in mathematics, an automatic semigroup is a finitely generated semigroup equipped with several regular languages over an alphabet representing a generating set. one of these languages determines " canonical forms " for the elements of the semigroup, the other languages determine if two canonical forms represent elements that differ by multiplication by a generator. formally, let s { \ displaystyle s } be a semigroup and a { \ displaystyle a } be a finite set of generators. then an automatic structure for s { \ displaystyle s } with respect to a { \ displaystyle a } consists of a regular language l { \ displaystyle l } over a { \ displaystyle a } such that every element of s { \ displaystyle s } has at least one representative in l { \ displaystyle l } and such that for each a ∈ a ∪ { ε } { \ displaystyle a \ in a \ cup \ { \ varepsilon \ } }, the relation consisting of pairs ( u, v ) { \ displaystyle ( u, v ) } with u a = v { \ displaystyle ua = v } is regular, viewed as a subset of ( a # × a # ) *. here a # is a augmented with a padding symbol. the concept of an automatic semigroup was generalized from automatic groups by campbell et al. ( 2001 ) unlike automatic groups ( see epstein et al. 1992 ), a semigroup may have an automatic structure with respect to one generating set, but not with respect to another. however, if an automatic semigroup has an identity, then it has an automatic structure with respect to any generating set ( duncan et al. 1999 ).


in the example shown below, a group that initially contains members { a, b, c } is asked to add { d, e, f }, but member c fails during the protocol. membership change requests are treated as a special kind of multicast and the sequence of events is the same. the example is thus nearly identical to the prior one, but now a series of new view events are delivered to the application. ( quorum size = 2, view1 = { a, b, c } ) member leader members application layer a a b c d e f a b c d e f | | | | | | | | | | | x - - - - - - - - > | | | | | | | | | | request ( " add d, e, f " ) | x - - - - - - - - - > | - > | - > | | | | | | | propose ( 1. 1 : " add d, e, f " ) | | | | * | | * | | |!!


in pseudocode : algorithm halton - sequence is inputs : index i { \ displaystyle i } base b { \ displaystyle b } output : result r { \ displaystyle r } f ← 1 { \ displaystyle f \ leftarrow 1 } r ← 0 { \ displaystyle r \ leftarrow 0 } while i > 0 { \ displaystyle i > 0 } do f ← f / b { \ displaystyle f \ leftarrow f / b } r ← r + f ∗ ( i mod b ) { \ displaystyle r \ leftarrow r + f * ( i \ operatorname { mod } b ) } i ← i / b { \ displaystyle i \ leftarrow \ lfloor i / b \ rfloor } return r { \ displaystyle r } an alternative implementation that produces subsequent numbers of a halton sequence for base b is given in the following generator function ( in python ). this algorithm uses only integer numbers internally, which makes it robust against round - off errors.


in the called subroutine, the first code executed is usually termed the subroutine prologue, since it does the necessary housekeeping before the code for the statements of the routine is begun. for instruction set architectures in which the instruction used to call a subroutine puts the return address into a register, rather than pushing it onto the stack, the prologue will commonly save the return address by pushing the value onto the call stack, although if the called subroutine does not call any other routines it may leave the value in the register. similarly, the current stack pointer and / or frame pointer values may be pushed.


variables often store simple data, like integers and literal strings, but some programming languages allow a variable to store values of other datatypes as well. such languages may also enable functions to be parametric polymorphic. these functions operate like variables to represent data of multiple types. for example, a function named length may determine the length of a list. such a length function may be parametric polymorphic by including a type variable in its type signature, since the number of elements in the list is independent of the elements'types.


the following theorems will aim to find the remaining nine axioms of standard pc within the " theorem - space " of frege's pc, showing that the theory of standard pc is contained within the theory of frege's pc. ( a theory, also called here, for figurative purposes, a " theorem - space ", is a set of theorems that are a subset of a universal set of well - formed formulas. the theorems are linked to each other in a directed manner by inference rules, forming a sort of dendritic network. at the roots of the theorem - space are found the axioms, which " generate " the theorem - space much like a generating set generates a group. )


in the fixed telephone network, a node may be a public or private telephone exchange, a remote concentrator or a computer providing some intelligent network service. in cellular communication, switching points and databases such as the base station controller, home location register, gateway gprs support node ( ggsn ) and serving gprs support node ( sgsn ) are examples of nodes. cellular network base stations are not considered to be nodes in this context. in cable television systems ( catv ), this term has assumed a broader context and is generally associated with a fiber optic node. this can be defined as those homes or businesses within a specific geographic area that are served from a common fiber optic receiver. a fiber optic node is generally described in terms of the number of " homes passed " that are served by that specific fiber node.


in one academic model, mobile banking is defined as : mobile banking refers to provision and availment of banking - and financial services with the help of mobile telecommunication devices. the scope of offered services may include facilities to conduct bank and stock market transactions, to administer accounts and to access customised information. " according to this model mobile banking can be said to consist of three inter - related concepts : mobile accounting mobile financial information servicesmost services in the categories designated accounting and brokerage are transaction - based. the non - transaction - based services of an informational nature are however essential for conducting transactions – for instance, balance inquiries might be needed before committing a money remittance.


in the course of studying the problem, church and his student stephen kleene introduced the notion of λ - definable functions, and they were able to prove that several large classes of functions frequently encountered in number theory were λ - definable. the debate began when church proposed to godel that one should define the " effectively computable " functions as the λ - definable functions. godel, however, was not convinced and called the proposal " thoroughly unsatisfactory ".


in telecommunication, continuous operation is an operation in which certain components, such as nodes, facilities, circuits, or equipment, are in an operational state at all times. continuous operation usually requires that there be fully redundant configuration, or at least a sufficient x out of y degree of redundancy for compatible equipment, where x is the number of spare components and y is the number of operational components. this article incorporates public domain material from federal standard 1037c. general services administration. ( in support of mil - std - 188 ).


intersection, union, complementation, and containment of elements is expressed in u. let v be the collection of n × n matrices that have entries taken from u. complementation of such a matrix is obtained by complementing each element. the intersection or union of two such matrices is obtained by applying the operation to entries of each pair of elements to obtain the corresponding matrix intersection or union. a matrix is contained in another if each entry of the first is contained in the corresponding entry of the second.


therefore, these fallacies, for pedagogic reasons, usually take the form of spurious proofs of obvious contradictions. although the proofs are flawed, the errors, usually by design, are comparatively subtle, or designed to show that certain steps are conditional, and are not applicable in the cases that are the exceptions to the rules. the traditional way of presenting a mathematical fallacy is to give an invalid step of deduction mixed in with valid steps, so that the meaning of fallacy is here slightly different from the logical fallacy.


in some cases, this type of arrangement is needed for boat cruises that do not return to the departure city. in other cases, the traveller wishes to explore between two points and using alternative transport ( e. g. buses, trains, ferries or flights on another ticket ). for example, a traveller might fly from london to bangkok, travel around thailand by public transport and fly back home to london from phuket. another example would be a traveller flying from new york city to london, travelling around different countries in europe by taking buses / trains or low - cost carrier flights, then returning from vilnius.


raju. ) the third column contains iso 15919 transliterations of the lines given in the second column. the digits encoded by the lines in second column are given in arabic numerals in the fourth column.


in the following, let g = ( v, e ) denote an undirected graph with a set of vertices v and a set of edges e ⊆ v × v. the set sizes are denoted by | v | = n and | e | = m. additionally, if not noted otherwise, the metric space [ 0, 1 ) d with the euclidean distance is considered, i. e. for any points x, y ∈ [ 0, 1 ) d { \ displaystyle x, y \ in [ 0, 1 ) ^ { d } } the euclidean distance of x and y is defined as d ( x, y ) = | | x − y | | 2 = i = 1 d ( x i − y i ) 2 { \ displaystyle d ( x, y ) = | | x - y | | _ { 2 } = { \ sqrt { \ sum _ { i = 1 } ^ { d } ( x _ { i } - y _ { i } ) ^ { 2 } } } }. a random geometric graph ( rgg ) is an undirected geometric graph with nodes randomly sampled from the uniform distribution of the underlying space [ 0, 1 ) d. two vertices p, q ∈ v are connected if, and only if, their distance is less than a previously specified parameter r ∈ ( 0, 1 ), excluding any loops. thus, the parameters r and n fully characterize a rgg.


in mathematics, specifically in computational geometry, geometric nonrobustness is a problem wherein branching decisions in computational geometry algorithms are based on approximate numerical computations, leading to various forms of unreliability including ill - formed output and software failure through crashing or infinite loops. for instance, algorithms for problems like the construction of a convex hull rely on testing whether certain " numerical predicates " have values that are positive, negative, or zero. if an inexact floating - point computation causes a value that is near zero to have a different sign than its exact value, the resulting inconsistencies can propagate through the algorithm causing it to produce output that is far from the correct output, or even to crash. one method for avoiding this problem involves using integers rather than floating point numbers for all coordinates and other quantities represented by the algorithm, and determining the precision required for all calculations to avoid integer overflow conditions.


