomorphic univalent functions, defined on the unit disk in the complex plane and fixing 0, can be formulated purely geometrically in terms of the limiting behaviour of the images of the functions. The kernel theorem has wide application in the theory of univalent functions and in particular provides the geometric basis for the Loewner differential equation.


In probability theory and statistics, given two jointly distributed random variables X {\displaystyle X} and Y {\displaystyle Y} , the conditional probability distribution of Y {\displaystyle Y} given X {\displaystyle X} is the probability distribution of Y {\displaystyle Y} when X {\displaystyle X} is known to be a particular value; in some cases the conditional probabilities may be expressed as functions containing the unspecified value x {\displaystyle x} of X {\displaystyle X} as a parameter. When both X {\displaystyle X} and Y {\displaystyle Y} are categorical variables, a conditional probability table is typically used to represent the conditional probability. The conditional distribution contrasts with the marginal distribution of a random variable, which is its distribution without reference to the value of the other variable.


That is, we want to find a channel code, which is able to partition the space of input X {\displaystyle X} into several cosets, where each coset has a unique syndrome associated with it. With a given coset and y {\displaystyle \mathbf {y} } , there is only one x {\displaystyle \mathbf {x} } that is possible to be the input given the correlation between two sources. In this example, we can use the ( 7 , 4 , 3 ) {\displaystyle (7,4,3)} binary Hamming Code C {\displaystyle \mathbf {C} } , with parity check matrix H {\displaystyle \mathbf {H} } .


In mathematics, negafibonacci coding is a universal code which encodes nonzero integers into binary code words. It is similar to Fibonacci coding, except that it allows both positive and negative integers to be represented. All codes end with "11" and have no "11" before the end.


In number theory, a Pierpont prime is a prime number of the form for some nonnegative integers u and v. That is, they are the prime numbers p for which p − 1 is 3-smooth. They are named after the mathematician James Pierpont, who used them to characterize the regular polygons that can be constructed using conic sections. The same characterization applies to polygons that can be constructed using ruler, compass, and angle trisector, or using paper folding. Except for 2 and the Fermat primes, every Pierpont prime must be 1 modulo 6. The first few Pierpont primes are: It has been conjectured that there are infinitely many Pierpont primes, but this remains unproven.


In the example 3 4 -, first the 3 is put onto the stack, then the 4; the 4 is now on top and the 3 below it. The subtraction operator removes the top two items from the stack, performs 3 - 4, and puts the result of -1 onto the stack. The common terminology is that added items are pushed on the stack and removed items are popped. The advantage of reverse Polish notation is that it removes the need for order of operations and parentheses that are required by infix notation and can be evaluated linearly, left-to-right. For example, the infix expression (3 × 4) + (5 × 6) becomes 3 4 × 5 6 × + in reverse Polish notation.


The axiom of choice is also rejected in most intuitionistic set theories, though in some versions it is accepted. In intuitionism, the term "explicit construction" is not cleanly defined, and that has led to criticisms. Attempts have been made to use the concepts of Turing machine or computable function to fill this gap, leading to the claim that only questions regarding the behavior of finite algorithms are meaningful and should be investigated in mathematics. This has led to the study of the computable numbers, first introduced by Alan Turing. Not surprisingly, then, this approach to mathematics is sometimes associated with theoretical computer science.


In mathematics, the concept of a residuated mapping arises in the theory of partially ordered sets. It refines the concept of a monotone function. If A, B are posets, a function f: A → B is defined to be monotone if it is order-preserving: that is, if x ≤ y implies f(x) ≤ f(y). This is equivalent to the condition that the preimage under f of every down-set of B is a down-set of A. We define a principal down-set to be one of the form ↓{b} = { b' ∈ B: b' ≤ b }.


In statistics, the residual sum of squares (RSS), also known as the sum of squared residuals (SSR) or the sum of squared estimate of errors (SSE), is the sum of the squares of residuals (deviations predicted from actual empirical values of data). It is a measure of the discrepancy between the data and an estimation model, such as a linear regression. A small RSS indicates a tight fit of the model to the data.


In real world application, one often observe only a few entries corrupted at least by a small amount of noise. For example, in the Netflix problem, the ratings are uncertain. Candès and Plan showed that it is possible to fill in the many missing entries of large low-rank matrices from just a few noisy samples by nuclear norm minimization. The noisy model assumes that we observe Y i j = M i j + Z i j , ( i , j ) ∈ Ω , {\displaystyle Y_{ij}=M_{ij}+Z_{ij},(i,j)\in \Omega ,} where Z i j: ( i , j ) ∈ Ω {\displaystyle {Z_{ij}:(i,j)\in \Omega }} is a noise term.


In physics, the Bekenstein bound (named after Jacob Bekenstein) is an upper limit on the thermodynamic entropy S, or Shannon entropy H, that can be contained within a given finite region of space which has a finite amount of energy—or conversely, the maximal amount of information required to perfectly describe a given physical system down to the quantum level. It implies that the information of a physical system, or the information necessary to perfectly describe that system, must be finite if the region of space and the energy are finite. In computer science this implies that non-finite models such as Turing machines are not realizable as finite devices.


Here, the original observations are effectively mapped into a higher dimensional non-linear space. Linear classification in this non-linear space is then equivalent to non-linear classification in the original space. The most commonly used example of this is the kernel Fisher discriminant.


A surface is a two-dimensional space; this means that a moving point on a surface may move in two directions (it has two degrees of freedom). In other words, around almost every point, there is a coordinate patch on which a two-dimensional coordinate system is defined. For example, the surface of the Earth resembles (ideally) a two-dimensional sphere, and latitude and longitude provide two-dimensional coordinates on it (except at the poles and along the 180th meridian). The concept of surface is widely used in physics, engineering, computer graphics, and many other disciplines, primarily in representing the surfaces of physical objects. For example, in analyzing the aerodynamic properties of an airplane, the central consideration is the flow of air along its surface.


In practice, one sets y 2 = x 3 + A x + B {\displaystyle y^{2}=x^{3}+Ax+B} , and then ψ 2 m + 1 ∈ Z  {\displaystyle \psi _{2m+1}\in \mathbb {Z} } and ψ 2 m ∈ 2 y Z  {\displaystyle \psi _{2m}\in 2y\mathbb {Z} } . The division polynomials form a generic elliptic divisibility sequence over the ring Q  / ( y 2 − x 3 − A x − B ) {\displaystyle \mathbb {Q} /(y^{2}-x^{3}-Ax-B)} . If an elliptic curve E {\displaystyle E} is given in the Weierstrass form y 2 = x 3 + A x + B {\displaystyle y^{2}=x^{3}+Ax+B} over some field K {\displaystyle K} , i.e. A , B ∈ K {\displaystyle A,B\in K} , one can use these values of A , B {\displaystyle A,B} and consider the division polynomials in the coordinate ring of E {\displaystyle E} . The roots of ψ 2 n + 1 {\displaystyle \psi _{2n+1}} are the x {\displaystyle x} -coordinates of the points of E  ∖ { O } {\displaystyle E\setminus \{O\}} , where E  {\displaystyle E} is the ( 2 n + 1 ) th {\displaystyle (2n+1)^{\text{th}}} torsion subgroup of E {\displaystyle E} .


Conceptual errors are a developer's misunderstanding of what the software must do. The resulting software may perform according to the developer's understanding, but not what is really needed. Other types:


In the continuing struggle to create the fastest and most advanced 3D accelerator, ATI came up with the RAGE 128. The chip was announced in two flavors, the RAGE 128 GL and the RAGE 128 VR. Aside from the VR chip's lower price, the main difference was that the former was a full 128-bit design, while the VR, still a 128-bit processor internally, used a 64-bit external memory interface. Magnum - A workstation board for OEMs with 32 MB SDRAM.


In the deques of this structure, elements are kept in sorted order according to their working set size. Formally, element x {\displaystyle x} lies after y {\displaystyle y} in deque Q i {\displaystyle Q_{i}} if and only if w ( x ) < w ( y ) {\displaystyle w(x)


In mathematics and computer science, a morphic word or substitutive word is an infinite sequence of symbols which is constructed from a particular class of endomorphism of a free monoid. Every automatic sequence is morphic.


In statistics and signal processing, the orthogonality principle is a necessary and sufficient condition for the optimality of a Bayesian estimator. Loosely stated, the orthogonality principle says that the error vector of the optimal estimator (in a mean square error sense) is orthogonal to any possible estimator. The orthogonality principle is most commonly stated for linear estimators, but more general formulations are possible. Since the principle is a necessary and sufficient condition for optimality, it can be used to find the minimum mean square error estimator.


