For example: the roots of numbers such as 10, 15, 20 which are not squares, the sides of numbers which are not cubes etc." In contrast to Euclid's concept of magnitudes as lines, Al-Mahani considered integers and fractions as rational magnitudes, and square roots and cube roots as irrational magnitudes. He also introduced an arithmetical approach to the concept of irrationality, as he attributes the following to irrational magnitudes: "their sums or differences, or results of their addition to a rational magnitude, or results of subtracting a magnitude of this kind from an irrational one, or of a rational magnitude from it." The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c.


In programming language theory, the associativity of an operator is a property that determines how operators of the same precedence are grouped in the absence of parentheses. If an operand is both preceded and followed by operators (for example, ^ 3 ^), and those operators have equal precedence, then the operand may be used as input to two different operations (i.e. the two operations indicated by the two operators). The choice of which operations to apply the operand to, is determined by the associativity of the operators. Operators may be associative (meaning the operations can be grouped arbitrarily), left-associative (meaning the operations are grouped from the left), right-associative (meaning the operations are grouped from the right) or non-associative (meaning operations cannot be chained, often because the output type is incompatible with the input types).


While the continuity assumption is usually met, the compactness assumption about the parameter space is often not, as the bounds of the true parameter values might be unknown. In that case, concavity of the likelihood function plays a key role. More specifically, if the likelihood function is twice continuously differentiable on the k-dimensional parameter space Θ {\displaystyle \Theta } assumed to be an open connected subset of R k , {\displaystyle \mathbb {R} ^{k}\,,} there exists a unique maximum θ ^ ∈ Θ {\displaystyle {\hat {\theta }}\in \Theta } if the matrix of second partials is negative definite for every θ ∈ Θ {\displaystyle \,\theta \in \Theta \,} at which the gradient ∇ L ≡  i = 1 n i {\displaystyle \;\nabla L\equiv \left_{i=1}^{n_{\mathrm {i} }}\;} vanishes, and if the likelihood function approaches a constant on the boundary of the parameter space, ∂ Θ , {\displaystyle \;\partial \Theta \;,} i.e., which may include the points at infinity if Θ {\displaystyle \,\Theta \,} is unbounded.


In the bandwidth-limited regime ( ρ > 2 b / 2 D {\displaystyle \rho >2~b/2D} , i.e. the domain of non-binary signaling), the effective coding gain γ e f f ( A ) {\displaystyle \gamma _{\mathrm {eff} }(A)} of a signal set A {\displaystyle A} at a given target error rate P s ( E ) {\displaystyle P_{s}(E)} is defined as the difference in dB between the S N R n o r m {\displaystyle SNR_{\mathrm {norm} }} required to achieve the target P s ( E ) {\displaystyle P_{s}(E)} with A {\displaystyle A} and the S N R n o r m {\displaystyle SNR_{\mathrm {norm} }} required to achieve the target P s ( E ) {\displaystyle P_{s}(E)} with M-PAM or (M×M)-QAM (i.e. no coding). The nominal coding gain γ c ( A ) {\displaystyle \gamma _{c}(A)} is defined as γ c ( A ) = ( 2 ρ − 1 ) d min 2 ( A ) 6 E s . {\displaystyle \gamma _{c}(A)={(2^{\rho }-1)d_{\min }^{2}(A) \over 6E_{s}}.} This definition is normalized so that γ c ( A ) = 1 {\displaystyle \gamma _{c}(A)=1} for M-PAM or (M×M)-QAM. The UBE becomes P s ( E ) ≈ K s ( A ) Q 3 γ c ( A ) S N R n o r m , {\displaystyle P_{s}(E)\approx K_{s}(A)Q{\sqrt {3\gamma _{c}(A)SNR_{\mathrm {norm} }}},} where K s ( A ) {\displaystyle K_{s}(A)} is the average number of nearest neighbors per two dimensions.


It follows from ZFC alone that Π 1 1 {\displaystyle {\boldsymbol {\Pi }}_{1}^{1}} and Σ 2 1 {\displaystyle {\boldsymbol {\Sigma }}_{2}^{1}} have the uniformization property. It follows from the existence of sufficient large cardinals that Π 2 n + 1 1 {\displaystyle {\boldsymbol {\Pi }}_{2n+1}^{1}} and Σ 2 n + 2 1 {\displaystyle {\boldsymbol {\Sigma }}_{2n+2}^{1}} have the uniformization property for every natural number n {\displaystyle n} . Therefore, the collection of projective sets has the uniformization property.


In multiple criteria decision aiding (MCDA), multicriteria classification (or sorting) involves problems where a finite set of alternative actions should be assigned into a predefined set of preferentially ordered categories (classes). For example, credit analysts classify loan applications into risk categories (e.g., acceptable/unacceptable applicants), customers rate products and classify them into attractiveness groups, candidates for a job position are evaluated and their applications are approved or rejected, technical systems are prioritized for inspection on the basis of their failure risk, clinicians classify patients according to the extent to which they have a complex disease or not, etc.


In some n-ary groups there exists an element e (called an n-ary identity or neutral element) such that any string of n-elements consisting of all e's, apart from one place, is mapped to the element at that place. E.g., in a quaternary group with identity e, eeae = a for every a. An n-ary group containing a neutral element is reducible. Thus, an n-ary group that is not reducible does not contain such elements.


