"Notable examples of attacks on internet connected facilities include the 2010 Stuxnet attack on Iran's Natanz nuclear facilities and the December 2015 Ukraine power grid cyberattack. “Today’s threats are a result of hybrid and blended attacks utilizing Information Technology (IT), physical infrastructure, and Operational Technology (OT) as the enemy avenue of approach," notes former CISA Assistant Director for Infrastructure Security Brian Harrell. "Highlighting this future threat landscape will ensure better situational awareness and a more rapid response.”


In some other programming languages it may be possible to achieve the same thing with less boilerplate, when the language has built-in support for such common constructs. For example, the equivalent of the above Java code can be expressed in Scala using just one line of code:


Again, unranked preferences have no value. In positional voting, ranked ballots with tied options are normally considered as invalid. The counting process is straightforward.


For example, the Economist provides a Big Mac Index that expresses the adjusted cost of a globally ubiquitous Big Mac as a percentage over or under the cost of a Big Mac in the U.S. in USD. Such indices can be used to help forecast currency values.


In the case where two objects are traveling in perpendicular directions, the relativistic relative velocity v → B | A {\displaystyle {\vec {v}}_{\mathrm {B|A} }} is given by the formula: v → B | A = v → B γ A − v → A {\displaystyle {\vec {v}}_{\mathrm {B|A} }={\frac {{\vec {v}}_{\mathrm {B} }}{\gamma _{\mathrm {A} }}}-{\vec {v}}_{\mathrm {A} }} where γ A = 1 1 − ( v A c ) 2 {\displaystyle \gamma _{\mathrm {A} }={\frac {1}{\sqrt {1-\left({\frac {v_{\mathrm {A} }}{c}}\right)^{2}}}}} The relative speed is given by the formula v B | A = c 4 − ( c 2 − v A 2 ) ( c 2 − v B 2 ) c {\displaystyle v_{\mathrm {B|A} }={\frac {\sqrt {c^{4}-\left(c^{2}-v_{\mathrm {A} }^{2}\right)\left(c^{2}-v_{\mathrm {B} }^{2}\right)}}{c}}}


In terms of how well a PC system follows a programmer’s intuition, it turns out that in properly synchronized systems, the outcomes of PC and SC are the same. Programmer’s intuition is essentially how the programmer expects the instructions to execute, usually in what is referred to as “program order.” Program order in a multiprocessor system is the execution of instructions resulting in the same outcome as a sequential execution. The fact that PC and SC both follow this expectation is a direct consequence of the fact that corresponding loads and stores in PC systems are still ordered with respect to each other. For example, in lock synchronization, the only operation whose behavior is not fully defined by PC is the lock-acquire store, where subsequent loads are in the critical section and their order affects the outcome. This operation, however, is usually implemented with a store conditional or atomic instruction, so that if the operation fails it will be repeated later and all the younger loads will also be repeated. All loads occurring before this store are still ordered with respect to the loads occurring in the critical section, and as such all the older loads have to complete before loads in the critical section can run.


There is also work on mapping some version of English (or another natural language) automatically to and from logic, as well as executing the logic directly. Examples are Attempto Controlled English, and Internet Business Logic, which do not seek to control the vocabulary or syntax. A feature of systems that support bidirectional English–logic mapping and direct execution of the logic is that they can be made to explain their results, in English, at the business or scientific level.


The similarity between languages is their very structural principle; the difference between languages is the carrying out of that principle in concreto. Both the similarity and the difference between languages lie, then, in language and in languages themselves, in their internal structure; and no similarity or difference between languages rests on any factor outside language." – Louis Hjelmslev


Societies are seen as coherent, bounded and fundamentally relational constructs that function like organisms, with their various (or social institutions) working together in an unconscious, quasi-automatic fashion toward achieving an overall social equilibrium. All social and cultural phenomena are therefore seen as functional in the sense of working together, and are effectively deemed to have "lives" of their own. They are primarily analyzed in terms of this function.


In software testing, conformance testing verifies that a product performs according to its specified standards. Compilers, for instance, are extensively tested to determine whether they meet the recognized standard for that language.


In the early 1980s, with the advent of 32-bit microprocessors such as the Motorola 68000, several new competitors appeared, including Apollo Computer and Sun Microsystems, with workstations based on 68000 and Unix. Meanwhile, DARPA's VLSI Project created several spinoff graphics products, such as the Silicon Graphics 3130. Target markets were differentiated, with Sun and Apollo considered to be network workstations and SGI as graphics workstations. RISC CPUs increased in the mid-1980s, typical of workstation vendors.Workstations often feature SCSI or Fibre Channel disk storage systems, high-end 3D accelerators, single or multiple 64-bit processors, large amounts of RAM, and well-designed cooling.


The precision of the measurement can exceed the least count of the instrument. used. The repetition method is used when high accuracy is required. For rough or approximate survey work, the ordinary method of measuring horizontal angles is used as it is less time consuming.


There may be more than one generalized Bayes rule, since there may be multiple choices of δ ( x ) {\displaystyle \delta (x)\,\!} that achieve the same expected loss. At first, this may appear rather different from the Bayes rule approach of the previous section, not a generalization.


In multiprocessor computer systems, software lockout is the issue of performance degradation due to the idle wait times spent by the CPUs in kernel-level critical sections. Software lockout is the major cause of scalability degradation in a multiprocessor system, posing a limit on the maximum useful number of processors. To mitigate the phenomenon, the kernel must be designed to have its critical sections as short as possible, therefore decomposing each data structure in smaller substructures.


This provides a concise and synthetic way for manipulating and studying systems of linear equations. Vector spaces are characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. This means that, for two vector spaces over a given field and with the same dimension, the properties that depend only on the vector-space structure are exactly the same (technically the vector spaces are isomorphic).


The machine executed 3 instructions and then a NOP (no op) to slow it down, as nearly every component was identical to the 90/30). Later a 90/40 model was added, with improved performance from a faster clock rate (cycle time of 500ns vs 600ns), pre-fetching of the next instruction, and greater maximum main memory capacity (1M vs 512K).The Sperry UNIVAC System 80 series: The entire 90/xx series was eventually replaced in 1981 by the System 80, models 4 and 6. More powerful System 80's (models 8, 10 and 20) were introduced in 1984. These were Sperry-badged, IBM/360-like mainframes actually developed and engineered by Mitsubishi in Japan. The final System 80 was the model 7E, released in 1990 by Unisys.


In (Costantini & Zacher 2004) it is shown that every finite simple group is a complemented group. Note that in the classification of finite simple groups, K-group is more used to mean a group whose proper subgroups only have composition factors amongst the known finite simple groups. An example of a group that is not complemented (in either sense) is the cyclic group of order p2, where p is a prime number. This group only has one nontrivial subgroup H, the cyclic group of order p, so there can be no other subgroup L to be the complement of H.


In set theory, the singular cardinals hypothesis (SCH) arose from the question of whether the least cardinal number for which the generalized continuum hypothesis (GCH) might fail could be a singular cardinal. According to Mitchell (1992), the singular cardinals hypothesis is: If κ is any singular strong limit cardinal, then 2κ = κ+.Here, κ+ denotes the successor cardinal of κ. Since SCH is a consequence of GCH, which is known to be consistent with ZFC, SCH is consistent with ZFC. The negation of SCH has also been shown to be consistent with ZFC, if one assumes the existence of a sufficiently large cardinal number. In fact, by results of Moti Gitik, ZFC + ¬SCH is equiconsistent with ZFC + the existence of a measurable cardinal κ of Mitchell order κ++.


In relation to the Japanese language and computers many adaptation issues arise, some unique to Japanese and others common to languages which have a very large number of characters. The number of characters needed in order to write in English is quite small, and thus it is possible to use only one byte (28=256 possible values) to encode each English character. However, the number of characters in Japanese is many more than 256 and thus cannot be encoded using a single byte - Japanese is thus encoded using two or more bytes, in a so-called "double byte" or "multi-byte" encoding. Problems that arise relate to transliteration and romanization, character encoding, and input of Japanese text.


This is an extension of Szemerédi's regularity lemma that partitions any given graph into bounded number parts such that edges between the parts behave almost randomly. Similarly, the hypergraph counting lemma is a generalization of the graph counting lemma that estimates number of copies of a fixed graph as a subgraph of a larger graph. There are several distinct formulations of the method, all of which imply the hypergraph removal lemma and a number of other powerful results, such as Szemerédi's theorem, as well as some of its multidimensional extensions. The following formulations are due to V. Rödl, B. Nagle, J. Skokan, M. Schacht, and Y. Kohayakawa, for alternative versions see Tao (2006), and Gowers (2007).


In number theory, the home prime HP(n) of an integer n greater than 1 is the prime number obtained by repeatedly factoring the increasing concatenation of prime factors including repetitions. The mth intermediate stage in the process of determining HP(n) is designated HPn(m). For instance, HP(10) = 773, as 10 factors as 2×5 yielding HP10(1) = 25, 25 factors as 5×5 yielding HP10(2) = HP25(1) = 55, 55 = 5×11 implies HP10(3) = HP25(2) = HP55(1) = 511, and 511 = 7×73 gives HP10(4) = HP25(3) = HP55(2) = HP511(1) = 773, a prime number. Some sources use the alternative notation HPn for the homeprime, leaving out parentheses.


Prohibiting courts from dismissing claims on the basis of the state secrets privilege until after they have reviewed all available evidence. Permitting the court to appoint an outside expert to scrutinize the evidence for national security content. Excluding illegal government action from the definition of "state secrets," or otherwise allowing the court to address the legality (instead of just the secrecy) of government conduct. This would prevent the government from using the state secrets privilege to conceal its illegal conduct.On January 22, 2008, Senators Edward Kennedy, Patrick Leahy and Arlen Specter introduced S. 2533, the State Secrets Protection Act.


This widely generalizes the Abel–Ruffini theorem, which asserts that a general polynomial of degree at least five cannot be solved by radicals. Galois theory has been used to solve classic problems including showing that two problems of antiquity cannot be solved as they were stated (doubling the cube and trisecting the angle), and characterizing the regular polygons that are constructible (this characterization was previously given by Gauss, but all known proofs that this characterization is complete require Galois theory).