( maximizing the weighted sum of covered elements ). subject to c ( s i ) ⋅ x i ≤ b { \ displaystyle \ sum { c ( s _ { i } ) \ cdot x _ { i } } \ leq b } ; ( the cost of the selected sets cannot exceed b { \ displaystyle b } ). e j ∈ s i x i ≥ y j { \ displaystyle \ sum _ { e _ { j } \ in s _ { i } } x _ { i } \ geq y _ { j } } ; ( if y j > 0 { \ displaystyle y _ { j } > 0 } then at least one set e j ∈ s i { \ displaystyle e _ { j } \ in s _ { i } } is selected ).


in the following c code, the value of zero is used to indicate a null pointer :


in probability theory, two sequences of probability measures are said to be contiguous if asymptotically they share the same support. thus the notion of contiguity extends the concept of absolute continuity to the sequences of measures. the concept was originally introduced by le cam ( 1960 ) as part of his foundational contribution to the development of asymptotic theory in mathematical statistics. he is best known for the general concepts of local asymptotic normality and contiguity.


clearly, summing the integers of a subset can be done in polynomial time, and the subset sum problem is therefore in np. the above example can be generalized for any decision problem. given any instance i of problem π { \ displaystyle \ pi } and witness w, if there exists a verifier v so that given the ordered pair ( i, w ) as input, v returns " yes " in polynomial time if the witness proves that the answer is " yes " or " no " in polynomial time otherwise, then π { \ displaystyle \ pi } is in np.


a minimum distance of 0. 9 m should also be maintained between any two beds in a dormitory, bedroom, or cubicle. in case students are provided with a cubicle, then each student must be provided with a window and a floor area of 5. 0 m2 at the least. a bedroom for a single student should be at least of the floor area of 6. 0 m2.


the minimum degree algorithm is derived from a method first proposed by markowitz in 1959 for non - symmetric linear programming problems, which is loosely described as follows. at each step in gaussian elimination row and column permutations are performed so as to minimize the number of off diagonal non - zeros in the pivot row and column. a symmetric version of markowitz method was described by tinney and walker in 1967 and rose later derived a graph theoretic version of the algorithm where the factorization is only simulated, and this was named the minimum degree algorithm.


in mathematics, an upper set ( also called an upward closed set, an upset, or an isotone set in x ) of a partially ordered set ( x, ≤ ) { \ displaystyle ( x, \ leq ) } is a subset s ⊆ x { \ displaystyle s \ subseteq x } with the following property : if s is in s and if x in x is larger than s ( that is, if s < x { \ displaystyle s


improvements in talkback and presfax have made the bbc's usage of cue dots rare. the prevalence of digital television and the accompanying delays have made the use of cue dots to communicate with outside broadcast obsolete. cue dots do have some other uses : presentation may be asked to " flash your dots " by an outside broadcast unit to confirm that their off - air check feed is correct, particularly when working regionally. the dots are also used during coverage of the wimbledon tennis championships to warn other broadcasters that the bbc feed will be cutting to an interview intended for the uk audience only, so they should be ready to go to something else.


in numerical analysis, the shooting method is a method for solving a boundary value problem by reducing it to an initial value problem. it involves finding solutions to the initial value problem for different initial conditions until one finds the solution that also satisfies the boundary conditions of the boundary value problem. in layman's terms, one " shoots " out trajectories in different directions from one boundary until one finds the trajectory that " hits " the other boundary condition.


in mathematics, there exist magmas that are commutative but not associative. a simple example of such a magma may be derived from the children's game of rock, paper, scissors. such magmas give rise to non - associative algebras. a magma which is both commutative and associative is a commutative semigroup.


similarly, new items also have the same problem. when new items are added to the system, they need to be rated by a substantial number of users before they could be recommended to users who have similar tastes to the ones who rated them. the new item problem does not affect content - based recommendations, because the recommendation of an item is based on its discrete set of descriptive qualities rather than its ratings.


in scientific visualization a tensor glyph is an object that can visualize all or most of the nine degrees of freedom, such as acceleration, twist, or shear – of a 3 × 3 { \ displaystyle 3 \ times 3 } matrix. it is used for tensor field visualization, where a data - matrix is available at every point in the grid. " glyphs, or icons, depict multiple data values by mapping them onto the shape, size, orientation, and surface appearance of a base geometric primitive. " tensor glyphs are a particular case of multivariate data glyphs. there are certain types of glyphs that are commonly used : ellipsoid cuboid cylindrical superquadricsaccording to thomas schultz and gordon kindlmann, specific types of tensor fields " play a central role in scientific and biomedical studies as well as in image analysis and feature - extraction methods. "


in mathematics, in particular linear algebra, the matrix determinant lemma computes the determinant of the sum of an invertible matrix a and the dyadic product, u vt, of a column vector u and a row vector vt.


in the field of criminal law, there are a variety of conditions that will tend to negate elements of a crime ( particularly the intent element ), known as defenses. the label may be apt in jurisdictions where the accused may be assigned some burden before a tribunal. however, in many jurisdictions, the entire burden to prove a crime is on the prosecution, which also must prove the absence of these defenses, where implicated. in other words, in many jurisdictions the absence of these so - called defenses is treated as an element of the crime. so - called defenses may provide partial or total refuge from punishment.


the individual is significant not in and of themselves, but rather in terms of their status, their position in patterns of social relations, and the behaviours associated with their status. therefore, the social structure is the network of statuses connected by associated roles. functionalism also has an anthropological basis in the work of theorists such as marcel mauss, bronisław malinowski and radcliffe - brown.


in mathematics, especially in topology, a topological group g { \ displaystyle g } is said to have no small subgroup if there exists a neighborhood u { \ displaystyle u } of the identity that contains no nontrivial subgroup of g. { \ displaystyle g. } an abbreviation'" nss "'is sometimes used.


in matroid theory, the closure of x is the largest superset of x that has the same rank as x. the transitive closure of a set. the algebraic closure of a field. the integral closure of an integral domain in a field that contains it.


in the above model, the marginal likelihood of the observations ( i. e. the joint distribution of the observations, with the prior parameter marginalized out ) is a dirichlet - multinomial distribution : p ( x α ) = p p ( x p ) p ( p α ) d p = γ ( k α k ) γ ( n + k α k ) k = 1 k γ ( c k + α k ) γ ( α k ) { \ displaystyle { \ begin { aligned } p ( \ mathbb { x } \ mid { \ boldsymbol { \ alpha } } ) & = \ int _ { \ mathbf { p } } p ( \ mathbb { x } \ mid \ mathbf { p } ) p ( \ mathbf { p } \ mid { \ boldsymbol { \ alpha } } ) { \ textrm { d } } \ mathbf { p } \ \ & = { \ frac { \ gamma \ left ( \ sum _ { k } \ alpha _ { k } \ right ) } { \ gamma \ left ( n + \ sum _ { k } \ alpha _ { k } \ right ) } } \ prod _ { k = 1 } ^ { k } { \ frac { \ gamma ( c _ { k } + \ alpha _ { k } ) } { \ gamma ( \ alpha _ { k } ) } } \ end { aligned } } } this distribution plays an important role in hierarchical bayesian models, because when doing inference over such models using methods such as gibbs sampling or variational bayes, dirichlet prior distributions are often marginalized out. see the article on this distribution for more details.


analysis of pocs and related methods attempt to show that the algorithm converges ( and if so, find the rate of convergence ), and whether it converges to the projection of the original point. these questions are largely known for simple cases, but a topic of active research for the extensions. there are also variants of the algorithm, such as dykstra's projection algorithm. see the references in the further reading section for an overview of the variants, extensions and applications of the pocs method ; a good historical background can be found in section iii of.


in the field called image processing, the focus of attention is formed by the operations that take ( at least ) one picture ( and potentially several secondary parameters that are not images ) and relate it to another picture. with these operations, we can define algorithms for improving the quality of images ( e. g., contrast reinforcement ), and procedures for extracting certain parts of an image ( e. g., edge finding ) or for stamping out pictorial patterns following a particular gestalt criterion ( e. g., blue screen technique ). compression algorithms for the efficient storing or transmitting of pictorial data also belong into this field.


arranging the numbers ( n 0 ), ( n 1 ), …, ( n n ) { \ displaystyle { \ tbinom { n } { 0 } }, { \ tbinom { n } { 1 } }, \ ldots, { \ tbinom { n } { n } } } in successive rows for n = 0, 1, 2, … { \ displaystyle n = 0, 1, 2, \ ldots } gives a triangular array called pascal's triangle, satisfying the recurrence relation ( n k ) = ( n − 1 k − 1 ) + ( n − 1 k ). { \ displaystyle { \ binom { n } { k } } = { \ binom { n - 1 } { k - 1 } } + { \ binom { n - 1 } { k } }. } the binomial coefficients occur in many areas of mathematics, and especially in combinatorics.


in modular arithmetic, the integers coprime ( relatively prime ) to n from the set { 0, 1, …, n − 1 } { \ displaystyle \ { 0, 1, \ dots, n - 1 \ } } of n non - negative integers form a group under multiplication modulo n, called the multiplicative group of integers modulo n. equivalently, the elements of this group can be thought of as the congruence classes, also known as residues modulo n, that are coprime to n. hence another name is the group of primitive residue classes modulo n. in the theory of rings, a branch of abstract algebra, it is described as the group of units of the ring of integers modulo n. here units refers to elements with a multiplicative inverse, which, in this ring, are exactly those coprime to n. this quotient group, usually denoted ( z / n z ) × { \ displaystyle ( \ mathbb { z } / n \ mathbb { z } ) ^ { \ times } }, is fundamental in number theory. it is used in cryptography, integer factorization, and primality testing. it is an abelian, finite group whose order is given by euler's totient function : | ( z / n z ) × | = φ ( n ). { \ displaystyle | ( \ mathbb { z } / n \ mathbb { z } ) ^ { \ times } | = \ varphi ( n ). } for prime n the group is cyclic, and in general the structure is easy to describe, but no simple general formula for finding generators is known.


in the field of bioinformatics, clustering is used for a number of applications. one use is as a pattern recognition technique to analyze gene expression data from rna - sequencing data or other technologies. in this case, genes with similar expression patterns are grouped into the same cluster, and different clusters display distinct, well - separated patterns of expression. use of clustering can provide insight into gene function and regulation.


this assertion, which uses set - builder notation, means that if the elements satisfying the property p ( x ) { \ displaystyle p ( x ) } are the same as the elements satisfying q ( x ), { \ displaystyle q ( x ), } then the two uses of the set - builder notation define the same set. this property is often expressed as " two sets that have the same elements are equal. " it is one of the usual axioms of set theory, called axiom of extensionality.


in much of probability, especially when conditional expectation is involved, one is concerned with sets that represent only part of all the possible information that can be observed. this partial information can be characterized with a smaller σ - algebra which is a subset of the principal σ - algebra ; it consists of the collection of subsets relevant only to and determined only by the partial information. a simple example suffices to illustrate this idea. imagine you and another person are betting on a game that involves flipping a coin repeatedly and observing whether it comes up heads ( h { \ displaystyle h } ) or tails ( t { \ displaystyle t } ).


in statistics, single - linkage clustering is one of several methods of hierarchical clustering. it is based on grouping clusters in bottom - up fashion ( agglomerative clustering ), at each step combining two clusters that contain the closest pair of elements not yet belonging to the same cluster as each other. this method tends to produce long thin clusters in which nearby elements of the same cluster have small distances, but elements at opposite ends of a cluster may be much farther from each other than two elements of other clusters. for some classes of data, this may lead to difficulties in defining classes that could usefully subdivide the data. however, it is popular in astronomy for analyzing galaxy clusters, which may often involve long strings of matter ; in this application, it is also known as the friends - of - friends algorithm.


often, the term linear equation refers implicitly to this particular case, in which the variable is sensibly called the unknown. in the case of two variables, each solution may be interpreted as the cartesian coordinates of a point of the euclidean plane. the solutions of a linear equation form a line in the euclidean plane, and, conversely, every line can be viewed as the set of all solutions of a linear equation in two variables.


similarly, the wmb maintains program order only among writes. the sparc v9 rmo model provides a membar instruction which can be customised to order previous reads and writes with respect to future read and write operations. there is no need for using read - modify - writes to achieve this order because the membar instruction can be used to order a write with respect to a succeeding read.