In the Turkish terms for the constructive and inflectional endings, three roots are involved: ek "supplement, affix" (notably Turkish has no prefixes) yap- "make" çek- "pull, draw"For the last two verbal roots, the constructive suffix -im can be added to form nouns for instances of the actions denoted by the roots: yapım "construction"; çekim " pull or draw" (or a "take" in cinema).Either of these nouns can be compounded with the noun ek, resulting in an indefinite compound (belirtisiz tamlama), the sign of which is the inflectional suffix -i attached to ek: yapım eki "structure-suffix"; çekim eki "inflection-suffix".The inflectional suffix -ler comes before the -i to form the plural, so yapım ekleri, çekim ekleri. Many words in Turkish— particularly many grammatical terms— are neologisms invented to replace earlier words borrowed from Arabic or Persian, which have largely been successful at permanently superseding the previously used foreign terms. (See the main article on Turkish language.) In some cases, the foreign term continues to be in use alongside the neologism.


In multiprocessing, omega networks may be used as connectors between the CPUs and their shared memory, in order to decrease the probability that the CPU-to-memory connection becomes a bottleneck. This class of networks has been built into the Illinois Cedar Multiprocessor, into the IBM RP3, and into the NYU Ultracomputer.


For example, for p =1, the latent variables of all entities i = 1 , . . .


In mathematical logic, computer programming, philosophy and linguistics fuzzy concepts can be analyzed and defined more accurately or comprehensively, by describing or modelling the concepts using the terms of fuzzy logic or other substructural logics. More generally, clarification techniques can be used such as: 1. Contextualizing the concept by defining the setting or situation in which the concept is used, or how it is used appropriately (context). 2.


It is distinct from renormalization, another technique to control infinities without assuming new physics, by adjusting for self-interaction feedback. Regularization was for many decades controversial even amongst its inventors, as it combines physical and epistemological claims into the same equations. However, it is now well understood and has proven to yield useful, accurate predictions.


It also depends on f {\displaystyle f} , α 0 {\displaystyle \alpha _{0}} , τ {\displaystyle \tau } and c {\displaystyle c} of course, although these dependencies can be left implicit if they are assumed to be fixed with respect to the optimization problem. The detailed steps are thus, see Armijo (1966), Bertsekas (2016): Choose an initial starting point x 0 {\displaystyle \mathbf {x} _{0}} and set the iteration counter n = 0 {\displaystyle n=0} . Until some stopping condition is satisfied, choose a descent direction p n {\displaystyle \mathbf {p} _{n}} , increment n {\displaystyle n} , and update the position to x n + 1 = x n + α ( x n , p n ) p n {\displaystyle \mathbf {x} _{n+1}=\mathbf {x} _{n}+\alpha (\mathbf {x} _{n},\mathbf {p} _{n})\,\mathbf {p} _{n}} .


In some languages, certain digraphs and trigraphs are counted as distinct letters in themselves, and assigned to a specific place in the alphabet, separate from that of the sequence of characters that composes them, for purposes of orthography and collation. For example: In the Gaj’s Latin alphabet used to write Serbo-Croatian, the digraphs ⟨dž⟩, ⟨lj⟩ and ⟨nj⟩, which correspond to the single Cyrillic letters ⟨џ⟩, ⟨љ⟩, ⟨њ⟩, are treated as distinct letters. In the Czech and Slovak alphabet, ⟨ch⟩ is treated as a distinct letter, coming after ⟨h⟩ in the alphabet. Also, in the Slovak alphabet the relatively rare digraphs ⟨dz⟩ and ⟨dž⟩ are treated as distinct letters.


Such experiments allow researchers to quantify the role of historical contingency and repeatability in network evolution, enabling predictions about the architecture and dynamics of large networks of interacting species. The inclusion of ecological interactions in digital systems enables new research avenues: investigations using self-replicating computer programs complement laboratory efforts by broadening the breadth of viable experiments focused on the emergence and diversification of coevolving interactions in complex communities. This cross-disciplinary research program provides fertile grounds for new collaborations between computer scientists and evolutionary biologists.


In number theory, a Sierpiński number is an odd natural number k such that k × 2 n + 1 {\displaystyle k\times 2^{n}+1} is composite for all natural numbers n. In 1960, Wacław Sierpiński proved that there are infinitely many odd integers k which have this property. In other words, when k is a Sierpiński number, all members of the following set are composite: { k ⋅ 2 n + 1: n ∈ N } . {\displaystyle \left\{\,k\cdot 2^{n}+1:n\in \mathbb {N} \,\right\}.} If the form is instead k × 2 n − 1 {\displaystyle k\times 2^{n}-1} , then k is a Riesel number.


This idea can be applied to any commutative monoid. On the other hand, the set of integers Z requires a more sophisticated argument for its hierarchical structure, since we can always solve the equation n + m = n ′ {\displaystyle n+m=n'} by writing m = ( n ′ − n ) {\displaystyle m=(n'-n)} .A mathematical hierarchy (a pre-ordered set) should not be confused with the more general concept of a hierarchy in the social realm, particularly when one is constructing computational models that are used to describe real-world social, economic or political systems. These hierarchies, or complex networks, are much too rich to be described in the category Set of sets.


{\displaystyle {\begin{pmatrix}0&0&2&0\\0&2&-1&1\\2&-1&2&-1\\0&1&-1&2\end{pmatrix}};\quad {\begin{pmatrix}0&0&2&0&0\\0&1&-1&2&0\\2&-1&-1&0&2\\0&0&2&0&0\\0&2&0&0&0\end{pmatrix}};\quad {\begin{pmatrix}0&0&0&2\\0&2&0&0\\2&-2&2&0\\0&2&0&0\end{pmatrix}};\quad {\begin{pmatrix}0&2&0&0\\0&0&0&2\\2&0&0&0\\0&0&2&0\end{pmatrix}}.} The set of HSASMs is a superset of the ASMs. The extreme points of the convex hull of the set of r-spin HSASMs are themselves integer multiples of the usual ASMs. == References ==


In the following, f k {\displaystyle f_{k}} denotes the value of the function f: { 0 , 1 } n → { 0 , 1 } {\displaystyle f:\{0,1\}^{n}\rightarrow \{0,1\}} when applied to an input vector of weight k {\displaystyle k} .


In mathematics, an aperiodic semigroup is a semigroup S such that every element is aperiodic, that is, for each x in S there exists a positive integer n such that xn = xn+1. An aperiodic monoid is an aperiodic semigroup which is a monoid.


. . , m a x d − 1 , + ∞ } {\displaystyle k_{max}=\{max_{0},+\infty ,max_{1},+\infty ,...,max_{d-1},+\infty \}}


In some cases the observations may be weighted—for example, they may not be equally reliable. In this case, one can minimize the weighted sum of squares: where wi > 0 is the weight of the ith observation, and W is the diagonal matrix of such weights. The weights should, ideally, be equal to the reciprocal of the variance of the measurement. (This implies that the observations are uncorrelated.


We use the potential function Φ = Σ 𝓁(u) (i.e. Φ is the sum of the labels of all active nodes). It is obvious that Φ is 0 initially and stays nonnegative throughout the execution of the algorithm. Both relabels and saturating pushes can increase Φ. However, the value of Φ must be equal to 0 at termination since there cannot be any remaining active nodes at the end of the algorithm's execution.


In such theories, the speech act of assertion is often analyzed as a proposal to add an additional proposition to the common ground. Similarly, presuppositions are taken to be licensed when they are already established in the common ground. While such approaches are typically construed as pragmatic, the framework of dynamic semantics treats the semantic denotations of sentences as functions which update the common ground. In many theories, the common ground is one of several elements of the conversational scoreboard.


Brent's method is due to Richard Brent and builds on an earlier algorithm by Theodorus Dekker. Consequently, the method is also known as the Brent–Dekker method. Modern improvements on Brent's method include Chandrupatla's method, which is simpler and faster for functions that are flat around their roots; Ridders' method, which performs exponential interpolations instead of quadratic providing a simpler closed formula for the iterations; and the ITP method which is a hybrid between regula-falsi and bisection that achieves optimal worst-case and asymptotic guarantees.


In probability theory and statistics, the hyperbolic secant distribution is a continuous probability distribution whose probability density function and characteristic function are proportional to the hyperbolic secant function. The hyperbolic secant function is equivalent to the reciprocal hyperbolic cosine, and thus this distribution is also called the inverse-cosh distribution. Generalisation of the distribution gives rise to the Meixner distribution, also known as the Natural Exponential Family - Generalised Hyperbolic Secant or NEF-GHS distribution.


If we restrict loops to be of a predictably finite size (like the FOR loop in BASIC), we can express all of the primitive recursive functions (Meyer and Ritchie, 1967). An example of such a machine is provided by the toy programming language PL-{GOTO} of Brainerd and Landweber (1974).


