In doctoral mathematics departments, however, only about 58% of statistics course instructors had at least a master’s degree in statistics or biostatistics as their highest degree earned. In master’s-level mathematics departments, the corresponding percentage was near 44%, and in bachelor’s-level departments only 19% of statistics course instructors had at least a master’s degree in statistics or biostatistics as their highest degree earned. As we expected, a large majority of instructors in statistics departments (83% for doctoral departments and 62% for master’s departments) held doctoral degrees in either statistics or biostatistics.


In more modern terms, a crypt is most often a stone chambered burial vault used to store the deceased. Placing a corpse into a crypt can be called immurement, and is a method of final disposition, as an alternative to, for example, cremation. Crypts are usually found in cemeteries and under public religious buildings, such as churches or cathedrals, but are also occasionally found beneath mausolea or chapels on personal estates. Wealthy or prestigious families will often have a 'family crypt' or 'vault,' in which all members of the family are interred.


According to the TCP specification, that first sequence number sent by an endpoint can be any value as decided by that endpoint. As the sequence number is chosen by the sender, returned by the recipient, and has no otherwise-defined internal structure, it can be overloaded to carry additional data. The following describes one possible implementation, however as there is no public standard to follow, the order, length, and semantics of the fields may differ between SYN cookie implementations.


When using a file transfer protocol, the packets themselves offer their own framing. The packets will always send a continuous stream of data, so the clock cannot "drift" in the same way that it could for data being sent by a user typing on a keyboard. By turning off these framing bits when operating on an error-corrected link, that 20% overhead can be eliminated.


In stratigraphic excavation, the goal is to remove some or, preferably, all archaeological deposits and features in the reverse order they were created and construct a Harris matrix as a chronological record or "sequence" of the site. This Harris matrix is used for interpretation and combining contexts into ever larger units of understanding. This stratigraphic removal of the site is crucial for understanding the chronology of events on site. Stratigraphic excavation involves a process of cleaning or "troweling back" the surface of the site and isolating contexts and edges which are definable as either: Discrete, discernible "edges" that are formed by being completely separated from the surrounding surface and therefore stratigraphically later than its surroundings Discrete, discernible "edges" (as in 1.)


In mathematics, an edge cycle cover (sometimes called simply cycle cover) of a graph is a family of cycles which are subgraphs of G and contain all edges of G. If the cycles of the cover have no vertices in common, the cover is called vertex-disjoint or sometimes simply disjoint cycle cover. In this case, the set of the cycles constitutes a spanning subgraph of G. If the cycles of the cover have no edges in common, the cover is called edge-disjoint or simply disjoint cycle cover.


In probability theory, the chain rule (also called the general product rule) describes how to calculate the probability of the intersection of, not necessarily independent, events or the joint distribution of random variables respectively, using conditional probabilities. The rule is notably used in the context of discrete stochastic processes and in applications, e.g. the study of Bayesian networks, which describe a probability distribution in terms of conditional probabilities.


Thus the number −1,234,567 is 7 digits wide and is encoded as: 0001 0010 0011 0100 0101 0110 0111 1101 1 2 3 4 5 6 7 − Like character strings, the first byte of the packed decimal – that with the most significant two digits – is usually stored in the lowest address in memory, independent of the endianness of the machine. In contrast, a 4-byte binary two's complement integer can represent values from −2,147,483,648 to +2,147,483,647. While packed BCD does not make optimal use of storage (using about 20% more memory than binary notation to store the same numbers), conversion to ASCII, EBCDIC, or the various encodings of Unicode is made trivial, as no arithmetic operations are required.


In programming language theory and type theory, polymorphism is the provision of a single interface to entities of different types or the use of a single symbol to represent multiple different types. The concept is borrowed from a principle in biology where an organism or species can have many different forms or stages.The most commonly recognized major classes of polymorphism are: Ad hoc polymorphism: defines a common interface for an arbitrary set of individually specified types. Parametric polymorphism: not specifying concrete types and instead use abstract symbols that can substitute for any type. Subtyping (also called subtype polymorphism or inclusion polymorphism): when a name denotes instances of many different classes related by some common superclass.


In recent years, considerable effort has been put into improving the performance of existing algorithms. Among them are CLARANS, and BIRCH. With the recent need to process larger and larger data sets (also known as big data), the willingness to trade semantic meaning of the generated clusters for performance has been increasing. This led to the development of pre-clustering methods such as canopy clustering, which can process huge data sets efficiently, but the resulting "clusters" are merely a rough pre-partitioning of the data set to then analyze the partitions with existing slower methods such as k-means clustering.


Though the transportation network does not depend on the power network to function, the communications network does. Thus the deactivation of a critical number of nodes in either the power network or the communication network can lead to a series of cascading failures across the system with potentially catastrophic repercussions. If the two networks were treated in isolation, this important feedback effect would not be seen and predictions of network robustness would be greatly overestimated.


In mathematics, given two groups, (G, ∗) and (H, ·), a group homomorphism from (G, ∗) to (H, ·) is a function h: G → H such that for all u and v in G it holds that h ( u ∗ v ) = h ( u ) ⋅ h ( v ) {\displaystyle h(u*v)=h(u)\cdot h(v)} where the group operation on the left side of the equation is that of G and on the right side that of H. From this property, one can deduce that h maps the identity element eG of G to the identity element eH of H, h ( e G ) = e H {\displaystyle h(e_{G})=e_{H}} and it also maps inverses to inverses in the sense that h ( u − 1 ) = h ( u ) − 1 . {\displaystyle h\left(u^{-1}\right)=h(u)^{-1}.\,} Hence one can say that h "is compatible with the group structure". Older notations for the homomorphism h(x) may be xh or xh, though this may be confused as an index or a general subscript. In automata theory, sometimes homomorphisms are written to the right of their arguments without parentheses, so that h(x) becomes simply x h {\displaystyle xh} .In areas of mathematics where one considers groups endowed with additional structure, a homomorphism sometimes means a map which respects not only the group structure (as above) but also the extra structure. For example, a homomorphism of topological groups is often required to be continuous.


In telecommunication, attack time is the time between the instant that a signal at the input of a device or circuit exceeds the activation threshold of the device or circuit and the instant that the device or circuit reacts in a specified manner, or to a specified degree, to the input. Attack time occurs in devices such as clippers, peak limiters, compressors, and voxes.


Equipment will search for a bit which has the correct pattern, and will align its framing based on that bit. The pattern sent is 12 bits long, so every group of 12 frames is called a superframe. The pattern used in the 193rd bit is 100011 011100.Each channel sends two bits of call supervision data during each superframe using robbed-bit signaling during frames 6 and 12 of the superframe. More specifically, after the 6th and 12th bit in the superframe pattern, the least significant data bit of each channel (bit 8; T1 data is sent big-endian and uses 1-origin numbering) is replaced by a "channel-associated signalling" bit (bits A and B, respectively).Superframe remained in service in many places through the turn of the century, replaced by the improved extended superframe (ESF) of the 1980s in applications where its additional features were desired.


The cases (x, y, z) = (n, n, 2) and all its permutations were proven for n ≥ 4 by Darmon and Merel in 1995 following work from Euler and Poonen. The cases (x, y, z) = (n, n, 3) and all its permutations were proven for n ≥ 3 by Édouard Lucas, Bjorn Poonen, and Darmon and Merel. The case (x, y, z) = (2n, 2n, 5) and all its permutations were proven for n ≥ 2 by Bennett in 2006.


The second expression is evaluated prior to each iteration and the loop is terminated if it evaluates to false. The third expression is evaluated after each iteration, prior to deciding whether to perform the next. This for loop is the only looping construct that can not have a continue block, but expr3 is functionally equivalent.


In practice, the two coordinate-free definitions described above are rarely used because in virtually all cases, the curl operator can be applied using some set of curvilinear coordinates, for which simpler representations have been derived. The notation ∇ × F has its origins in the similarities to the 3-dimensional cross product, and it is useful as a mnemonic in Cartesian coordinates if ∇ is taken as a vector differential operator del. Such notation involving operators is common in physics and algebra. Expanded in 3-dimensional Cartesian coordinates (see Del in cylindrical and spherical coordinates for spherical and cylindrical coordinate representations),∇ × F is, for F composed of  (where the subscripts indicate the components of the vector, not partial derivatives): ∇ × F = | ı ^ ȷ ^ k ^ ∂ ∂ x ∂ ∂ y ∂ ∂ z F x F y F z | {\displaystyle \nabla \times \mathbf {F} ={\begin{vmatrix}{\boldsymbol {\hat {\imath }}}&{\boldsymbol {\hat {\jmath }}}&{\boldsymbol {\hat {k}}}\\{\dfrac {\partial }{\partial x}}&{\dfrac {\partial }{\partial y}}&{\dfrac {\partial }{\partial z}}\\F_{x}&F_{y}&F_{z}\end{vmatrix}}} where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively.


In the eighth chapter titled "The explanatory power of linguistic theory", Chomsky writes a linguistic theory cannot content itself by just generating valid grammatical sentences. It also has to account for other structural phenomena at different levels of linguistic representation. At a certain linguistic level, there can be two items which can be understood having different meanings but they are structurally indistinguishable within that level. This is called a “constructional homonymity” .


{ x ∣ P ( x ) } = { x ∣ Q ( x ) } {\displaystyle \{x\mid P(x)\}=\{x\mid Q(x)\}} if and only if P ( x ) ⇔ Q ( x ) . {\displaystyle P(x)\Leftrightarrow Q(x).}


In probability theory, the Doob–Dynkin lemma, named after Joseph L. Doob and Eugene Dynkin, characterizes the situation when one random variable is a function of another by the inclusion of the σ {\displaystyle \sigma } -algebras generated by the random variables. The usual statement of the lemma is formulated in terms of one random variable being measurable with respect to the σ {\displaystyle \sigma } -algebra generated by the other. The lemma plays an important role in the conditional expectation in probability theory, where it allows replacement of the conditioning on a random variable by conditioning on the σ {\displaystyle \sigma } -algebra that is generated by the random variable.