( roughly speaking, girard found pure systems in which one can express the sentence " the types form a type ". ) furthermore, all known examples of pure type systems that are not strongly normalizing are not even ( weakly ) normalizing : they contain expressions that do not have normal forms, just like the untyped lambda calculus. it is a major open problem in the field whether this is always the case, i. e. whether a ( weakly ) normalizing pts always has the strong normalization property. this is known as the barendregt – geuvers – klop conjecture ( named after henk barendregt, herman geuvers, and jan willem klop ).


in probability theory, kolmogorov's three - series theorem, named after andrey kolmogorov, gives a criterion for the almost sure convergence of an infinite series of random variables in terms of the convergence of three different series involving properties of their probability distributions. kolmogorov's three - series theorem, combined with kronecker's lemma, can be used to give a relatively easy proof of the strong law of large numbers.


a wide variety of mathematical theories to understand and analyze message - passing systems are available, including the actor model, and various process calculi. message passing can be efficiently implemented via symmetric multiprocessing, with or without shared memory cache coherence. shared memory and message passing concurrency have different performance characteristics. typically ( although not always ), the per - process memory overhead and task switching overhead is lower in a message passing system, but the overhead of message passing is greater than for a procedure call. these differences are often overwhelmed by other performance factors.


in probability, a discrete - time markov chain ( dtmc ) is a sequence of random variables, known as a stochastic process, in which the value of the next variable depends only on the value of the current variable, and not any variables in the past. for instance, a machine may have two states, a and e. when it is in state a, there is a 40 % chance of it moving to state e and a 60 % chance of it remaining in state a. when it is in state e, there is a 70 % chance of it moving to a and a 30 % chance of it staying in e. the sequence of states of the machine is a markov chain. if we denote the chain by x 0, x 1, x 2,.


since the minimum hamming weight of ( 7, 4, 3 ) { \ displaystyle ( 7, 4, 3 ) } hamming code is 3, d h ( x 1, x 2 ) ≥ 3 { \ displaystyle d _ { h } ( \ mathbf { x _ { 1 } }, \ mathbf { x _ { 2 } } ) \ geq 3 }. therefore, the input x { \ displaystyle \ mathbf { x } } can be recovered since d h ( x, y ) ≤ 1 { \ displaystyle d _ { h } ( \ mathbf { x }, \ mathbf { y } ) \ leq 1 }. similarly, the bits distribution with r x = 7 { \ displaystyle r _ { x } = 7 }, r y = 3 { \ displaystyle r _ { y } = 3 } can be achieved by reversing the roles of x { \ displaystyle x } and y { \ displaystyle y }.


a different ontology arises if we need to attend to the electrodynamics in the device : here signals propagate at finite speed and an object ( like a resistor ) that was previously viewed as a single component with an i / o behavior may now have to be thought of as an extended medium through which an electromagnetic wave flows. ontologies can of course be written down in a wide variety of languages and notations ( e. g., logic, lisp, etc. ) ; the essential information is not the form of that language but the content, i. e., the set of concepts offered as a way of thinking about the world. simply put, the important part is notions like connections and components, not the choice between writing them as predicates or lisp constructs.


s = ∅, t = ∅ s = ∅, t = ∅, s ⊂ t s = ∅, t = ∅ s = ∅, t = ∅, t ⊂ s s = ∅, t = ∅ s = ∅, t = ∅, t = s s = ∅, t = ∅, s ∩ t = ∅ s = ∅, t = ∅, s ∩ t = ∅, ¬ ( s ⊆ t ), ¬ ( t ⊆ s ), s = t { \ displaystyle { \ begin { array } { l | l } s = \ emptyset, t = \ emptyset & s \ neq \ emptyset, t \ neq \ emptyset, s \ subset t \ \ \ hline s = \ emptyset, t \ neq \ emptyset & s \ neq \ emptyset, t \ neq \ emptyset, t \ subset s \ \ \ hline s \ neq \ emptyset, t = \ emptyset & s \ neq \ emptyset, t \ neq \ emptyset, t = s \ \ \ hline s \ neq \ emptyset, t \ neq \ emptyset, s \ cap t = \ emptyset & s \ neq \ emptyset, t \ neq \ emptyset, s \ cap t \ neq \ emptyset, \ lnot ( s \ subseteq t ), \ lnot ( t \ subseteq s ), s \ neq t \ end { array } } } as can be noticed, standard partitions might change according to how much testing the engineer wants to perform. sub - domain propagation ( sdp ). this tactic is applied to expressions containing : two or more mathematical operators for which there are already defined standard partitions, or mathematical operators which are defined in terms of other mathematical operators. in any of these cases, the standard partitions of the operators appearing in the expression or in the definition of a complex one, are combined to produce a partition for the expression.


in the united states, most cities have zoning codes that set the minimum size for a housing unit ( often 400 square feet ) as well as the number of non - related persons who can live together in one unit. in june 2016, new york city got its first microapartment building, carmel place, with 55 units that are as small as 250 square feet ( 23 m2 ) and ceilings from nine to ten feet ( 2. 7 to 3. 0 m ). common's williamsburg in brooklyn rents single rooms where tenants share a kitchen for $ 2, 050 per month ; the guardian states that " ingle room occupancy housing is obviously not a new concept, however, the genius of late capitalism is that it has made it desirable " to high - income renters ". in 2017, california passed a law that encourages development of " efficiency units " of at least 150 sq ft by disallowing localities from limiting their numbers near public universities and public transportation. : 1 in san francisco, starcity is converting unused parking garages, commercial spaces and offices into single room residential units, where tenants ( tech professionals are the typical renter ) get a furnished bedroom and access to wifi, janitor services and common kitchens and lounges for $ 1, 400 to $ 2, 400 per month, an approach that has been called " dorm living for grown ups ". boston's first microapartment building opened in august 2016, on commonwealth avenue in packard's corner. as the largest microapartment building in the united states, the building is currently being leased by boston university to house 341 students during the renovation of another university residence.


in percolation theory one examines a finite or infinite graph and removes edges ( or links ) randomly. thus the erdos – renyi process is in fact unweighted link percolation on the complete graph. ( one refers to percolation in which nodes and / or links are removed with heterogeneous weights as weighted percolation ). as percolation theory has much of its roots in physics, much of the research done was on the lattices in euclidean spaces.


note that this implies that p { \ displaystyle p } is contained in β { \ displaystyle \ beta } and γ { \ displaystyle \ gamma } ( with opposite polarity ) and q { \ displaystyle q } is contained in δ { \ displaystyle \ delta } and α { \ displaystyle \ alpha } ( also with opposite polarity ). the table below shows the rewriting rules proposed by simone et al.. the idea of the algorithm is to reduce proof size by opportunistically applying these rules. the first five rules were introduced in an earlier paper.


in mathematics and computer science, a splicing rule is a transformation on formal languages which formalises the action of gene splicing in molecular biology. a splicing language is a language generated by iterated application of a splicing rule : the splicing languages form a proper subset of the regular languages.


in the context of healthcare in the united states, a pre - existing condition is a medical condition that started before a person's health insurance went into effect. before 2014, some insurance policies would not cover expenses due to pre - existing conditions. these exclusions by the insurance industry were meant to cope with adverse selection by potential customers. such exclusions have been prohibited since january 1, 2014, by the patient protection and affordable care act. according to the kaiser family foundation, more than a quarter of adults below the age of 65 ( approximately 52 million people ) had pre - existing conditions in 2016.


fourth, the " trialability " of an innovation ; that is, whether it can be tested without commitment for a period of time. an innovation that has an available trial period provides less uncertainty to the group member who will be trying it. lastly, whether there are observable results with use of the innovation. the more positive and visible results, the higher the likelihood it gets adopted as a permanent idea for the group.


in mathematics, given a collection s of subsets of a set x, an exact hitting set x * is a subset of x such that each subset in s contains exactly one element in x *. one says that each subset in s is hit by exactly one element in x *. in computer science, the exact hitting set problem is a decision problem to find an exact hitting set or else determine none exists. the exact hitting set problem is an abstract exact cover problem.


the technique is also known as wiener – kolmogorov prediction, after norbert wiener and andrey kolmogorov. the theoretical basis for the method was developed by the french mathematician georges matheron based on the master's thesis of danie g. krige, the pioneering plotter of distance - weighted average gold grades at the witwatersrand reef complex in south africa. krige sought to estimate the most likely distribution of gold based on samples from a few boreholes.