Furthermore, the objective should be to obtain an experts' carefully considered judgment based on a systematic consideration of all relevant evidence. For this reason one should take care to adopt strategies designed to help the expert being interviewed to avoid overlooking relevant evidence. Additionally, vocabulary used should face intense scrutiny; qualitative uncertainty words such as "likely" and "unlikely" are not sufficient and can lead to confusion. Such words can mean very different things to different people, or to the same people in different situations.


In some parts of the world, including the United Kingdom, 2G remains widely used for older feature phones and for internet of things (IoT) devices such as smart meters, eCall systems and vehicle trackers to avoid the high patent licensing cost of newer technologies. Terminating 2G services could leave vulnerable people who rely on 2G infrastructure unable to communicate even with emergency contacts, causing harm and possibly deaths.


In optimization and other branches of mathematics, and in search algorithms (a topic in computer science), a candidate solution is a member of the set of possible solutions in the feasible region of a given problem. A candidate solution does not have to be a likely or reasonable solution to the problem—it is simply in the set that satisfies all constraints; that is, it is in the set of feasible solutions. Algorithms for solving various types of optimization problems often narrow the set of candidate solutions down to a subset of the feasible solutions, whose points remain as candidate solutions while the other feasible solutions are henceforth excluded as candidates.


Most GSM/UMTS phones support all four bands, while most CDMA2000/1xRTT phones (mostly North America and voice transmission only) do not, and so are considered only dual-band devices. A few phones support both of the domestic frequencies but only one foreign one for limited roaming, making them tri-band phones.The term penta-band describes a device that supports a fifth frequency band, commonly the 1700/2100 MHz band in much of the world. The Advanced Wireless Services (AWS) 1700 MHz band is also seeing increased usage.


2D spatial directions are numerically equivalent to points on the unit circle and spatial directions in 3D are equivalent to a point on the unit sphere. The normalized vector û of a non-zero vector u is the unit vector in the direction of u, i.e., u ^ = u ‖ u ‖ {\displaystyle \mathbf {\hat {u}} ={\frac {\mathbf {u} }{\|\mathbf {u} \|}}} where ‖u‖ is the norm (or length) of u. The term normalized vector is sometimes used as a synonym for unit vector. Unit vectors are often chosen to form the basis of a vector space, and every vector in the space may be written as a linear combination of unit vectors.


In standard matrix notation, each element of Rn is typically written as a column vector and sometimes as a row vector: The coordinate space Rn may then be interpreted as the space of all n × 1 column vectors, or all 1 × n row vectors with the ordinary matrix operations of addition and scalar multiplication. Linear transformations from Rn to Rm may then be written as m × n matrices which act on the elements of Rn via left multiplication (when the elements of Rn are column vectors) and on elements of Rm via right multiplication (when they are row vectors). The formula for left multiplication, a special case of matrix multiplication, is: Any linear transformation is a continuous function (see below). Also, a matrix defines an open map from Rn to Rm if and only if the rank of the matrix equals to m.


In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: 7 = 71, 9 = 32 and 64 = 26 are prime powers, while 6 = 2 × 3, 12 = 22 × 3 and 36 = 62 = 22 × 32 are not. The sequence of prime powers begins: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 243, 251, … (sequence A246655 in the OEIS). The prime powers are those positive integers that are divisible by exactly one prime number; in particular, the number 1 is not a prime power. Prime powers are also called primary numbers, as in the primary decomposition.


In software engineering, profiling ("program profiling", "software profiling") is a form of dynamic program analysis that measures, for example, the space (memory) or time complexity of a program, the usage of particular instructions, or frequency and duration of function calls. The most common use of profiling information is to aid program optimization. Profiling is achieved by instrumenting either the program source code or its binary executable form using a tool called a profiler (or code profiler). A number of different techniques may be used by profilers, such as event-based, statistical, instrumented, and simulation methods.


In robust statistics, the convex hull provides one of the key components of a bagplot, a method for visualizing the spread of two-dimensional sample points. The contours of Tukey depth form a nested family of convex sets, with the convex hull outermost, and the bagplot also displays another polygon from this nested family, the contour of 50% depth.In statistical decision theory, the risk set of a randomized decision rule is the convex hull of the risk points of its underlying deterministic decision rules.


Beta Langfi pointed out that "it is not only to study parts and processes in isolation, but also to study the interaction of various parts. The organism should be considered as a whole or system. Environmental protection requires systematic thinking. Landscape planning is required for overall planning.


Typically, people try to systematically "retrace their steps" to determine all of the possible places where the item might be located. Based on the role that context plays in determining recall, it is not at all surprising that individuals often quite easily discover the lost item upon returning to the correct context. This concept is heavily related to the encoding specificity principle.


The selection of central office names was conducted in a careful manner to avoid misunderstanding of the verbal requests.For automatic telephone service, impulse senders (dials) were installed on customer telephones, so that subscribers did not need operators to initiate a call, but simply dial the directory number themselves. This required a digit or letter identification of central offices, so that the central office for the recipient could be dialed before the line number. Telephone dials were typically supplemented with letters next to the numerals on the dial, as seen in the right-hand photo, so that a name could be dialed by its first letter, or by multiple letters.


using Bayes' theorem). Having made explicit the expected loss for each given x {\displaystyle x\,\!} separately, we can define a decision rule δ {\displaystyle \delta \,\!}


In the context of digital privacy, communication privacy is the notion that individuals should have the freedom, or right, to communicate information digitally with the expectation that their communications are secure—meaning that messages and communications will only be accessible to the sender's original intended recipient.However, communications can be intercepted or delivered to other recipients without the sender's knowledge, in a multitude of ways. Communications can be intercepted directly through various hacking methods, such as the man-in-the-middle attack (MITM). Communications can also be delivered to recipients unbeknown to the sender due to false assumptions made regarding the platform or medium that was used to send information.


In the design of algorithms, partition refinement is a technique for representing a partition of a set as a data structure that allows the partition to be refined by splitting its sets into a larger number of smaller sets. In that sense it is dual to the union-find data structure, which also maintains a partition into disjoint sets but in which the operations merge pairs of sets. In some applications of partition refinement, such as lexicographic breadth-first search, the data structure maintains as well an ordering on the sets in the partition. Partition refinement forms a key component of several efficient algorithms on graphs and finite automata, including DFA minimization, the Coffman–Graham algorithm for parallel scheduling, and lexicographic breadth-first search of graphs.


In functional programming languages, and many others, it provides a way of automatically managing how arguments are passed to functions and exceptions. In theoretical computer science, it provides a way to study functions with multiple arguments in simpler theoretical models which provide only one argument. The most general setting for the strict notion of currying and uncurrying is in the closed monoidal categories, which underpins a vast generalization of the Curry–Howard correspondence of proofs and programs to a correspondence with many other structures, including quantum mechanics, cobordisms and string theory.


The S-box used is derived from the multiplicative inverse over GF(28), known to have good non-linearity properties. To avoid attacks based on simple algebraic properties, the S-box is constructed by combining the inverse function with an invertible affine transformation. The S-box is also chosen to avoid any fixed points (and so is a derangement), i.e., S ( a i , j ) ≠ a i , j {\displaystyle S(a_{i,j})\neq a_{i,j}} , and also any opposite fixed points, i.e., S ( a i , j ) ⊕ a i , j ≠ FF 16 {\displaystyle S(a_{i,j})\oplus a_{i,j}\neq {\text{FF}}_{16}} . While performing the decryption, the InvSubBytes step (the inverse of SubBytes) is used, which requires first taking the inverse of the affine transformation and then finding the multiplicative inverse.


In quantum computing, a graph state is a special type of multi-qubit state that can be represented by a graph. Each qubit is represented by a vertex of the graph, and there is an edge between every interacting pair of qubits. In particular, they are a convenient way of representing certain types of entangled states. Graph states are useful in quantum error-correcting codes, entanglement measurement and purification and for characterization of computational resources in measurement based quantum computing models. A graph state is a particular case of a 2-uniform hypergraph state, a generalization where the edges have N cardinality.


The challenge problems do not require the formalisation of large programming languages, but they do require sophistication in reasoning about: Binding Most programming languages have some form of binding, ranging in complexity from the simple binders of simply typed lambda calculus to complex, potentially infinite binders needed in the treatment of record patterns. Induction Properties such as subject reduction and strong normalisation often require complex induction arguments. Reuse Furthering collaboration being a key aim of the challenge, the solutions are expected to contain reusable components that would allow researchers to share language features and designs without requiring them to start from scratch every time.


In software engineering, a monolithic application describes a software application that is designed as a single service. Multiple services can be desirable in certain scenarios as it can facilitate maintenance by allowing repair or replacement of parts of the application without requiring wholesale replacement. Modularity is achieved to various extents by different modular programming approaches.