In telephony, ringdown is a method of signaling an operator in which telephone ringing current is sent over the line to operate a lamp or cause the operation of a self-locking relay known as a drop. Ringdown is used in manual operation, and is distinguished from automatic signaling by dialing a number. The signal consists of a continuous or pulsed alternating current (AC) signal transmitted over the line.


The process also involves taking geographic coordinates of the ground resolution cell with GPS technology and comparing those with the coordinates of the "pixel" being studied provided by the remote sensing software to understand and analyze the location errors and how it may affect a particular study. Ground truth is important in the initial supervised classification of an image. When the identity and location of land cover types are known through a combination of field work, maps, and personal experience these areas are known as training sites.


In superscalar designs, the number of execution units is invisible to the instruction set. Each instruction encodes one operation only. For most superscalar designs, the instruction width is 32 bits or fewer. In contrast, one VLIW instruction encodes multiple operations, at least one operation for each execution unit of a device.


In mathematics, a multiplicative character (or linear character, or simply character) on a group G is a group homomorphism from G to the multiplicative group of a field (Artin 1966), usually the field of complex numbers. If G is any group, then the set Ch(G) of these morphisms forms an abelian group under pointwise multiplication. This group is referred to as the character group of G. Sometimes only unitary characters are considered (characters whose image is in the unit circle); other such homomorphisms are then called quasi-characters.


In mathematics, dimension theory is the study in terms of commutative algebra of the notion dimension of an algebraic variety (and by extension that of a scheme). The need of a theory for such an apparently simple notion results from the existence of many definitions of dimension that are equivalent only in the most regular cases (see Dimension of an algebraic variety). A large part of dimension theory consists in studying the conditions under which several dimensions are equal, and many important classes of commutative rings may be defined as the rings such that two dimensions are equal; for example, a regular ring is a commutative ring such that the homological dimension is equal to the Krull dimension. The theory is simpler for commutative rings that are finitely generated algebras over a field, which are also quotient rings of polynomial rings in a finite number of indeterminates over a field.


In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. This particular method relies on event B occurring with some sort of relationship with another event A. In this event, the event B can be analyzed by a conditional probability with respect to A. If the event of interest is A and the event B is known or assumed to have occurred, "the conditional probability of A given B", or "the probability of A under the condition B", is usually written as P(A|B) or occasionally PB(A). This can also be understood as the fraction of probability B that intersects with A, or the ratio of the probabilities of both events happening to the "given" one happening (how many times A occurs rather than not assuming B has occurred): P ( A ∣ B ) = P ( A ∩ B ) P ( B ) {\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}}} .For example, the probability that any given person has a cough on any given day may be only 5%. But if we know or assume that the person is sick, then they are much more likely to be coughing.


In programming languages and type theory, a product of types is another, compounded, type in a structure. The "operands" of the product are types, and the structure of a product type is determined by the fixed order of the operands in the product. An instance of a product type retains the fixed order, but otherwise may contain all possible instances of its primitive data types. The expression of an instance of a product type will be a tuple, and is called a "tuple type" of expression.


In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism.


In the context of the Release 8 of 3GPP standards, VCC was replaced by a wider concept that covers all services provided by IMS. This work resulted in the specification of IMS Service Continuity and IMS Centralized Services (ICS), which are meant to be used in particular to provide the continuity of voice calls between LTE and legacy 2G/3G networks.


In signal processing, signal subspace methods are empirical linear methods for dimensionality reduction and noise reduction. These approaches have attracted significant interest and investigation recently in the context of speech enhancement, speech modeling, and speech classification research. The signal subspace is also used in radio direction finding using the MUSIC (algorithm).Essentially the methods represent the application of a principal components analysis (PCA) approach to ensembles of observed time-series obtained by sampling, for example sampling an audio signal. Such samples can be viewed as vectors in a high-dimensional vector space over the real numbers.


Multiplication is modulo irreducible polynomial x 8 + x 4 + x 3 + x + 1 {\displaystyle x^{8}+x^{4}+x^{3}+x+1} . If processed bit by bit, then, after shifting, a conditional XOR with 1B16 should be performed if the shifted value is larger than FF16 (overflow must be corrected by subtraction of generating polynomial). These are special cases of the usual multiplication in GF  ( 2 8 ) {\displaystyle \operatorname {GF} (2^{8})} .


The Baire space is defined to be the Cartesian product of countably infinitely many copies of the set of natural numbers, and is given the product topology (where each copy of the set of natural numbers is given the discrete topology). The Baire space is often represented using the tree of finite sequences of natural numbers. The Baire space can be contrasted with Cantor space, the set of infinite sequences of binary digits.


The buffer pointer is a proxy for the memory address 0xB8000000. Functional programming languages based on lambda-calculus reify the concept of a procedure abstraction and procedure application in the form of the Lambda expression. The Scheme programming language reifies continuations (approximately, the call stack).


On the song's third week on the chart it reached its peak position of 16, where it remained for three weeks. It has since been certified gold by the Australian Recording Industry Association (ARIA) for sales of 35,000 units.In New Zealand the single entered at number 38 on the New Zealand Singles Chart. The following week, it rose to position 21 and then proceeded to fall off the charts.


One purpose of XML Metadata Interchange (XMI) is to enable easy interchange of metadata between UML-based modeling tools and MOF-based metadata repositories in distributed heterogeneous environments. XMI is also commonly used as the medium by which models are passed from modeling tools to software generation tools as part of model-driven engineering. Examples of XMI, and lists of the XML tags that make up XMI-formatted files, are available in the version 2.5.1 specification document.