in mathematics, the notion of cancellativity ( or cancellability ) is a generalization of the notion of invertibility. an element a in a magma ( m, ∗ ) has the left cancellation property ( or is left - cancellative ) if for all b and c in m, a ∗ b = a ∗ c always implies that b = c. an element a in a magma ( m, ∗ ) has the right cancellation property ( or is right - cancellative ) if for all b and c in m, b ∗ a = c ∗ a always implies that b = c. an element a in a magma ( m, ∗ ) has the two - sided cancellation property ( or is cancellative ) if it is both left - and right - cancellative. a magma ( m, ∗ ) has the left cancellation property ( or is left - cancellative ) if all a in the magma are left cancellative, and similar definitions apply for the right cancellative or two - sided cancellative properties. a left - invertible element is left - cancellative, and analogously for right and two - sided. if a⁻¹ is the inverse of a, then a ∗ b = a ∗ c implies a⁻¹ ∗ a ∗ b = a⁻¹ ∗ a ∗ c which implies b = c. for example, every quasigroup, and thus every group, is cancellative.


in natural language processing, a sentence embedding refers to a numeric representation of a sentence in the form of a vector of real numbers which encodes meaningful semantic information. state of the art embeddings are based on the learned hidden layer representation of dedicated sentence transformer models. bert pioneered an approach involving the use of a dedicated token preprended to the beginning of each sentence inputted into the model ; the final hidden state vector of this token encodes information about the sentence and can be fine - tuned for use in sentence classification tasks. in practice however, bert's sentence embedding with the token achieves poor performance, often worse than simply averaging non - contextual word embeddings. sbert later achieved superior sentence embedding performance by fine tuning bert's token embeddings through the usage of a siamese neural network architecture on the snli dataset.


in number theory, quadratic integers are a generalization of the usual integers to quadratic fields. quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form x2 + bx + c = 0with b and c ( usual ) integers. when algebraic integers are considered, the usual integers are often called rational integers. common examples of quadratic integers are the square roots of rational integers, such as √2, and the complex number i = √−1, which generates the gaussian integers.


adolescents and adults who know the rule are faster than those who do not. in the learning of a second language the correction of errors remains a controversial topic with many differing schools of thought. throughout the last century much advancement has been made in research on the correction of students'errors.


in mathematics and group theory, the term multiplicative group refers to one of the following concepts : the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referred to as multiplication. in the case of a field f, the group is ( f { 0 }, • ), where 0 refers to the zero element of f and the binary operation • is the field multiplication, the algebraic torus gl ( 1 )..


in network science, reciprocity is a measure of the likelihood of vertices in a directed network to be mutually linked. like the clustering coefficient, scale - free degree distribution, or community structure, reciprocity is a quantitative measure used to study complex networks.


showing this with hundreds of pages of hand analysis, appel and haken concluded that no smallest counterexample exists because any must contain, yet do not contain, one of these 1, 936 maps. this contradiction means there are no counterexamples at all and that the theorem is therefore true. initially, their proof was not accepted by mathematicians at all because the computer - assisted proof was infeasible for a human to check by hand. however, the proof has since then gained wider acceptance, although doubts still remain.


in mathematics parity can refer to the evenness or oddness of an integer, which, when written in its binary form, can be determined just by examining only its least significant bit. in information technology parity refers to the evenness or oddness, given any set of binary digits, of the number of those bits with value one. because parity is determined by the state of every one of the bits, this property of parity — being dependent upon all the bits and changing its value from even to odd parity if any one bit changes — allows for its use in error detection and correction schemes. in telecommunications the parity referred to by some protocols is for error - detection.


the four dimensions ( 4d ) of spacetime consist of events that are not absolutely defined spatially and temporally, but rather are known relative to the motion of an observer. minkowski space first approximates the universe without gravity ; the pseudo - riemannian manifolds of general relativity describe spacetime with matter and gravity. 10 dimensions are used to describe superstring theory ( 6d hyperspace + 4d ), 11 dimensions can describe supergravity and m - theory ( 7d hyperspace + 4d ), and the state - space of quantum mechanics is an infinite - dimensional function space.


in order to achieve the highest level of privacy and protection when calculating and transmitting sensitive information, biometric tokenization leverages existing encryption algorithms, authentication protocols, as well as hardware trust zones. combining some or all of these methods maximizes the level of protection needed to uphold the integrity of the process and security of data that could otherwise expose users to a breach of trust on a mass scale.


it must be sensitive to differences between mt systems and reliable in that mt systems that score similarly should be expected to perform similarly. finally, the metric must be general, that is it should work with different text domains, in a wide range of scenarios and mt tasks. the aim of this subsection is to give an overview of the state of the art in automatic metrics for evaluating machine translation.


people interested in building a small home can encounter institutional “ discrimination ” when building codes require minimum size well above the size of a small home. also, neighbors may be hostile because they fear negative impacts on their property values and have concerns about increased taxes. more broadly, these sentiments of " othering " homeless and unhoused persons have culminated into a broader movement of nimby - ism, or " not in my backyard. "


the use of this feature by programmers led to the gate a20 compatibility issues in later cpu generations, where the linear address space was expanded past 20 bits. in 16 - bit real mode, enabling applications to make use of multiple memory segments ( in order to access more memory than available in any one 64k - segment ) is quite complex, but was viewed as a necessary evil for all but the smallest tools ( which could do with less memory ). the root of the problem is that no appropriate address - arithmetic instructions suitable for flat addressing of the entire memory range are available.


in modular arithmetic, the set of available numbers is restricted to a finite subset of the integers, and addition " wraps around " when reaching a certain value, called the modulus. for example, the set of integers modulo 12 has twelve elements ; it inherits an addition operation from the integers that is central to musical set theory. the set of integers modulo 2 has just two elements ; the addition operation it inherits is known in boolean logic as the " exclusive or " function. a similar " wrap around " operation arises in geometry, where the sum of two angle measures is often taken to be their sum as real numbers modulo 2π. this amounts to an addition operation on the circle, which in turn generalizes to addition operations on many - dimensional tori.


in other programming languages such as pascal, indices may start at 1, so indexing in a block of memory can be changed to fit a start - at - 1 addressing scheme by a simple linear transformation – in this scheme, the memory location of the ith element with base address b and element size s is b + ( i − 1 ) s.


in statistical classification, the fisher kernel, named after ronald fisher, is a function that measures the similarity of two objects on the basis of sets of measurements for each object and a statistical model. in a classification procedure, the class for a new object ( whose real class is unknown ) can be estimated by minimising, across classes, an average of the fisher kernel distance from the new object to each known member of the given class. the fisher kernel was introduced in 1998. it combines the advantages of generative statistical models ( like the hidden markov model ) and those of discriminative methods ( like support vector machines ) : generative models can process data of variable length ( adding or removing data is well - supported ) discriminative methods can have flexible criteria and yield better results.


furthermore, the warranty of habitability generally cannot be disclaimed. contractual language can also limit the remedies available for breach of an implied warranty such as by capping recoverable damages or limiting the legal remedy to a replacement of a defective item. however, such a term can be found to be unconscionable. for example, if a defective product causes a personal injury, a contractual provision limiting recovery in such a case will be deemed prima facie unconscionable. ucc § 2 - 719 ( 3 )


in model checking, a field of computer science, a difference bound matrix ( dbm ) is a data structure used to represent some convex polytopes called zones. this structure can be used to efficiently implement some geometrical operations over zones, such as testing emptyness, inclusion, equality, and computing the intersection and the sum of two zones. it is, for example, used in the uppaal model checker ; where it is also distributed as an independent library. more precisely, there is a notion of canonical dbm ; there is a one - to - one relation between canonical dbms and zones and from each dbm a canonical equivalent dbm can be efficiently computed. thus, equality of zone can be tested by checking for equality of canonical dbms.


in probability theory, events e1, e2,..., en are said to be mutually exclusive if the occurrence of any one of them implies the non - occurrence of the remaining n − 1 events. therefore, two mutually exclusive events cannot both occur. formally said, the intersection of each two of them is empty ( the null event ) : a ∩ b = ∅.


.., n + 1, { \ displaystyle v _ { k } = \ mathrm { constant } \ \ forall \ k = 0,..., n + 1, } and this corresponds to eigenvalue 0 { \ displaystyle 0 }.


होना ہونا ( hona ) is the only verb in hindi - urdu to have the present indicative, imperfect indicative, presumptive mood and the present subjunctive conjugations, and all the other verbs in hindi - urdu lack them. the verb होना / ہونا ( hona ) can be translated as " to be ", " to exist ", " to happen " or " to have " depending on the context, and when used in the third person it could also be translated as " there is / are ". many verbs conjugations in hindi - urdu are derived from participles and hence are gendered and numbered, and they agree with either the object or the subject of the sentence depending on the grammatical case of the subject of the sentence. when the subject is in the ergative or the dative case ( seeː dative construction & quirky subject ) the verb agrees in gender and number with the object of the sentence and with the subject when the subject is in the nominative case.


in the above no - tyranny scenario, suppose no floor federation, but ( only ) a room with some local governance. suppose that the gym room is not used by all, but there is a " community " of regulars, there is a grouping of voters by its activity as speed - cyclists ( illustrated as spiked hair ), that have the gym room key for some activities on sundays. they are acting collectively to preserve the gym room for a local cyclists group. in this situation the following facts hold : there is a subset of voters and some collective action, uniting them, making them a cohesive group.


in the case of a speech signal, inputs are spectral coefficients over time. in order to learn critical acoustic - phonetic features ( for example formant transitions, bursts, frication, etc. ) without first requiring precise localization, the tdnn is trained time - shift - invariantly. time - shift invariance is achieved through weight sharing across time during training : time shifted copies of the tdnn are made over the input range ( from left to right in fig. 1 ). backpropagation is then performed from an overall classification target vector ( see tdnn diagram, three phoneme class targets ( / b /, / d /, / g / ) are shown in the output layer ), resulting in gradients that will generally vary for each of the time - shifted network copies.


in haskell, there is no variable assignment ; but operations similar to assignment ( like assigning to a field of an array or a field of a mutable data structure ) usually evaluate to the unit type, which is represented as ( ). this type has only one possible value, therefore containing no information. it is typically the type of an expression that is evaluated purely for its side effects.