In mathematics, Welch bounds are a family of inequalities pertinent to the problem of evenly spreading a set of unit vectors in a vector space. The bounds are important tools in the design and analysis of certain methods in telecommunication engineering, particularly in coding theory. The bounds were originally published in a 1974 paper by L. R. Welch.


In mathematics, the Wiener–Wintner theorem, named after Norbert Wiener and Aurel Wintner, is a strengthening of the ergodic theorem, proved by Wiener and Wintner (1941).


In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contains an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system. A formal theory is an axiomatic system (usually formulated within model theory) that describes a set of sentences that is closed under logical implication. A formal proof is a complete rendition of a mathematical proof within a formal system.


Each work item consists of a series of instructions, to be executed sequentially, but in the course of its execution, a work item may also spawn new work items that can feasibly be executed in parallel with its other work. These new items are initially put on the queue of the processor executing the work item. When a processor runs out of work, it looks at the queues of the other processors and "steals" their work items.


{\displaystyle \|T\|_{1}\geq \|T\|_{p}\geq \|T\|_{p'}\geq \|T\|_{\infty }.} Duality: Let H 1 , H 2 {\displaystyle H_{1},H_{2}} be finite-dimensional Hilbert spaces, p ∈  {\displaystyle p\in } and q {\displaystyle q} such that 1 p + 1 q = 1 {\displaystyle {\frac {1}{p}}+{\frac {1}{q}}=1} , then ‖ S ‖ p = sup { | ⟨ S , T ⟩ | ∣ ‖ T ‖ q = 1 } , {\displaystyle \|S\|_{p}=\sup \lbrace |\langle S,T\rangle |\mid \|T\|_{q}=1\rbrace ,} where ⟨ S , T ⟩ = t r ( S ∗ T ) {\displaystyle \langle S,T\rangle =\mathrm {tr} (S^{*}T)} denotes the Hilbert–Schmidt inner product. Let ( e k ) k , ( f k ′ ) k ′ {\displaystyle (e_{k})_{k},(f_{k'})_{k'}} be two orthonormal basis of the Hilbert spaces H 1 , H 2 {\displaystyle H_{1},H_{2}} , then for p = 1 {\displaystyle p=1} ‖ T ‖ 1 ≤ ∑ k , k ′ | T k , k ′ | {\displaystyle \|T\|_{1}\leq \sum _{k,k'}\left|T_{k,k'}\right|} .


In statistics, the generalized canonical correlation analysis (gCCA), is a way of making sense of cross-correlation matrices between the sets of random variables when there are more than two sets. While a conventional CCA generalizes principal component analysis (PCA) to two sets of random variables, a gCCA generalizes PCA to more than two sets of random variables. The canonical variables represent those common factors that can be found by a large PCA of all of the transformed random variables after each set underwent its own PCA.


For example, it can be used about primes in an arithmetic progression of the form a n + b {\displaystyle an+b} , where a and b are coprime which according to Dirichlet's theorem on arithmetic progressions contains infinitely many primes, along with infinitely many composites. For integer k ≥ 3, an AP-k (also called PAP-k) is any sequence of k primes in arithmetic progression. An AP-k can be written as k primes of the form a·n + b, for fixed integers a (called the common difference) and b, and k consecutive integer values of n. An AP-k is usually expressed with n = 0 to k − 1. This can always be achieved by defining b to be the first prime in the arithmetic progression.


In queueing theory, a discipline within the mathematical theory of probability, an M/D/c queue represents the queue length in a system having c servers, where arrivals are determined by a Poisson process and job service times are fixed (deterministic). The model name is written in Kendall's notation. Agner Krarup Erlang first published on this model in 1909, starting the subject of queueing theory. The model is an extension of the M/D/1 queue which has only a single server.


In network science, ingredient-flavor networks are networks describing the sharing of flavor compounds of culinary ingredients. In the bipartite form, an ingredient-flavor network consist of two different types of nodes: the ingredients used in the recipes and the flavor compounds that contributes to the flavor of each ingredients. The links connecting different types of nodes are undirected, represent certain compound occur in each ingredients. The ingredient-flavor network can also be projected in the ingredient or compound space where nodes are ingredients or compounds, links represents the sharing of the same compounds to different ingredients or the coexistence in the same ingredient of different compounds.


There are a finite number of possible configurations in the section between two walls so the automaton must eventually start repeating inside each section, though the period may be very long if the section is wide enough. These walls will form with probability 1 for completely random initial conditions.


In order to assign a digital representation to an entity, the attributing party must trust that the claim of an attribute (such as name, location, role as an employee, or age) is correct and associated with the person or thing presenting the attribute. Conversely, the individual claiming an attribute may only grant selective access to its information (e.g., proving identity in a bar or PayPal authentication for payment at a website). In this way, digital identity is better understood as a particular viewpoint within a mutually-agreed relationship than as an objective property.


In mathematics, the goal of lattice basis reduction is to find a basis with short, nearly orthogonal vectors when given an integer lattice basis as input. This is realized using different algorithms, whose running time is usually at least exponential in the dimension of the lattice.


A typical pole stage plantation tree is 7–30 cm in diameter at breast height (dbh). Such trees are sometimes not suitable for timber, but are used as pulp for paper and particleboard, and as chips for oriented strand board. As the trees grow and become dense and crowded again, the thinning process is repeated.


When employed in parallel connected electric distribution systems, closed-core transformers finally made it technically and economically feasible to provide electric power for lighting in homes, businesses and public spaces. Bláthy had suggested the use of closed cores, Zipernowsky had suggested the use of parallel shunt connections, and Déri had performed the experiments; In early 1885, the three engineers also eliminated the problem of eddy current losses with the invention of the lamination of electromagnetic cores.Transformers today are designed on the principles discovered by the three engineers. They also popularized the word 'transformer' to describe a device for altering the EMF of an electric current although the term had already been in use by 1882. In 1886, the ZBD engineers designed, and the Ganz factory supplied electrical equipment for, the world's first power station that used AC generators to power a parallel connected common electrical network, the steam-powered Rome-Cerchi power plant.


For an example, given set y {\displaystyle y} , let Q ( n ) {\displaystyle Q(n)} denote the existential statement that a certain function space set exist, ∃ h . h ≃ y { 0 , 1 , … , n − 1 } {\displaystyle \exists h.h\simeq y^{\{0,1,\dots ,n-1\}}} . Here the existential quantifier is not merely one over natural numbers or bounded by any other set.


In statistics, specifically regression analysis, a binary regression estimates a relationship between one or more explanatory variables and a single output binary variable. Generally the probability of the two alternatives is modeled, instead of simply outputting a single value, as in linear regression. Binary regression is usually analyzed as a special case of binomial regression, with a single outcome ( n = 1 {\displaystyle n=1} ), and one of the two alternatives considered as "success" and coded as 1: the value is the count of successes in 1 trial, either 0 or 1. The most common binary regression models are the logit model (logistic regression) and the probit model (probit regression).


The multinational control of the export of cryptography on the Western side of the cold war divide was done via the mechanisms of CoCom. By the 1960s, however, financial organizations were beginning to require strong commercial encryption on the rapidly growing field of wired money transfer. The U.S. Government's introduction of the Data Encryption Standard in 1975 meant that commercial uses of high quality encryption would become common, and serious problems of export control began to arise. Generally these were dealt with through case-by-case export license request proceedings brought by computer manufacturers, such as IBM, and by their large corporate customers.


This approach is good solely for instructive purposes. The Euler–Lagrange equation will now be used to find the extremal function f ( x ) {\displaystyle f(x)} that minimizes the functional A  . {\displaystyle A.}


When clock speeds of CPUs began to move into the 3–5 GHz range, CPU power dissipation and other problems became more important. The ability of industry to produce ever-faster single CPU systems (linked to Moore's Law about the periodic doubling of transistor counts) began to be threatened. In the early 21st century, many flavors of parallel computing began to proliferate, including multi-core architectures at the low-end and massively parallel processing at the high end.


The simple logistic function varies by less than 0.01 from the cumulative normal ogive across the range, given an arbitrary scale factor. In the BTL model, the probability that object j is judged to have more of an attribute than object i is: Pr { X j i = 1 } = e δ j − δ i 1 + e δ j − δ i = σ ( δ j − δ i ) , {\displaystyle \Pr\{X_{ji}=1\}={\frac {e^{{\delta _{j}}-{\delta _{i}}}}{1+e^{{\delta _{j}}-{\delta _{i}}}}}=\sigma (\delta _{j}-\delta _{i}),} where δ i {\displaystyle \delta _{i}} is the scale location of object i {\displaystyle i} ; σ {\displaystyle \sigma } is the logistic function (the inverse of the logit). For example, the scale location might represent the perceived quality of a product, or the perceived weight of an object.