A stronger notion is that of strongly exposed point of C {\displaystyle C} which is an exposed point x ∈ C {\displaystyle x\in C} such that some exposing functional f {\displaystyle f} of x {\displaystyle x} attains its strong maximum over C {\displaystyle C} at x {\displaystyle x} , i.e. for each sequence ( x n ) ⊂ C {\displaystyle (x_{n})\subset C} we have the following implication: f ( x n ) → max f ( C ) ⟹ ‖ x n − x ‖ → 0 {\displaystyle f(x_{n})\to \max f(C)\Longrightarrow \|x_{n}-x\|\to 0} . The set of all strongly exposed points of C {\displaystyle C} is usually denoted str  exp  ( C ) {\displaystyle \operatorname {str} \exp(C)} . There are two weaker notions, that of extreme point and that of support point of C {\displaystyle C} .


In social network theory, social relationships are viewed in terms of nodes and ties. Nodes are the individual actors within the networks, and ties are the relationships between the actors. There can be many kinds of ties between the nodes. In its simplest form, a social network is a map of all of the relevant ties between the nodes being studied.


In the Japanese kantenji braille, the standard 8-dot Braille patterns 37, 137, 347, and 1347 are the patterns related to Braille pattern dots-23, since the two additional dots of kantenji patterns 023, 237, and 0237 are placed above the base 6-dot cell, instead of below, as in standard 8-dot braille.


During batch processing, several different programs were loaded in the computer memory, and the first one began to run. When the first program reached an instruction waiting for a peripheral, the context of this program was stored away, and the second program in memory was given a chance to run. The process continued until all programs finished running.The use of multiprogramming was enhanced by the arrival of virtual memory and virtual machine technology, which enabled individual programs to make use of memory and operating system resources as if other concurrently running programs were, for all practical purposes, nonexistent.Multiprogramming gives no guarantee that a program will run in a timely manner.


In the Internet protocol suite, the application layer contains the communications protocols and interface methods used in process-to-process communications across an Internet Protocol (IP) computer network. The application layer only standardizes communication and depends upon the underlying transport layer protocols to establish host-to-host data transfer channels and manage the data exchange in a client–server or peer-to-peer networking model. Though the TCP/IP application layer does not describe specific rules or data formats that applications must consider when communicating, the original specification (in RFC 1123) does rely on and recommend the robustness principle for application design.


In statistics and data mining, affinity propagation (AP) is a clustering algorithm based on the concept of "message passing" between data points. Unlike clustering algorithms such as k-means or k-medoids, affinity propagation does not require the number of clusters to be determined or estimated before running the algorithm. Similar to k-medoids, affinity propagation finds "exemplars," members of the input set that are representative of clusters.


The C programming language uses an integer type, where relational expressions like i > j and logical expressions connected by && and || are defined to have value 1 if true and 0 if false, whereas the test parts of if, while, for, etc., treat any non-zero value as true. Indeed, a Boolean variable may be regarded (and implemented) as a numerical variable with one binary digit (bit), or as a bit string of length one, which can store only two values. The implementation of Booleans in computers are most likely represented as a full word, rather than a bit; this is usually due to the ways computers transfer blocks of information.


More generally, | Z: n Z | = n {\displaystyle |\mathbb {Z} :n\mathbb {Z} |=n} for any positive integer n. When G is finite, the formula may be written as | G: H | = | G | / | H | {\displaystyle |G:H|=|G|/|H|} , and it implies Lagrange's theorem that | H | {\displaystyle |H|} divides | G | {\displaystyle |G|} . When G is infinite, | G: H | {\displaystyle |G:H|} is a nonzero cardinal number that may be finite or infinite. For example, | Z: 2 Z | = 2 {\displaystyle |\mathbb {Z} :2\mathbb {Z} |=2} , but | R: Z | {\displaystyle |\mathbb {R} :\mathbb {Z} |} is infinite. If N is a normal subgroup of G, then | G: N | {\displaystyle |G:N|} is equal to the order of the quotient group G / N {\displaystyle G/N} , since the underlying set of G / N {\displaystyle G/N} is the set of cosets of N in G.


Intermediate angles give intermediate correlations in a way that, on careful analysis, proves inconsistent with the idea that each particle has a definite, independent probability of producing the observed measurements (the correlations violate Bell's inequality). This subtle dependence of one measurement on the other holds even when measurements are made simultaneously and a great distance apart, which gives the appearance of a superluminal communication taking place between the two electrons. Put simply, how can Bob's electron "know" what Alice measured on hers, so that it can adjust its own behavior accordingly?


Comprehensive test procedures which outline the steps, and their expected results clearly identify what is to be seen as a result of performing the step. After the step or set of steps is completed the last step's expected result will call out what has been seen and then identify what requirement or requirements have been verified (identified by number). The requirement number, title and verbiage are tied together in another location in the test document.


In some cases, the problem may also be simplified by subtracting a point from both sides of the equation and taking the norm of the result. For example, to solvefor ξ 3 {\textstyle \xi _{3}} , where ξ 1 {\textstyle \xi _{1}} and ξ 2 {\textstyle \xi _{2}} intersect at the point q {\textstyle q} , both sides of the equation may be applied to a point p {\textstyle p} that is not on the axis of ξ 3 {\textstyle \xi _{3}} . Subtracting q {\textstyle q} and taking the norm of both sides yields This may be solved using Subproblem 3.