in the matrix equation case in which a is a square matrix, eigenvalues λ { \ displaystyle \ lambda } may be found by setting the determinant of the matrix equal to zero, i. e. finding where the matrix has no inverse. ( such a square matrix is said to be singular. ) finite matrices have only a finite number of eigenvalues / eigenvectors, whereas linear operators can have a countably infinite number of eigenvalues / eigenfunctions ( in confined regions ) or uncountably infinite ( continuous ) spectra of solutions, as in unbounded regions.


alternatively, the elements of a free abelian group may be thought of as signed multisets containing finitely many elements of b { \ displaystyle b }, with the multiplicity of an element in the multiset equal to its coefficient in the formal sum. another way to represent an element of a free abelian group is as a function from b { \ displaystyle b } to the integers with finitely many nonzero values ; for this functional representation, the group operation is the pointwise addition of functions. every set b { \ displaystyle b } has a free abelian group with b { \ displaystyle b } as its basis.


in the microsoft document " output content protection and windows vista ", it is claimed that : " the security level achieved for typical video data is estimated to be approaching that of regular aes. this assertion is being tested by intel putting its cascaded cipher out to the cryptography community to get their security assessment — that is, to see if they can break it. " the security of the system requires that it is impossible to recover the currently active inner key from the output of the reduced round serpent encrypted video stream. furthermore, the security of this method is highly sensitive to the number of rounds used in serpent, the mode of operation described in the patent application, and the number of times the inner key is reused.


in number theory, given a positive integer n and an integer a coprime to n, the multiplicative order of a modulo n is the smallest positive integer k such that a k ≡ 1 ( mod n ) { \ textstyle a ^ { k } \ \ equiv \ 1 { \ pmod { n } } }. in other words, the multiplicative order of a modulo n is the order of a in the multiplicative group of the units in the ring of the integers modulo n. the order of a modulo n is sometimes written as ord n ( a ) { \ displaystyle \ operatorname { ord } _ { n } ( a ) }.


in number theory, the dedekind psi function is the multiplicative function on the positive integers defined by ψ ( n ) = n p | n ( 1 + 1 p ), { \ displaystyle \ psi ( n ) = n \ prod _ { p | n } \ left ( 1 + { \ frac { 1 } { p } } \ right ), } where the product is taken over all primes p { \ displaystyle p } dividing n. { \ displaystyle n. } ( by convention, ψ ( 1 ) { \ displaystyle \ psi ( 1 ) }, which is the empty product, has value 1. ) the function was introduced by richard dedekind in connection with modular functions.


in mathematics, the feit – thompson conjecture is a conjecture in number theory, suggested by walter feit and john g. thompson ( 1962 ). the conjecture states that there are no distinct prime numbers p and q such that p q − 1 p − 1 { \ displaystyle { \ frac { p ^ { q } - 1 } { p - 1 } } } divides q p − 1 q − 1 { \ displaystyle { \ frac { q ^ { p } - 1 } { q - 1 } } }. if the conjecture were true, it would greatly simplify the final chapter of the proof ( feit & thompson 1963 ) of the feit – thompson theorem that every finite group of odd order is solvable. a stronger conjecture that the two numbers are always coprime was disproved by stephens ( 1971 ) with the counterexample p = 17 and q = 3313 with common factor 2pq + 1 = 112643. it is known that the conjecture is true for q = 3 ( le 2012 ). informal probability arguments suggest that the " expected " number of counterexamples to the feit – thompson conjecture is very close to 0, suggesting that the feit – thompson conjecture is likely to be true.


additionally, in the united states, progress reports may be issued to track a student's performance in between report cards. they are typically issued at the midpoint of a grading period, ( for example : 4½ weeks into a nine - week grading period, or three weeks into a six - week grading period ) and contain virtually the same information as the report card. these reports allow students and their parents to see if school performance is slipping and if intervention is required to bring up the grade.


these logarithms are equally spaced along a vertical line in the complex plane. a complex - valued function log : u → c { \ displaystyle \ log \ colon u \ to \ mathbb { c } }, defined on some subset u { \ displaystyle u } of the set c ∗ { \ displaystyle \ mathbb { c } ^ { * } } of nonzero complex numbers, satisfying e log z = z { \ displaystyle e ^ { \ log z } = z } for all z { \ displaystyle z } in u { \ displaystyle u }. such complex logarithm functions are analogous to the real logarithm function ln : r > 0 → r { \ displaystyle \ ln \ colon \ mathbb { r } _ { > 0 } \ to \ mathbb { r } }, which is the inverse of the real exponential function and hence satisfies eln x = x for all positive real numbers x. complex logarithm functions can be constructed by explicit formulas involving real - valued functions, by integration of 1 / z { \ displaystyle 1 / z }, or by the process of analytic continuation. there is no continuous complex logarithm function defined on all of c ∗ { \ displaystyle \ mathbb { c } ^ { * } }. ways of dealing with this include branches, the associated riemann surface, and partial inverses of the complex exponential function. the principal value defines a particular complex logarithm function log : c ∗ → c { \ displaystyle \ operatorname { log } \ colon \ mathbb { c } ^ { * } \ to \ mathbb { c } } that is continuous except along the negative real axis ; on the complex plane with the negative real numbers and 0 removed, it is the analytic continuation of the ( real ) natural logarithm.


in the early 20th century, it was common for weather maps to be hand drawn. the symbols for cloud cover on these maps, like the modern symbols, were drawn inside the circle marking the position of the weather station making the measurements. unlike the modern symbols, these ones consisted of straight lines only rather than filled in blocks which would have been less practical on a hand drawing. a reduced set of these symbols were used on teleprinters used for distributing weather information and warnings. these machines were 5 - bit teleprinters using a modified version of the baudot - murray code.


in signal processing, undersampling or bandpass sampling is a technique where one samples a bandpass - filtered signal at a sample rate below its nyquist rate ( twice the upper cutoff frequency ), but is still able to reconstruct the signal. when one undersamples a bandpass signal, the samples are indistinguishable from the samples of a low - frequency alias of the high - frequency signal. such sampling is also known as bandpass sampling, harmonic sampling, if sampling, and direct if - to - digital conversion.


in some situations, one can avoid using the transpose a { \ displaystyle a ^ { \ top } }, as proposed by mikhail lavrentyev. for example, if a { \ displaystyle a } is symmetric positive definite, i. e. a = a > 0 { \ displaystyle a = a ^ { \ top } > 0 }, so is its inverse a − 1 { \ displaystyle a ^ { - 1 } }, which can thus be used to set up the weighted norm squared ‖ x ‖ p 2 = x a − 1 x { \ displaystyle \ | x \ | _ { p } ^ { 2 } = x ^ { \ top } a ^ { - 1 } x } in the generalized tikhonov regularization, leading to minimizing ‖ a x − b ‖ a − 1 2 + ‖ x − x 0 ‖ q 2 { \ displaystyle \ | ax - b \ | _ { a ^ { - 1 } } ^ { 2 } + \ | x - x _ { 0 } \ | _ { q } ^ { 2 } } or, equivalently up to a constant term, x ( a + q ) x − 2 x ( b + q x 0 ) { \ displaystyle x ^ { \ top } ( a + q ) x - 2x ^ { \ top } ( b + qx _ { 0 } ) }. this minimization problem has an optimal solution x ∗ { \ displaystyle x ^ { * } } which can be written explicitly using the formula x ∗ = ( a + q ) − 1 ( b + q x 0 ) { \ displaystyle x ^ { * } = ( a + q ) ^ { - 1 } ( b + qx _ { 0 } ) }, which is nothing but the solution of the generalized tikhonov problem where a = a = p − 1. { \ displaystyle a = a ^ { \ top } = p ^ { - 1 }. } the lavrentyev regularization, if applicable, is advantageous to the original tikhonov regularization, since the lavrentyev matrix a + q { \ displaystyle a + q } can be better conditioned, i. e., have a smaller condition number, compared to the tikhonov matrix a a + γ γ. { \ displaystyle a ^ { \ top } a + \ gamma ^ { \ top } \


gamma. }


in mathematics, a noncommutative ring is a ring whose multiplication is not commutative ; that is, there exist a and b in the ring such that ab and ba are different. equivalently, a noncommutative ring is a ring that is not a commutative ring. noncommutative algebra is the part of ring theory devoted to study of properties of the noncommutative rings, including the properties that apply also to commutative rings.


in the exploit, bidirectional characters are abused to visually reorder text in source code so that later execution occurs in a different order. bidirectional characters can be inserted in areas of source code where string literals are allowed. this often applies to documentation, variables, or comments.


in the philippines, the philippine atmospheric, geophysical and astronomical services administration ( pagasa ) branch of the department of science and technology ( dost ) issues general flood advisory ( for non - telemetered river basins whenever there is a significant amount of rainfall recorded ).


in some programming languages, the maximum size of the call stack is much less than the space available in the heap, and recursive algorithms tend to require more stack space than iterative algorithms. consequently, these languages sometimes place a limit on the depth of recursion to avoid stack overflows ; python is one such language. note the caveat below regarding the special case of tail recursion.


creating a measure or scale of the degree to which the concept applies ( metrology ). 18. examining the distribution patterns or distributional frequency of ( possibly different ) uses of the concept ( statistics ).


in some circumstances the natural key that uniquely identifies a tuple in a relation may be cumbersome to use for software development. for example, it may involve multiple columns or large text fields. in such cases, a surrogate key can be used instead as the primary key. in other situations there may be more than one candidate key for a relation, and no candidate key is obviously preferred.


the processors ran at 20 mhz in the integer units and 40 mhz for the fpus, with the intention being to increase this to 50 mhz by the time it shipped. at about 12 mflops peak per cu, the machine as a whole would deliver up to 1. 5 gflops, although due to the memory latencies this was typically closer to 250 mflops. while this was fast for a cmos machine processor of the time, it was hardly competitive for a supercomputer.


in the 1990s the chip design and fabrication process grew to the point where it was possible to build a commodity processor with every potential feature built into it. to improve performance, cpu designs started adding internal parallelism, becoming " superscalar ". in any program there are instructions that work on unrelated data, so by adding more functional units these instructions can be run at the same time. a new portion of the cpu, the scheduler, looks for these independent instructions and feeds them into the units, taking their outputs and re - ordering them so externally it appears they ran in succession.