They were introduced in the course of Stanley's enumeration of the reduced decompositions of permutations, and in particular his proof that the permutation w0 = n(n − 1)...21 (written here in one-line notation) has exactly ( n 2 ) ! 1 n − 1 ⋅ 3 n − 2 ⋅ 5 n − 3 ⋯ ( 2 n − 3 ) 1 {\displaystyle {\frac {{\binom {n}{2}}! }{1^{n-1}\cdot 3^{n-2}\cdot 5^{n-3}\cdots (2n-3)^{1}}}} reduced decompositions. (Here ( n 2 ) {\displaystyle {\binom {n}{2}}} denotes the binomial coefficient n(n − 1)/2 and ! denotes the factorial.)


Subscribers are often provided with several set-top boxes as part of their subscription, and can give or sell unneeded activated boxes to neighboring nonsubscribers who can use them in their own residences, though a provider using IP location using the cable modem within a set-top box featuring advanced two-way features can avert this situation. This system is dependent on the security of the encryption system chosen by the cable company in question. Old cable equipment used an analog signal that was scrambled by tuning the signal so the picture was unsteady, just as Macrovision does at an attempt to copy a video.


In order to gain access to the software, the Scientologists must first sign a contract. Section 7 of this contract states that the members must agree to "use the specific Internet Filter Program that CSI has provided to you which allows you freedom to view other sites on Dianetics, Scientology or its principals without threat of accessing sites deemed to be using the Marks or Works in an unauthorized fashion or deemed to be improper or discreditable to the Scientology religion. "The program works by preventing the user from accessing sites with certain keywords which Scientology has identified as being objectionable material for viewing by their members. This use of keywords functions as a way to prevent members from learning of guarded Scientology doctrine, such as Xenu, OT III, and other material relating to Space opera in Scientology scripture.


But then, by the same argument as before, 2 divides b2, so b must be even. However, if a and b are both even, they have 2 as a common factor. This contradicts our previous statement that a and b have no common factor, so we must conclude that 2 {\displaystyle {\sqrt {2}}} is an irrational number.To paraphrase: if one could write 2 {\displaystyle {\sqrt {2}}} as a fraction, this fraction could never be written in lowest terms, since 2 could always be factored from numerator and denominator.


Part of the overhead is transmitted, then part of the payload, then the next part of the overhead, then the next part of the payload, until the entire frame has been transmitted. In the case of an STS-1, the frame is 810 octets in size, while the STM-1/STS-3c frame is 2,430 octets in size. For STS-1, the frame is transmitted as three octets of overhead, followed by 87 octets of payload.


In mathematics, Legendre's equation is the Diophantine equation The equation is named for Adrien-Marie Legendre who proved in 1785 that it is solvable in integers x, y, z, not all zero, if and only if −bc, −ca and −ab are quadratic residues modulo a, b and c, respectively, where a, b, c are nonzero, square-free, pairwise relatively prime integers, not all positive or all negative .


In 1819, Charles Babbage showed the same congruence modulo p2, which holds for p ≥ 3 {\displaystyle p\geq 3} . An equivalent formulation is the congruence ( a p b p ) ≡ ( a b ) ( mod p 3 ) {\displaystyle {ap \choose bp}\equiv {a \choose b}{\pmod {p^{3}}}} for p ≥ 5 {\displaystyle p\geq 5} , which is due to Wilhelm Ljunggren (and, in the special case b = 1 {\displaystyle b=1} , to J. W. L. Glaisher) and is inspired by Lucas' theorem. No known composite numbers satisfy Wolstenholme's theorem and it is conjectured that there are none (see below).


A read from one such memory can happen before the write to this memory completes. Therefore, the data read can be stale. Thus, a processor under PC can execute a younger load when an older store needs to be stalled.


In the following section it is shown that for 455 spheres the sausage packing is non-optimal, and that there instead exists a special cluster packing that occupies a smaller volume. The volume of a convex hull of a sausage packing with n {\displaystyle n} spheres of radius r {\displaystyle r} is calculable with elementary geometry. The middle part of the hull is a cylinder with length h = 2 r ⋅ ( n − 1 ) {\displaystyle h=2r\cdot (n-1)} while the caps at the end are half-spheres with radius r {\displaystyle r} . The total volume V W {\displaystyle V_{W}} is therefore given by.


One of the most notably cyber attacks that had a physical impact, causing significant degradation of a target system, were the Stuxnet and Aurora worms. The Stuxnet worm was first revealed in 2010 and specially targeted weaknesses in Programmable Logic Controllers (PLCs), devices in the SCADA category of systems. Though it was never positivity attributed, it is widely believed that the malicious software was developed jointly by the United States and Israel to disrupt the Iranian nuclear enrichment facility at Natanz.


In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if n and a are coprime positive integers, and φ ( n ) {\displaystyle \varphi (n)} is Euler's totient function, then a raised to the power φ ( n ) {\displaystyle \varphi (n)} is congruent to 1 modulo n; that is a φ ( n ) ≡ 1 ( mod n ) . {\displaystyle a^{\varphi (n)}\equiv 1{\pmod {n}}.} In 1736, Leonhard Euler published a proof of Fermat's little theorem (stated by Fermat without proof), which is the restriction of Euler's theorem to the case where n is a prime number. Subsequently, Euler presented other proofs of the theorem, culminating with his paper of 1763, in which he proved a generalization to the case where n is not prime.The converse of Euler's theorem is also true: if the above congruence is true, then a {\displaystyle a} and n {\displaystyle n} must be coprime.


In the field of recorded music, a hidden track (sometimes called a ghost track, secret track or unlisted track) is a song or a piece of audio that has been placed on a CD, audio cassette, LP record, or other recorded medium, in such a way as to avoid detection by the casual listener. In some cases, the piece of music may simply have been left off the track listing, while in other cases, more elaborate methods are used. In rare cases, a 'hidden track' is actually the result of an error that occurred during the mastering stage production of the recorded media. However, since the rise of digital and streaming services such as iTunes and Spotify in the late 2000s and early 2010s, the inclusion of hidden tracks has declined on studio albums.


It is the starting point for understanding the structure of semigroups. It serves as a counterexample in illuminating many situations. For example, the semigroup with one element is the only semigroup in which 0 = 1, that is, the zero element and the identity element are equal. Further, if S is a semigroup with one element, the semigroup obtained by adjoining an identity element to S is isomorphic to the semigroup obtained by adjoining a zero element to S. The semigroup with one element is also a group. In the language of category theory, any semigroup with one element is a terminal object in the category of semigroups.


2 S ( n + 1 , j + 1 ) S ( k + 1 , j + 1 ) , {\displaystyle B_{n}^{(-k)}=\sum _{j=0}^{\min(n,k)}(j! )^{2}S(n+1,j+1)S(k+1,j+1),} where S ( n , k ) {\displaystyle S(n,k)} is the number of ways to partition a size n {\displaystyle n} set into k {\displaystyle k} non-empty subsets (the Stirling number of the second kind). A combinatorial interpretation is that the poly-Bernoulli numbers of negative index enumerate the set of n {\displaystyle n} by k {\displaystyle k} (0,1)-matrices uniquely reconstructible from their row and column sums.


In mathematical optimization, a feasible region, feasible set, search space, or solution space is the set of all possible points (sets of values of the choice variables) of an optimization problem that satisfy the problem's constraints, potentially including inequalities, equalities, and integer constraints. This is the initial set of candidate solutions to the problem, before the set of candidates has been narrowed down. For example, consider the problem of minimizing the function x 2 + y 4 {\displaystyle x^{2}+y^{4}} with respect to the variables x {\displaystyle x} and y , {\displaystyle y,} subject to 1 ≤ x ≤ 10 {\displaystyle 1\leq x\leq 10} and 5 ≤ y ≤ 12. {\displaystyle 5\leq y\leq 12.\,} Here the feasible set is the set of pairs (x, y) in which the value of x is at least 1 and at most 10 and the value of y is at least 5 and at most 12.


In probability theory and statistics, the normal-inverse-gamma distribution (or Gaussian-inverse-gamma distribution) is a four-parameter family of multivariate continuous probability distributions. It is the conjugate prior of a normal distribution with unknown mean and variance.


In order to implement general-purpose coroutines, a second call stack must be obtained, which is a feature not directly supported by the C language. A reliable (albeit platform-specific) way to achieve this is to use a small amount of inline assembly to explicitly manipulate the stack pointer during initial creation of the coroutine. This is the approach recommended by Tom Duff in a discussion on its relative merits vs. the method used by Protothreads. On platforms which provide the POSIX sigaltstack system call, a second call stack can be obtained by calling a springboard function from within a signal handler to achieve the same goal in portable C, at the cost of some extra complexity.