There are also some similar definitions for functions mapping α {\displaystyle \alpha } to α {\displaystyle \alpha }: A function mapping α {\displaystyle \alpha } to α {\displaystyle \alpha } is α {\displaystyle \alpha } -recursively-enumerable, or α {\displaystyle \alpha } -partial recursive, iff its graph is Σ 1 {\displaystyle \Sigma _{1}} -definable on ( L α , ∈ ) {\displaystyle (L_{\alpha },\in )} . A function mapping α {\displaystyle \alpha } to α {\displaystyle \alpha } is α {\displaystyle \alpha } -recursive iff its graph is Δ 1 {\displaystyle \Delta _{1}} -definable on ( L α , ∈ ) {\displaystyle (L_{\alpha },\in )} . Additionally, a function mapping α {\displaystyle \alpha } to α {\displaystyle \alpha } is α {\displaystyle \alpha } -arithmetical iff there exists some n ∈ ω {\displaystyle n\in \omega } such that the function's graph is Σ n {\displaystyle \Sigma _{n}} -definable on ( L α , ∈ ) {\displaystyle (L_{\alpha },\in )} .Additional connections between recursion theory and α recursion theory can be drawn, although explicit definitions may not have yet been written to formalize them: The functions Δ 0 {\displaystyle \Delta _{0}} -definable in ( L α , ∈ ) {\displaystyle (L_{\alpha },\in )} play a role similar to those of the primitive recursive functions.We say R is a reduction procedure if it is α {\displaystyle \alpha } recursively enumerable and every member of R is of the form ⟨ H , J , K ⟩ {\displaystyle \langle H,J,K\rangle } where H, J, K are all α-finite.


In network science, the Matthew effect is used to describe the preferential attachment of earlier nodes in a network, which explains that these nodes tend to attract more links early on. "Because of preferential attachment, a node that acquires more connections than another one will increase its connectivity at a higher rate, and thus an initial difference in the connectivity between two nodes will increase further as the network grows, while the degree of individual nodes will grow proportional with the square root of time." The Matthew Effect therefore explains the growth of some nodes in vast networks such as the Internet.


The Confidentiality, Integrity and Availability (C, I, A) metrics were updated to have scores consisting of None, Low, or High, rather than the None, Partial, Complete of CVSSv2. This allows more flexibility in determining the impact of a vulnerability on CIA metrics. Access Complexity was renamed Attack Complexity (AC) to make clear that access privileges were moved to a separate metric. This metric now describes how repeatable exploit of this vulnerability may be; AC is High if the attacker requires perfect timing or other circumstances (other than user interaction, which is also a separate metric) which may not be easily duplicated on future attempts. Attack Vector (AV) saw the inclusion of a new metric value of Physical (P), to describe vulnerabilities that require physical access to the device or system to perform.


In the clinical context, "invisible light" medical imaging is generally equated to radiology or "clinical imaging". "Visible light" medical imaging involves digital video or still pictures that can be seen without special equipment. Dermatology and wound care are two modalities that use visible light imagery.


In regression analysis, the distinction between errors and residuals is subtle and important, and leads to the concept of studentized residuals. Given an unobservable function that relates the independent variable to the dependent variable – say, a line – the deviations of the dependent variable observations from this function are the unobservable errors. If one runs a regression on some data, then the deviations of the dependent variable observations from the fitted function are the residuals. If the linear model is applicable, a scatterplot of residuals plotted against the independent variable should be random about zero with no trend to the residuals.


In the context of SQL, data definition or data description language (DDL) is a syntax for creating and modifying database objects such as tables, indices, and users. DDL statements are similar to a computer programming language for defining data structures, especially database schemas. Common examples of DDL statements include CREATE, ALTER, and DROP.


In mathematics, Graeffe's method or Dandelin–Lobachesky–Graeffe method is an algorithm for finding all of the roots of a polynomial. It was developed independently by Germinal Pierre Dandelin in 1826 and Lobachevsky in 1834. In 1837 Karl Heinrich Gräffe also discovered the principal idea of the method.


Due to other concerns, this correspondence is inexact: for example, SP (Space) and 0 (zero) both have low bits 00000 (to ease collation for space and conversion to/from binary-coded decimal for 0), preventing 0 from lining up with ) (right parenthesis), its conventional value, and thus instead () corresponded to 89, instead of 90 as on typewriters. Further, while digits were placed in column 3, the characters ,-./ (conventionally unshifted) were placed in column 2, to ease collation, due to being used as separators, and the characters ;: (conventionally paired) were both placed in column 3. Other symbols also did not line up with their conventional digit pair, as detailed below.


In peer-to-peer swarming systems, a swarm is self-sustaining if all the blocks of its files are available among peers (excluding seeds and publishers).


In queueing theory, a discipline within the mathematical theory of probability, the M/M/∞ queue is a multi-server queueing model where every arrival experiences immediate service and does not wait. In Kendall's notation it describes a system where arrivals are governed by a Poisson process, there are infinitely many servers, so jobs do not need to wait for a server. Each job has an exponentially distributed service time. It is a limit of the M/M/c queue model where the number of servers c becomes very large. The model can be used to model bound lazy deletion performance.


In order to achieve linear time complexity, Re-Pair requires the following data structures A sequence representing the input string. Position i {\displaystyle i} of the sequence contains the i-th symbol of the input string plus two references to other positions in the sequence. These references point to the next/previous positions, say k {\displaystyle k} and m {\displaystyle m} , such that the same substring begins at w  {\displaystyle w} , w  {\displaystyle w} and w  {\displaystyle w} and all three occurrences are captured by the same reference (i.e. there is a variable in the grammar generating the string). A priority queue.


In the following subsections, a selection of significant tests (by no means exhaustive) is listed, representative of the testing effort in each nuclear country.


One can define a sesquilinear form over any *-ring. Also, one can define *-versions of algebraic objects, such as ideal and subring, with the requirement to be *-invariant: x ∈ I ⇒ x* ∈ I and so on. *-rings are unrelated to star semirings in the theory of computation.


In the C family of languages, the following operators are unary: Increment: ++x, x++ Decrement: −−x, x−− Address: &x Indirection: *x Positive: +x Negative: −x Ones' complement: ~x Logical negation: !x Sizeof: sizeof x, sizeof(type-name) Cast: (type-name) cast-expression


In several countries, a digital signature has a status somewhat like that of a traditional pen and paper signature, as in the 1999 EU digital signature directive and 2014 EU follow-on legislation. Generally, these provisions mean that anything digitally signed legally binds the signer of the document to the terms therein. For that reason, it is often thought best to use separate key pairs for encrypting and signing. Using the encryption key pair, a person can engage in an encrypted conversation (e.g., regarding a real estate transaction), but the encryption does not legally sign every message he or she sends.


As a result, students inherit all the attributes common to all persons. Additionally, students have unique attributes that other people do not have. Object-oriented languages model subset/superset relationships using inheritance.


In the 1960s, computer design was based on mounting electronic components (transistors, resistors, etc.) on circuit boards. Several boards formed a discrete logic element of the machine, known as a module. Overall machine cycle speed is strongly related to the signal path—the length of the wiring—requiring high-speed computers to make their modules as small as possible. This was at odds with the need to make the modules themselves more complex to increase functionality.


In the 1980s, backpropagation did not work well for deep FNNs and RNNs. Here the word "deep" refers to the number of layers through which the data is transformed. More precisely, deep learning systems have a substantial credit assignment path (CAP) depth. The CAP is the chain of transformations from input to output.


If a change to a property's use is made within a class, no permission for alterations is needed. If permission is granted, it may be unconditional, or under section 106 the local authority can attach conditions that the landowner must follow. Sections 171A to 196C contain the enforcement rules, which include allowing local authorities to charge people with offences for breach of planning rules, and to have a right of entry to property.


In modular arithmetic, Barrett reduction is a reduction algorithm introduced in 1986 by P.D. Barrett.A naive way of computing c = a mod n {\displaystyle c=a\,{\bmod {\,}}n\,} would be to use a fast division algorithm. Barrett reduction is an algorithm designed to optimize this operation assuming n {\displaystyle n} is constant, and a < n 2 {\displaystyle a


Alternatively, link signs may be figured as dimensions themselves, e.g. G = ( V , E , D ) {\displaystyle G=(V,E,D)} where D = { − 1 , 0 , 1 } {\displaystyle D=\{-1,0,1\}} and E = { ( u , v , d ) ; u , v ∈ V , d ∈ D } {\displaystyle E=\{(u,v,d);u,v\in V,d\in D\}} This approach has particular value when considering unweighted networks. This conception of dimensionality can be expanded should attributes in multiple dimensions need specification. In this instance, links are n-tuples e = ( u , v , d 1 … d n − 2 ) {\displaystyle e=(u,v,d_{1}\dots d_{n-2})} . Such an expanded formulation, in which links may exist within multiple dimensions, is uncommon but has been used in the study of multidimensional time-varying networks.


In the 1950s, it was found that there could be electrical coupling between the unencrypted side of a "RED" signal inside a secure communications facility, and either the conductor carrying the "BLACK" encrypted signal, or possibly the electrical ground(s) of the system. TEMPEST protective measures work against the situation when the frequency of the RED and BLACK signals are the same. The RED signal, at a low power level, may be intercepted directly, or there may be intermodulation between the RED and BLACK signals.


In number theory, a natural number is called k-almost prime if it has k prime factors. More formally, a number n is k-almost prime if and only if Ω(n) = k, where Ω(n) is the total number of primes in the prime factorization of n (can be also seen as the sum of all the primes' exponents): Ω ( n ) := ∑ a i if n = ∏ p i a i . {\displaystyle \Omega (n):=\sum a_{i}\qquad {\mbox{if}}\qquad n=\prod p_{i}^{a_{i}}.} A natural number is thus prime if and only if it is 1-almost prime, and semiprime if and only if it is 2-almost prime.


This gives rise to marked differences in the ways the modes are deployed and used in different parts of the world. Recently, there is a trend towards integrating the modes through intermodality and linking the modes ever more closely into production and distribution activities. At the same time; however, passenger and freight activity is becoming increasingly separated across most modes."


In mathematics, a function f is logarithmically convex or superconvex if log ∘ f {\displaystyle {\log }\circ f} , the composition of the logarithm with f, is itself a convex function.


In neuroimaging, the most common variants are voxel-based morphometry, deformation-based morphometry and surface-based morphometry of the brain.