Therefore, in the case of drawing a red card or a king, drawing any of a red king, a red non-king, or a black king is considered a success. In a standard 52-card deck, there are twenty-six red cards and four kings, two of which are red, so the probability of drawing a red or a king is 26/52 + 4/52 – 2/52 = 28/52. Events are collectively exhaustive if all the possibilities for outcomes are exhausted by those possible events, so at least one of those outcomes must occur.


In mathematical logic, Goodstein's theorem is a statement about the natural numbers, proved by Reuben Goodstein in 1944, which states that every Goodstein sequence eventually terminates at 0. Laurence Kirby and Jeff Paris showed that it is unprovable in Peano arithmetic (but it can be proven in stronger systems, such as second-order arithmetic). This was the third example of a true statement that is unprovable in Peano arithmetic, after the examples provided by Gödel's incompleteness theorem and Gerhard Gentzen's 1943 direct proof of the unprovability of ε0-induction in Peano arithmetic. The Paris–Harrington theorem gave another example.


For example, the torus is different from the sphere: the torus has a "hole"; the sphere doesn't. However, since continuity (the basic notion of topology) only deals with the local structure, it can be difficult to formally define the obvious global difference. The homotopy groups, however, carry information about the global structure.


In the example program only normal calls are used, so all the information will be on a single stack. For asynchronous calls, a separate stack is initiated for each asynchronous process so that the processes share data but run asynchronously.


In order for virtual memory compression to provide measurable performance improvements, the throughput of the virtual memory system must be improved when compared to the uncompressed equivalent. Thus, the additional amount of processing introduced by the compression must not increase the overall latency. However, in I/O-bound systems or applications with highly compressible data sets, the gains can be substantial.


In plumbing fittings, the "M" or "F" usually comes at the beginning rather than the end of the abbreviated designation. For example: MIPT denotes male iron pipe thread; FIPT denotes female iron pipe thread.A short length of pipe having an MIP thread at both ends is sometimes called a nipple. A short pipe fitting having an FIP thread at both ends is sometimes called a coupling.


In mathematical optimization theory, duality or the duality principle is the principle that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. If the primal is a minimization problem then the dual is a maximization problem (and vice versa). Any feasible solution to the primal (minimization) problem is at least as large as any feasible solution to the dual (maximization) problem. Therefore, the solution to the primal is an upper bound to the solution of the dual, and the solution of the dual is a lower bound to the solution of the primal.


In the following code, neither f nor g functions is reentrant. In the above, f() depends on a non-constant global variable v; thus, if f() is interrupted during execution by an ISR which modifies v, then reentry into f() will return the wrong value of v. The value of v and, therefore, the return value of f, cannot be predicted with confidence: they will vary depending on whether an interrupt modified v during f's execution. Hence, f is not reentrant. Neither is g, because it calls f, which is not reentrant.


In linear algebra, the n × n {\displaystyle n\times n} identity matrix I {\displaystyle \mathbf {I} } has entries equal to the Kronecker delta: where i {\displaystyle i} and j {\displaystyle j} take the values 1 , 2 , ⋯ , n {\displaystyle 1,2,\cdots ,n} , and the inner product of vectors can be written as Here the Euclidean vectors are defined as n-tuples: a = ( a 1 , a 2 , … , a n ) {\displaystyle \mathbf {a} =(a_{1},a_{2},\dots ,a_{n})} and b = ( b 1 , b 2 , . . . , b n ) {\displaystyle \mathbf {b} =(b_{1},b_{2},...,b_{n})} and the last step is obtained by using the values of the Kronecker delta to reduce the summation over j {\displaystyle j} . It is common for i and j to be restricted to a set of the form {1, 2, ..., n} or {0, 1, ..., n − 1}, but the Kronecker delta can be defined on an arbitrary set.


In many less-developed countries, such as Spain, Mexico, Brazil, and Egypt, calls were placed at a central office the caller went to, filled out a paper slip, sometimes paid in advance for the call, and then waited for it to be connected. In Spain these were known as locutorios, literally "a place to talk". In towns too small to support a phone office, placing long-distance calls was a sideline for some businesses with telephones, such as pharmacies.


Internally, the CPU of the computer is built up from a number of separate parts dedicated to a single task, for instance, adding a number, or fetching from memory. Normally, as the instruction flows through the machine, only one part is active at any given time. This means that each sequential step of the entire process must complete before a result can be saved.


A bijective function is also called a bijection. That is, combining the definitions of injective and surjective, ∀ y ∈ Y , ∃ ! x ∈ X such that y = f ( x ) , {\displaystyle \forall y\in Y,\exists !x\in X{\text{ such that }}y=f(x),} where ∃ !


In mathematics, particularly abstract algebra, a binary operation • on a set is flexible if it satisfies the flexible identity: a ∙ ( b ∙ a ) = ( a ∙ b ) ∙ a {\displaystyle a\bullet \left(b\bullet a\right)=\left(a\bullet b\right)\bullet a} for any two elements a and b of the set. A magma (that is, a set equipped with a binary operation) is flexible if the binary operation with which it is equipped is flexible. Similarly, a nonassociative algebra is flexible if its multiplication operator is flexible. Every commutative or associative operation is flexible, so flexibility becomes important for binary operations that are neither commutative nor associative, e.g. for the multiplication of sedenions, which are not even alternative. In 1954, Richard D. Schafer examined the algebras generated by the Cayley–Dickson process over a field and showed that they satisfy the flexible identity.