in order to use the above mentioned models efficiently, it is important to have a large amount of data. however, as users, we are stuck with limited data i. e. the original image. in order to compensate for these issues, we employ tricks such as random cropping. random cropping refers to the act of randomly choosing certain sub images from the existing original image.


in mathematics, symbolic dynamics is the practice of modeling a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics ( evolution ) given by the shift operator. formally, a markov partition is used to provide a finite cover for the smooth system ; each set of the cover is associated with a single symbol, and the sequences of symbols result as a trajectory of the system moves from one covering set to another.


in statistical mechanics, the radial distribution function, ( or pair correlation function ) g ( r ) { \ displaystyle g ( r ) } in a system of particles ( atoms, molecules, colloids, etc. ), describes how density varies as a function of distance from a reference particle. if a given particle is taken to be at the origin o, and if ρ = n / v { \ displaystyle \ rho = n / v } is the average number density of particles, then the local time - averaged density at a distance r { \ displaystyle r } from o is ρ g ( r ) { \ displaystyle \ rho g ( r ) }. this simplified definition holds for a homogeneous and isotropic system. a more general case will be considered below.


however, issues may arise that complicate the process. in 1992, christel and kang identified problems that indicate the challenges for requirements elicitation :'problems of scope '. the boundary of the system is ill - defined or the customers / users specify unnecessary technical details that may confuse, rather than clarify, overall system objectives.


in mathematics, a family f { \ displaystyle { \ mathcal { f } } } of sets is of finite character if for each a { \ displaystyle a }, a { \ displaystyle a } belongs to f { \ displaystyle { \ mathcal { f } } } if and only if every finite subset of a { \ displaystyle a } belongs to f { \ displaystyle { \ mathcal { f } } }. that is, for each a ∈ f { \ displaystyle a \ in { \ mathcal { f } } }, every finite subset of a { \ displaystyle a } belongs to f { \ displaystyle { \ mathcal { f } } }. if every finite subset of a given set a { \ displaystyle a } belongs to f { \ displaystyle { \ mathcal { f } } }, then a { \ displaystyle a } belongs to f { \ displaystyle { \ mathcal { f } } }.


in category theory, the empty semigroup is always admitted. it is the unique initial object of the category of semigroups. a semigroup with no elements is an inverse semigroup, since the necessary condition is vacuously satisfied.


for constant k, this is in the same complexity class as the lucas - lehmer test. in practice however, the cost of doing many iterations and other differences leads to worse performance for miller – rabin. the most efficient deterministic primality test for any n - digit number, the aks primality test, requires o ( n6 ) bit operations in its best known variant and is extremely slow even for relatively small values.


in mathematics, and disciplines in which mathematics plays a major role, hand - waving refers to either absence of formal proof or methods that do not meet mathematical rigor. in practice, it often involves the use of unrepresentative examples, unjustified assumptions, key omissions and faulty logic, and while these may be useful in expository papers and seminar presentations, they ultimately fall short of the standard of proof needed to establish a result. the mathematical profession tends to be receptive to informed critiques from any listener, and a claimant to a new result is expected to be able to answer any such question with a logical argument, up to a full proof. should a speaker apparently fail to give such an answer, anyone in the audience who can supply the needed demonstration may sometimes upstage the speaker.


in mathematics and economics, a corner solution is a special solution to an agent's maximization problem in which the quantity of one of the arguments in the maximized function is zero. in non - technical terms, a corner solution is when the chooser is either unwilling or unable to make a trade - off between goods.


in relational grammar, constituents that serve as the arguments to predicates are numbered in what is called the grammatical relations ( gr ) hierarchy. this numbering system corresponds loosely to the notions of subject, direct object and indirect object. the numbering scheme is subject → ( 1 ), direct object → ( 2 ) and indirect object → ( 3 ).


in practice, many screens that almost comply with part f of the standard deviate in various minor ways, and most brands of compliant brackets are designed to handle these deviations with little or no trouble for the end user : non - square patterns, such as 600 × 200 mm and 600 x 400 mm and 800 × 400 mm. these were apparently permitted by the version 1 ( 2002 ) standard. : vii – viii the labeling for these may have been " mis - f, width, height " e. g. " mis - f, 600, 200 " for 600 × 200 mm. a somewhat strange pattern of 280 × 150 mm. various protrusions on the screen extending a few millimeters further back than the mounting surfaces.


( fuzzy extractors convert biometric data into secret, uniformly random, and reliably reproducible random strings. ) these techniques can also have other broader applications for other type of noisy inputs such as approximative data from human memory, images used as passwords, and keys from quantum channels. fuzzy extractors also have applications in the proof of impossibility of the strong notions of privacy with regard to statistical databases.


in number theory, the sum of two squares theorem relates the prime decomposition of any integer n > 1 to whether it can be written as a sum of two squares, such that n = a2 + b2 for some integers a, b. an integer greater than one can be written as a sum of two squares if and only if its prime decomposition contains no factor pk, where prime p ≡ 3 ( mod 4 ) { \ displaystyle p \ equiv 3 { \ pmod { 4 } } } and k is odd. in writing a number as a sum of two squares, it is allowed for one of the squares to be zero, or for both of them to be equal to each other, so all squares and all doubles of squares are included in the numbers that can be represented in this way. this theorem supplements fermat's theorem on sums of two squares which says when a prime number can be written as a sum of two squares, in that it also covers the case for composite numbers. a number may have multiple representations as a sum of two squares, counted by the sum of squares function ; for instance, every pythagorean triple a 2 + b 2 = c 2 { \ displaystyle a ^ { 2 } + b ^ { 2 } = c ^ { 2 } } gives a second representation for c 2 { \ displaystyle c ^ { 2 } } beyond the trivial representation c 2 + 0 2 { \ displaystyle c ^ { 2 } + 0 ^ { 2 } }.


bg2 kd4 2. kd2 ke5 3. ke3 kf5 4.


when an individual algorithm is profiled, as with complexity analysis, memory and cache considerations are often more significant than instruction counts or clock cycles ; however, the profiler's findings can be considered in light of how the algorithm accesses data rather than the number of instructions it uses. profiling may provide intuitive insight into an algorithm's behavior by revealing performance findings as a visual representation. performance profiling has been applied, for example, during the development of algorithms for matching wildcards. early algorithms for matching wildcards, such as rich salz'wildmat algorithm, typically relied on recursion, a technique criticized on grounds of performance.


in probability theory, a fair coin is defined as a probability space ( ω, f, p ) { \ displaystyle ( \ omega, { \ mathcal { f } }, p ) }, which is in turn defined by the sample space, event space, and probability measure. using h { \ displaystyle h } for heads and t { \ displaystyle t } for tails, the sample space of a coin is defined as : the event space for a coin includes all sets of outcomes from the sample space which can be assigned a probability, which is the full power set 2 ω { \ displaystyle 2 ^ { \ omega } }. thus, the event space is defined as : { } { \ displaystyle \ { \ } } is the event where neither outcome happens ( which is impossible and can therefore be assigned 0 probability ), and { h, t } { \ displaystyle \ { h, t \ } } is the event where either outcome happens, ( which is guaranteed and can be assigned 1 probability ). because the coin is fair, the possibility of any single outcome is 50 - 50.


in some languages, infinitives may be marked for grammatical categories like voice, aspect, and to some extent tense. this may be done by inflection, as with the latin perfect and passive infinitives, or by periphrasis ( with the use of auxiliary verbs ), as with the latin future infinitives or the english perfect and progressive infinitives. latin has present, perfect and future infinitives, with active and passive forms of each. for details see latin conjugation § infinitives.


a complete version of the theorem with a self - contained proof was given by m. w. liebeck, cheryl praeger and jan saxl. the theorem is now a standard part of textbooks on permutation groups.


this is signified by the'always false'proposition, called the " falsum ", denoted " ". since the consequence is false, at least one of the antecedents must be false. thus for example,'a1, a2'means that at least one of the antecedents a1 and a2 must be false.


in metaphysics, ontology is the philosophical study of being. it investigates what types of entities exist, how they are grouped into categories, and how they are related to one another on the most fundamental level. ontologists often try to determine what the categories or highest kinds are and how they form a system of categories that encompasses the classification of all entities. commonly proposed categories include substances, properties, relations, states of affairs, and events.


consequently, a four - pole generator could output twice the voltage of a two - pole generator, a six - pole generator could output three times the voltage of a two - pole, and so forth. this allows output voltage to increase without also increasing the rotational rate. in a multipolar generator, the armature and field magnets are surrounded by a circular frame or " ring yoke " to which the field magnets are attached. this has the advantages of strength, simplicity, symmetrical appearance, and minimum magnetic leakage, since the pole pieces have the least possible surface and the path of the magnetic flux is shorter than in a two - pole design.


the first base 10 pandigital prime is 10123457689 ; oeis : a050288 lists more. for different reasons, redundant digits are also required for a pandigital number ( in any base except unary ) to also be a palindromic number in that base. the smallest pandigital palindromic number in base 10 is 1023456789876543201.


the concepts of the conceptual model can be mapped into physical design or implementation constructs using either manual or automated code generation approaches. the realization of conceptual models of many domains can be combined to a coherent platform. a conceptual model can be described using various notations, such as uml, orm or omt for object modelling, ite, or idef1x for entity relationship modelling.


in mathematical analysis, epi - convergence is a type of convergence for real - valued and extended real - valued functions. epi - convergence is important because it is the appropriate notion of convergence with which to approximate minimization problems in the field of mathematical optimization. the symmetric notion of hypo - convergence is appropriate for maximization problems. mosco convergence is a generalization of epi - convergence to infinite dimensional spaces.


in system software, a job queue ( a. k. a. batch queue, input queue ), is a data structure maintained by job scheduler software containing jobs to run. users submit their programs that they want executed, " jobs ", to the queue for batch processing. the scheduler software maintains the queue as the pool of jobs available for it to run. multiple batch queues might be used by the scheduler to differentiate types of jobs depending on parameters such as : job priority estimated execution time resource requirementsthe use of a batch queue gives these benefits : sharing of computer resources among many users time - shifts job processing to when the computer is less busy avoids idling the compute resources without minute - by - minute human supervision allows around - the - clock high utilization of expensive computing resourcesany process that comes to the cpu should wait in a queue.