In public key infrastructure, a validation authority (VA) is an entity that provides a service used to verify the validity or revocation status of a digital certificate per the mechanisms described in the X.509 standard and RFC 5280 (page 69).The dominant method used for this purpose is to host a certificate revocation list (CRL) for download via the HTTP or LDAP protocols. To reduce the amount of network traffic required for certificate validation, the OCSP protocol may be used instead. While a validation authority is capable of responding to a network-based request for a CRL, it lacks the ability to issue or revoke certificates. It must be continuously updated with current CRL information from a certificate authority which issued the certificates contained within the CRL.


It shows that it is impossible to express the square root of 2 as a ratio of two integers. Another consequential proof of impossibility was Ferdinand von Lindemann's proof in 1882, which showed that the problem of squaring the circle cannot be solved because the number π is transcendental (i.e., non-algebraic), and that only a subset of the algebraic numbers can be constructed by compass and straightedge. Two other classical problems—trisecting the general angle and doubling the cube—were also proved impossible in the 19th century, and all of these problems gave rise to research into more complicated mathematical structures.


In mathematics and computing, the hexadecimal (also base-16 or simply hex) numeral system is a positional numeral system that represents numbers using a radix (base) of sixteen. Unlike the decimal system representing numbers using ten symbols, hexadecimal uses sixteen distinct symbols, most often the symbols "0"–"9" to represent values 0 to 9, and "A"–"F" (or alternatively "a"–"f") to represent values from ten to fifteen. Software developers and system designers widely use hexadecimal numbers because they provide a human-friendly representation of binary-coded values. Each hexadecimal digit represents four bits (binary digits), also known as a nibble (or nybble).


Previous studies have been conducted, focusing on priming effects having a rapid rise time and a hasty decay time. For example, an experiment by Donald Foss researched the decay time of semantic facilitation in lists and sentences. Three experiments were conducted and it was found that semantic relationships within words differs when words occur in sentences rather than lists. Thus, supporting the ongoing discourse model.


In order to abstract the concept of orientation on the edges of a graph to sets, one needs the ability to assign "direction" to the elements of a set. The way this achieved is with the following definition of signed sets. A signed set, X {\displaystyle X} , combines a set of objects, X _ {\displaystyle {\underline {X}}} , with a partition of that set into two subsets: X + {\displaystyle X^{+}} and X − {\displaystyle X^{-}} .The members of X + {\displaystyle X^{+}} are called the positive elements; members of X − {\displaystyle X^{-}} are the negative elements.The set X _ = X + ∪ X − {\displaystyle {\underline {X}}=X^{+}\cup X^{-}} is called the support of X {\displaystyle X} . The empty signed set, ∅ {\displaystyle \emptyset } , is defined as the empty set ∅ _ {\displaystyle {\underline {\emptyset }}} combined with a "partition" of it into two empty sets: ∅ + {\displaystyle \emptyset ^{+}} and ∅ − {\displaystyle \emptyset ^{-}} .


In multiprocessor systems, a processor may send an interrupt request to another processor via inter-processor interrupts (IPI).


MDP also used packet-level forward error correction coding concepts as a repair mechanism. Encoded parity repair packets were generally sent in response to repair requests by receivers. In this way no additional protocol overhead was added above selective retransmission methods.


In telecommunications, a channel bank is a device that performs multiplexing or demultiplexing ("demux") of a group of communications channels, such as analog or digital telephone lines, into one channel of higher bandwidth or higher digital bit rate, such as a DS-1 (T1) circuit, so that all the channels can be sent simultaneously over a single cable called a trunkline. A channel bank may be located in a telephone exchange, or in an enterprise's telephone closet or enclosure where it "breaks out" individual telephone lines from a high-capacity telephone trunk line connected to the central telephone office, or the enterprise's PBX system.


In one of Google's promotional videos for search published in the summer of 2010, the majority of the links available were reported to be produced at content farms. In late February 2011, Google announced it was adjusting search algorithms significantly to "provide better rankings for high-quality sites—sites with original content and information such as research, in-depth reports, thoughtful analysis and so on." This was reported to be a reaction to content farms and an attempt to reduce their effectiveness in manipulating search result rankings.The privacy-focused search engine DuckDuckGo does not show results from content farms.


Construct the shortest-path tree using the edges between each node and its parent.The above algorithm guarantees the existence of shortest-path trees. Like minimum spanning trees, shortest-path trees in general are not unique. In graphs for which all edge weights are equal, shortest path trees coincide with breadth-first search trees.


There is also Bell's theorem: no physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics. While an impossibility assertion in natural science can never be absolutely proved, it could be refuted by the observation of a single counterexample. Such a counterexample would require that the assumptions underlying the theory that implied the impossibility be re-examined.


In telecommunication, the term terminal equipment has the following meanings: Communications equipment at either end of a communications link, used to permit the stations involved to accomplish the mission for which the link was established. In radio-relay systems, equipment used at points where data are inserted or derived, as distinct from equipment used only to relay a reconstituted signal. Telephone and telegraph switchboards and other centrally located equipment at which communications circuits are terminated.


For the source and destination of every flight i, one adds two nodes to V, node si as the source and node di as the destination node of flight i. One also adds the following edges to E: An edge with capacity  between s and each si. An edge with capacity  between each di and t. An edge with capacity  between each pair of si and di. An edge with capacity  between each di and sj, if source sj is reachable with a reasonable amount of time and cost from the destination of flight i. An edge with capacity  between s and t.In the mentioned method, it is claimed and proved that finding a flow value of k in G between s and t is equal to finding a feasible schedule for flight set F with at most k crews.Another version of airline scheduling is finding the minimum needed crews to perform all the flights.


In mathematics, a surface is a mathematical model of the common concept of a surface. It is a generalization of a plane, but, unlike a plane, it may be curved; this is analogous to a curve generalizing a straight line. There are several more precise definitions, depending on the context and the mathematical tools that are used for the study. The simplest mathematical surfaces are planes and spheres in the Euclidean 3-space.


As a result, it is common for researchers working in the majority of languages with only one type or the other to simply use the alveolar symbols indifferently for both types, unless they specifically want to call attention to the distinction. The most common sounds are the stops  and . More generally, several kinds are distinguished: , voiceless dental plosive , voiced dental plosive , dental ejective , voiced dental implosive == Notes ==


In mathematics, a ternary operation is an n-ary operation with n = 3. A ternary operation on a set A takes any given three elements of A and combines them to form a single element of A. In computer science, a ternary operator is an operator that takes three arguments as input and returns one output.


Thus, in each convolutional layer, each neuron takes input from a larger area in the input than previous layers. This is due to applying the convolution over and over, which takes the value of a pixel into account, as well as its surrounding pixels.


In semantics, the most well-known types of semantic equivalence are dynamic equivalence and formal equivalence (coined by Eugene Nida) are associated with two dissimilar translation approaches that are employed to achieve different levels of literalness between the source and target text, as evidenced in biblical translation. The two have been understood basically, with dynamic equivalence as sense-for-sense translation (translating the meanings of phrases or whole sentences) with readability in mind, and with formal equivalence as word-for-word translation (translating the meanings of words and phrases in a more literal way), keeping literal fidelity.


In software engineering and hardware engineering, serviceability (also known as supportability) is one of the -ilities or aspects (from IBM's RAS(U) (Reliability, Availability, Serviceability, and Usability)). It refers to the ability of technical support personnel to install, configure, and monitor computer products, identify exceptions or faults, debug or isolate faults to root cause analysis, and provide hardware or software maintenance in pursuit of solving a problem and restoring the product into service. Incorporating serviceability facilitating features typically results in more efficient product maintenance and reduces operational costs and maintains business continuity. Examples of features that facilitate serviceability include: Help desk notification of exceptional events (e.g., by electronic mail or by sending text to a pager) Network monitoring Documentation Event logging / Tracing (software) Logging of program state, such as Execution path and/or local and global variables Procedure entry and exit, optionally with incoming and return variable values (see: subroutine) Exception block entry, optionally with local state (see: exception handling) Software upgrade Graceful degradation, where the product is designed to allow recovery from exceptional events without intervention by technical support staff Hardware replacement or upgrade planning, where the product is designed to allow efficient hardware upgrades with minimal computer system downtime (e.g., hotswap components.


In probability theory and statistics, the Zipf–Mandelbrot law is a discrete probability distribution. Also known as the Pareto–Zipf law, it is a power-law distribution on ranked data, named after the linguist George Kingsley Zipf who suggested a simpler distribution called Zipf's law, and the mathematician Benoit Mandelbrot, who subsequently generalized it. The probability mass function is given by: f ( k ; N , q , s ) = 1 / ( k + q ) s H N , q , s {\displaystyle f(k;N,q,s)={\frac {1/(k+q)^{s}}{H_{N,q,s}}}} where H N , q , s {\displaystyle H_{N,q,s}} is given by: H N , q , s = ∑ i = 1 N 1 ( i + q ) s {\displaystyle H_{N,q,s}=\sum _{i=1}^{N}{\frac {1}{(i+q)^{s}}}} which may be thought of as a generalization of a harmonic number. In the formula, k {\displaystyle k} is the rank of the data, and q {\displaystyle q} and s {\displaystyle s} are parameters of the distribution.


In the UML Superstructure 2.4.1 Specification Document the following definition is given: The notion of power type was inspired by the notion of power set. A power set is defined as a set whose instances are subsets. In essence, then, a power type is a class whose instances are subclasses. == References ==


In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings.