In statistics, the monotone likelihood ratio property is a property of the ratio of two probability density functions (PDFs). Formally, distributions ƒ(x) and g(x) bear the property if for every x 1 > x 0 , f ( x 1 ) g ( x 1 ) ≥ f ( x 0 ) g ( x 0 ) {\displaystyle {\text{for every }}x_{1}>x_{0},\quad {\frac {f(x_{1})}{g(x_{1})}}\geq {\frac {f(x_{0})}{g(x_{0})}}} that is, if the ratio is nondecreasing in the argument x {\displaystyle x} . If the functions are first-differentiable, the property may sometimes be stated ∂ ∂ x ( f ( x ) g ( x ) ) ≥ 0 {\displaystyle {\frac {\partial }{\partial x}}\left({\frac {f(x)}{g(x)}}\right)\geq 0} For two distributions that satisfy the definition with respect to some argument x, we say they "have the MLRP in x." For a family of distributions that all satisfy the definition with respect to some statistic T(X), we say they "have the MLR in T(X)."


In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime (no two divisors share a common factor other than 1). For example, if we know that the remainder of n divided by 3 is 2, the remainder of n divided by 5 is 3, and the remainder of n divided by 7 is 2, then without knowing the value of n, we can determine that the remainder of n divided by 105 (the product of 3, 5, and 7) is 23. Importantly, this tells us that if n is a natural number less than 105, then 23 is the only possible value of n. The earliest known statement of the theorem is by the Chinese mathematician Sunzi in the Sunzi Suanjing in the 3rd century CE.


Job j {\displaystyle j} on machine i {\displaystyle i} takes time p j / s i {\displaystyle p_{j}/s_{i}} . R: Unrelated-machines scheduling. There are m {\displaystyle m} parallel machines, and they are unrelated – Job j {\displaystyle j} on machine i {\displaystyle i} takes time p i j {\displaystyle p_{ij}} .These letters might be followed by the number of machines, which is then fixed.


Their specific use as metadata is left to the developer and can cover a wide range of types of information about any given application, classes and members that is not instance-specific. The decision to expose any given attribute as a property is also left to the developer as is the decision to use them as part of a larger application framework. Attributes are implemented as classes that are derived from System.Attribute.


Unlike Peano arithmetic, Skolem arithmetic is a decidable theory. This means it is possible to effectively determine, for any sentence in the language of Skolem arithmetic, whether that sentence is provable from the axioms of Skolem arithmetic. The asymptotic running-time computational complexity of this decision problem is triply exponential.


The process responds at the next time step by randomly moving into a new state s ′ {\displaystyle s'} , and giving the decision maker a corresponding reward R a ( s , s ′ ) {\displaystyle R_{a}(s,s')} . The probability that the process moves into its new state s ′ {\displaystyle s'} is influenced by the chosen action. Specifically, it is given by the state transition function P a ( s , s ′ ) {\displaystyle P_{a}(s,s')} .


In the SQUOZE encoding, identifiers in the symbol table were represented in a 50-character alphabet, allowing a 36-bit machine word to represent six alphanumeric characters plus two flag bits, thus saving two bits per six characters, because the six bits normally allocated for each character could store up to 64 states rather than only the 50 states needed to represent the 50 letters of the alphabet, and 506 < 234. Using base 50 already saves a single bit every three characters, so it was used in two three-character chunks. The manual has a formula for encoding six characters ABCDEF: ( A ∗ 50 2 + B ∗ 50 + C ) ∗ 2 17 + ( D ∗ 50 2 + E ∗ 50 + F ) {\displaystyle (A*50^{2}+B*50+C)*2^{17}+(D*50^{2}+E*50+F)} For example "SQUOZE", normally 36 bits: 35 33 37 31 44 17(base 8) would be encoded in two 17-bit pieces to fit in the 34 bits as ( 0o220231 << 17 ) | 0o175473 == 0o110114575473. A simpler example of the same logic would be how a three-digit BCD number would take up 12 bits, such as 987: 9 8 7(base 16) 1001 1000 0111(base 2), but any such value could be stored in 10 bits directly, saving two bits, such as 987: 3db(base 16) 11 1101 1011(base 2).


In order to state the hypergraph regularity and counting lemmas formally, we need to define several rather technical terms to formalize appropriate notions of pseudo-randomness (random-likeness) and boundedness, as well as to describe the random-like blocks and partitions. Notation K j ( k ) {\displaystyle K_{j}^{(k)}} denotes a k {\displaystyle k} -uniform clique on j {\displaystyle j} vertices. G ( j ) {\displaystyle {\mathcal {G}}^{(j)}} is an l {\displaystyle l} -partite j {\displaystyle j} -graph on vertex partition G ( 1 ) = V 1 ⊔ … ⊔ V l {\displaystyle {\mathcal {G}}^{(1)}=V_{1}\sqcup \ldots \sqcup V_{l}} . K j ( G ( i ) ) {\displaystyle {\mathcal {K}}_{j}({\mathcal {G}}^{(i)})} is the family of all j {\displaystyle j} -element vertex sets that span the clique K j ( i ) {\displaystyle K_{j}^{(i)}} in G ( i ) {\displaystyle {\mathcal {G}}^{(i)}} .