currently, simmons'algorithm is the only approximation algorithm for envy - free cake - cutting with connected pieces. simmons'algorithm is one of the few fair division algorithms which have been implemented and put online. one nice thing about the algorithm is that the queries it asks the partners are very simple : they just have to decide, in each partition, which piece they prefer. this is in contrast to other algorithm, which ask numerical queries such as " cut a piece with a value of 1 / 3 " etc.


in the field of security engineering, an oracle attack is an attack that exploits the availability of a weakness in a system that can be used as an " oracle " to give a simple go / no go indication to inform attackers how close they are to their goals. the attacker can then combine the oracle with a systematic search of the problem space to complete their attack. the padding oracle attack, and compression oracle attacks such as breach, are examples of oracle attacks, as was the practice of " crib - dragging " in the cryptanalysis of the enigma machine. an oracle need not be 100 % accurate : even a small statistical correlation with the correct go / no go result can frequently be enough for a systematic automated attack.


associativity is not the same as commutativity, which addresses whether the order of two operands affects the result. for example, the order does not matter in the multiplication of real numbers, that is, a × b = b × a, so we say that the multiplication of real numbers is a commutative operation. however, operations such as function composition and matrix multiplication are associative, but not ( generally ) commutative.


in part one of the book, d'alembert provides a general introduction to the origin of knowledge, which led to the works found in the encyclopedie. he asserts that the " existence of our senses " is " indisputable, " and that these senses are thus the principle of all knowledge. he links this idea to a chain of thinking and reflection that eventually leads to the need to communicate, which sets another chain of events in effect. one of his arguments for the origin of communication is that it was necessary for people to protect themselves from the evils of the world and to benefit from each other's knowledge.


variables are available as if they are registers and some c expressions are allowed. arm compiler used to have a similar facility. the two families of extensions represent different understandings of division of labor in processing inline assembly. the gcc form preserves the overall syntax of the language and compartmentizes what the compiler needs to know : what is needed and what is changed.


in mathematical optimization, constrained optimization ( in some contexts called constraint optimization ) is the process of optimizing an objective function with respect to some variables in the presence of constraints on those variables. the objective function is either a cost function or energy function, which is to be minimized, or a reward function or utility function, which is to be maximized. constraints can be either hard constraints, which set conditions for the variables that are required to be satisfied, or soft constraints, which have some variable values that are penalized in the objective function if, and based on the extent that, the conditions on the variables are not satisfied.


a q – q plot is used to compare the shapes of distributions, providing a graphical view of how properties such as location, scale, and skewness are similar or different in the two distributions. q – q plots can be used to compare collections of data, or theoretical distributions. the use of q – q plots to compare two samples of data can be viewed as a non - parametric approach to comparing their underlying distributions.


in the area of mathematical logic and computer science known as type theory, a kind is the type of a type constructor or, less commonly, the type of a higher - order type operator. a kind system is essentially a simply typed lambda calculus " one level up ", endowed with a primitive type, denoted ∗ { \ displaystyle * } and called " type ", which is the kind of any data type which does not need any type parameters. a kind is sometimes confusingly described as the " type of a ( data ) type ", but it is actually more of an arity specifier. syntactically, it is natural to consider polymorphic types to be type constructors, thus non - polymorphic types to be nullary type constructors.


such a network of co - authorship is not unusual in narrowly defined areas of science. when allegations of plagiarism in this section were examined, wegman said that material in it had been " basically copied and pasted " by a student who was the " most knowledgeable " person about such analyses on his team, as she had taken a one - week course on network analysis with kathleen carley of carnegie mellon university. carley described the paper based on this section as " more of an opinion piece ", lacking the data needed to support its argument. the paper based on this social network analysis, published by wegman and his former student said, reached the conclusion that eminent scientists ought not work together, and that the findings of studies would be less biased when a " principal author tends to co - author papers with younger colleagues who were his students ".


in physics, chemistry and biology, a potential gradient is the local rate of change of the potential with respect to displacement, i. e. spatial derivative, or gradient. this quantity frequently occurs in equations of physical processes because it leads to some form of flux.


in mathematics, an explicit reciprocity law is a formula for the hilbert symbol of a local field. the name " explicit reciprocity law " refers to the fact that the hilbert symbols of local fields appear in hilbert's reciprocity law for the power residue symbol. the definitions of the hilbert symbol are usually rather roundabout and can be hard to use directly in explicit examples, and the explicit reciprocity laws give more explicit expressions for the hilbert symbol that are sometimes easier to use. there are also several explicit reciprocity laws for various generalizations of the hilbert symbol to higher local fields, p - divisible groups, and so on.


in operations research, the cutting - stock problem is the problem of cutting standard - sized pieces of stock material, such as paper rolls or sheet metal, into pieces of specified sizes while minimizing material wasted. it is an optimization problem in mathematics that arises from applications in industry. in terms of computational complexity, the problem is an np - hard problem reducible to the knapsack problem. the problem can be formulated as an integer linear programming problem.


in text retrieval, full - text search refers to techniques for searching a single computer - stored document or a collection in a full - text database. full - text search is distinguished from searches based on metadata or on parts of the original texts represented in databases ( such as titles, abstracts, selected sections, or bibliographical references ). in a full - text search, a search engine examines all of the words in every stored document as it tries to match search criteria ( for example, text specified by a user ).


they must be able to notate and reference any problems they find in detailed reports, meet deadlines with assignments and have the skill level to complete the game titles on their most difficult settings. most of the time the position of game tester is a highly stressful and competitive position with little pay yet is highly sought after for it serves as a doorway into the industry. game testers are observant individuals and can spot minor defects in the game build. a common misconception is that all game testers enjoy alpha or beta version of the game and report occasionally found bugs. in contrast, game testing is highly focused on finding bugs using established and often tedious methodologies before alpha version.


in probability theory and statistics, the exponential - logarithmic ( el ) distribution is a family of lifetime distributions with decreasing failure rate, defined on the interval [ 0, ∞ ). this distribution is parameterized by two parameters p ∈ ( 0, 1 ) { \ displaystyle p \ in ( 0, 1 ) } and β > 0 { \ displaystyle \ beta > 0 }.


in the airline industry a major problem is the scheduling of the flight crews. the airline scheduling problem can be considered as an application of extended maximum network flow. the input of this problem is a set of flights f which contains the information about where and when each flight departs and arrives.


in number theory, a regular prime is a special kind of prime number, defined by ernst kummer in 1850 to prove certain cases of fermat's last theorem. regular primes may be defined via the divisibility of either class numbers or of bernoulli numbers. the first few regular odd primes are : 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, 71, 73, 79, 83, 89, 97, 107, 109, 113, 127, 137, 139, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199,... ( sequence a007703 in the oeis ).


n 3!. { \ displaystyle { n \ choose n _ { 1 } } { n - n _ { 1 } \ choose n _ { 2 } } { n - n _ { 1 } - n _ { 2 } \ choose n _ { 3 } } \ cdots = { \ frac { n! } { ( n - n _ { 1 } )! n _ { 1 }!


in mathematical logic, predicate functor logic ( pfl ) is one of several ways to express first - order logic ( also known as predicate logic ) by purely algebraic means, i. e., without quantified variables. pfl employs a small number of algebraic devices called predicate functors ( or predicate modifiers ) that operate on terms to yield terms. pfl is mostly the invention of the logician and philosopher willard quine.


in programming language theory, the call - by - push - value ( cbpv ) paradigm, inspired by monads, allows writing semantics for lambda - calculus without writing two variants to deal with the difference between call - by - name and call - by - value. to do so, cbpv introduces a term language that distinguishes computations and values, according to the slogan a value is, a computation does ; this term language has a single evaluation order. however, to evaluate a lambda - calculus term according to either the call - by - name ( cbn ) or call - by - value ( cbv ) reduction strategy, one can translate the term to cbpv using a call - by - name or call - by - value translation strategy, which give rise to different terms. evaluating the result of the call - by - value translation corresponds to evaluating the original term with the call - by - value strategy ; evaluating the result of the call - by - name translation corresponds instead to evaluating the original term with the call - by - name strategy.


in telecommunications, an information - bearer channel is one of : a communication channel capable of transmitting all the information required for communication, such as user data, synchronizing sequences, and control signals. the information - bearer channel may operate at a higher data rate than that required for user data alone. a basic communications channel with the necessary bandwidth but without enhanced or value - added services.


in parallel algorithms, the list ranking problem involves determining the position, or rank, of each item in a linked list. that is, the first item in the list should be assigned the number 1, the second item in the list should be assigned the number 2, etc. although it is straightforward to solve this problem efficiently on a sequential computer, by traversing the list in order, it is more complicated to solve in parallel. as anderson & miller ( 1990 ) wrote, the problem was viewed as important in the parallel algorithms community both for its many applications and because solving it led to many important ideas that could be applied in parallel algorithms more generally.


in mathematics, statistics, finance, computer science, particularly in machine learning and inverse problems, regularization is a process that changes the result answer to be " simpler ". it is often used to obtain results for ill - posed problems or to prevent overfitting. although regularization procedures can be divided in many ways, the following delineation is particularly helpful : explicit regularization is regularization whenever one explicitly adds a term to the optimization problem. these terms could be priors, penalties, or constraints. explicit regularization is commonly employed with ill - posed optimization problems.


information obtained from the atkin primes permits a further improvement which is asymptotically negligible but can be quite important in practice. the modification of schoof's algorithm to use elkies and atkin primes is known as the schoof – elkies – atkin ( sea ) algorithm. the status of a particular prime ℓ { \ displaystyle \ ell } depends on the elliptic curve e / f q { \ displaystyle e / \ mathbb { f } _ { q } }, and can be determined using the modular polynomial ψ ℓ ( x, y ) { \ displaystyle \ psi _ { \ ell } ( x, y ) }.