In symbols, one writes φ i ( n ) = { φ ( n ) , if i = 1 φ ( φ i − 1 ( n ) ) , if i ≥ 2 {\displaystyle \varphi ^{i}(n)={\begin{cases}\varphi (n),&{\text{ if }}i=1\\\varphi (\varphi ^{i-1}(n)),&{\text{ if }}i\geq 2\end{cases}}} for the iterated totient function. Then if c is the integer such that φ c ( n ) = 2 , {\displaystyle \displaystyle \varphi ^{c}(n)=2,} one has that n is a perfect totient number if n = ∑ i = 1 c + 1 φ i ( n ) . {\displaystyle n=\sum _{i=1}^{c+1}\varphi ^{i}(n).}


In software engineering, Canonical Schema is a design pattern, applied within the service-orientation design paradigm, which aims to reduce the need for performing data model transformation when services exchange messages that reference the same data model.


The small Veblen ordinal θ Ω ω ( 0 ) {\displaystyle \theta _{\Omega ^{\omega }}(0)} or ψ ( Ω Ω ω ) {\displaystyle \psi (\Omega ^{\Omega ^{\omega }})} is the limit of ordinals that can be described using a version of Veblen functions with finitely many arguments. It is the ordinal that measures the strength of Kruskal's theorem. It is also the ordinal type of a certain ordering of rooted trees (Jervell 2005).


Lovelace went on to write one of history's earliest computer programs that aided Babbage in the design of the Analytical Engine, a machine capable of running programs automatically. Lovelace's work with Babbage positioned her as a significant pioneer in the field of cybersecurity. Kevin Kelly, the co-founder of Wired magazine, once stated that Lovelace played a major role in the invention of computer science, and there are more lines of code written in languages that she created than in C++, Java, JavaScript, or Python combined.


In a priority interrupt system, the FLIH also (briefly) masks other interrupts of equal or lesser priority. A SLIH completes long interrupt processing tasks similarly to a process. SLIHs either have a dedicated kernel thread for each handler, or are executed by a pool of kernel worker threads.


Extended real-valued function for which the minimization problem is not solved by any one of these three trivial cases are exactly those that are called proper. Many (although not all) results whose hypotheses require that the function be proper add this requirement specifically to exclude these trivial cases. If the problem is instead a maximization problem (which would be clearly indicated, such as by the function being concave rather than convex) then the definition of "proper" is defined in an analogous (albeit technically different) manner but with the same goal: to exclude cases where the maximization problem can be answered immediately. Specifically, a concave function g {\displaystyle g} is called proper if its negation − g , {\displaystyle -g,} which is a convex function, is proper in the sense defined above.


In quantum information theory, the identity channel is a noise-free quantum channel. That is, the channel outputs exactly what was put in.The identity channel is commonly denoted as I {\displaystyle I} , i d {\displaystyle {\mathsf {id}}} or I {\displaystyle \mathbb {I} } . == References ==


In the case of the original Grätzel and O'Regan design, the cell has 3 primary parts. On top is a transparent anode made of fluoride-doped tin dioxide (SnO2:F) deposited on the back of a (typically glass) plate. On the back of this conductive plate is a thin layer of titanium dioxide (TiO2), which forms into a highly porous structure with an extremely high surface area. The (TiO2) is chemically bound by a process called sintering.


In principle FSK can be implemented by using completely independent free-running oscillators, and switching between them at the beginning of each symbol period. In general, independent oscillators will not be at the same phase and therefore the same amplitude at the switch-over instant, causing sudden discontinuities in the transmitted signal. In practice, many FSK transmitters use only a single oscillator, and the process of switching to a different frequency at the beginning of each symbol period preserves the phase. The elimination of discontinuities in the phase (and therefore elimination of sudden changes in amplitude) reduces sideband power, reducing interference with neighboring channels.


with Since f {\displaystyle f} does not appear explicitly in L , {\displaystyle L,} the first term in the Euler–Lagrange equation vanishes for all f ( x ) {\displaystyle f(x)} and thus, Substituting for L {\displaystyle L} and taking the derivative, Thus for some constant c . {\displaystyle c.} Then where Solving, we get which implies that is a constant and therefore that the shortest curve that connects two points ( x 1 , y 1 ) {\displaystyle \left(x_{1},y_{1}\right)} and ( x 2 , y 2 ) {\displaystyle \left(x_{2},y_{2}\right)} is and we have thus found the extremal function f ( x ) {\displaystyle f(x)} that minimizes the functional A  {\displaystyle A} so that A  {\displaystyle A} is a minimum.


In mathematical optimization, Bland's rule (also known as Bland's algorithm, Bland's anti-cycling rule or Bland's pivot rule) is an algorithmic refinement of the simplex method for linear optimization. With Bland's rule, the simplex algorithm solves feasible linear optimization problems without cycling.The original simplex algorithm starts with an arbitrary basic feasible solution, and then changes the basis in order to decrease the minimization target and find an optimal solution. Usually, the target indeed decreases in every step, and thus after a bounded number of steps an optimal solution is found. However, there are examples of degenerate linear programs, on which the original simplex algorithm cycles forever.


In number theory, Brocard's conjecture is the conjecture that there are at least four prime numbers between (pn)2 and (pn+1)2, where pn is the nth prime number, for every n ≥ 2. The conjecture is named after Henri Brocard. It is widely believed that this conjecture is true.


In mathematics and physics, vector notation is a commonly used notation for representing vectors, which may be Euclidean vectors, or more generally, members of a vector space. For representing a vector, the common typographic convention is lower case, upright boldface type, as in v. The International Organization for Standardization (ISO) recommends either bold italic serif, as in v, or non-bold italic serif accented by a right arrow, as in v → {\displaystyle {\vec {v}}} .In advanced mathematics, vectors are often represented in a simple italic type, like any variable.


The member also has duties to the tax authorities, notably of compliance with the law and the honest presentation of his client's circumstances. It is the taxpayer's responsibility to ensure that returns made to the tax authorities are correct and complete. It is for the member to assist him to decide on the extent and manner of disclosure of facts in relation to his tax affairs.


"Phenotypic correlations between psychological traits show significant and substantial genetic mediation." "The heritability of intelligence increases throughout development." "Age-to-age stability is mainly due to genetics."


This does not allow for full cross-referencing of information and, according to P. Simon and J. Stavo-Debauge, produces the opposite effect by creating a quasi-hierarchy, with foreigners from outside the European Union making up the group with the most difficulties. Paradoxically, a highly aggregated country classification is more evocative of ethnic discrimination than a country-level classification. Other organizations redact variables such as nationality, necessary for the proper processing of their files, when publishing data.


In some text messaging software products, an ellipsis is displayed while the interlocutor is typing characters. The feature has been referred to as a "typing awareness indicator", for which patents have been filed since the 1990s.


In mathematics, the classification of finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six or twenty-seven exceptions, called sporadic. The proof consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004. Simple groups can be seen as the basic building blocks of all finite groups, reminiscent of the way the prime numbers are the basic building blocks of the natural numbers.


When the commutative diagrams of a category are interpreted as a typed equational theory whose objects are the types, a codiscrete posetal category corresponds to an inconsistent theory understood as one satisfying the axiom x = y at all types. Viewing a 2-category as an enriched category whose hom-objects are categories, the hom-objects of any extension of a posetal category to a 2-category having the same 1-cells are monoids. Some lattice-theoretic structures are definable as posetal categories of a certain kind, usually with the stronger assumption of being skeletal.


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


s . {\displaystyle \mathbb {E} \left=0,a.s.}


In mathematics, a dyadic rational or binary rational is a number that can be expressed as a fraction whose denominator is a power of two. For example, 1/2, 3/2, and 3/8 are dyadic rationals, but 1/3 is not. These numbers are important in computer science because they are the only ones with finite binary representations.


Subshells are usually identified by their n {\displaystyle n} - and ℓ {\displaystyle \ell } -values. n {\displaystyle n} is represented by its numerical value, but ℓ {\displaystyle \ell } is represented by a letter as follows: 0 is represented by 's', 1 by 'p', 2 by 'd', 3 by 'f', and 4 by 'g'. For instance, one may speak of the subshell with n = 2 {\displaystyle n=2} and ℓ = 0 {\displaystyle \ell =0} as a '2s subshell'.


In mathematics, the sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit. It does so by iteratively marking as composite (i.e., not prime) the multiples of each prime, starting with the first prime number, 2. The multiples of a given prime are generated as a sequence of numbers starting from that prime, with constant difference between them that is equal to that prime. This is the sieve's key distinction from using trial division to sequentially test each candidate number for divisibility by each prime.


. (  (  (  (  (  P +  P ) +  P ) +  P ) + … ) ⋯ +  P ) +  P =  P +  P + ⋯ +  P +  P {\displaystyle (....(((((P+P)+P)+P)+\dots )\dots +P)+P=P+P+\dots +P+P} . Note that, this simple algorithm takes at most 2l steps and each step consists of a doubling and (if ki ≠ 0) adding two points. So, this is one of the reasons why addition and doubling formulas are defined. Furthermore, this method is applicable to any group and if the group law is written multiplicatively, the double-and-add algorithm is instead called square-and-multiply algorithm.


In mathematics, an associative algebra A is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field K. The addition and multiplication operations together give A the structure of a ring; the addition and scalar multiplication operations together give A the structure of a vector space over K. In this article we will also use the term K-algebra to mean an associative algebra over the field K. A standard first example of a K-algebra is a ring of square matrices over a field K, with the usual matrix multiplication. A commutative algebra is an associative algebra that has a commutative multiplication, or, equivalently, an associative algebra that is also a commutative ring. In this article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called unital associative algebras for clarification. In some areas of mathematics this assumption is not made, and we will call such structures non-unital associative algebras.


In statistics, the Lehmann–Scheffé theorem is a prominent statement, tying together the ideas of completeness, sufficiency, uniqueness, and best unbiased estimation. The theorem states that any estimator which is unbiased for a given unknown quantity and that depends on the data only through a complete, sufficient statistic is the unique best unbiased estimator of that quantity. The Lehmann–Scheffé theorem is named after Erich Leo Lehmann and Henry Scheffé, given their two early papers.If T is a complete sufficient statistic for θ and E(g(T)) = τ(θ) then g(T) is the uniformly minimum-variance unbiased estimator (UMVUE) of τ(θ).