In most cases, we want to find out the relationships between social data and another event or we want to get interesting results from social data analyses to predict some events. There are some outstanding articles in this field, including Twitter Mood Predicts The Stock Market, Predicting The Present With Google Trends etc. In order to accomplish these goals, we need the appropriate methods to do the analyses. Usually, we use statistic methods, methods of machine learning or methods of data mining to do the analyses. Universities all over the world are opening graduate program in Social Data Analysis.


The client decides which commands to execute at which points. To execute a command, it passes the command object to the invoker object.


, y 1 ) . {\displaystyle P(y_{i}|x_{i},x_{i-1},x_{i-2},...,x_{1},y_{i-1},y_{i-2},...,y_{1}).} .


For small samples the chi-squared approximation is overly sensitive, often rejecting the null hypothesis when it is true. Furthermore, the distribution of p-values departs from a uniform distribution and becomes a right-skewed unimodal distribution, especially for small p-values. This leads to a large Type I error rate.


In set theory, the notation Y X {\displaystyle Y^{X}} is used to denote the set of functions from the set X {\displaystyle X} to the set Y {\displaystyle Y} . Currying is the natural bijection between the set A B × C {\displaystyle A^{B\times C}} of functions from B × C {\displaystyle B\times C} to A {\displaystyle A} , and the set ( A C ) B {\displaystyle (A^{C})^{B}} of functions from B {\displaystyle B} to the set of functions from C {\displaystyle C} to A {\displaystyle A} . In symbols: A B × C ≅ ( A C ) B {\displaystyle A^{B\times C}\cong (A^{C})^{B}} Indeed, it is this natural bijection that justifies the exponential notation for the set of functions. As is the case in all instances of currying, the formula above describes an adjoint pair of functors: for every fixed set C {\displaystyle C} , the functor B ↦ B × C {\displaystyle B\mapsto B\times C} is left adjoint to the functor A ↦ A C {\displaystyle A\mapsto A^{C}} . In the category of sets, the object Y X {\displaystyle Y^{X}} is called the exponential object.


The various problems, algorithms, and tools of cost distance analysis operate over an unconstrained two-dimensional space, meaning that a path could be of any shape. Similar cost optimization problems can also arise in a constrained space, especially a one-dimensional linear network such as a road or telecommunications network. Although they are similar in principle, the problems in network space require very different (usually simpler) algorithms to solve, largely adopted from graph theory. The collection of GIS tools for solving these problems are called network analysis.


In modulus 12, one can assert that: 38 ≡ 14 ( mod 12 ) {\displaystyle 38\equiv 14{\pmod {12}}} because 38 − 14 = 24, which is a multiple of 12. Another way to express this is to say that both 38 and 14 have the same remainder 2, when divided by 12. The definition of congruence also applies to negative values. For example: 2 ≡ − 3 ( mod 5 ) − 8 ≡ 7 ( mod 5 ) − 3 ≡ − 8 ( mod 5 ) . {\displaystyle {\begin{aligned}2&\equiv -3{\pmod {5}}\\-8&\equiv 7{\pmod {5}}\\-3&\equiv -8{\pmod {5}}.\end{aligned}}}


In mathematics, the Birch and Swinnerton-Dyer conjecture (often called the Birch–Swinnerton-Dyer conjecture) describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems. It is named after mathematicians Bryan John Birch and Peter Swinnerton-Dyer, who developed the conjecture during the first half of the 1960s with the help of machine computation. As of 2023, only special cases of the conjecture have been proven.


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Renaming truth as a product of the will cannot help it solve the problems of the intellect, according to Bittle. Bittle cited what he saw as contradictions in pragmatism, such as using objective facts to prove that truth does not emerge from objective fact; this reveals that pragmatists do recognize truth as objective fact, and not, as they claim, what is useful. Bittle argued there are also some statements that cannot be judged on human welfare at all.


In the Japanese kantenji braille, the standard 8-dot Braille patterns 2, 12, 24, and 124 are the 8-dot braille patterns related to Braille pattern dots-1, since the two additional dots of kantenji patterns 01, 17, and 017 are placed above the base 6-dot cell, instead of below, as in standard 8-dot braille.


In mathematics, quantales are certain partially ordered algebraic structures that generalize locales (point free topologies) as well as various multiplicative lattices of ideals from ring theory and functional analysis (C*-algebras, von Neumann algebras). Quantales are sometimes referred to as complete residuated semigroups.


This can be significantly faster, as setjmp and longjmp must conservatively store all registers which may be in use according to the ABI, whereas the clobber method allows the compiler to store (by spilling to the stack) only what it knows is actually in use. Due to the lack of direct language support, many authors have written their own libraries for coroutines which hide the above details.


For example, 29/30 could be written as 1 2 4 2 3 5 {\displaystyle {\tfrac {1\,\,2\,\,4}{2\,\,3\,\,5}}} , representing the value 4 5 + 2 3 × 5 + 1 2 × 3 × 5 {\displaystyle {\tfrac {4}{5}}+{\tfrac {2}{3\times 5}}+{\tfrac {1}{2\times 3\times 5}}} . This can be viewed as a form of mixed radix notation, and was very convenient for dealing with traditional systems of weights, measures, and currency. For instance, for units of length, a foot is 1/3 of a yard, and an inch is 1/12 of a foot, so a quantity of 5 yards, 2 feet, and 7 3 4 {\displaystyle 7{\tfrac {3}{4}}} inches could be represented as a composite fraction: 3 7 2 4 12 3 5 {\displaystyle {\tfrac {3\ \,7\,\,2}{4\,\,12\,\,3}}\,5} yards.