formally, satisfiability is studied with respect to a fixed logic defining the syntax of allowed symbols, such as first - order logic, second - order logic or propositional logic. rather than being syntactic, however, satisfiability is a semantic property because it relates to the meaning of the symbols, for example, the meaning of + { \ displaystyle + } in a formula such as x + 1 = x { \ displaystyle x + 1 = x }. formally, we define an interpretation ( or model ) to be an assignment of values to the variables and an assignment of meaning to all other non - logical symbols, and a formula is said to be satisfiable if there is some interpretation which makes it true.


these numbers are equal to or slightly smaller than ( n lg n − 2 lg n + 1 ), which is between ( n lg n − n + 1 ) and ( n lg n + n + o ( lg n ) ). merge sort's best case takes about half as many iterations as its worst case. for large n and a randomly ordered input list, merge sort's expected ( average ) number of comparisons approaches α · n fewer than the worst case, where α = − 1 + k = 0 ∞ 1 2 k + 1 ≈ 0. 2645. { \ displaystyle \ alpha = - 1 + \ sum _ { k = 0 } ^ { \ infty } { \ frac { 1 } { 2 ^ { k } + 1 } } \ approx 0. 2645. }


then, we have the following bound on the probability that the empirical measure p ^ x n { \ displaystyle { \ hat { p } } _ { x ^ { n } } } of the samples falls within the set a : q n ( p ^ x n ∈ a ) ≤ ( n + 1 ) | x | 2 − n d k l ( p ∗ | | q ) { \ displaystyle q ^ { n } ( { \ hat { p } } _ { x ^ { n } } \ in a ) \ leq ( n + 1 ) ^ { | x | } 2 ^ { - nd _ { \ mathrm { kl } } ( p ^ { * } | | q ) } }, where q n { \ displaystyle q ^ { n } } is the joint probability distribution on x n { \ displaystyle x ^ { n } }, and p ∗ { \ displaystyle p ^ { * } } is the information projection of q onto a. in words, the probability of drawing an atypical distribution is bounded by a function of the kl divergence from the true distribution to the atypical one ; in the case that we consider a set of possible atypical distributions, there is a dominant atypical distribution, given by the information projection. furthermore, if a is the closure of its interior, lim n → ∞ 1 n log q n ( p ^ x n ∈ a ) = − d k l ( p ∗ | | q ). { \ displaystyle \ lim _ { n \ to \ infty } { \ frac { 1 } { n } } \ log q ^ { n } ( { \ hat { p } } _ { x ^ { n } } \ in a ) = - d _ { \ mathrm { kl } } ( p ^ { * } | | q ). }


the algorithm stops when it finds the minimum, determined when no progress is made after a direction reset ( i. e. in the steepest descent direction ), or when some tolerance criterion is reached. within a linear approximation, the parameters α { \ displaystyle \ displaystyle \ alpha } and β { \ displaystyle \ displaystyle \ beta } are the same as in the linear conjugate gradient method but have been obtained with line searches. the conjugate gradient method can follow narrow ( ill - conditioned ) valleys, where the steepest descent method slows down and follows a criss - cross pattern.


instead, the java platform provides a comprehensive set of its own standard class libraries containing many of the same reusable functions commonly found in modern operating systems. most of the system library is also written in java. for instance, the swing library paints the user interface and handles the events itself, eliminating many subtle differences between how different platforms handle components.


in software, a spell checker ( or spelling checker or spell check ) is a software feature that checks for misspellings in a text. spell - checking features are often embedded in software or services, such as a word processor, email client, electronic dictionary, or search engine.


in object - oriented programming, in languages such as c + +, and object pascal, a virtual function or virtual method is an inheritable and overridable function or method for which dynamic dispatch is facilitated. this concept is an important part of the ( runtime ) polymorphism portion of object - oriented programming ( oop ). in short, a virtual function defines a target function to be executed, but the target might not be known at compile time. most programming languages, such as javascript, php and python, treat all methods as virtual by default and do not provide a modifier to change this behavior. however, some languages provide modifiers to prevent methods from being overridden by derived classes ( such as the final and private keywords in java and php ).


string?? = string? ).


in mathematics and theoretical physics, an invariant differential operator is a kind of mathematical map from some objects to an object of similar type. these objects are typically functions on r n { \ displaystyle \ mathbb { r } ^ { n } }, functions on a manifold, vector valued functions, vector fields, or, more generally, sections of a vector bundle. in an invariant differential operator d { \ displaystyle d }, the term differential operator indicates that the value d f { \ displaystyle df } of the map depends only on f ( x ) { \ displaystyle f ( x ) } and the derivatives of f { \ displaystyle f } in x { \ displaystyle x }.


in number theory, rosser's theorem states that the n { \ displaystyle n } th prime number is greater than n log n { \ displaystyle n \ log n }, where log { \ displaystyle \ log } is the natural logarithm function. it was published by j. barkley rosser in 1939. its full statement is : let p n { \ displaystyle p _ { n } } be the n { \ displaystyle n } th prime number. then for n ≥ 1 { \ displaystyle n \ geq 1 } p n > n log n. { \ displaystyle p _ { n } > n \ log n. } in 1999, pierre dusart proved a tighter lower bound : p n > n ( log n + log log n − 1 ). { \ displaystyle p _ { n } > n ( \ log n + \ log \ log n - 1 ). }


in number theory, there are many integer factoring algorithms that heuristically have expected running time l n = e ( 1 + o ( 1 ) ) ( log n ) ( log log n ) { \ displaystyle l _ { n } \ left = e ^ { ( 1 + o ( 1 ) ) { \ sqrt { ( \ log n ) ( \ log \ log n ) } } } } in little - o and l - notation. some examples of those algorithms are the elliptic curve method and the quadratic sieve. another such algorithm is the class group relations method proposed by schnorr, seysen, and lenstra, which they proved only assuming the unproved generalized riemann hypothesis ( grh ).


masking depends on the spectral composition of both masker and masking signal, and on other variations with time. the basic block diagram of a perceptual coding system is shown in the figure. the input signal is decomposed into subsampled spectral components.


a pc cannot currently contain 4 pebibytes of memory ( due to the physical size of the memory chips ), but amd envisioned large servers, shared memory clusters, and other uses of physical address space that might approach this in the foreseeable future. thus the 52 - bit physical address provides ample room for expansion while not incurring the cost of implementing full 64 - bit physical addresses. similarly, the 48 - bit virtual address space was designed to provide 65, 536 ( 216 ) times the 32 - bit limit of 4 gib ( 4 × 10243 bytes ), allowing room for later expansion and incurring no overhead of translating full 64 - bit addresses.


this means that the code is always executed first and then the expression or test condition is evaluated. this process is repeated as long as the expression evaluates to true. if the expression is false the loop terminates.


in statistics, simple linear regression is a linear regression model with a single explanatory variable. that is, it concerns two - dimensional sample points with one independent variable and one dependent variable ( conventionally, the x and y coordinates in a cartesian coordinate system ) and finds a linear function ( a non - vertical straight line ) that, as accurately as possible, predicts the dependent variable values as a function of the independent variable. the adjective simple refers to the fact that the outcome variable is related to a single predictor. it is common to make the additional stipulation that the ordinary least squares ( ols ) method should be used : the accuracy of each predicted value is measured by its squared residual ( vertical distance between the point of the data set and the fitted line ), and the goal is to make the sum of these squared deviations as small as possible.


in the context of integers, subtraction of one also plays a special role : for any integer a, the integer ( a − 1 ) is the largest integer less than a, also known as the predecessor of a.


its cumulant generating function ( logarithm of the characteristic function ) is the inverse of the cumulant generating function of a gaussian random variable. to indicate that a random variable x is inverse gaussian - distributed with mean μ and shape parameter λ we write x ig ( μ, λ ) { \ displaystyle x \ sim \ operatorname { ig } ( \ mu, \ lambda ) \, \! }.


in mathematics, a sequence transformation is an operator acting on a given space of sequences ( a sequence space ). sequence transformations include linear mappings such as convolution with another sequence, and resummation of a sequence and, more generally, are commonly used for series acceleration, that is, for improving the rate of convergence of a slowly convergent sequence or series. sequence transformations are also commonly used to compute the antilimit of a divergent series numerically, and are used in conjunction with extrapolation methods.


faster image ingest : it will take no more than a few seconds to transfer a high - resolution raw file from a memory card vs many minutes to scan film with a high - quality scanner. flash : using flash in images can provide a different look such as the lighting of the image. higher image quantity : which enables longer photography sessions without changing film rolls.


in mathematical optimization, the ordered subset expectation maximization ( osem ) method is an iterative method that is used in computed tomography. in applications in medical imaging, the osem method is used for positron emission tomography, for single photon emission computed tomography, and for x - ray computed tomography. the osem method is related to the expectation maximization ( em ) method of statistics. the osem method is also related to methods of filtered back projection.


{ \ displaystyle x = ( { \ textbf { a } } ^ { \ mathrm { t } } { \ textbf { a } } ) ^ { - 1 } { \ textbf { a } } ^ { \ mathrm { t } } \ phi. } here the matrix a can be based on any set of functions mutually independent ( not necessarily orthogonal ) when evaluated at the sample times ; functions used for spectral analysis are typically sines and cosines evenly distributed over the frequency range of interest.


1149. 1 - 1990 after many years of initial use. in the same year, intel released their first processor with jtag ( the 80486 ) which led to quicker industry adoption by all manufacturers. in 1994, a supplement that contains a description of the boundary scan description language ( bsdl ) was added.


interpersonal relations may be regulated by law, custom, or mutual agreement, and form the basis of social groups and societies. they appear when people communicate or act with each other within specific social contexts, and they thrive on equitable and reciprocal compromises. interdisciplinary analysis of relationships draws heavily upon the other social sciences, including, but not limited to : anthropology, linguistics, sociology, economics, political science, communication, mathematics, social work, communication, and cultural studies. this scientific analysis had evolved during the 1990s and has become " relationship science ", through the researches of ellen berscheid and elaine hatfield. this interdisciplinary science attempts to provide evidence - based conclusions through the use of data analysis.