, a n ( R ) {\displaystyle \Pi _{a_{1},...,a_{n}}(R)} , where R {\displaystyle R} is a relation and a 1 , . . .


At a higher logical level, the MCH also sees two channels, each with four ranks. In contrast, banks, while similar from a logical perspective to ranks, are implemented quite differently in physical hardware. Banks are sub-units inside a single memory chip, while ranks are sub-units composed of a subset of the chips on a module. Similar to chip select, banks are selected by bank select bits, which are part of the memory interface.


In telecommunication, the term bit-count integrity (BCI) has the following meanings: In message communications, the preservation of the exact number of bits that are in the original message. In connection-oriented services, preservation of the number of bits per unit time.Note: Bit-count integrity is not the same as bit integrity, which requires that the delivered bits correspond exactly with the original bits. Source: from Federal Standard 1037C and from MIL-STD-188


In sequential code it is possible to control the flow of the program using if-then-else statements and various forms of loops. Such flow control structures have only recently been added to GPUs. Conditional writes could be performed using a properly crafted series of arithmetic/bit operations, but looping and conditional branching were not possible. Recent GPUs allow branching, but usually with a performance penalty. Branching should generally be avoided in inner loops, whether in CPU or GPU code, and various methods, such as static branch resolution, pre-computation, predication, loop splitting, and Z-cull can be used to achieve branching when hardware support does not exist.


Social hierarchies within the 17th century were highly regarded, as architecture was able to epitomize the servants and the upper class. More privacy is offered to the occupant as Pratt further claims, "the ordinary servants may never publicly appear in passing to and fro for their occasions there." This social divide between rich and poor favored the physical integration of the corridor into housing by the 19th century. Sociologist Witold Rybczynski wrote, "the subdivision of the house into day and night uses, and into formal and informal areas, had begun." Rooms were changed from public to private as single entryways forced notions of entering a room with a specific purpose.


Since classical algorithms for NP-complete problems require exponentially many steps, and Grover's algorithm provides at most a quadratic speedup over the classical solution for unstructured search, this suggests that Grover's algorithm by itself will not provide polynomial-time solutions for NP-complete problems (as the square root of an exponential function is an exponential, not polynomial, function).Unlike other quantum algorithms, which may provide exponential speedup over their classical counterparts, Grover's algorithm provides only a quadratic speedup. However, even quadratic speedup is considerable when N {\displaystyle N} is large, and Grover's algorithm can be applied to speed up broad classes of algorithms. Grover's algorithm could brute-force a 128-bit symmetric cryptographic key in roughly 264 iterations, or a 256-bit key in roughly 2128 iterations. It may not be the case that Grover's algorithm poses a significantly increased risk to encryption over existing classical algorithms, however.


{\displaystyle {\boldsymbol {S}}=J~{\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\sigma }}\cdot {\boldsymbol {F}}^{-T}~.} In index notation with respect to an orthonormal basis, S I L = J F I k − 1 F L m − 1 σ k m = J ∂ X I ∂ x k ∂ X L ∂ x m σ k m {\displaystyle S_{IL}=J~F_{Ik}^{-1}~F_{Lm}^{-1}~\sigma _{km}=J~{\cfrac {\partial X_{I}}{\partial x_{k}}}~{\cfrac {\partial X_{L}}{\partial x_{m}}}~\sigma _{km}\!\,\!} This tensor, a one-point tensor, is symmetric. If the material rotates without a change in stress state (rigid rotation), the components of the second Piola–Kirchhoff stress tensor remain constant, irrespective of material orientation. The second Piola–Kirchhoff stress tensor is energy conjugate to the Green–Lagrange finite strain tensor.


259) but offers no firm resolution. He asserts: "In general a RPT operation could not be an instruction in the finite-state part of the machine ... this might exhaust any particular amount of storage allowed in the finite part of the computer . RPT operations require infinite registers of their own."


In the example on the right, the rows restricted to the first three columns contain the 8 possible ordered triples consisting of 0's and 1's, each appearing once. The same holds for any other choice of three columns. Thus this is an orthogonal array of strength 3.


In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and the class of compound Poisson–gamma distributions which have positive mass at zero, but are otherwise continuous. Tweedie distributions are a special case of exponential dispersion models and are often used as distributions for generalized linear models.The Tweedie distributions were named by Bent Jørgensen after Maurice Tweedie, a statistician and medical physicist at the University of Liverpool, UK, who presented the first thorough study of these distributions in 1984.


In the Japanese kantenji braille, the standard 8-dot Braille patterns 23678, 123678, 234678, and 1234678 are the patterns related to Braille pattern dots-12356, since the two additional dots of kantenji patterns 012356, 123567, and 0123567 are placed above the base 6-dot cell, instead of below, as in standard 8-dot braille.


In mathematics, the spectral radius of a square matrix is the maximum of the absolute values of its eigenvalues. More generally, the spectral radius of a bounded linear operator is the supremum of the absolute values of the elements of its spectrum. The spectral radius is often denoted by ρ(·).


This control structure can be known as a post-test loop. This means the do-while loop is an exit-condition loop. However a while loop will test the condition before the code within the block is executed.


In semantics, pragmatics, and philosophy of language, the common ground of a conversation is the set of propositions that the interlocutors have agreed to treat as true. For a proposition to be in the common ground, it must be common knowledge in the conversational context. The set of possible worlds compatible with the common ground is often called the context set.The concept is fundamental to many theories of discourse.


In the following diagram, only one instance (or "execution") of the basic Paxos protocol, with an initial Leader (a Proposer), is shown. Note that a Multi-Paxos consists of several instances of the basic Paxos protocol. Client Proposer Acceptor Learner | | | | | | | --- First Request --- X-------->| | | | | | Request | X--------->|->|->| | | Prepare(N) | |<---------X--X--X | | Promise(N,I,{Va,Vb,Vc}) | X--------->|->|->| | | Accept! (N,I,V) | |<---------X--X--X------>|->| Accepted(N,I,V) |<---------------------------------X--X Response | | | | | | | where V = last of (Va, Vb, Vc).


In a compression oracle attack the use of adaptive data compression on a mixture of chosen plaintext and unknown plaintext can result in content-sensitive changes in the length of the compressed text that can be detected even though the content of the compressed text itself is then encrypted. This can be used in protocol attacks to detect when the injected known plaintext is even partially similar to the unknown content of a secret part of the message, greatly reducing the complexity of a search for a match for the secret text. The CRIME and BREACH attacks are examples of protocol attacks using this phenomenon.


In the field of psychometrics, the Standards for Educational and Psychological Testing place standards about validity and reliability, along with errors of measurement and issues related to the accommodation of individuals with disabilities. The third and final major topic covers standards related to testing applications, credentialing, plus testing in program evaluation and public policy.


The largest known prime number, 282,589,933 − 1, is a Mersenne prime. Since 1997, all newly found Mersenne primes have been discovered by the Great Internet Mersenne Prime Search, a distributed computing project. In December 2020, a major milestone in the project was passed after all exponents below 100 million were checked at least once.


In most deployments of Paxos, each participating process acts in three roles; Proposer, Acceptor and Learner. This reduces the message complexity significantly, without sacrificing correctness: In Paxos, clients send commands to a leader. During normal operation, the leader receives a client's command, assigns it a new command number i {\displaystyle i} , and then begins the i {\displaystyle i} th instance of the consensus algorithm by sending messages to a set of acceptor processes. By merging roles, the protocol "collapses" into an efficient client-master-replica style deployment, typical of the database community. The benefit of the Paxos protocols (including implementations with merged roles) is the guarantee of its safety properties. A typical implementation's message flow is covered in the section Multi-Paxos.


In mathematics, especially in algebraic topology, an induced homomorphism is a homomorphism derived in a canonical way from another map. For example, a continuous map from a topological space X to a topological space Y induces a group homomorphism from the fundamental group of X to the fundamental group of Y. More generally, in category theory, any functor by definition provides an induced morphism in the target category for each morphism in the source category. For example, fundamental groups, higher homotopy groups, singular homology, and De Rham cohomology are algebraic structures that are functorial, meaning that their definition provides a functor from (e.g.) the category of topological spaces to (e.g.) the category of groups or rings. This means that each space is associated with an algebraic structure, while each continuous map between spaces is associated with a structure-preserving map between structures, called an induced homomorphism.


In statistics, least-angle regression (LARS) is an algorithm for fitting linear regression models to high-dimensional data, developed by Bradley Efron, Trevor Hastie, Iain Johnstone and Robert Tibshirani.Suppose we expect a response variable to be determined by a linear combination of a subset of potential covariates. Then the LARS algorithm provides a means of producing an estimate of which variables to include, as well as their coefficients. Instead of giving a vector result, the LARS solution consists of a curve denoting the solution for each value of the L1 norm of the parameter vector. The algorithm is similar to forward stepwise regression, but instead of including variables at each step, the estimated parameters are increased in a direction equiangular to each one's correlations with the residual.