In regard to the future of information policy, it should be flexible and changing to different circumstances as the ability to access, store, and share information grows. Galvin suggests that information policy might include setting a boundary to the uncertainty in this field. As information policy becomes a larger and more important topic, it will also become a greater subject to governmental regulation in regards to the future of technology as well. It will also include the studies of these subjects: information science, communications, library science, and technology studies.The information policies will earn more advantages for national and organizational, such as getting the best from development of Web 2.0 nationally and in organization, influence people for paying attention to the socio aspect and socio-technical system, for securing preservation of digital content, bringing out information product, also respecting all users and making thinking time respectable.


This is approximately equal to log 2  ( n ) − 1 {\displaystyle \log _{2}(n)-1} iterations. When the target element is not in the array, binary search makes ⌊ log 2  ( n ) ⌋ + 2 − 2 ⌊ log 2  ( n ) ⌋ + 1 / ( n + 1 ) {\displaystyle \lfloor \log _{2}(n)\rfloor +2-2^{\lfloor \log _{2}(n)\rfloor +1}/(n+1)} iterations on average, assuming that the range between and outside elements is equally likely to be searched.In the best case, where the target value is the middle element of the array, its position is returned after one iteration.In terms of iterations, no search algorithm that works only by comparing elements can exhibit better average and worst-case performance than binary search.


In mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality is always true in elementary algebra. For example, in elementary arithmetic, one has Therefore, one would say that multiplication distributes over addition. This basic property of numbers is part of the definition of most algebraic structures that have two operations called addition and multiplication, such as complex numbers, polynomials, matrices, rings, and fields. It is also encountered in Boolean algebra and mathematical logic, where each of the logical and (denoted ∧ {\displaystyle \,\land \,} ) and the logical or (denoted ∨ {\displaystyle \,\lor \,} ) distributes over the other.


Generalizations are in two directions. Firstly, to geometric questions about that morphism, for example the local Torelli theorem. Secondly, to other period mappings. A case that has been investigated deeply is for K3 surfaces (by Viktor S. Kulikov, Ilya Pyatetskii-Shapiro, Igor Shafarevich and Fedor Bogomolov) and hyperkähler manifolds (by Misha Verbitsky, Eyal Markman and Daniel Huybrechts).


395). On the other hand their RASP model P'0 equipped with an "index register" (indirect addressing) can compute all the "partial recursive sequential functions" (the mu recursive functions) (p.


Several languages like Tidore and Kuuk Thaayorre lack a general term meaning 'body'. On the basis of such data it has been argued that the highest level in the partonomy of body part terms would be the word for 'person'.Some other examples of proposed linguistic universals in semantics include the idea that all languages possess words with the meaning '(biological) mother' and 'you (second person singular pronoun)' as well as statistical tendencies of meanings of basic color terms in relation to the number of color terms used by a respective language. For example, if a languages possesses only two terms for describing color, their respective meanings will be 'black' and 'white' (or perhaps 'dark' and 'light').


In telecommunications, return loss is a measure in relative terms of the power of the signal reflected by a discontinuity in a transmission line or optical fiber. This discontinuity can be caused by a mismatch between the termination or load connected to the line and the characteristic impedance of the line. It is usually expressed as a ratio in decibels (dB); R L ( d B ) = 10 log 10  P i P r {\displaystyle RL(\mathrm {dB} )=10\log _{10}{P_{\mathrm {i} } \over P_{\mathrm {r} }}} where RL(dB) is the return loss in dB, Pi is the incident power and Pr is the reflected power.Return loss is related to both standing wave ratio (SWR) and reflection coefficient (Γ).


"The Roman Rite of the Catholic Church also mentions use of blessed salt. The 1962 Rituale Romanum includes salt as component in three rites: Baptism: Before the candidates enter the church or baptistry, salt is blessed with an exorcism, and a pinch can be put in the mouth of the candidates. However, in modern practice this can be skipped. Reconsecration of an altar: In one rite for the reconsecration of an altar which has been disturbed, salt is exorcized, blessed, and mixed with ashes, water and wine, the resulting mixture being used to make the mortar with which the altar is resealed. Blessing holy water: Salt is added to water in silence after a prayer in which God is asked to bless the salt, recalling the blessed salt "scattered over the water by the prophet Elisha" and invoking the protective powers of salt and water, that they may "drive away the power of evil".An additional rite provides for the blessing of salt for animals.Blessed salt is also used in prayer services of Pentecostal churches, such as the Apostolic Church Fullness of God's Throne in Brasil.


In mathematical logic, a Boolean-valued model is a generalization of the ordinary Tarskian notion of structure from model theory. In a Boolean-valued model, the truth values of propositions are not limited to "true" and "false", but instead take values in some fixed complete Boolean algebra. Boolean-valued models were introduced by Dana Scott, Robert M. Solovay, and Petr Vopěnka in the 1960s in order to help understand Paul Cohen's method of forcing. They are also related to Heyting algebra semantics in intuitionistic logic.


Much of the data stored and manipulated on computers, including text and images, can be represented as points in a high-dimensional space (see vector space model for the case of text). However, the essential algorithms for working with such data tend to become bogged down very quickly as dimension increases. It is therefore desirable to reduce the dimensionality of the data in a way that preserves its relevant structure. The Johnson–Lindenstrauss lemma is a classic result in this vein. Also, the lemma is tight up to a constant factor, i.e. there exists a set of points of size m that needs dimension Ω ( log  ( m ) ε 2 ) {\displaystyle \Omega \left({\frac {\log(m)}{\varepsilon ^{2}}}\right)} in order to preserve the distances between all pairs of points within a factor of ( 1 ± ε ) {\displaystyle (1\pm \varepsilon )} .


Nephology is the science of clouds, which is undertaken in the cloud physics branch of meteorology. There are two methods of naming clouds in their respective layers of the homosphere, Latin and common name. Genus types in the troposphere, the atmospheric layer closest to Earth's surface, have Latin names because of the universal adoption of Luke Howard's nomenclature that was formally proposed in 1802.


A particularly important application of Dirichlet processes is as a prior probability distribution in infinite mixture models. The Dirichlet process was formally introduced by Thomas Ferguson in 1973. It has since been applied in data mining and machine learning, among others for natural language processing, computer vision and bioinformatics.


In some languages, e.g., Rexx, no control expression is allowed and each alternative begins with a WHEN clause containing a boolean expression and a match occurs for the first case for which that expression evaluates to true.Each alternative begins with the particular value, or list of values (see below), that the control variable may match and which will cause the control to goto the corresponding sequence of statements. The value (or list/range of values) is usually separated from the corresponding statement sequence by a colon or by an implication arrow. In many languages, every case must also be preceded by a keyword such as case or when.


In non-synapsids, such as reptiles and crocodiles, teeth similar to canines may be termed "caniniform" ("canine-shaped") teeth.


In set theory, Cantor's paradox states that there is no set of all cardinalities. This is derived from the theorem that there is no greatest cardinal number. In informal terms, the paradox is that the collection of all possible "infinite sizes" is not only infinite, but so infinitely large that its own infinite size cannot be any of the infinite sizes in the collection.


In the case of two nested square roots, the following theorem completely solves the problem of denesting.If a and c are rational numbers and c is not the square of a rational number, there are two rational numbers x and y such that if and only if a 2 − c {\displaystyle a^{2}-c~} is the square of a rational number d. If the nested radical is real, x and y are the two numbers and a − d 2 , {\displaystyle ~{\frac {a-d}{2}}~,~} where d = a 2 − c {\displaystyle ~d={\sqrt {a^{2}-c}}~} is a rational number. In particular, if a and c are integers, then 2x and 2y are integers. This result includes denestings of the form as z may always be written z = ± z 2 , {\displaystyle z=\pm {\sqrt {z^{2}}},} and at least one of the terms must be positive (because the left-hand side of the equation is positive).


In mathematics, a matroid polytope, also called a matroid basis polytope (or basis matroid polytope) to distinguish it from other polytopes derived from a matroid, is a polytope constructed via the bases of a matroid. Given a matroid M {\displaystyle M} , the matroid polytope P M {\displaystyle P_{M}} is the convex hull of the indicator vectors of the bases of M {\displaystyle M} .


The expression for vector { X } {\displaystyle \{\ X\}} may be written as { X } = { X ′ } + { Δ X } {\displaystyle \{\ X\}=\{\ X'\}+\{\Delta \ X\}} The vector { X } {\displaystyle \{\ X\}} determination is related to minimization of the quadratic form Π {\displaystyle \ \Pi } by incremental vector { Δ X } {\displaystyle \{\Delta \ X\}} , i.e. ∂ Π ∂ Δ X k l = 0 {\displaystyle {\frac {\partial \Pi }{\partial \Delta X_{kl}}}=0} where l {\displaystyle \ l} - is the number of interior node of the area, k {\displaystyle \ k} - the number of co-ordinateAfter all transformations we may write the following two independent systems of linear algebraic equations  { Δ X 1 } = { B 1 } {\displaystyle \{\Delta X_{1}\}=\{\ B_{1}\}}  { Δ X 2 } = { B 2 } {\displaystyle \{\Delta X_{2}\}=\{\ B_{2}\}} where  {\displaystyle } - symmetrical matrix in the banded form similar to global stiffness matrix of FEM assemblage, { Δ X 1 } {\displaystyle \{\Delta \ X_{1}\}} and { Δ X 2 } {\displaystyle \{\Delta \ X_{2}\}} - incremental vectors of co-ordinates of all nodes at axes 1, 2, { B 1 } {\displaystyle \{\ B_{1}\}} and { B 2 } {\displaystyle \{\ B_{2}\}} - the right part vectors that are combined by co-ordinates of all nodes in axes 1, 2.The solution of both systems, keeping all boundary nodes conservative, obtains new interior node positions corresponding to a non-distorted mesh with pseudo-regular elements. For example, Fig. 2 presents the rectangular area covered by a triangular mesh. The initial auto mesh possesses some degenerative triangles (left mesh).


In number theory, Issai Schur showed in 1912 that for every nonconstant polynomial p(x) with integer coefficients, if S is the set of all nonzero values { p ( n ) ≠ 0: n ∈ N } {\displaystyle {\begin{Bmatrix}p(n)\neq 0:n\in \mathbb {N} \end{Bmatrix}}} , then the set of primes that divide some member of S is infinite.