In nonlinear optimization, quasiconvex programming studies iterative methods that converge to a minimum (if one exists) for quasiconvex functions. Quasiconvex programming is a generalization of convex programming. Quasiconvex programming is used in the solution of "surrogate" dual problems, whose biduals provide quasiconvex closures of the primal problem, which therefore provide tighter bounds than do the convex closures provided by Lagrangian dual problems. In theory, quasiconvex programming and convex programming problems can be solved in reasonable amount of time, where the number of iterations grows like a polynomial in the dimension of the problem (and in the reciprocal of the approximation error tolerated); however, such theoretically "efficient" methods use "divergent-series" stepsize rules, which were first developed for classical subgradient methods. Classical subgradient methods using divergent-series rules are much slower than modern methods of convex minimization, such as subgradient projection methods, bundle methods of descent, and nonsmooth filter methods.


It takes inspiration from NUSAP, a method used to qualify the worth (quality) of quantitative information with the generation of `Pedigrees' of numbers. Likewise, sensitivity auditing has been developed to provide pedigrees of models and model-based inferences.


The notion of a replicating portfolio is fundamental to rational pricing, which assumes that market prices are arbitrage-free – concretely, arbitrage opportunities are exploited by constructing a replicating portfolio. In practice, replicating portfolios are seldom, if ever, exact replications. Most significantly, unless they are claims against the same counterparties, there is credit risk. Further, dynamic replication is invariably imperfect, since actual price movements are not infinitesimal – they may in fact be large – and transaction costs to change the hedge are not zero.


In the context of biometric systems, presentation attacks may also be called "spoofing attacks". As per the recent ISO/IEC 30107 standard, presentation attacks are defined as "presentation to the biometric capture subsystem with the goal of interfering with the operation of the biometric system". These attacks can be either impersonation or obfuscation attacks.


In mathematics and computer science, a class of objects or methods exhibits recursive behavior when it can be defined by two properties: A simple base case (or cases) — a terminating scenario that does not use recursion to produce an answer A recursive step — a set of rules that reduces all successive cases toward the base case.For example, the following is a recursive definition of a person's ancestor. One's ancestor is either: One's parent (base case), or One's parent's ancestor (recursive step).The Fibonacci sequence is another classic example of recursion: Fib(0) = 0 as base case 1,Fib(1) = 1 as base case 2,For all integers n > 1, Fib(n) = Fib(n − 1) + Fib(n − 2).Many mathematical axioms are based upon recursive rules. For example, the formal definition of the natural numbers by the Peano axioms can be described as: "Zero is a natural number, and each natural number has a successor, which is also a natural number."


In the early 2000s, scholars noted a lack of theory and conceptual frameworks to inform and guide research and teacher preparation in technology integration. The classic definition of PCK proposed by Shulman included one dynamic and complex relationship between two different knowledge bodies: content knowledge and pedagogical knowledge. Shulman defined PCK as the blend between content and pedagogy, highlighting the teacher's comprehension of how specific topics are organized, adapted, and represented according to students' diverse interests and capabilities. For five years, Mishra & Koehler participated in a design experiment whose focus was to understand P-20 educators’ professional development of rich technology uses as well as helping them develop their teaching with technology.


In the 1970s and early 1980s, the cost of making a phone call decreased and more business communication was done by phone. As corporations grew and labor rates increased, the ratio of secretaries to employees decreased. The initial solution to the phone communication problem for businesses was the “message center.” A message center or “message desk” was a centralized, manual answering service inside a company manned by a few people answering everyone's phones.


In planar graphs, the maximum independent set may be approximated to within any approximation ratio c < 1 in polynomial time; similar polynomial-time approximation schemes exist in any family of graphs closed under taking minors.


In order to fully specify the Bayesian network and thus fully represent the joint probability distribution, it is necessary to specify for each node X the probability distribution for X conditional upon X's parents. The distribution of X conditional upon its parents may have any form. It is common to work with discrete or Gaussian distributions since that simplifies calculations.


In other parts of the world, similar concepts have been created to define standards for electronic signatures. In Switzerland, the digital signing standard ZertES has comparable standards that address the conformity and regulation of trust service providers who product digital certificates.In the United States, the NIST Digital Signature Standard (DSS) does not provide a comparable standard for regulating qualified certificates that would address non-repudiation of a signatory's qualified certificate. An amendment to NIST DSS is currently being discussed that would be more in-line with how eIDAS and ZertES handle trusted services.


In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655, John Wallis first used the notation ∞ {\displaystyle \infty } for such a number in his De sectionibus conicis, and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of 1 ∞ . {\displaystyle {\tfrac {1}{\infty }}.}


In the EQP phase of SLQP, the search direction d k {\displaystyle d_{k}} of the step is obtained by solving the following equality-constrained quadratic program: min d f ( x k ) + ∇ f ( x k ) T d + 1 2 d T ∇ x x 2 L ( x k , λ k , σ k ) d s . t . b A k ( x k ) + ∇ b A k ( x k ) T d = 0 c A k ( x k ) + ∇ c A k ( x k ) T d = 0. {\displaystyle {\begin{array}{rl}\min \limits _{d}&f(x_{k})+\nabla f(x_{k})^{T}d+{\tfrac {1}{2}}d^{T}\nabla _{xx}^{2}{\mathcal {L}}(x_{k},\lambda _{k},\sigma _{k})d\\\mathrm {s.t.} &b_{{\cal {A}}_{k}}(x_{k})+\nabla b_{{\cal {A}}_{k}}(x_{k})^{T}d=0\\&c_{{\cal {A}}_{k}}(x_{k})+\nabla c_{{\cal {A}}_{k}}(x_{k})^{T}d=0.\end{array}}} Note that the term f ( x k ) {\displaystyle f(x_{k})} in the objective functions above may be left out for the minimization problems, since it is constant.


Each element of the queue is a pair of symbols (terminals or previously defined pairs) that occur consecutively in the sequence. The priority of a pair is given by the number of occurrences of the pair in the remaining sequence. Each time a new pair is created, the priority queue is updated.


In symmetric case, what we want is equal bitrate for the two sources: 5 bits each with separate encoder and joint decoder. We still use linear codes for this system, as we used for asymmetric case. The basic idea is similar, but in this case, we need to do coset partition for both sources, while for a pair of received syndromes (corresponds to one coset), only one pair of input variables are possible given the correlation between two sources. Suppose we have a pair of linear code C 1 {\displaystyle \mathbf {C_{1}} } and C 2 {\displaystyle \mathbf {C_{2}} } and an encoder-decoder pair based on linear codes which can achieve symmetric coding.


In the case of its application to a disjunction, consider the following claim: "it is false that either of A or B is true", which is written as: ¬ ( A ∨ B ) . {\displaystyle \neg (A\lor B).} In that it has been established that neither A nor B is true, then it must follow that both A is not true and B is not true, which may be written directly as: ( ¬ A ) ∧ ( ¬ B ) .


Selecting the references can be done manually ( Φ 1 , Φ 2 , Φ 5 , . . .


(See also the A-theory and the B-theory of time.) Chalmers also considers negative facts. For example, a statement like "there do not exist nonphysical angels." If in fact true, it does not seem that this logically follows from any of the physical facts by themselves; but, he argues, it would follow if one added a "That is all" statement at the end of the list of all the physical facts.


In statistics, a unimodal probability distribution or unimodal distribution is a probability distribution which has a single peak. The term "mode" in this context refers to any peak of the distribution, not just to the strict definition of mode which is usual in statistics. If there is a single mode, the distribution function is called "unimodal". If it has more modes it is "bimodal" (2), "trimodal" (3), etc., or in general, "multimodal".


In scores for sporting events, in particular tennis and association football, the number 0 has the very specialized names "love" and "nil". This can cause difficulty for radio and television newsreaders, because the reader must be aware of which name to use, when the score is often written as the digit "0" in the script. (McLeish recommends to readers that they write the number out on the script in words if necessary.) In cricket, a batsman who is out without scoring is said to have scored "a duck", but "duck" is not used as a synonym for zero in the same way that "love" or "nil" are: it is always accompanied by the indefinite article and is not usually used in a formal reading of a team's scoresheet.


In nonlinear programming, the constraints are not necessarily linear. Nonetheless, many of the same principles apply. To ensure that the global maximum of a non-linear problem can be identified easily, the problem formulation often requires that the functions be convex and have compact lower level sets. This is the significance of the Karush–Kuhn–Tucker conditions.


In the United States, film classification is a voluntary process with the ratings issued by the Motion Picture Association (MPA) via the Classification and Rating Administration (CARA). The system was established in 1968, but the version listed below is the most recent revision, having been in effect since 1990. An unrated film is often informally denoted by "NR" in newspapers and so forth. G (General Audiences) – All ages admitted.


are: write performance increases linearly with the number of connected devices in the cluster. while the storage cluster is partitioned, all parts remain responsive. There is a risk of conflicting updates.


Since the method employs a selector variable (a factor introduced for each element to permit a counting procedure) the method is also known as the Darwin–Fowler method of selector variables. Note that a distribution function is not the same as the probability – cf. Maxwell–Boltzmann distribution, Bose–Einstein distribution, Fermi–Dirac distribution. Also note that the distribution function f i {\displaystyle f_{i}} which is a measure of the fraction of those states which are actually occupied by elements, is given by f i = n i / g i {\displaystyle f_{i}=n_{i}/g_{i}} or n i = f i g i {\displaystyle n_{i}=f_{i}g_{i}} , where g i {\displaystyle g_{i}} is the degeneracy of energy level i {\displaystyle i} of energy ε i {\displaystyle \varepsilon _{i}} and n i {\displaystyle n_{i}} is the number of elements occupying this level (e.g. in Fermi–Dirac statistics 0 or 1).


Folders can be named just as files can (except for the root folder, which often does not have a name). The use of folders makes it easier to organize files in a logical way. When a computer allows the use of folders, each file and folder has not only a name of its own, but also a path, which identifies the folder or folders in which a file or folder resides.