In statistics, factor analysis of mixed data or factorial analysis of mixed data (FAMD, in the French original: AFDM or Analyse Factorielle de Données Mixtes), is the factorial method devoted to data tables in which a group of individuals is described both by quantitative and qualitative variables. It belongs to the exploratory methods developed by the French school called Analyse des données (data analysis) founded by Jean-Paul Benzécri. The term mixed refers to the use of both quantitative and qualitative variables. Roughly, we can say that FAMD works as a principal components analysis (PCA) for quantitative variables and as a multiple correspondence analysis (MCA) for qualitative variables.


With stack machines, in contrast, results can be stored in one of two ways. Firstly, results can be stored using a temporary variable in memory. Storing and subsequent retrievals cost additional instructions and additional data cache cycles.


Around 2007, LSTM trained by Connectionist Temporal Classification (CTC) started to outperform traditional speech recognition in certain applications. In 2015, Google's speech recognition reportedly experienced a dramatic performance jump of 49% through CTC-trained LSTM, which is now available through Google Voice to all smartphone users. Transformers, a type of neural network based on solely on attention, have been widely adopted in computer vision and language modeling, sparking the interest of adapting such models to new domains, including speech recognition.


In programming languages that have non-hygienic macro systems, it is possible for existing variable bindings to be hidden from a macro by variable bindings that are created during its expansion. In C, this problem can be illustrated by the following fragment: Running the above through the C preprocessor produces: The variable a declared in the top scope is shadowed by the a variable in the macro, which introduces a new scope. As a result, a is never altered by the execution of the program, as the output of the compiled program shows: a is now 4, b is now 9


In parallel France and Germany signed a joint development agreement in 1984 and were joined by Italy and the UK in 1986. In 1986, the MTS and European Commission proposed reserving the 900 MHz spectrum band for GSM. It was long believed that the former Finnish prime minister Harri Holkeri made the world's first GSM call on 1 July 1991, MTS calling Kaarina Suonio (deputy mayor of the city of Tampere) using a network built by Nokia and Siemens and operated by Radiolinja. In 2021 a former Nokia engineer Pekka Lonka revealed to Helsingin Sanomat making a test call just a couple of hours earlier.


In mutual authentication schemes that require a user's input password as part of the verification process, there is a higher vulnerability to hackers because the password is human-made rather than a computer-generated certificate. While applications could simply require users to use a computer-generated password, it is inconvenient for people to remember. User-made passwords and the ability to change one's password are important for making an application user-friendly, so many schemes work to accommodate the characteristic.


In neuropsychology, linguistics, and philosophy of language, a natural language or ordinary language is any language that occurs naturally in a human community by a process of use, repetition, and change without conscious planning or premeditation. It can take different forms, namely either a spoken language or a sign language. Natural languages are distinguished from constructed and formal languages such as those used to program computers or to study logic.


Second, synchronization in many concurrent programs in a finite state, and therefore can be adequately described by regular expressions. For precisely the same reasons, path expressions are useful for controlling the behavior of complicated asynchronous circuits. In fact, the finite state assumption may be even more reasonable at the hardware level than at the monitor level. Path expressions provide a high level of descriptive synchronization that aids in the prevention and detection of design errors in complex systems and overcomes some of the dangers, such as certain forms of coding errors.


In paging the memory address space or segment is divided into equal-sized blocks called pages. Using virtual memory hardware, each page can reside in any location at a suitable boundary of the computer's physical memory, or be flagged as being protected. Virtual memory makes it possible to have a linear virtual memory address space and to use it to access blocks fragmented over physical memory address space. Most computer architectures which support paging also use pages as the basis for memory protection.


In the above example, the RLI mark (right-to-left isolate) forces the following text to be interpreted in the reverse order: the triple-quote is first (ending the string), followed by a semicolon (starting a new line), and finally with the premature return (returning None and ignoring any code below it). The new line terminates the RLI mark, preventing it from flowing into the below code. Because of the Bidi character, some source code editors and IDEs rearrange the code for display without any visual indication that the code has been rearranged, so a human code reviewer would not normally detect them.


If we remove the vertex v, we can four-color the remaining vertices. We can set the colors as (in clockwise order) red, yellow, blue, and green. In this situation, there can be a Kempe chain joining the red and blue neighbors or a Kempe chain joining the green and yellow neighbors, but not both, since these two paths would necessarily intersect, and the vertex where they intersect cannot be colored.


In mathematics, a regular semigroup is a semigroup S in which every element is regular, i.e., for each element a in S there exists an element x in S such that axa = a. Regular semigroups are one of the most-studied classes of semigroups, and their structure is particularly amenable to study via Green's relations.


Infinity can also be used to describe infinite series, as follows: ∑ i = 0 ∞ f ( i ) = a {\displaystyle \sum _{i=0}^{\infty }f(i)=a} means that the sum of the infinite series converges to some real value a . {\displaystyle a.}


Although languages that are described as having imperfective and perfective aspects agree in most cases in their use of these aspects, they may not agree in every situation. For example: Some languages have additional grammatical aspects. Spanish and Ancient Greek, for example, have a perfect (not the same as the perfective), which refers to a state resulting from a previous action (also described as a previous action with relevance to a particular time, or a previous action viewed from the perspective of a later time).