(e.g. (a * b) * c = a * (b * c)). Many programming language manuals provide a table of operator precedence and associativity; see, for example, the table for C and C++.


In practical implementations, especially with high dimensional data (large p), the naive covariance method is rarely used because it is not efficient due to high computational and memory costs of explicitly determining the covariance matrix. The covariance-free approach avoids the np2 operations of explicitly calculating and storing the covariance matrix XTX, instead utilizing one of matrix-free methods, for example, based on the function evaluating the product XT(X r) at the cost of 2np operations.


Again, a measure of distance between random variables may relate to the extent of dependence between them, rather than to their individual values. Statistical distance measures are not typically metrics, and they need not be symmetric. Some types of distance measures, which generalize squared distance, are referred to as (statistical) divergences.


In telecommunication, a Berger code is a unidirectional error detecting code, named after its inventor, J. M. Berger. Berger codes can detect all unidirectional errors. Unidirectional errors are errors that only flip ones into zeroes or only zeroes into ones, such as in asymmetric channels. The check bits of Berger codes are computed by counting all the zeroes in the information word, and expressing that number in natural binary.


In practice, many commercial recommender systems are based on large datasets. As a result, the user-item matrix used for collaborative filtering could be extremely large and sparse, which brings about challenges in the performance of the recommendation. One typical problem caused by the data sparsity is the cold start problem. As collaborative filtering methods recommend items based on users' past preferences, new users will need to rate a sufficient number of items to enable the system to capture their preferences accurately and thus provides reliable recommendations.


The above property may then be stated as ∀ ( a ∈ A ) . ∃ !


Minimum excluded values of subclasses of the ordinal numbers are used in combinatorial game theory to assign nim-values to impartial games. According to the Sprague–Grundy theorem, the nim-value of a game position is the minimum excluded value of the class of values of the positions that can be reached in a single move from the given position.Minimum excluded values are also used in graph theory, in greedy coloring algorithms. These algorithms typically choose an ordering of the vertices of a graph and choose a numbering of the available vertex colors. They then consider the vertices in order, for each vertex choosing its color to be the minimum excluded value of the set of colors already assigned to its neighbors.


Also, an explicit matrix for Qkℓ is rarely computed; instead, auxiliary values are computed and A is updated in an efficient and numerically stable way. However, for reference, we may write the matrix as Q k ℓ =  . {\displaystyle Q_{k\ell }={\begin{bmatrix}1&&&&&&\\&\ddots &&&&0&\\&&c&\cdots &s&&\\&&\vdots &\ddots &\vdots &&\\&&-s&\cdots &c&&\\&0&&&&\ddots &\\&&&&&&1\end{bmatrix}}.}


In probability theory and statistics, the negative multinomial distribution is a generalization of the negative binomial distribution (NB(x0, p)) to more than two outcomes.As with the univariate negative binomial distribution, if the parameter x 0 {\displaystyle x_{0}} is a positive integer, the negative multinomial distribution has an urn model interpretation. Suppose we have an experiment that generates m+1≥2 possible outcomes, {X0,...,Xm}, each occurring with non-negative probabilities {p0,...,pm} respectively. If sampling proceeded until n observations were made, then {X0,...,Xm} would have been multinomially distributed. However, if the experiment is stopped once X0 reaches the predetermined value x0 (assuming x0 is a positive integer), then the distribution of the m-tuple {X1,...,Xm} is negative multinomial. These variables are not multinomially distributed because their sum X1+...+Xm is not fixed, being a draw from a negative binomial distribution.


Choosing large public key: Replace e {\displaystyle e} by e ′ {\displaystyle e'} , where e ′ = e + k ⋅ λ ( N ) {\displaystyle e'=e+k\cdot \lambda (N)} for some large of k {\displaystyle k} . When e ′ {\displaystyle e'} is large enough, i.e. e ′ > N 3 2 {\displaystyle e'>N^{\frac {3}{2}}} , then Wiener’s attack can not be applied regardless of how small d {\displaystyle d} is. Using the Chinese Remainder Theorem: Suppose one chooses d such that both d p = d mod ( p − 1 ) {\displaystyle d_{p}=d{\bmod {(}}p-1)} and d q = d mod ( q − 1 ) {\displaystyle d_{q}=d{\bmod {(}}q-1)} are small but d {\displaystyle d} itself is not, then a fast decryption of C {\displaystyle C} can be done as follows: 1.


In telecommunications and computer networks, a channel access method or multiple access method allows more than two terminals connected to the same transmission medium to transmit over it and to share its capacity. Examples of shared physical media are wireless networks, bus networks, ring networks and point-to-point links operating in half-duplex mode. A channel access method is based on multiplexing, that allows several data streams or signals to share the same communication channel or transmission medium.


He also describes the way that entities will be allowed in or out in the future. In describing this collective, Latour draws attention to the role of the spokesperson, who must be doubted but who must speak for otherwise mute things in order to ensure that the collective involves both "humans and non-humans". This is also an important aspect of Actor-network theory (ANT) that can be found in his main sociological works. The book includes a short summary at the end and a glossary of terms.


For example, it is possible to model and measure geometrically the distance between a particular pitch and a particular key, both represented as points in the spiral array space. To preserve pitch spelling, because musically A# ≠ Bb in their function and usage, the spiral array does not assume enharmonic equivalence, i.e. it does not fold into a torus. The spatial relationships between pitches, between chords, and between keys agree with those in other representations of tonal space.The model and its real-time algorithms have been implemented in the tonal visualization software MuSA.RT (Music on the Spiral Array . Real-Time) and a free app, MuSA_RT, both of which have been used in music education videos and in live performance.


In the early 19th century, Sophie Germain developed several novel approaches to prove Fermat's Last Theorem for all exponents. First, she defined a set of auxiliary primes θ {\displaystyle \theta } constructed from the prime exponent p {\displaystyle p} by the equation θ = 2 h p + 1 {\displaystyle \theta =2hp+1} , where h {\displaystyle h} is any integer not divisible by three. She showed that, if no integers raised to the p t h {\displaystyle p^{\mathrm {th} }} power were adjacent modulo θ {\displaystyle \theta } (the non-consecutivity condition), then θ {\displaystyle \theta } must divide the product x y z {\displaystyle xyz} . Her goal was to use mathematical induction to prove that, for any given p {\displaystyle p} , infinitely many auxiliary primes θ {\displaystyle \theta } satisfied the non-consecutivity condition and thus divided x y z {\displaystyle xyz} ; since the product x y z {\displaystyle xyz} can have at most a finite number of prime factors, such a proof would have established Fermat's Last Theorem.


In the field of data compression, Shannon–Fano coding, named after Claude Shannon and Robert Fano, is a name given to two different but related techniques for constructing a prefix code based on a set of symbols and their probabilities (estimated or measured). Shannon's method chooses a prefix code where a source symbol i {\displaystyle i} is given the codeword length l i = ⌈ − log 2  p i ⌉ {\displaystyle l_{i}=\lceil -\log _{2}p_{i}\rceil } . One common way of choosing the codewords uses the binary expansion of the cumulative probabilities.


In number theory, an odd integer n is called an Euler–Jacobi probable prime (or, more commonly, an Euler probable prime) to base a, if a and n are coprime, and a ( n − 1 ) / 2 ≡ ( a n ) ( mod n ) {\displaystyle a^{(n-1)/2}\equiv \left({\frac {a}{n}}\right){\pmod {n}}} where ( a n ) {\displaystyle \left({\frac {a}{n}}\right)} is the Jacobi symbol. If n is an odd composite integer that satisfies the above congruence, then n is called an Euler–Jacobi pseudoprime (or, more commonly, an Euler pseudoprime) to base a.


In predicate transformers semantics, expressions are restricted to terms of the logic (see above). However, this restriction seems too strong for most existing programming languages, where expressions may have side effects (call to a function having a side effect), may not terminate or abort (like division by zero). There are many proposals to extend weakest-preconditions or strongest-postconditions for imperative expression languages and in particular for monads. Among them, Hoare Type Theory combines Hoare logic for a Haskell-like language, separation logic and type theory. This system is currently implemented as a Coq library called Ynot. In this language, evaluation of expressions corresponds to computations of strongest-postconditions.


In mathematics, logic, philosophy, and formal systems, a primitive notion is a concept that is not defined in terms of previously-defined concepts. It is often motivated informally, usually by an appeal to intuition and everyday experience. In an axiomatic theory, relations between primitive notions are restricted by axioms. Some authors refer to the latter as "defining" primitive notions by one or more axioms, but this can be misleading.


In a true scientific process, no consensus does exist and no consensus can exist as the process is conducted scientifically in the pursuit of knowledge. If any party involved in the process stands to personally lose or gain from the result, the process will be flawed and unscientific. In a true scientific process, a theory is formed after a scientist - amateur or professional - has observed a phenomenon and has asked "why?"


For a binary channel the symbols (e.g. integers { 1 , … , 9 } {\displaystyle \{1,\ldots ,9\}} ) have to be mapped onto base 2. The binary erasure channel model however is not applicable because it erases only individual bits with some probability and not Sudoku symbols. If the symbols of the Sudoku are sent in packets the channel can be described as a packet erasure channel model.


Each processor also included a floating point unit from Weitek. For marketing purposes, each processor was called a "computational unit", and a card-cage populated with 16 was referred to as a "processor". This allowed favorable per-processor performance comparisons with other supercomputers of the era.


Phones (and often also phonemes) are commonly represented by using symbols of the International Phonetic Alphabet (IPA). For example, the English word spin consists of four phones, , ,  and  and so the word has the phonetic representation . The word pin has three phones.


In order to determine the actual impacts of search customization on end users, researchers at Northeastern University determined in a study with logged in users vs. a control group that 11.7% of results show differences due to personalization. The research showed that this result varies widely by search query and result ranking position.In the following example, the Portent Team performed a search query for 'JavaScript' (shown on the right) and then performed a search for 'Programming Textbooks' and 'Books on HTML' prior to searching for 'JavaScript, which changed the search results by bringing in three book listings that were not part of the original set of results. The study showed that of the various factors being tested, the two with the most measurable impact were whether the user was logged in with a Google account and the IP address of searching users. This same study also investigated the impact of the 11.7% personalization by utilizing Amazon Mechanical Turk (AMT) (a crowdsourcing Internet Marketplace and a part of Amazon Web Services) vs. a control group to determine the difference between the two. The results showed that the top ranked URLs are less likely to change based on personalization, and that the most personalization is taking place at lower ranks of the resulting pages.