In the branch of mathematics known as order theory, a semimodular lattice, is a lattice that satisfies the following condition: Semimodular law a ∧ b <: a implies b <: a ∨ b.The notation a <: b means that b covers a, i.e. a < b and there is no element c such that a < c < b. An atomistic semimodular bounded lattice is called a matroid lattice because such lattices are equivalent to (simple) matroids. An atomistic semimodular bounded lattice of finite length is called a geometric lattice and corresponds to a matroid of finite rank.Semimodular lattices are also known as upper semimodular lattices; the dual notion is that of a lower semimodular lattice. A finite lattice is modular if and only if it is both upper and lower semimodular. A finite lattice, or more generally a lattice satisfying the ascending chain condition or the descending chain condition, is semimodular if and only if it is M-symmetric. Some authors refer to M-symmetric lattices as semimodular lattices.A semimodular lattice is one kind of algebraic lattice.


All the nodes on average are the same in degree therefore attacking one shouldn't cause anymore damage than attacking another. As the number of attacks go up and more nodes are removed, we observe that S decreases non-linearly and acts as if a threshold exists when a fraction of the nodes (f) has been removed, (f≈.28). At this point, S goes to zero. The average size of the isolated clusters behaves opposite, increasing exponentially to = 2, also approaching the threshold line f≈.28, except decreases back to 1 after. This model was tested for a large range of nodes and proven to maintain the same pattern.


In telecommunication, effective data transfer rate is the average number of units of data, such as bits, characters, blocks, or frames, transferred per unit time from a source and accepted as valid by a sink. Note: The effective data transfer rate is usually expressed in bits, characters, blocks, or frames per second. The effective data transfer rate may be averaged over a period of seconds, minutes, or hours.


The amount of parallelism that can be extracted in superscalar designs is limited by the number of instructions that the scheduler can examine for interdependencies. Examining a greater number of instructions can improve the chance of finding an instruction that can be run in parallel, but only at the cost of increasing the complexity of the scheduler itself. Despite massive efforts, CPU designs using classic RISC or CISC ISA's have plateaued at about three or four functional units.


The anode is specially designed to dissipate the heat and wear resulting from this intense focused barrage of electrons. The anode is precisely angled at 1-20 degrees off perpendicular to the electron current so as to allow the escape of some of the X-ray photons which are emitted perpendicular to the direction of the electron current. The anode is usually made out of tungsten or molybdenum. The tube has a window designed for escape of the generated X-ray photons. The power of a Coolidge tube usually ranges from 0.1 to 18 kW.


Idle wait is instead not convenient in case of a kernel critical section for device management, present in monolithic kernels only. A microkernel instead falls on just the first two of the above cases. In a multiprocessor system, most of the conflicts are kernel-level conflicts, due to the access to the kernel level critical sections, and thus the idle wait periods generated by them have a major impact in performance degradation. This idle wait time increases the average number of idle processors and thus decreases scalability and relative efficiency.


In reality, radio propagation changes along with the weather and tropospheric ducting, and occasionally along with other upper-atmospheric phenomena like sunspots and even meteor showers. Thus, while a broadcasting authority might fix the range to an area with exact boundaries (defined as a series of vectors), this is rarely if ever true. When a broadcast reaches well outside of its intended range due to unusual conditions, DXing is possible. The local terrain can also play a major role in limiting broadcast range.


Programmable tools vs. ‘fixed function’ tools. Some tools can be altered to get varying amounts of data, at varying times. Some tools have only a fixed function.


In talent scheduling problem, we can prove that is NP-hard by a reduction from the optimal linear arrangement(OLA) problem. And in this problem, even we restrict each actor is needed for just two days and all actors' salaries are 1, it's still polynomially reducible to the OLA problem. Thus, this problem is unlikely to have pseudo-polynomial algorithm.


The radiographer (also known as a radiologic technologist) is usually responsible for acquiring medical images of diagnostic quality; although other professionals may train in this area, notably some radiological interventions performed by radiologists are done so without a radiographer. As a field of scientific investigation, medical imaging constitutes a sub-discipline of biomedical engineering, medical physics or medicine depending on the context: Research and development in the area of instrumentation, image acquisition (e.g., radiography), modeling and quantification are usually the preserve of biomedical engineering, medical physics, and computer science; Research into the application and interpretation of medical images is usually the preserve of radiology and the medical sub-discipline relevant to medical condition or area of medical science (neuroscience, cardiology, psychiatry, psychology, etc.) under investigation. Many of the techniques developed for medical imaging also have scientific and industrial applications.


The physical addresses are 32-bit on the 386, but can be larger on newer processors which support Physical Address Extension. The 80386 also introduced two new general-purpose data segment registers, FS and GS, to the original set of four segment registers (CS, DS, ES, and SS). A 386 CPU can be put back into real mode by clearing a bit in the CR0 control register, however this is a privileged operation in order to enforce security and robustness. By way of comparison, a 286 could only be returned to real mode by forcing a processor reset, e.g. by a triple fault or using external hardware.


In the United Kingdom, the Financial Conduct Authority functions as the national competent authority for the regulation of financial markets; the definition in its Handbook of the term "security" applies only to equities, debentures, alternative debentures, government and public securities, warrants, certificates representing certain securities, units, stakeholder pension schemes, personal pension schemes, rights to or interests in investments, and anything that may be admitted to the Official List. In the United States, a "security" is a tradable financial asset of any kind. Securities can be broadly categorized into: debt securities (e.g., banknotes, bonds, and debentures) equity securities (e.g., common stocks) derivatives (e.g., forwards, futures, options, and swaps).The company or other entity issuing the security is called the issuer.


Developing enough shared vocabulary to communicate can often take a while. The same knowledge can be included in different domain knowledge. Knowledge which may be applicable across a number of domains is called domain-independent knowledge, for example logics and mathematics. Operations on domain knowledge are performed by meta-knowledge.


The high-order byte of the instruction specifies the operation. Bits 9 through 15 are the op-code, and bit 8 is the value of the condition code calculation which results in the branch being taken. The low-order byte is a signed word offset relative to the current location of the program counter.


In programming language syntax, spaces are frequently used to explicitly separate tokens. In most languages multiple whitespace characters are treated the same as a single whitespace character (outside of quoted strings); such languages are called free-form. In a few languages, including Haskell, occam, ABC, and Python, whitespace and indentation are used for syntactical purposes. In the satirical language called Whitespace, whitespace characters are the only valid characters for programming, while any other characters are ignored.


In the 1970s, most real-time applications did not use operating systems because the latter were perceived as adding too much overhead. Typical computers of the time had barely enough computing power to handle the tasks at hand. Moreover, most software of this type was written in assembly language. As a consequence, real-time systems were classic examples of spaghetti code: complex masses of assembly language software using all sorts of machine-dependent tricks to achieve maximum performance.


One of the processor actually controls the exchange, but other is synchronized with the former but does not participate in the exchange control. If a fault is detected by the comparator the processors are decoupled and a check-out program is run independently to find faulty processor. This process runs without disturbing the call processing which is suspended temporarily.


In order to use Dantzig–Wolfe decomposition, the constraint matrix of the linear program must have a specific form. A set of constraints must be identified as "connecting", "coupling", or "complicating" constraints wherein many of the variables contained in the constraints have non-zero coefficients. The remaining constraints need to be grouped into independent submatrices such that if a variable has a non-zero coefficient within one submatrix, it will not have a non-zero coefficient in another submatrix. This description is visualized below: The D matrix represents the coupling constraints and each Fi represents the independent submatrices. Note that it is possible to run the algorithm when there is only one F submatrix.


In telecommunications, 6b/8b is a line code that expands 6-bit codes to 8-bit symbols for the purposes of maintaining DC-balance in a communications system.The 6b/8b encoding is a balanced code -- each 8-bit output symbol contains 4 zero bits and 4 one bits. So the code can, like a parity bit, detect all single-bit errors. The number of 8-bit patterns with 4 bits set is the binomial coefficient ( 8 4 ) {\displaystyle {\tbinom {8}{4}}} = 70. Further excluding the patterns 11110000 and 00001111, this allows 68 coded patterns: 64 data codes, plus 4 additional control codes.


Thus it can be expressed as the vector ⟨ x 11 , x 22 , x 12 ⟩ {\displaystyle \langle x_{11},x_{22},x_{12}\rangle } .As another example: The stress tensor (in matrix notation) is given as σ =  . {\displaystyle {\boldsymbol {\sigma }}=\left.} In Voigt notation it is simplified to a 6-dimensional vector: σ ~ = ( σ x x , σ y y , σ z z , σ y z , σ x z , σ x y ) ≡ ( σ 1 , σ 2 , σ 3 , σ 4 , σ 5 , σ 6 ) .


In the Universal Mobile Telecommunications System (UMTS) and 3GPP Long Term Evolution (LTE), user equipment (UE) is any device used directly by an end-user to communicate. It can be a hand-held telephone, a laptop computer equipped with a mobile broadband adapter, or any other device. It connects to the base station Node B/eNodeB as specified in the ETSI 125/136-series and 3GPP 25/36-series of specifications. It roughly corresponds to the mobile station (MS) in GSM systems. The radio interface between the UE and the Node B is called Uu.


In software engineering, dependency injection is a programming technique in which an object or function receives other objects or functions that it requires, as opposed to creating them internally. Dependency injection aims to separate the concerns of constructing objects and using them, leading to loosely coupled programs. The pattern ensures that an object or function which wants to use a given service should not have to know how to construct those services.


The geometric mean is often used for a set of numbers whose values are meant to be multiplied together or are exponential in nature, such as a set of growth figures: values of the human population or interest rates of a financial investment over time. It also applies to benchmarking, where it is particularly useful for computing means of speedup ratios: since the mean of 0.5x (half as fast) and 2x (twice as fast) will be 1 (i.e., no speedup overall). The geometric mean can be understood in terms of geometry.