In mathematics, the infinity symbol is used more often to represent a potential infinity, rather than an actually infinite quantity as included in the cardinal numbers and the ordinal numbers (which use other notations, such as ℵ 0 {\displaystyle \,\aleph _{0}\,} and ω, for infinite values). For instance, in mathematical expressions with summations and limits such as the infinity sign is conventionally interpreted as meaning that the variable grows arbitrarily large towards infinity, rather than actually taking an infinite value, although other interpretations are possible.The infinity symbol may also be used to represent a point at infinity, especially when there is only one such point under consideration. This usage includes, in particular, the infinite point of a projective line, and the point added to a topological space to form its one-point compactification.


In the area of mathematics known as graph theory, a tree is said to be starlike if it has exactly one vertex of degree greater than 2. This high-degree vertex is the root and a starlike tree is obtained by attaching at least three linear graphs to this central vertex.


Computer frameworks may have a few audit trails each gave to a specific sort of action. Related to proper apparatuses and systems, audit trails can help with distinguishing security infringement, execution issues and application issues. Routine log audits and investigation are valuable for distinguishing security episodes, approach infringement, fake movement, and operational issues soon after they have happened, and for giving information valuable to settling such issues.


Capturing two opposing tokens by occupying the single square separating them, also known as interception. Declaring an "attack" on an opposing token, and then determining the outcome of the attack, either in a deterministic way by the game rules (e.g. Stratego, Illuminati), or by using a randomising method (e.g. Illuminati: New World Order). Surrounding a token or region with one's own tokens in some manner (e.g. go), also known as enclosure.


Programming languages commonly associated with buffer overflows include C and C++, which provide no built-in protection against accessing or overwriting data in any part of memory and do not automatically check that data written to an array (the built-in buffer type) is within the boundaries of that array. Bounds checking can prevent buffer overflows, but requires additional code and processing time. Modern operating systems use a variety of techniques to combat malicious buffer overflows, notably by randomizing the layout of memory, or deliberately leaving space between buffers and looking for actions that write into those areas ("canaries").


For each integer n {\displaystyle n} the smallest fusible number that is greater than n {\displaystyle n} has the form n + 1 / 2 k {\displaystyle n+1/2^{k}} . The existence of k {\displaystyle k} for each n {\displaystyle n} cannot be proven in Peano arithmetic, and k {\displaystyle k} grows so rapidly as a function of n {\displaystyle n} that for n = 3 {\displaystyle n=3} it is (in Knuth's up-arrow notation for large numbers) already larger than 2 ↑ 9 16 {\displaystyle 2\uparrow ^{9}16} .The usual proof of Urysohn's lemma utilizes the dyadic fractions for constructing the separating function from the lemma. == References ==


The upper bound on the fraction of agents that can be guaranteed 1 of their best c items (a property weaker than 1-out-of-c MMS) is ( 1 − 1 / 2 c ) {\displaystyle (1-1/2^{c})} . For c = 2 {\displaystyle c=2} , the lower bound for 1-out-of-best-c allocation can be improved from 1/2 to 3/5; it is an open question whether the upper bound of 3/4 can always be attained. It is NP-hard to decide if a given instance admits an allocation that gives each agent a positive utility.


In metadata, the term data element is an atomic unit of data that has precise meaning or precise semantics. A data element has: An identification such as a data element name A clear data element definition One or more representation terms Optional enumerated values Code (metadata) A list of synonyms to data elements in other metadata registries Synonym ringData elements usage can be discovered by inspection of software applications or application data files through a process of manual or automated Application Discovery and Understanding. Once data elements are discovered they can be registered in a metadata registry. In telecommunication, the term data element has the following components: A named unit of data that, in some contexts, is considered indivisible and in other contexts may consist of data items.


: 87 As Quine points out: 'The basis combination in which general and singular terms find their contrasting roles is that of predication. ': 87 Predication combines general terms with singular terms, in a sentence that is true or false just as the general term ('F') is true or false of the object to which the singular term ('a') refers. Predication is thus logically represented as 'Fa'.


In the application of integrated circuits, process control monitoring (PCM) is the procedure followed to obtain detailed information about the process used. PCM is associated with designing and fabricating special structures that can monitor technology specific parameters such as Vth in CMOS and Vbe in bipolars. These structures are placed across the wafer at specific locations along with the chip produced so that a closer look into the process variation is possible.


In mathematics, the Besicovitch inequality is a geometric inequality relating volume of a set and distances between certain subsets of its boundary. The inequality was first formulated by Abram Besicovitch.Consider the n-dimensional cube  n {\displaystyle ^{n}} with a Riemannian metric g {\displaystyle g} . Let denote the distance between opposite faces of the cube.


In the 1980s and early 1990s, when the 8088 and 80286 were still in common use, the term x86 usually represented any 8086-compatible CPU. Today, however, x86 usually implies a binary compatibility also with the 32-bit instruction set of the 80386. This is due to the fact that this instruction set has become something of a lowest common denominator for many modern operating systems and probably also because the term became common after the introduction of the 80386 in 1985. A few years after the introduction of the 8086 and 8088, Intel added some complexity to its naming scheme and terminology as the "iAPX" of the ambitious but ill-fated Intel iAPX 432 processor was tried on the more successful 8086 family of chips, applied as a kind of system-level prefix.


In mathematics, and in particular, algebra, a generalized inverse (or, g-inverse) of an element x is an element y that has some properties of an inverse element but not necessarily all of them. The purpose of constructing a generalized inverse of a matrix is to obtain a matrix that can serve as an inverse in some sense for a wider class of matrices than invertible matrices. Generalized inverses can be defined in any mathematical structure that involves associative multiplication, that is, in a semigroup. This article describes generalized inverses of a matrix A {\displaystyle A} .


In the Japanese kantenji braille, the standard 8-dot Braille patterns 36, 136, 346, and 1346 are the patterns related to Braille pattern dots-25, since the two additional dots of kantenji patterns 025, 257, and 0257 are placed above the base 6-dot cell, instead of below, as in standard 8-dot braille.


In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin for use in deformation theory. Intuitively, schemes are given by gluing together affine schemes using the Zariski topology, while algebraic spaces are given by gluing together affine schemes using the finer étale topology. Alternatively one can think of schemes as being locally isomorphic to affine schemes in the Zariski topology, while algebraic spaces are locally isomorphic to affine schemes in the étale topology. The resulting category of algebraic spaces extends the category of schemes and allows one to carry out several natural constructions that are used in the construction of moduli spaces but are not always possible in the smaller category of schemes, such as taking the quotient of a free action by a finite group (cf. the Keel–Mori theorem).


In philosophy, counterexamples are usually used to argue that a certain philosophical position is wrong by showing that it does not apply in certain cases. Alternatively, the first philosopher can modify their claim so that the counterexample no longer applies; this is analogous to when a mathematician modifies a conjecture because of a counterexample. For example, in Plato's Gorgias, Callicles, trying to define what it means to say that some people are "better" than others, claims that those who are stronger are better. But Socrates replies that, because of their strength of numbers, the class of common rabble is stronger than the propertied class of nobles, even though the masses are prima facie of worse character.


In probability theory and statistics Chow–Liu tree is an efficient method for constructing a second-order product approximation of a joint probability distribution, first described in a paper by Chow & Liu (1968). The goals of such a decomposition, as with such Bayesian networks in general, may be either data compression or inference.


This is useful if the user wants to do a prior art search (e.g. finding an existing approach to complete a specific task, finding a document that discloses a system that exhibits a procedural behavior collaboratively conducted by several components and links between these components). Web search engines which support proximity search via an explicit proximity operator in their query language include Walhello, Exalead, Yandex, Yahoo!, Altavista, and Bing: When using the Walhello search-engine, the proximity can be defined by the number of characters between the keywords. The search engine Exalead allows the user to specify the required proximity, as the maximum number of words between keywords.


In statistics and, in particular, in the fitting of linear or logistic regression models, the elastic net is a regularized regression method that linearly combines the L1 and L2 penalties of the lasso and ridge methods.


In the 1930s, J. H. C. Whitehead claimed a proof but then retracted it. In the process, he discovered some examples of simply-connected (indeed contractible, i.e. homotopically equivalent to a point) non-compact 3-manifolds not homeomorphic to R 3 {\displaystyle \mathbb {R} ^{3}} , the prototype of which is now called the Whitehead manifold. In the 1950s and 1960s, other mathematicians attempted proofs of the conjecture only to discover that they contained flaws. Influential mathematicians such as Georges de Rham, R. H. Bing, Wolfgang Haken, Edwin E. Moise, and Christos Papakyriakopoulos attempted to prove the conjecture.