The set of n {\displaystyle n} -tuples of real numbers ( X 1 , … X n ) {\displaystyle (X_{1},\dots X_{n})} for which F ( X 1 , … X n ) {\displaystyle F(X_{1},\dots X_{n})} is true is called a semialgebraic set, so the decision problem for the existential theory of the reals can equivalently be rephrased as testing whether a given semialgebraic set is nonempty.In determining the time complexity of algorithms for the decision problem for the existential theory of the reals, it is important to have a measure of the size of the input. The simplest measure of this type is the length of a sentence: that is, the number of symbols it contains. However, in order to achieve a more precise analysis of the behavior of algorithms for this problem, it is convenient to break down the input size into several variables, separating out the number of variables to be quantified, the number of polynomials within the sentence, and the degree of these polynomials.


Further, in On similar arcs, he commented on Ptolemy's Karpos (or Centiloquium); many scholars believe that ibn Yusuf was in fact the true author of that work. He also wrote a book on the astrolabe. He invented methods to solve tax problems that were later presented in Fibonacci's Liber Abaci. He was also quoted by mathematicians such as Thomas Bradwardine, Jordanus de Nemore and Luca Pacioli.


The cluster of nuclear profiles was calculated based on their similarity to each other using a k-means clustering method. To begin the process, three nuclear profiles were chosen at random as the ‘centers’ of the cluster. After the centers were chosen at random, every other nuclear profile is assigned to a cluster based on its distance from each center using a calculated distance value.


In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self-dual condition, Modular law a ≤ b implies a ∨ (x ∧ b) = (a ∨ x) ∧ bwhere x, a, b are arbitrary elements in the lattice, ≤ is the partial order, and ∨ and ∧ (called join and meet respectively) are the operations of the lattice. This phrasing emphasizes an interpretation in terms of projection onto the sublattice , a fact known as the diamond isomorphism theorem. An alternative but equivalent condition stated as an equation (see below) emphasizes that modular lattices form a variety in the sense of universal algebra. Modular lattices arise naturally in algebra and in many other areas of mathematics.


In telecommunication and information theory, the code rate (or information rate) of a forward error correction code is the proportion of the data-stream that is useful (non-redundant). That is, if the code rate is k / n {\displaystyle k/n} for every k bits of useful information, the coder generates a total of n bits of data, of which n − k {\displaystyle n-k} are redundant. If R is the gross bit rate or data signalling rate (inclusive of redundant error coding), the net bit rate (the useful bit rate exclusive of error correction codes) is ≤ R ⋅ k / n {\displaystyle \leq R\cdot k/n} .


, p + q ) {\displaystyle (\alpha ^{p}\smile \beta ^{q})(\sigma )=\alpha ^{p}(\sigma \circ \iota _{0,1,...p})\cdot \beta ^{q}(\sigma \circ \iota _{p,p+1,...,p+q})} where σ is a singular (p + q) -simplex and ι S , S ⊂ { 0 , 1 , . . .


In mathematical terms, a MOLP can be written as: min x P x s.t. a ≤ B x ≤ b , ℓ ≤ x ≤ u {\displaystyle \min _{x}Px\quad {\text{s.t. }}\quad a\leq Bx\leq b,\;\ell \leq x\leq u} where B {\displaystyle B} is an ( m × n ) {\displaystyle (m\times n)} matrix, P {\displaystyle P} is a ( q × n ) {\displaystyle (q\times n)} matrix, a {\displaystyle a} is an m {\displaystyle m} -dimensional vector with components in R ∪ { − ∞ } {\displaystyle \mathbb {R} \cup \{-\infty \}} , b {\displaystyle b} is an m {\displaystyle m} -dimensional vector with components in R ∪ { + ∞ } {\displaystyle \mathbb {R} \cup \{+\infty \}} , ℓ {\displaystyle \ell } is an n {\displaystyle n} -dimensional vector with components in R ∪ { − ∞ } {\displaystyle \mathbb {R} \cup \{-\infty \}} , u {\displaystyle u} is an n {\displaystyle n} -dimensional vector with components in R ∪ { + ∞ } {\displaystyle \mathbb {R} \cup \{+\infty \}}


In probability theory, the central limit theorem states that, under certain circumstances, the probability distribution of the scaled mean of a random sample converges to a normal distribution as the sample size increases to infinity. Under stronger assumptions, the Berry–Esseen theorem, or Berry–Esseen inequality, gives a more quantitative result, because it also specifies the rate at which this convergence takes place by giving a bound on the maximal error of approximation between the normal distribution and the true distribution of the scaled sample mean. The approximation is measured by the Kolmogorov–Smirnov distance. In the case of independent samples, the convergence rate is n−1/2, where n is the sample size, and the constant is estimated in terms of the third absolute normalized moment.


In one of the early works, Chang and Roberts proposed a uniform algorithm in which a processor with the highest ID is selected as the leader. Each processor sends its ID in a clockwise direction. A process receiving a message and compares it with its own. If it is bigger, it passes it through, otherwise it will discard the message. They show that this algorithm uses at most O ( n 2 ) {\displaystyle O(n^{2})} messages and O ( n log  n ) {\displaystyle O(n\log n)} in the average case.Hirschberg and Sinclair improved this algorithm with O ( n log  n ) {\displaystyle O(n\log n)} message complexity by introducing a 2 directional message passing scheme allowing the processors to send messages in both directions.


In terms of the singular value decomposition (SVD) of A {\displaystyle A} , A = W Σ V ∗ {\displaystyle A=W\Sigma V^{*}} , one haswhere U {\displaystyle U} , V {\displaystyle V} , and W {\displaystyle W} are unitary matrices (called orthogonal matrices if the field is the reals R {\displaystyle \mathbb {R} } ). This confirms that P {\displaystyle P} is positive-definite and U {\displaystyle U} is unitary. Thus, the existence of the SVD is equivalent to the existence of polar decomposition. One can also decompose A {\displaystyle A} in the formHere U {\displaystyle U} is the same as before and P ′ {\displaystyle P'} is given byThis is known as the left polar decomposition, whereas the previous decomposition is known as the right polar decomposition. Left polar decomposition is also known as reverse polar decomposition. The polar decomposition of a square invertible real matrix A {\displaystyle A} is of the form where | A | = ( A A T ) 1 2 {\displaystyle |A|=\left(AA^{\textsf {T}}\right)^{\frac {1}{2}}} is a positive-definite matrix and R = | A | − 1 A {\displaystyle R=|A|^{-1}A} is an orthogonal matrix.


Nevertheless, numerous early counter-mapping projects successfully utilised manual techniques, and many still use them. For instance, in recent years, the use of simple sketch mapping approaches has been revitalised, whereby maps are made on the ground, using natural materials. Similarly, the use of scale model constructions and felt boards, as means of representing cartographic claims of different groups, have become increasingly popular. Consequently, Wood et al. assert that counter-mappers can "make gateau out of technological crumbs".


In number theory, Kaprekar's routine is an iterative algorithm named after its inventor, Indian mathematician D. R. Kaprekar. Each iteration starts with a number, sorts the digits into descending and ascending order, and calculates the difference between the two new numbers. As an example, starting with the number 8991 in base 10: 9981 – 1899 = 8082 8820 – 0288 = 8532 8532 – 2358 = 6174 7641 – 1467 = 61746174, known as Kaprekar's constant, is a fixed point of this algorithm. Any four-digit number (in base 10) with at least two distinct digits will reach 6174 within seven iterations. The algorithm runs on any natural number in any given number base.


x}}\right)} uniformly in θ {\displaystyle \theta } .


In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in v ^ {\displaystyle {\hat {\mathbf {v} }}} (pronounced "v-hat").


It also includes reciprocals of some numbers of more than six places, such as 323 (2 1 4 8 3 0 7 in sexagesimal), whose reciprocal has 17 sexagesimal digits. Noting the difficulty of both calculating these numbers and sorting them, Donald Knuth in 1972 hailed Inaqibıt-Anu as "the first man in history to solve a computational problem that takes longer than one second of time on a modern electronic computer!" (Two tables are also known giving approximations of reciprocals of non-regular numbers, one of which gives reciprocals for all the numbers from 56 to 80.


In the real number system, square numbers are non-negative. A non-negative integer is a square number when its square root is again an integer. For example, 9 = 3 , {\displaystyle {\sqrt {9}}=3,} so 9 is a square number.


In number theory, the Hasse norm theorem states that if L/K is a cyclic extension of number fields, then if a nonzero element of K is a local norm everywhere, then it is a global norm. Here to be a global norm means to be an element k of K such that there is an element l of L with N L / K ( l ) = k {\displaystyle \mathbf {N} _{L/K}(l)=k} ; in other words k is a relative norm of some element of the extension field L. To be a local norm means that for some prime p of K and some prime P of L lying over K, then k is a norm from LP; here the "prime" p can be an archimedean valuation, and the theorem is a statement about completions in all valuations, archimedean and non-archimedean. The theorem is no longer true in general if the extension is abelian but not cyclic. Hasse gave the counterexample that 3 is a local norm everywhere for the extension Q ( − 3 , 13 ) / Q {\displaystyle {\mathbf {Q} }({\sqrt {-3}},{\sqrt {13}})/{\mathbf {Q} }} but is not a global norm.


In mathematics, nonlinear programming (NLP) is the process of solving an optimization problem where some of the constraints or the objective function are nonlinear. An optimization problem is one of calculation of the extrema (maxima, minima or stationary points) of an objective function over a set of unknown real variables and conditional to the satisfaction of a system of equalities and inequalities, collectively termed constraints. It is the sub-field of mathematical optimization that deals with problems that are not linear.


In the Agile development process, unit testing is done per user story and comes in the later half of the sprint after requirements gathering and development are complete. Typically, the developers or other members from the development team, such as consultants, will write step-by-step 'test scripts' for the developers to execute in the tool. Test scripts are generally written to prove the effective and technical operation of specific developed features in the tool, as opposed to full fledged business processes that would be interfaced by the end user, which is typically done during user acceptance testing. If the test-script can be fully executed from start to finish without incident, the unit test is considered to have "passed", otherwise errors are noted and the user story is moved back to development in an 'in-progress' state. User stories that successfully pass unit tests are moved on to the final steps of the sprint - Code review, peer review, and then lastly a 'show-back' session demonstrating the developed tool to stakeholders.