Similarly, a high-priority task has a high priority because it is more likely to be subject to strict time constraints—it may be providing data to an interactive user, or acting subject to real-time response guarantees. Because priority inversion results in the execution of a lower-priority task blocking the high-priority task, it can lead to reduced system responsiveness or even the violation of response time guarantees. A similar problem called deadline interchange can occur within earliest deadline first scheduling (EDF).


In the Alpha architecture, a byte is defined as an 8-bit datum (octet), a word as a 16-bit datum, a longword as a 32-bit datum, a quadword as a 64-bit datum, and an octaword as a 128-bit datum. The Alpha architecture originally defined six data types: Quadword (64-bit) integer Longword (32-bit) integer IEEE T-floating-point (double precision, 64-bit) IEEE S-floating-point (single precision, 32-bit)To maintain a level of compatibility with the VAX, the 32-bit architecture that preceded the Alpha, two other floating-point data types are included: VAX G-floating point (double precision, 64-bit) VAX F-floating point (single precision, 32-bit) VAX H-floating point (quad precision, 128-bit) was not supported, but another 128-bit floating-point option, X-floating point, is available on Alpha, but not VAX.H and X have been described as similar, but not identical. Software emulation for H-floating is available from DEC, as is a source-code level converter named DECmigrate.


For instance, the typeface "Bauer Bodoni" (sample shown here) includes fonts "Roman" (or "Regular"), "Bold" and "Italic"; each of these exists in a variety of sizes. The term "font" is correctly applied to any one of these alone but may be seen used loosely to refer to the whole typeface. When used in computers, each style is in a separate digital "font file". In both traditional typesetting and computing, the word "font" refers to the delivery mechanism of the typeface. In traditional typesetting, the font would be made from metal or wood type: to compose a page may require multiple fonts or even multiple typefaces.


Perhaps the simplest model of distributed computing is a synchronous system where all nodes operate in a lockstep fashion. This model is commonly known as the LOCAL model. During each communication round, all nodes in parallel (1) receive the latest messages from their neighbours, (2) perform arbitrary local computation, and (3) send new messages to their neighbors.


For example, when viewing content on the Internet (the channel), a web browser (a communicating party) would use the HTTP (the communication protocol) to request a web page from the server (another communicating party), and then render the returned data into its visual form. This is how the request–response messaging pattern operates. Alternatively, in computer networking, we have the UDP network protocol. It is used with the one-way messaging pattern, where the sending party is not interested whether the message arrives to any receiving party, nor it expects any of the receiving parties to produce an "answering" message.


In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is known as a van der Waals profile. It is a special case of the inverse-gamma distribution. It is a stable distribution.


In most telecommunications organizations, a virtual channel is a method of remapping the program number as used in H.222 Program Association Tables and Program Mapping Tables to a channel number that can be entered as digits on a receiver's remote control. Often, virtual channels are implemented in digital television to help users to go to channels easily and in general to ease the transition from analogue to digital broadcasting. Assigning virtual channels is most common in parts of the world where TV stations were colloquially named after the RF channel they were transmitting on ("Channel 6 Springfield"), as was common in North America during the analogue TV era. In other parts of the world, such as Europe, virtual channels are rarely used or needed, because TV stations there identify themselves by name, not by RF channel or callsign.


In numerical mathematics, the Uzawa iteration is an algorithm for solving saddle point problems. It is named after Hirofumi Uzawa and was originally introduced in the context of concave programming.


Annotea is part of the W3C Semantic Web efforts. An example implementation of Annotea is W3C's Amaya editor/browser. The current Amaya user interface for annotations is presented in the Amaya documentation. Other projects consists of Plugins for Firefox/Mozilla or Annotatio Client which interacts with most browsers per JavaScript.


In the expressions in this article, is the standard normal probability density function, is the corresponding cumulative distribution function (where erf is the error function), and is Owen's T function. Owen has an extensive list of Gaussian-type integrals; only a subset is given below.


They are especially important in the study of optimization problems where they are distinguished by a number of convenient properties. For instance, a strictly convex function on an open set has no more than one minimum. Even in infinite-dimensional spaces, under suitable additional hypotheses, convex functions continue to satisfy such properties and as a result, they are the most well-understood functionals in the calculus of variations. In probability theory, a convex function applied to the expected value of a random variable is always bounded above by the expected value of the convex function of the random variable. This result, known as Jensen's inequality, can be used to deduce inequalities such as the arithmetic–geometric mean inequality and Hölder's inequality.


The definition of theorems as sentences of a formal language is useful within proof theory, which is a branch of mathematics that studies the structure of formal proofs and the structure of provable formulas. It is also important in model theory, which is concerned with the relationship between formal theories and structures that are able to provide a semantics for them through interpretation. Although theorems may be uninterpreted sentences, in practice mathematicians are more interested in the meanings of the sentences, i.e. in the propositions they express.


In mathematics and computer science, the probabilistic automaton (PA) is a generalization of the nondeterministic finite automaton; it includes the probability of a given transition into the transition function, turning it into a transition matrix. Thus, the probabilistic automaton also generalizes the concepts of a Markov chain and of a subshift of finite type. The languages recognized by probabilistic automata are called stochastic languages; these include the regular languages as a subset.


In modular programming, modularity refers to the compartmentalization and interrelation of the parts of a software package. In software design, modularity refers to a logical partitioning of the "software design" that allows complex software to be manageable for the purpose of implementation and maintenance. The logic of partitioning may be based on related functions, implementation considerations, data links, or other criteria. In self-reconfiguring modular robotics, modularity refers to the ability of the robotic system to automatically achieve different morphologies to execute the task at hand.


The limit of 17 planes is due to UTF-16, which can encode 220 code points (16 planes) as pairs of words, plus the BMP as a single word. UTF-8 was designed with a much larger limit of 231 (2,147,483,648) code points (32,768 planes), and would still be able to encode 221 (2,097,152) code points (32 planes) even under the current limit of 4 bytes.The 17 planes can accommodate 1,114,112 code points. Of these, 2,048 are surrogates (used to make the pairs in UTF-16), 66 are non-characters, and 137,468 are reserved for private use, leaving 974,530 for public assignment.


After receiving Alice's qubit, operating on the pair and measuring both, Bob obtains two classical bits of information. It is worth stressing that if Alice and Bob do not pre-share entanglement, then the superdense protocol is impossible, as this would violate Holevo's theorem. Superdense coding is the underlying principle of secure quantum secret coding. The necessity of having both qubits to decode the information being sent eliminates the risk of eavesdroppers intercepting messages.


A black box algorithm is one which uses only these oracles. Hence, straight-line programs for black box groups are black box algorithms. Explicit straight-line programs are given for a wealth of finite simple groups in the online ATLAS of Finite Groups.


In recent years, however, confusion of the term "dollar cost averaging" with what Vanguard call a systematic implementation plan has arisen. The term "dollar cost averaging" is used to describe a delayed and staged investment strategy used in the situation where the investor has a windfall gain such as an insurance payout or inheritance, as opposed to the immediate investment of the entire sum. The delayed, staged strategy seems preferable for the investor who is concerned with avoiding timing risk (the risk of missing out in beneficial movements in price due to an error in market timing) then instead of investing the entire sum immediately, or waiting for the (mythical) ideal time to invest the entire sum, the investor spreads their investment of the windfall sum into the market over time in a staged way, which appears similar to dollar cost averaging. This behaviour is driven by the fear that volatility in the market could cause a significant drop in the value of the investment immediately after the investment is made.


Functional programming languages often allow the subtyping of records. Consequently, simply typed lambda calculus extended with record types is perhaps the simplest theoretical setting in which a useful notion of subtyping may be defined and studied. Because the resulting calculus allows terms to have more than one type, it is no longer a "simple" type theory.


In probability theory and statistics, the discrete uniform distribution is a symmetric probability distribution wherein a finite number of values are equally likely to be observed; every one of n values has equal probability 1/n. Another way of saying "discrete uniform distribution" would be "a known, finite number of outcomes equally likely to happen". A simple example of the discrete uniform distribution is throwing a fair die. The possible values are 1, 2, 3, 4, 5, 6, and each time the die is thrown the probability of a given score is 1/6.


Substituting into the first equation relating P ( s 1 ) , P ( s 2 ) {\displaystyle P(s_{1}),\;P(s_{2})}: P ( s 1 ) P ( s 2 ) = e − E ( s 1 ) / k T e − E ( s 2 ) / k T , {\displaystyle {\frac {P(s_{1})}{P(s_{2})}}={\frac {e^{-E(s_{1})/kT}}{e^{-E(s_{2})/kT}}},} which implies, for any state s of the system P ( s ) = 1 Z e − E ( s ) / k T , {\displaystyle P(s)={\frac {1}{Z}}e^{-E(s)/kT},} where Z is an appropriately chosen "constant" to make total probability 1. (Z is constant provided that the temperature T is invariant.) Z = ∑ s e − E ( s ) / k T , {\displaystyle Z=\sum _{s}e^{-E(s)/kT},} where the index s runs through all microstates of the system.


In order to get more stable results and use all valuable data for training, a data set can be repeatedly split into several training and a validation datasets. This is known as cross-validation. To confirm the model's performance, an additional test data set held out from cross-validation is normally used.


In packed pixel or chunky framebuffer organization, the bits defining each pixel are clustered and stored consecutively. For example, if there are 16 bits per pixel, each pixel is represented in two consecutive (contiguous) 8-bit bytes in the framebuffer. If there are 4 bits per pixel, each framebuffer byte defines two pixels, one in each nibble. The latter example is as opposed to storing a single 4-bit pixel in a byte, leaving 4 bits of the byte unused.