In order for the minimization problem to have a well-defined solution, we have to place constraints on the set H {\displaystyle {\mathcal {H}}} of hypotheses being considered. If H {\displaystyle {\mathcal {H}}} is a normed space (as is the case for SVM), a particularly effective technique is to consider only those hypotheses f {\displaystyle f} for which ‖ f ‖ H < k {\displaystyle \lVert f\rVert _{\mathcal {H}}


All these results are employed in probability and statistics with a particular importance in the theory of point processes and queueing theory as well as the related fields stochastic geometry, continuum percolation theory, and spatial statistics.Another result by the name of Campbell's theorem is specifically for the Poisson point process and gives a method for calculating moments as well as the Laplace functional of a Poisson point process. The name of both theorems stems from the work by Norman R. Campbell on thermionic noise, also known as shot noise, in vacuum tubes, which was partly inspired by the work of Ernest Rutherford and Hans Geiger on alpha particle detection, where the Poisson point process arose as a solution to a family of differential equations by Harry Bateman. In Campbell's work, he presents the moments and generating functions of the random sum of a Poisson process on the real line, but remarks that the main mathematical argument was due to G. H. Hardy, which has inspired the result to be sometimes called the Campbell–Hardy theorem.


In statistics, the explained sum of squares (ESS), alternatively known as the model sum of squares or sum of squares due to regression (SSR – not to be confused with the residual sum of squares (RSS) or sum of squares of errors), is a quantity used in describing how well a model, often a regression model, represents the data being modelled. In particular, the explained sum of squares measures how much variation there is in the modelled values and this is compared to the total sum of squares (TSS), which measures how much variation there is in the observed data, and to the residual sum of squares, which measures the variation in the error between the observed data and modelled values.


To prevent critical races and inconsistency, only one processor (CPU) at a given time is allowed to access a particular data structure (a memory portion), while other CPUs trying to access at the same time are locked-out, waiting in idle status.Three cases can be distinguished when this idle wait is either necessary, convenient, or not convenient. The idle wait is necessary when the access is to a ready list for a low level scheduling operation. The idle wait is not necessary but convenient in the case of a critical section for synchronization/IPC operations, which require less time than a context switch (executing another process to avoid idle wait).


Gödel's incompleteness theorems were other examples that uncovered fundamental limitations in the provability of formal systems.In computational complexity theory, techniques like relativization (the addition of an oracle) allow for "weak" proofs of impossibility, in that proofs techniques that are not affected by relativization cannot resolve the P versus NP problem. Another technique is the proof of completeness for a complexity class, which provides evidence for the difficulty of problems by showing them to be just as hard to solve as any other problem in the class. In particular, a complete problem is intractable if one of the problems in its class is.


In the Japanese kantenji braille, the standard 8-dot Braille patterns 278, 1278, 2478, and 12478 are the patterns related to Braille pattern dots-136, since the two additional dots of kantenji patterns 0136, 1367, and 01367 are placed above the base 6-dot cell, instead of below, as in standard 8-dot braille.


In the field of transportation, bilevel optimization commonly appears in the toll-setting problem. Consider a network of highways that is operated by the government. The government wants to maximize its revenues by choosing the optimal toll setting for the highways. However, the government can maximize its revenues only by taking the highway users' problem into account.


Let G denote the subcollection of F consisting of all balls from the cN disjoint families A1,...,AcN. The less precise following statement is clearly true: every point x ∈ RN belongs to at most cN different balls from the subcollection G, and G remains a cover for E (every point y ∈ E belongs to at least one ball from the subcollection G). This property gives actually an equivalent form for the theorem (except for the value of the constant). There exists a constant bN depending only on the dimension N with the following property: Given any Besicovitch cover F of a bounded set E, there is a subcollection G of F such that G is a cover of the set E and every point x ∈ E belongs to at most bN different balls from the subcover G.In other words, the function SG equal to the sum of the indicator functions of the balls in G is larger than 1E and bounded on RN by the constant bN, 1 E ≤ S G := ∑ B ∈ G 1 B ≤ b N . {\displaystyle \mathbf {1} _{E}\leq S_{\mathbf {G} }:=\sum _{B\in \mathbf {G} }\mathbf {1} _{B}\leq b_{N}.}


Proposer Acceptor Learner | | | | | | | X--------->|->|->| | | Prepare(1) |<---------X--X--X | | Promise(1,{null,null,null}) x--------->|->| | | | Accept! (1,V1) | | X--X--------->|->| Accepted(1,V1) ! | | | | | | !!


In statistical mechanics, probability theory, graph theory, etc. the random cluster model is a random graph that generalizes and unifies the Ising model, Potts model, and percolation model. It is used to study random combinatorial structures, electrical networks, etc. It is also referred to as the RC model or sometimes the FK representation after its founders Cees Fortuin and Piet Kasteleyn.


Conclusions that assume limiting operations do 'commute' are called formal. The analyst tries to delineate conditions under which such conclusions are valid; in other words mathematical rigour is established by the specification of some set of sufficient conditions for the formal analysis to hold. This approach justifies, for example, the notion of uniform convergence.


By comparison, the Sun Microsystems SPARC architecture provides simultaneous visibility into four sets of eight registers each. Three sets of eight registers each are "windowed". Eight registers (i0 through i7) form the input registers to the current procedure level.


In mathematics, a polycyclic group is a solvable group that satisfies the maximal condition on subgroups (that is, every subgroup is finitely generated). Polycyclic groups are finitely presented, which makes them interesting from a computational point of view.


In telecommunications, a transverse redundancy check (TRC) or vertical redundancy check is a redundancy check for synchronized parallel bits applied once per bit time, across the bit streams. This requires additional parallel channels for the check bit or bits. The term usually applies to a single parity bit, although it could also be used to refer to a larger Hamming code.


This brings one to idea of using approximate factorization LU of A as the iteration matrix M. A version of incomplete lower-upper decomposition method was proposed by Stone in 1968. This method is designed for equation system arising from discretisation of partial differential equations and was firstly used for a pentadiagonal system of equations obtained while solving an elliptic partial differential equation in a two-dimensional space by a finite difference method. The LU approximate decomposition was looked in the same pentadiagonal form as the original matrix (three diagonals for L and three diagonals for U) as the best match of the seven possible equations for the five unknowns for each row of the matrix.


A chain of responsibility can be used to allow native code to better inter-operate with the JVM. On Windows platforms, Structured Exception Handling (SEH) may be employed to wrap native code in SEH try/catch blocks so as to capture machine (CPU/FPU) generated software interrupts (such as NULL pointer access violations and divide-by-zero operations), and to handle these situations before the interrupt is propagated back up into the JVM (i.e. Java side code), in all likelihood resulting in an unhandled exception.The encoding used for the NewStringUTF, GetStringUTFLength, GetStringUTFChars, ReleaseStringUTFChars and GetStringUTFRegion functions is "modified UTF-8", which is not valid UTF-8 for all inputs, but a different encoding really. The null character (U+0000) and codepoints not on the Basic Multilingual Plane (greater than or equal to U+10000, i.e. those represented as surrogate pairs in UTF-16) are encoded differently in modified UTF-8. Many programs actually use these functions incorrectly and treat the UTF-8 strings returned or passed into the functions as standard UTF-8 strings instead of modified UTF-8 strings. Programs should use the NewString, GetStringLength, GetStringChars, ReleaseStringChars, GetStringRegion, GetStringCritical and ReleaseStringCritical functions, which use UTF-16LE encoding on little-endian architectures and UTF-16BE on big-endian architectures, and then use a UTF-16 to UTF-8 conversion routine.


Diophantine approximations and transcendental number theory are very close areas that share many theorems and methods. Diophantine approximations also have important applications in the study of Diophantine equations. The 2022 Fields Medal was awarded to James Maynard for his work on Diophantine approximation.


There is, however, the Information Commissioner's Office (ICO), an independent public body set up to promote access to official information and protect personal information. They do this by promoting good practice, ruling on eligible complaints, giving information to individuals and organisations, and taking action when the law is broken. The relevant UK laws include: Data Protection Act 1998; Freedom of Information Act 2000; Environmental Information Regulations 2004; Privacy and Electronic Communications Regulations 2003. The ICO has also provided a "Personal Information Toolkit" online which explains in more detail the various ways of protecting privacy online.


For every representation ( ρ , V ) {\displaystyle (\rho ,V)} of a group G {\displaystyle G} we define V G := { v ∈ V: ρ ( s ) v = v ∀ s ∈ G } . {\displaystyle V^{G}:=\{v\in V:\rho (s)v=v\,\,\,\,\forall \,s\in G\}.} In general, ρ ( s ): V → V {\displaystyle \rho (s):V\to V} is not G {\displaystyle G} -linear.