In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: 1 − 2 is not a natural number, although both 1 and 2 are. Similarly, a subset is said to be closed under a collection of operations if it is closed under each of the operations individually.


In quicksort, one of the critical operations is choosing the pivot: the element around which the list is partitioned. The simplest pivot selection algorithm is to take the first or the last element of the list as the pivot, causing poor behavior for the case of sorted or nearly sorted input. Niklaus Wirth's variant uses the middle element to prevent these occurrences, degenerating to O(n2) for contrived sequences. The median-of-3 pivot selection algorithm takes the median of the first, middle, and last elements of the list; however, even though this performs well on many real-world inputs, it is still possible to contrive a median-of-3 killer list that will cause dramatic slowdown of a quicksort based on this pivot selection technique.


In mathematical optimization, the revised simplex method is a variant of George Dantzig's simplex method for linear programming. The revised simplex method is mathematically equivalent to the standard simplex method but differs in implementation. Instead of maintaining a tableau which explicitly represents the constraints adjusted to a set of basic variables, it maintains a representation of a basis of the matrix representing the constraints. The matrix-oriented approach allows for greater computational efficiency by enabling sparse matrix operations.


In the case of a binary repetition code, there exist two code words - all ones and all zeros - which have a length of n {\displaystyle n} . Therefore, the minimum Hamming distance of the code equals its length n {\displaystyle n} . This gives the repetition code an error correcting capacity of n − 1 2 {\displaystyle {\tfrac {n-1}{2}}} (i.e. it will correct up to n − 1 2 {\displaystyle {\tfrac {n-1}{2}}} errors in any code word). If the length of a binary repetition code is odd, then it's a perfect code. The binary repetition code of length n is equivalent to the (n,1)-Hamming code.


The addition of an instruction pipeline changes this. In such machines the CPU will "look ahead" and begin fetching succeeding instructions while the current instruction is still being processed. In this assembly line fashion any one instruction still requires as long to complete, but as soon as it finishes executing, the next instruction is right behind it, with most of the steps required for its execution already completed.


In the Communications Decency Act of 1996, section 230 states that "No provider or user of an interactive computer service shall be treated as the publisher or speaker of any information provided by another information content provider". Section 230 protects the media from liabilities or being sued of third-party content, such as illegal activity from a user. However, this approach reduces a company's incentive to remove harmful content or misinformation. This loophole has allowed social media companies to maximize profits through pushing radical content without legal risks.


Since this happens O(p) times, the total time complexity is O(p3). A more efficient multiplication algorithm is the Schönhage–Strassen algorithm, which is based on the Fast Fourier transform. It only requires O(p log p log log p) time to square a p-bit number.


In metal typesetting some fonts have default increased or decreased leading. To achieve this, a smaller font face is cast on the body of a larger font or vice versa. Such fonts are usually called "bastard" fonts or types. In the notation they are usually written with the face and the body size separated by a slash, like 10/12 that is a 10-point font face on a 12-point body, or 12/10, a 12-point font face on a 10-point body.


Plans that include nationwide long distance and/or nationwide roaming at no additional charge over "local" outgoing calls are popular. Mobile networks in Europe, Asia (except Hong Kong, Macau (Macao) and Singapore), Australia, and Argentina only charge their subscribers for outgoing calls. Incoming calls are free to the mobile subscriber with the exception of receiving a call while the subscriber is roaming as described below.


The current cellular location of the phone (i.e., which BTS it is at) is entered into the VLR record and will be used during a process called paging when the GSM network wishes to locate the mobile phone. Every SIM card contains a secret key, called the Ki, which is used to provide authentication and encryption services. This is useful to prevent theft of service, and also to prevent "over the air" snooping of a user's activity.


Intuitively, when two programs of equal size are mated intimately, the complexity is not additive, but multiplicative, because the connections between the program halves multiply out of control. Another lesson from the Z80 days, which was corrected on the 80386 compiler, was to write as much of the code as possible into Pascal itself, even the support library. Having the 80386 support code all written in Pascal has made it so modular and portable that most of it was moved out of the operating system specific area and into the "common code" library section, a section reserved for code that never changes for each machine and operating system.


In modern linguistic theory, tense is understood as a category that expresses (grammaticalizes) time reference; namely one which, using grammatical means, places a state or action in time. Nonetheless, in many descriptions of languages, particularly in traditional European grammar, the term "tense" is applied to verb forms or constructions that express not merely position in time, but also additional properties of the state or action – particularly aspectual or modal properties. The category of aspect expresses how a state or action relates to time – whether it is seen as a complete event, an ongoing or repeated situation, etc. Many languages make a distinction between perfective aspect (denoting complete events) and imperfective aspect (denoting ongoing or repeated situations); some also have other aspects, such as a perfect aspect, denoting a state following a prior event. Some of the traditional "tenses" express time reference together with aspectual information.


primes including 2 are the only primes that exist. If either pn# + 1 or pn# − 1 is a primorial prime, it means that there are larger primes than the nth prime (if neither is a prime, that also proves the infinitude of primes, but less directly; each of these two numbers has a remainder of either p − 1 or 1 when divided by any of the first n primes, and hence all its prime factors are larger than pn).


In the context of quantum computing, the quantum walk search is a quantum algorithm for finding a marked node in a graph.The concept of a quantum walk is inspired by classical random walks, in which a walker moves randomly through a graph or lattice. In a classical random walk, the position of the walker can be described using a probability distribution over the different nodes of the graph. In a quantum walk, on the other hand, the walker is represented by a quantum state, which can be in a superposition of several locations simultaneously.Search algorithms based on quantum walks have the potential to find applications in various fields, including optimization, machine learning, cryptography, and network analysis. The efficiency and probability of success of a quantum walk search depend heavily on the structure of the search space. In general, quantum walk search algorithms offer an asymptotic quadratic speedup similar to that of Grover's algorithm.One of the first works on the application of quantum walk to search problems was proposed by Neil Shenvi, Julia Kempe, and K. Birgitta Whaley.


In the case of private persons, calls and conversations may be recorded by any active participant. There is no requirement to make other parties aware of the recording, but the use of recordings, depending on their content, may be subject to various laws, such as data protection (privacy) legislation, libel laws, laws governing trade and national secrets, and any agreements, such as non-disclosure agreements.Recording of calls by a company or an employer is subject to data protection legislation and, as a general rule, requires informing the participants prior to recording.


In the Unicode Standard v. 5.1 (4 April 2008), 152 medieval and classical glyphs were given specific locations outside of the Private Use Area. Specifically, they are located in the charts "Combining Diacritical Marks Supplement" (26 characters), "Latin Extended Additional" (10 characters), "Supplemental Punctuation" (15 characters), "Ancient Symbols" (12 characters) and especially "Latin Extended-D" (89 characters). These consist in both precomposed characters and modifiers for other characters, called combining diacritical marks (such as writing in LaTeX or using overstrike in MS Word). Characters are "the smallest components of written language that have semantic value" but glyphs are "the shapes that characters can have when they are rendered or displayed".


In personal computers, the IGP (integrated graphics processors) are mostly manufactured by Intel and AMD and are integrated onto their CPUs. They are commonly known as: Intel HD and Iris Graphics - also called HD series and Iris series AMD Accelerated Processing Unit (APU) - also formerly known as: fusion


In statistical hypothesis testing, there are various notions of so-called type III errors (or errors of the third kind), and sometimes type IV errors or higher, by analogy with the type I and type II errors of Jerzy Neyman and Egon Pearson. Fundamentally, type III errors occur when researchers provide the right answer to the wrong question, i.e. when the correct hypothesis is rejected but for the wrong reason. Since the paired notions of type I errors (or "false positives") and type II errors (or "false negatives") that were introduced by Neyman and Pearson are now widely used, their choice of terminology ("errors of the first kind" and "errors of the second kind"), has led others to suppose that certain sorts of mistakes that they have identified might be an "error of the third kind", "fourth kind", etc.None of these proposed categories have been widely accepted. The following is a brief account of some of these proposals.


In number theory, reversing the digits of a number n sometimes produces another number m that is divisible by n. This happens trivially when n is a palindromic number; the nontrivial reverse divisors are 1089, 2178, 10989, 21978, 109989, 219978, 1099989, 2199978, ... (sequence A008919 in the OEIS).For instance, 1089 × 9 = 9801, the reversal of 1089, and 2178 × 4 = 8712, the reversal of 2178. The multiples produced by reversing these numbers, such as 9801 or 8712, are sometimes called palintiples.


In statistics, an optimality criterion provides a measure of the fit of the data to a given hypothesis, to aid in model selection. A model is designated as the "best" of the candidate models if it gives the best value of an objective function measuring the degree of satisfaction of the criterion used to evaluate the alternative hypotheses. The term has been used to identify the different criteria that are used to evaluate a phylogenetic tree. For example, in order to determine the best topology between two phylogenetic trees using the maximum likelihood optimality criterion, one would calculate the maximum likelihood score of each tree and choose the one that had the better score.


In the area of communications research and informetrics, the concept of self-organizing systems appears in mid-1990s research related to scientific communications. Scientometrics and bibliometrics are areas of research in which discrete data are available, as are several other areas of social communications research such as sociolinguistics. Social complexity is also a concept used in semiotics.


In specifying disk drive capacities, manufacturers have always used conventional decimal SI prefixes representing powers of 10. Storage in a rotating disk drive is organized in platters and tracks whose sizes and counts are determined by mechanical engineering constraints so that the capacity of a disk drive has hardly ever been a simple multiple of a power of 2. For example, the first commercially sold disk drive, the IBM 350 (1956), had 50 physical disk platters containing a total of 50000 sectors of 100 characters each, for a total quoted capacity of 5 million characters.Moreover, since the 1960s, many disk drives used IBM's disk format, where each track was divided into blocks of user-specified size; and the block sizes were recorded on the disk, subtracting from the usable capacity. For example, the|IBM 3336]] disk pack was quoted to have a 200-megabyte capacity, achieved only with a single 13030-byte block in each of its 808 x 19 tracks.