Modularity—the ability to define boundaries around specific domains and problem spaces—is essential for these languages because as stated by Tom Gruber, "Every ontology is a treaty- a social agreement among people with common motive in sharing." There are always many competing and differing views that make any general-purpose ontology impossible. A general-purpose ontology would have to be applicable in any domain and different areas of knowledge need to be unified.There is a long history of work attempting to build ontologies for a variety of task domains, e.g., an ontology for liquids, the lumped element model widely used in representing electronic circuits (e.g.,), as well as ontologies for time, belief, and even programming itself.


In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers. Because there are infinitely many prime numbers, there are also infinitely many semiprimes. Semiprimes are also called biprimes.


In standard multiple regression, each predictor is multiplied by a number that is called the beta weight, regression weight or weighted regression coefficients (denoted βW or BW). The prediction is obtained by adding these products along with a constant. When the weights are chosen to give the best prediction by some criterion, the model referred to as a proper linear model. Therefore, multiple regression is a proper linear model. By contrast, unit-weighted regression is called an improper linear model.


All matrices are assumed to have coefficients in the complex numbers. For the equation to make sense, the matrices must have appropriate sizes, for example they could all be square matrices of the same size. But more generally, A and B must be square matrices of sizes n and m respectively, and then X and C both have n rows and m columns.


In some settings it is natural to consider primitive recursive functions that take as inputs tuples that mix numbers with truth values (that is t for true and f for false), or that produce truth values as outputs. This can be accomplished by identifying the truth values with numbers in any fixed manner. For example, it is common to identify the truth value t with the number 1 and the truth value f with the number 0. Once this identification has been made, the characteristic function of a set A, which always returns 1 or 0, can be viewed as a predicate that tells whether a number is in the set A. Such an identification of predicates with numeric functions will be assumed for the remainder of this article.


This is used in order to create input at exactly a specific time, seen as how one can find a particular moment simply by checking every frame at one's leisure. Glitch An unintentional feature in a game — usually considered an error.


Related to the diatonic modes are the eight church modes or Gregorian modes, in which authentic and plagal forms of scales are distinguished by ambitus and tenor or reciting tone. Although both diatonic and gregorian modes borrow terminology from ancient Greece, the Greek tonoi do not otherwise resemble their mediaeval/modern counterparts. In the Middle Ages the term modus was used to describe both intervals and rhythm.


In the United States, anonymous work is legally defined as "a work on the copies or phonorecords of which no natural person is identified as author."


In one rearrangement proof, two squares are used whose sides have a measure of a + b {\displaystyle a+b} and which contain four right triangles whose sides are a, b and c, with the hypotenuse being c. In the square on the right side, the triangles are placed such that the corners of the square correspond to the corners of the right angle in the triangles, forming a square in the center whose sides are length c. Each outer square has an area of ( a + b ) 2 {\displaystyle (a+b)^{2}} as well as 2 a b + c 2 {\displaystyle 2ab+c^{2}} , with 2 a b {\displaystyle 2ab} representing the total area of the four triangles. Within the big square on the left side, the four triangles are moved to form two similar rectangles with sides of length a and b. These rectangles in their new position have now delineated two new squares, one having side length a is formed in the bottom-left corner, and another square of side length b formed in the top-right corner. In this new position, this left side now has a square of area ( a + b ) 2 {\displaystyle (a+b)^{2}} as well as 2 a b + a 2 + b 2 {\displaystyle 2ab+a^{2}+b^{2}} . Since both squares have the area of ( a + b ) 2 {\displaystyle (a+b)^{2}} it follows that the other measure of the square area also equal each other such that 2 a b + c 2 {\displaystyle 2ab+c^{2}} = 2 a b + a 2 + b 2 {\displaystyle 2ab+a^{2}+b^{2}} .


{\displaystyle ~\operatorname {\mathbb {P} } \left(\mathbf {x} \right)_{0}=N!\,\prod _{i=1}^{k}{\frac {\pi _{i}^{x_{i}}}{x_{i}!}}~.} The significance probability for the test is the probability of occurrence of the data set observed, or of a data set less likely than that observed, if the null hypothesis is true. Using an exact test, this is calculated as p  = ∑ y: P  ( y ) ≤ P  ( x ) 0 P  ( y ) {\displaystyle ~p_{\mathcal {}}=\sum _{\mathbf {y} \,:\;\operatorname {\mathbb {P} } \left(\mathbf {y} \right)\,\leq \,\operatorname {\mathbb {P} } \left(\mathbf {x} \right)_{0}}\operatorname {\mathbb {P} } \left(\mathbf {y} \right)~} where the sum ranges over all outcomes as likely as, or less likely than, that observed.


In this way, the tester determines whether the particular device under test meets the device specifications. While packaged as a wafer, automatic test equipment (ATE) can connect to the individual units using a set of microscopic needles. Once the chips are sawn apart and packaged, test equipment can connect to the chips using ZIF sockets (sometimes called contactors).


These maximal hypotheses essentially constitute a (optimistic) claim that the true concept is defined just by the negative data already observed: Thus, if a novel (never-before-seen) data point is observed, it should be assumed to be positive. (I.e., if data has not previously been ruled out, then it's ruled in.) Thus, during learning, the version space (which itself is a set – possibly infinite – containing all consistent hypotheses) can be represented by just its lower and upper bounds (maximally general and maximally specific hypothesis sets), and learning operations can be performed just on these representative sets. After learning, classification can be performed on unseen examples by testing the hypothesis learned by the algorithm. If the example is consistent with multiple hypotheses, a majority vote rule can be applied.


In mathematics, a filter or order filter is a special subset of a partially ordered set (poset), describing "large" or "eventual" elements. Filters appear in order and lattice theory, but also topology, whence they originate. The notion dual to a filter is an order ideal. Special cases of filters include ultrafilters, which are filters that cannot be enlarged, and describe nonconstructive techniques in mathematical logic.


In older literature on mathematical logic, the period glyph used to indicate how expressions should be bracketed (see Glossary of Principia Mathematica).


It was characteristic of logical positivism to consider a scientific theory to be nothing more than a set of sentences, subdivided into the class of theoretical sentences, the class of observational sentences, and the class of mixed sentences. The first class contains terms which refer to theoretical entities, that is to entities not directly observable such as electrons, atoms and molecules; the second class contains terms which denote quantities or observable entities, and the third class consists of precisely the coordinative definitions which contain both types of terms because they connect the theoretical terms with empirical procedures of measurement or with observable entities. For example, the interpretation of "the geodesic between two points" as correspondent to "the path of a light ray in a vacuum" provides a coordinative definition.


The deterministic behavior is desired and expected in compiling programming languages. In natural language processing, it was thought for a long time that deterministic parsing is impossible due to ambiguity inherent in natural languages (many sentences have more than one plausible parse). Thus, non-deterministic approaches such as the chart parser had to be applied. However, Mitch Marcus proposed in 1978 the Parsifal parser that was able to deal with ambiguities while still keeping the deterministic behavior.


For example, the inverse heat equation, deducing a previous distribution of temperature from final data, is not well-posed in that the solution is highly sensitive to changes in the final data. Continuum models must often be discretized in order to obtain a numerical solution. While solutions may be continuous with respect to the initial conditions, they may suffer from numerical instability when solved with finite precision, or with errors in the data.


In plain English, a hierarchy can be thought of as a set in which: No element is superior to itself, and One element, the (apex or hierarch), is superior to all of the other elements in the set.The first requirement is also interpreted to mean that a hierarchy can have no circular relationships; the association between two objects is always transitive. The second requirement asserts that a hierarchy must have a leader or root that is common to all of the objects.


In predicate logic, existential instantiation (also called existential elimination) is a rule of inference which says that, given a formula of the form ( ∃ x ) ϕ ( x ) {\displaystyle (\exists x)\phi (x)} , one may infer ϕ ( c ) {\displaystyle \phi (c)} for a new constant symbol c. The rule has the restrictions that the constant c introduced by the rule must be a new term that has not occurred earlier in the proof, and it also must not occur in the conclusion of the proof. It is also necessary that every instance of x {\displaystyle x} which is bound to ∃ x {\displaystyle \exists x} must be uniformly replaced by c. This is implied by the notation P ( a ) {\displaystyle P\left({a}\right)} , but its explicit statement is often left out of explanations. In one formal notation, the rule may be denoted by ∃ x P ( x ) ⟹ P ( a ) {\displaystyle \exists xP\left({x}\right)\implies P\left({a}\right)} where a is a new constant symbol that has not appeared in the proof.


In the MixColumns step, the four bytes of each column of the state are combined using an invertible linear transformation. The MixColumns function takes four bytes as input and outputs four bytes, where each input byte affects all four output bytes. Together with ShiftRows, MixColumns provides diffusion in the cipher.


In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event. An optimization problem seeks to minimize a loss function. An objective function is either a loss function or its opposite (in specific domains, variously called a reward function, a profit function, a utility function, a fitness function, etc.), in which case it is to be maximized.


In predicate calculus, a Henkin witness for a sentence ∃ x φ ( x ) {\displaystyle \exists x\,\varphi (x)} in a theory T is a term c such that T proves φ(c) (Hinman 2005:196). The use of such witnesses is a key technique in the proof of Gödel's completeness theorem presented by Leon Henkin in 1949.


Moreover, it's worth mentioning that if we carry forward the same line of thinking, something fascinating emerges. When we take infinity and divide it by a regular number like 10, the result still holds true: it's infinity. This adds another layer of insight to our mathematical journey, underscoring the depth of what we're uncovering here.


Thus the total demand for page file-backed virtual memory must exceed 250% of the computer's physical memory before the page file will expand. The fragmentation of the page file that occurs when it expands is temporary. As soon as the expanded regions are no longer in use (at the next reboot, if not sooner) the additional disk space allocations are freed and the page file is back to its original state.