Let the interest r follow a Markov process with probability transition function Q ( r , d μ r ) {\displaystyle Q(r,d\mu _{r})} where d μ r {\displaystyle d\mu _{r}} denotes the probability measure governing the distribution of interest rate next period if current interest rate is r {\displaystyle r} . In this model the consumer decides their current period consumption after the current period interest rate is announced. Rather than simply choosing a single sequence { c t } {\displaystyle \{{\color {OliveGreen}c_{t}}\}} , the consumer now must choose a sequence { c t } {\displaystyle \{{\color {OliveGreen}c_{t}}\}} for each possible realization of a { r t } {\displaystyle \{r_{t}\}} in such a way that their lifetime expected utility is maximized: max { c t } t = 0 ∞ E ( ∑ t = 0 ∞ β t u ( c t ) ) .


In order to define A {\displaystyle A} , we define an objective function describing classification accuracy in the transformed space and try to determine A ∗ {\displaystyle A^{*}} such that this objective function is maximized. A ∗ = argmax A f ( A ) {\displaystyle A^{*}={\mbox{argmax}}_{A}f(A)}


In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is acyclic. A polytree is an example of an oriented graph. The term polytree was coined in 1987 by Rebane and Pearl.


In medicine, a sequence is a series of ordered consequences due to a single cause.It differs from a syndrome in that seriality is more predictable: if A causes B, and B causes C, and C causes D, then D would not be seen if C is not seen. However, in less formal contexts, the term "syndrome" is sometimes used instead of sequence. Examples include: oligohydramnios sequence (also known as Potter sequence) Pierre Robin sequence Poland sequence == References ==


This is a general advantage of the axiomatic approach in mathematics. The axiomatic approach to Kolmogorov complexity was further developed in the book (Burgin 2005) and applied to software metrics (Burgin and Debnath, 2003; Debnath and Burgin, 2003). In information theory, information fluctuation complexity is the fluctuation of information about information entropy.


It is very similar to the z-score but with the difference that t-statistic is used when the sample size is small or the population standard deviation is unknown. For example, the t-statistic is used in estimating the population mean from a sampling distribution of sample means if the population standard deviation is unknown. It is also used along with p-value when running hypothesis tests where the p-value tells us what the odds are of the results to have happened.


In mathematics, a submodular set function (also known as a submodular function) is a set function that, informally, describes the relationship between a set of inputs and an output, where adding more of one input has a decreasing additional benefit (diminishing returns). The natural diminishing returns property which makes them suitable for many applications, including approximation algorithms, game theory (as functions modeling user preferences) and electrical networks. Recently, submodular functions have also found immense utility in several real world problems in machine learning and artificial intelligence, including automatic summarization, multi-document summarization, feature selection, active learning, sensor placement, image collection summarization and many other domains.


In the event of accumulation in the long-term, the capacity at which the bottlenecked machine is running could be so slow that the accumulated resources that are in the queue need to be stored. The cost of storing resources is significant as it takes resources to transport the materials back and forth as well as requiring space, another potential cost.


Programming languages may, however, share the syntax with markup languages if a computational semantics is defined. XSLT, for example, is a Turing complete language entirely using XML syntax. Moreover, LaTeX, which is mostly used for structuring documents, also contains a Turing complete subset.


In telecommunications, the term system integrity has the following meanings: That condition of a system wherein its mandated operational and technical parameters are within the prescribed limits. The quality of an AIS when it performs its intended function in an unimpaired manner, free from deliberate or inadvertent unauthorized manipulation of the system. The state that exists when there is complete assurance that under all conditions an IT system is based on the logical correctness and reliability of the operating system, the logical completeness of the hardware and software that implement the protection mechanisms, and data integrity.


Corners are detected through the simplest implementation which literally extracts a ring of 16 pixels and compares the intensity values with an appropriate threshold. For candidate p, each location on the circle x ∈ {1, 2, 3, ..., 16} can be denoted by p→x. The state of each pixel, Sp→x must be in one of the following three states: d, Ip→x ≤ Ip - t (darker) s, Ip - t ≤ Ip→x ≤ Ip + t (similar) b, Ip→x≥ Ip + t (brighter)Then choosing an x (same for all p) partitions P (the set of all pixels of all training images) into 3 different subsets, Pd, Ps, Pb where: Pd = {p ∈ P: Sp→x = d } Ps = {p ∈ P: Sp→x = s } Pb = {p ∈ P: Sp→x = b }Secondly, a decision tree algorithm, the ID3 algorithm is applied to the 16 locations in order to achieve the maximum information gain.


In recent years, it has become a popular subject of discourse among test-takers on various social media networks. Many of them poke fun at passages or questions in the PSAT that they find strange or amusing. The level of discussion is so significant that in 2013, the hashtag #PSAT reached trending status on Twitter near its administration date. This is despite the fact that since 2012, test participants have been required to copy and sign a statement agreeing to the test regulations, which include not discussing the test. Previously, that statement had to be written in cursive, a requirement that had drawn ire from both students and teachers, as many students found writing the statement in cursive to be difficult. However, in 2015, the requirement to write the statement in cursive was removed.


In the case that X and Y are both finite-dimensional (i.e. linearly isomorphic to Rm and Rn for some natural numbers m and n) then writing out equation (L) in matrix form shows that λ is the usual Lagrange multiplier vector; in the case n = 1, λ is the usual Lagrange multiplier, a real number.


In the below table, Columns I, II, III, IV, and V show the code; the Let. and Fig. columns show the letters and numbers for the Continental and UK versions; and the sort keys present the table in the order: alphabetical, Gray and UK Baudot developed his first multiplexed telegraph in 1872 and patented it in 1874. In 1876, he changed from a six-bit code to a five-bit code, as suggested by Carl Friedrich Gauss and Wilhelm Weber in 1834, with equal on and off intervals, which allowed for transmission of the Roman alphabet, and included punctuation and control signals.


So, given an integral domain R, it is often very useful to know that R has a Euclidean function: in particular, this implies that R is a PID. However, if there is no "obvious" Euclidean function, then determining whether R is a PID is generally a much easier problem than determining whether it is a Euclidean domain. Euclidean domains appear in the following chain of class inclusions: rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields


For tossing two coins, the sample space is { H H , H T , T H , T T } {\displaystyle \{HH,HT,TH,TT\}} , where the outcome is H H {\displaystyle HH} if both coins are heads, H T {\displaystyle HT} if the first coin is heads and the second is tails, T H {\displaystyle TH} if the first coin is tails and the second is heads, and T T {\displaystyle TT} if both coins are tails. The event that at least one of the coins is heads is given by E = { H H , H T , T H } {\displaystyle E=\{HH,HT,TH\}} . For tossing a single six-sided die one time, where the result of interest is the number of pips facing up, the sample space is { 1 , 2 , 3 , 4 , 5 , 6 } {\displaystyle \{1,2,3,4,5,6\}} .A well-defined, non-empty sample space S {\displaystyle S} is one of three components in a probabilistic model (a probability space).


In particular, support vector machines find a hyperplane that separates the feature space into two classes with the maximum margin. If the problem is not originally linearly separable, the kernel trick can be used to turn it into a linearly separable one, by increasing the number of dimensions. Thus a general hypersurface in a small dimension space is turned into a hyperplane in a space with much larger dimensions. Neural networks try to learn the decision boundary which minimizes the empirical error, while support vector machines try to learn the decision boundary which maximizes the empirical margin between the decision boundary and data points.


{\displaystyle K/p.} The ring K  / ( p ) {\displaystyle K/(p)} is a field if and only if p is an irreducible polynomial. In fact, if p is irreducible, every nonzero polynomial q of lower degree is coprime with p, and Bézout's identity allows computing r and s such that sp + qr = 1; so, r is the multiplicative inverse of q modulo p. Conversely, if p is reducible, then there exist polynomials a, b of degrees lower than deg(p) such that ab = p ; so a, b are nonzero zero divisors modulo p, and cannot be invertible.


This is due, in part, because two ISPs may be connected through multiple connections. In choosing the single router-level path, it is common practice for each ISP to employ hot-potato routing: sending traffic along the path that minimizes the distance through the ISP's own network—even if that path lengthens the total distance to the destination. For example, consider two ISPs, A and B. Each has a presence in New York, connected by a fast link with latency 5 ms—and each has a presence in London connected by a 5 ms link.


In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named after the symbol he used to denote them, the Hebrew letter aleph ( ℵ {\displaystyle \,\aleph \,} ).The cardinality of the natural numbers is ℵ 0 {\displaystyle \,\aleph _{0}\,} (read aleph-nought or aleph-zero; the term aleph-null is also sometimes used), the next larger cardinality of a well-ordered set is aleph-one ℵ 1 , {\displaystyle \,\aleph _{1}\;,} then ℵ 2 {\displaystyle \,\aleph _{2}\,} and so on. Continuing in this manner, it is possible to define a cardinal number ℵ α {\displaystyle \,\aleph _{\alpha }\,} for every ordinal number α , {\displaystyle \,\alpha \;,} as described below. The concept and notation are due to Georg Cantor, who defined the notion of cardinality and realized that infinite sets can have different cardinalities. The aleph numbers differ from the infinity ( ∞ {\displaystyle \,\infty \,} ) commonly found in algebra and calculus, in that the alephs measure the sizes of sets, while infinity is commonly defined either as an extreme limit of the real number line (applied to a function or sequence that "diverges to infinity" or "increases without bound"), or as an extreme point of the extended real number line.


In practice, the most common form of pre-emption right is the right of existing shareholders to acquire new shares issued by a company in a rights issue, usually a public offering. In this context, the pre-emptive right is also called subscription right or subscription privilege. It is the right but not the obligation of existing shareholders to buy the new shares before they are offered to the public. In that way, existing shareholders can maintain their proportional ownership of the company and thus prevent stock dilution.


However, only two of these symbols were used for letters, making it largely binary. The third symbol only appeared in control characters.


In the Java programming language, the constant interface pattern describes the use of an interface solely to define constants, and having classes implement that interface in order to achieve convenient syntactic access to those constants. However, since constants are very often merely an implementation detail, and the interfaces implemented by a class are part of its exported API, this practice amounts to putting implementations details into the API, which was considered inappropriate by, e.g., Java designer Joshua Bloch. In general, collecting system constants into classes independent of behaviour might create a poor object-oriented design because it is often a sign of low cohesion.


Similar lists exist for several other types of products. They vary and often deviate significantly from any geometric series in order to accommodate traditional sizes when feasible. Adjacent package sizes in these lists differ typically by factors 2⁄3 or 3⁄4, in some cases even 1⁄2, 4⁄5, or some other ratio of two small integers.


In the Japanese kantenji braille, the standard 8-dot Braille patterns 257, 1257, 2457, and 12457 are the patterns related to Braille pattern dots-134, since the two additional dots of kantenji patterns 0134, 1347, and 01347 are placed above the base 6-dot cell, instead of below, as in standard 8-dot braille.


Myanmar North Korea – The US Bureau of Diplomatic Security advises visitors that they have "no right to privacy in North Korea and should assume your communications are monitored" which excludes the possibility of satellite phone technology. Russia – in 2012, new regulations governing the use of satellite phones inside Russia or its territories were developed with the stated aim of fighting terrorism by enabling the Russian government to intercept calls. These regulations allow non-Russian visitors to register their SIM cards for use within Russian territory for up to six months.


However, this is not the case in a strong algorithm. In the past few years a number of research articles have addressed the development of strong multi-time-step algorithms.


For instance, the use of object-oriented programming often results in a greater number of "smaller" calls, which can be accommodated by increasing the windows from eight to sixteen for instance. This was the approach used in the SPARC, which has included more register windows with newer generations of the architecture. The end result is fewer slow register window spill and fill operations because the register windows overflow less often.


There are times when rumors are created with malicious intent, but shared by unknowing users. With the large audiences that can be reached and the experts on various subjects on social media, some believe social media could also be the key to correcting misinformation.Agent-based models and other computational models have been used by researchers to explain how false beliefs spread through networks. Epistemic network analysis is one example of a computational method for evaluating connections in data shared in a social media network or similar network. In The Misinformation Age: How False Beliefs Spread, a trade book by philosopher Cailin O'Connor and physicist James Owen Weatherall, the authors used a combination of case studies and agent-based models to show how false beliefs spread on social media and scientific networks. This book analyses the social nature of scientific research; the nature of information flow between scientists, propagandists, and politicians; and the spread of false beliefs among the general population.


Some examples: All Greeks are human and all humans are mortal; therefore, all Greeks are mortal. : Valid argument; if the premises are true the conclusion must be true. Some Greeks are logicians and some logicians are tiresome; therefore, some Greeks are tiresome.


If the observations are correlated, the expression S = ∑ k ∑ j r k W k j r j {\textstyle S=\sum _{k}\sum _{j}r_{k}W_{kj}r_{j}\,} applies. In this case the weight matrix should ideally be equal to the inverse of the variance-covariance matrix of the observations). The normal equations are then: This method is used in iteratively reweighted least squares.


The rules above are to be obeyed for large-scale maps. If the map being drawn is a small-scale map (less than 1:500 000 according to Imhof), rules may be relaxed in order to obtain a more suggestive representation.


In mathematics, specifically statistics and information geometry, a Bregman divergence or Bregman distance is a measure of difference between two points, defined in terms of a strictly convex function; they form an important class of divergences. When the points are interpreted as probability distributions – notably as either values of the parameter of a parametric model or as a data set of observed values – the resulting distance is a statistical distance. The most basic Bregman divergence is the squared Euclidean distance. Bregman divergences are similar to metrics, but satisfy neither the triangle inequality (ever) nor symmetry (in general).


In statistical mechanics, the mean squared displacement (MSD, also mean square displacement, average squared displacement, or mean square fluctuation) is a measure of the deviation of the position of a particle with respect to a reference position over time. It is the most common measure of the spatial extent of random motion, and can be thought of as measuring the portion of the system "explored" by the random walker. In the realm of biophysics and environmental engineering, the Mean Squared Displacement is measured over time to determine if a particle is spreading slowly due to diffusion, or if an advective force is also contributing.


The modulo operation is the operation that produces such a remainder when given a dividend and divisor. Alternatively, a remainder is also what is left after subtracting one number from another, although this is more precisely called the difference. This usage can be found in some elementary textbooks; colloquially it is replaced by the expression "the rest" as in "Give me two dollars back and keep the rest." However, the term "remainder" is still used in this sense when a function is approximated by a series expansion, where the error expression ("the rest") is referred to as the remainder term.


In matrix form, Oja's rule can be written d w ( t ) d t = w ( t ) Q − d i a g  w ( t ) {\displaystyle \,{\frac {{\text{d}}w(t)}{{\text{d}}t}}~=~w(t)Q-\mathrm {diag} w(t)} ,and the Gram-Schmidt algorithm is Δ w ( t ) = − l o w e r  w ( t ) {\displaystyle \,\Delta w(t)~=~-\mathrm {lower} w(t)} ,where w(t) is any matrix, in this case representing synaptic weights, Q = η x xT is the autocorrelation matrix, simply the outer product of inputs, diag is the function that diagonalizes a matrix, and lower is the function that sets all matrix elements on or above the diagonal equal to 0. We can combine these equations to get our original rule in matrix form, Δ w ( t ) = η ( t ) ( y ( t ) x ( t ) T − L T  w ( t ) ) {\displaystyle \,\Delta w(t)~=~\eta (t)\left(\mathbf {y} (t)\mathbf {x} (t)^{\mathrm {T} }-\mathrm {LT} w(t)\right)} ,where the function LT sets all matrix elements above the diagonal equal to 0, and note that our output y(t) = w(t) x(t) is a linear neuron.


The let expression is a conjunction within an existential quantifier. ( ∃ x E ∧ F ) ⟺ let  x: E in  F {\displaystyle (\exists xE\land F)\iff \operatorname {let} x:E\operatorname {in} F} where E and F are of type Boolean. The let expression allows the substitution to be applied to another expression.


In mathematics, a permutation group G acting on a non-empty finite set X is called primitive if G acts transitively on X and the only partitions the G-action preserves are the trivial partitions into either a single set or into |X| singleton sets. Otherwise, if G is transitive and G does preserve a nontrivial partition, G is called imprimitive. While primitive permutation groups are transitive, not all transitive permutation groups are primitive.


Hence cost is the metric used in the standard modeling paradigm applied to economic processes. Costs (pl.) are often further described based on their timing or their applicability.


In salt bath nitriding the nitrogen donating medium is a nitrogen-containing salt such as cyanide salt. The salts used also donate carbon to the workpiece surface making salt bath a nitrocarburizing process. The temperature used is typical of all nitrocarburizing processes: 550 to 570 °C. The advantages of salt nitriding is that it achieves higher diffusion in the same period of time compared to any other method.


This is also true for several other infinite sets, such as any n-dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} (see space filling curve). That is, The smallest infinite cardinal number is ℵ 0 {\displaystyle \aleph _{0}} (aleph-null). The second smallest is ℵ 1 {\displaystyle \aleph _{1}} (aleph-one). The continuum hypothesis, which asserts that there are no sets whose cardinality is strictly between ℵ 0 {\displaystyle \aleph _{0}} and c {\displaystyle {\mathfrak {c}}} , means that c = ℵ 1 {\displaystyle {\mathfrak {c}}=\aleph _{1}} . The truth or falsity of this hypothesis is undecidable and cannot be proven within the widely used Zermelo–Fraenkel set theory with axiom of choice (ZFC).


In the Japanese kantenji braille, the standard 8-dot Braille patterns 28, 128, 248, and 1248 are the patterns related to Braille pattern dots-16, since the two additional dots of kantenji patterns 016, 167, and 0167 are placed above the base 6-dot cell, instead of below, as in standard 8-dot braille.


In order to understand the performance paradox, it is helpful to first have a basic understanding of performance appraisals. Performance appraisals, also known as performance evaluations, are assessments that many organizations use to measure individuals' productivity, ability and talent in their respective job positions. The goal of these appraisals is not only to measure each person's performance, but also to align all of the employee's values, goals and motivations and become a better performing organization as a whole. While the implementation of performance evaluations has been characterized as beneficial and even essential for organizational success, many of these performance evaluations have also become more ineffective over time due to both the excessive number of evaluation measures and employee reactivity to these evaluations.


In mathematics, a singular trace is a trace on a space of linear operators of a separable Hilbert space that vanishes on operators of finite rank. Singular traces are a feature of infinite-dimensional Hilbert spaces such as the space of square-summable sequences and spaces of square-integrable functions. Linear operators on a finite-dimensional Hilbert space have only the zero functional as a singular trace since all operators have finite rank. For example, matrix algebras have no non-trivial singular traces and the matrix trace is the unique trace up to scaling.


In quantum computing, the Brassard-Høyer-Tapp algorithm or BHT algorithm is a quantum algorithm that solves the collision problem. In this problem, one is given n and an r-to-1 function f: { 1 , … , n } → { 1 , … , n } {\displaystyle f:\,\{1,\ldots ,n\}\rightarrow \{1,\ldots ,n\}} and needs to find two inputs that f maps to the same output. The BHT algorithm only makes O ( n 1 / 3 ) {\displaystyle O(n^{1/3})} queries to f, which matches the lower bound of Ω ( n 1 / 3 ) {\displaystyle \Omega (n^{1/3})} in the black box model.The algorithm was discovered by Gilles Brassard, Peter Høyer, and Alain Tapp in 1997. It uses Grover's algorithm, which was discovered the year before.


In this case, the space requirements are no longer guaranteed to be constant since it requires storing all previous data points, but the solution may take less time to compute with the addition of a new data point, as compared to batch learning techniques. A common strategy to overcome the above issues is to learn using mini-batches, which process a small batch of b ≥ 1 {\displaystyle b\geq 1} data points at a time, this can be considered as pseudo-online learning for b {\displaystyle b} much smaller than the total number of training points. Mini-batch techniques are used with repeated passing over the training data to obtain optimized out-of-core versions of machine learning algorithms, for example, stochastic gradient descent. When combined with backpropagation, this is currently the de facto training method for training artificial neural networks.


A typically diverse area is in the representation of sounds not present in most varieties of Romani. For example, the centralised vowel phonemes of several varieties of Vlax and Xaladitka, when they are indicated separately from the non-centralised vowels, can be represented using ə, ъ or ă. Another particularly variant area is the representation of palatalised consonants, which are absent from a number of dialects. Some variant graphemes for /tʲ/ include tj, ty, ć, čj and t᾿. Finally, the representation of the second rhotic, which in several dialects has been merged with /r/, tends to vary between ř, rr, and rh, and sometimes even gh, with the first two being the most frequently found variants.