In fact, a stronger result is true: the number of permutations of length n with major index k and i inversions is the same as the number of permutations of length n with major index i and k inversions, that is, the two statistics are equidistributed. For example, the number of permutations of length 4 with given major index and number of inversions is given in the table below. 0 1 2 3 4 5 6 0 1 0 0 0 0 0 0 1 0 1 1 1 0 0 0 2 0 1 2 1 1 0 0 3 0 1 1 2 1 1 0 4 0 0 1 1 2 1 0 5 0 0 0 1 1 1 0 6 0 0 0 0 0 0 1 {\displaystyle {\begin{array}{c|ccccccc}&0&1&2&3&4&5&6\\\hline 0&1&0&0&0&0&0&0\\1&0&1&1&1&0&0&0\\2&0&1&2&1&1&0&0\\3&0&1&1&2&1&1&0\\4&0&0&1&1&2&1&0\\5&0&0&0&1&1&1&0\\6&0&0&0&0&0&0&1\end{array}}}


In social psychology, signed graphs have been used to model social situations, with positive edges representing friendships and negative edges enmities between nodes, which represent people. Then, for example, a positive 3-cycle is either three mutual friends, or two friends with a common enemy; while a negative 3-cycle is either three mutual enemies, or two enemies who share a mutual friend. According to balance theory, positive cycles are balanced and supposed to be stable social situations, whereas negative cycles are unbalanced and supposed to be unstable. According to the theory, in the case of three mutual enemies, this is because sharing a common enemy is likely to cause two of the enemies to become friends.


In the context of networks, social capital exists where people have an advantage because of their location in a network. Contacts in a network provide information, opportunities and perspectives that can be beneficial to the central player in the network. Most social structures tend to be characterized by dense clusters of strong connections. Information within these clusters tends to be rather homogeneous and redundant.


In social cataloging much like social bookmarking, this software is aimed towards academics. It allows the user to post a citation for an article found on the internet or a website, online database like Academic Search Premier or LexisNexis Academic University, a book found in a library catalog and so on. These citations can be organized into predefined categories, or a new category defined by the user through the use of tags. This method allows academics researching or interested in similar areas to connect and share resources.


The Wasserstein distance between two measures is, roughly speaking, the cost of transporting one to the other. The set of all m by n matrices over some field is a metric space with respect to the rank distance d ( A , B ) = r a n k ( B − A ) {\displaystyle d(A,B)=\mathrm {rank} (B-A)} . The Helly metric in game theory measures the difference between strategies in a game.


From this definition the following standard definition of a let expression, as used in a functional language may be derived. x ∉ FV  ( y ) ⟹ ( let  x: x = y in  z ) = z  = ( λ x . z ) y {\displaystyle x\not \in \operatorname {FV} (y)\implies (\operatorname {let} x:x=y\operatorname {in} z)=z=(\lambda x.z)\ y} For simplicity the marker specifying the existential variable, x: {\displaystyle x:} , will be omitted from the expression where it is clear from the context. x ∉ FV  ( y ) ⟹ ( let  x = y in  z ) = z  = ( λ x . z ) y {\displaystyle x\not \in \operatorname {FV} (y)\implies (\operatorname {let} x=y\operatorname {in} z)=z=(\lambda x.z)\ y}


Investigations into home primes make up a minor side issue in number theory. Its questions have served as test fields for the implementation of efficient algorithms for factoring composite numbers, but the subject is really one in recreational mathematics. The outstanding computational problem as of 2016 is whether HP(49) = HP(77) can be calculated in practice.


Line 2: Second matrix-valued set of conditional probabilities. By definition p ( y i | c k ) = ∑ j p ( y i | x j ) p ( x j | c k ) = ∑ j p ( y i | x j ) p ( x j , c k ) / p ( c k ) = ∑ j p ( y i | x j ) p ( c k | x j ) p ( x j ) / p ( c k ) {\displaystyle {\begin{aligned}p(y_{i}|c_{k})&=\sum _{j}p(y_{i}|x_{j})p(x_{j}|c_{k})\\&=\sum _{j}p(y_{i}|x_{j})p(x_{j},c_{k}){\big /}p(c_{k})\\&=\sum _{j}p(y_{i}|x_{j})p(c_{k}|x_{j})p(x_{j}){\big /}p(c_{k})\\\end{aligned}}} where the Bayes identities p ( a , b ) = p ( a | b ) p ( b ) = p ( b | a ) p ( a ) {\displaystyle p(a,b)=p(a|b)p(b)=p(b|a)p(a)\,} are used. Line 3: this line finds the marginal distribution of the clusters c {\displaystyle c\,} p ( c i ) = ∑ j p ( c i , x j ) = ∑ j p ( c i | x j ) p ( x j ) {\displaystyle {\begin{aligned}p(c_{i})&=\sum _{j}p(c_{i},x_{j})&=\sum _{j}p(c_{i}|x_{j})p(x_{j})\end{aligned}}} This is a standard result.


Traditional single-dispatch OO languages make it trivial to add new datatypes but not new functions; traditional functional languages tend to have the opposite effect, and multiple dispatch, if implemented correctly, allows both. It is desirable for an implementation of multiple dispatch to have the following properties: It is possible to define different "cases" of a multi-method from within different packages without modifying the source of a base package. Inclusion of another package in the program should not change the behavior of a given multi-method call, when the call does not use any datatypes defined in the package. Conversely, if a datatype is defined in a given package, and a multi-method extension using that type is also defined in the same package, and a value of that type is passed (through a base type reference or into a generic function) into another package with no dependency on that package, and then the multi-method is invoked with that value as an argument, the multi-method case defined in the package which includes the type should be employed. To put it another way--within a given program, the same multi-method invoked with the same set of arguments should resolve to the same implementation, regardless of the location of the call site, and whether or not a given definition is "in scope" or "visible" at the point of the method call.


An examination of the development in artificial intelligence that has followed reveals that the learning machine did take the abstract path suggested by Turing as in the case of Deep Blue, a chess playing computer developed by IBM and one which defeated the world champion Garry Kasparov (though, this too is controversial) and the numerous computer chess games which can outplay most amateurs. As for the second suggestion Turing makes, it has been likened by some authors as a call to finding a simulacrum of human cognitive development. And such attempts at finding the underlying algorithms by which children learn of the features of the world around them are only beginning to be made.


In mathematics, the Johnson–Lindenstrauss lemma is a result named after William B. Johnson and Joram Lindenstrauss concerning low-distortion embeddings of points from high-dimensional into low-dimensional Euclidean space. The lemma states that a set of points in a high-dimensional space can be embedded into a space of much lower dimension in such a way that distances between the points are nearly preserved. The map used for the embedding is at least Lipschitz, and can even be taken to be an orthogonal projection. The lemma has applications in compressed sensing, manifold learning, dimensionality reduction, and graph embedding.


In the C, C++, and D programming languages, a type qualifier is a keyword that is applied to a type, resulting in a qualified type. For example, const int is a qualified type representing a constant integer, while int is the corresponding unqualified type, simply an integer. In D these are known as type constructors, by analogy with constructors in object-oriented programming. Type qualifiers are a way of expressing additional information about a value through the type system, and ensuring correctness in the use of the data. Type qualifiers are not generally used outside the C/C++ family of languages: many languages have a notation of constants, but express this by the name binding being constant (a "variable that doesn't vary"), rather than through the type system; see alternatives, below.


In some contexts one considers the more general notion of order-indiscernibles, and the term sequence of indiscernibles often refers implicitly to this weaker notion. In our example of binary formulas, to say that the triple (a, b, c) of distinct elements is a sequence of indiscernibles implies (  ∨  ) ∧ (  ∨  ) . {\displaystyle (\lor )\land (\lor )\,.}


This is because models which depend linearly on their unknown parameters are easier to fit than models which are non-linearly related to their parameters and because the statistical properties of the resulting estimators are easier to determine. Linear regression has many practical uses. Most applications fall into one of the following two broad categories: If the goal is error reduction in prediction or forecasting, linear regression can be used to fit a predictive model to an observed data set of values of the response and explanatory variables.


The improved coloring with ColorSort algorithm takes in consideration the above observation. Each vertex is assigned to a color class Ck. If k < |Qmax| − |Q| + 1 the vertex is moved to R (behind the last vertex in R).


The fastest 3G-based standard in the UMTS family is the HSPA+ standard, which has been commercially available since 2009 and offers 21 Mbit/s downstream (11 Mbit/s upstream) without MIMO, i.e. with only one antenna, and in 2011 accelerated up to 42 Mbit/s peak bit rate downstream using either DC-HSPA+ (simultaneous use of two 5 MHz UMTS carriers) or 2x2 MIMO. In theory speeds up to 672 Mbit/s are possible, but have not been deployed yet. The fastest 3G-based standard in the CDMA2000 family is the EV-DO Rev. B, which is available since 2010 and offers 15.67 Mbit/s downstream.


In mathematics, a square root of a number x is a number y such that y 2 = x {\displaystyle y^{2}=x} ; in other words, a number y whose square (the result of multiplying the number by itself, or y ⋅ y {\displaystyle y\cdot y} ) is x. For example, 4 and −4 are square roots of 16 because 4 2 = ( − 4 ) 2 = 16 {\displaystyle 4^{2}=(-4)^{2}=16} . Every nonnegative real number x has a unique nonnegative square root, called the principal square root or simply the square root (with a definite article, see below), which is denoted by x , {\displaystyle {\sqrt {x}},} where the symbol " {\displaystyle {\sqrt {~^{~}}}} " is called the radical sign or radix. For example, to express the fact that the principal square root of 9 is 3, we write 9 = 3 {\displaystyle {\sqrt {9}}=3} . The term (or number) whose square root is being considered is known as the radicand.


In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of the property that says something like "3 + 4 = 4 + 3" or "2 × 5 = 5 × 2", the property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, "3 − 5 ≠ 5 − 3"); such operations are not commutative, and so are referred to as noncommutative operations.


The above results may be summarized in the following table. Each cell represents a subshell, and lists the values of m ℓ {\displaystyle m_{\ell }} available in that subshell. Empty cells represent subshells that do not exist.


Teamwork offers different perspectives, and each member may have a different way of handling the project. By sharing ideas, teams can produce better quality work than if the project was done by an individual.