In order to get 128 bits of security for hash based signatures to sign 1 million messages using the fractal Merkle tree method of Naor Shenhav and Wool the public and private key sizes are roughly 36,000 bits in length.


y-axis ( ↑ {\displaystyle \uparrow } ): terms that can bind types, corresponding to polymorphism. z-axis ( ↗ {\displaystyle \nearrow } ): types that can bind types, corresponding to (binding) type operators.The different ways to combine these three dimensions yield the 8 vertices of the cube, each corresponding to a different kind of typed system. The λ-cube can be generalized into the concept of a pure type system.


(This bound is tight, as a sequence of m − 1 {\displaystyle m-1} zeroes and m − 1 {\displaystyle m-1} ones cannot have any subset of size m {\displaystyle m} summing to zero.) There are known proofs of this result using the Cauchy-Davenport theorem, Fermat's little theorem, or the Chevalley–Warning theorem.Generalizing this result, one can define for any abelian group G the minimum quantity E G Z ( G ) {\displaystyle EGZ(G)} of elements of G such that there must be a subsequence of o ( G ) {\displaystyle o(G)} elements (where o ( G ) {\displaystyle o(G)} is the order of the group) which adds to zero. It is known that E G Z ( G ) ≤ 2 o ( G ) − 1 {\displaystyle EGZ(G)\leq 2o(G)-1} , and that this bound is strict if and only if G = Z m {\displaystyle G=\mathbb {Z} _{m}} .


In mathematics, the regula falsi, method of false position, or false position method is a very old method for solving an equation with one unknown; this method, in modified form, is still in use. In simple terms, the method is the trial and error technique of using test ("false") values for the variable and then adjusting the test value according to the outcome. This is sometimes also referred to as "guess and check". Versions of the method predate the advent of algebra and the use of equations.


In the United States, common telephone lines are designed to transmit frequencies between 300 and 3400 Hz. Since electric power in the United States is distributed at 60 Hz, it normally does not interfere with telephone communications because its frequency is too low.


Unsatisfiable statements, both through negation and affirmation, are known formally as contradictions. A formula that is neither a tautology nor a contradiction is said to be logically contingent.


In marketing, contact center telephony is the communication and collaboration system used by businesses to either manage high volumes of inbound queries or outbound telephone calls keeping their workforce or agents productive and in control to serve or acquire customers. This business communication system is an extension of computer telephony integration (CTI).


{\displaystyle f\circ g_{1}=f\circ g_{2}\implies g_{1}=g_{2}.} Monomorphisms are a categorical generalization of injective functions (also called "one-to-one functions"); in some categories the notions coincide, but monomorphisms are more general, as in the examples below. In the setting of posets intersections are idempotent: the intersection of anything with itself is itself.


In mistrustful cryptography the participating parties do not trust each other. For example, Alice and Bob collaborate to perform some computation where both parties enter some private inputs. But Alice does not trust Bob and Bob does not trust Alice. Thus, a secure implementation of a cryptographic task requires that after completing the computation, Alice can be guaranteed that Bob has not cheated and Bob can be guaranteed that Alice has not cheated either.


Derived from the causal order is the differential structure and the conformal metric of a manifold. A probability is assigned to a causal set becoming embedded in a manifold; thus there can be a transition from a discrete Planck scale fundamental unit of volume to a classical large scale continuous space.Random graphs by Antonsen Spacetime is described by dynamical graphs with points (associated with vertices) and links (of unit length) that are created or annihilated according to probability calculations. The parameterization of graphs in a metaspace gives rise to time.Bootstrap universe by Cahill and Klinger An iterative map composed of monads and the relations between them becomes a tree-graph of nodes and links.


The same cannot be done with the other two deductions systems: as context is changed in some of their rules of inferences, they cannot be formalized so that hypothetical judgments could be avoided—not even if we want to use them just for proving derivability of tautologies. This basic diversity among the various calculi allows such difference, that the same basic thought (e.g. deduction theorem) must be proven as a metatheorem in Hilbert-style deduction system, while it can be declared explicitly as a rule of inference in natural deduction. In type theory, some analogous notions are used as in mathematical logic (giving rise to connections between the two fields, e.g. Curry–Howard correspondence). The abstraction in the notion of judgment in mathematical logic can be exploited also in foundation of type theory as well.


In the 1970s, analysis of high-level languages indicated compilers produced some complex corresponding machine language. It was determined that new instructions could improve performance. Some instructions were added that were never intended to be used in assembly language but fit well with compiled high-level languages.


In mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups. A simple group is a group G that does not have any normal subgroups except for the trivial group and G itself. The classification theorem states that the list of finite simple groups consists of 18 countably infinite families plus 26 exceptions that do not follow such a systematic pattern.


In neutral prosody, Turkish verb phrases are primarily head-final, as the verb comes after its complement. Variation in object-verb ordering is not strictly rigid. However, constructions where the verb precedes the object are less common. ]


In these cases it can happen that ab ≠ ba; then "inverse" typically implies that an element is both a left and right inverse. The notation f −1 is sometimes also used for the inverse function of the function f, which is for most functions not equal to the multiplicative inverse. For example, the multiplicative inverse 1/(sin x) = (sin x)−1 is the cosecant of x, and not the inverse sine of x denoted by sin−1 x or arcsin x. The terminology difference reciprocal versus inverse is not sufficient to make this distinction, since many authors prefer the opposite naming convention, probably for historical reasons (for example in French, the inverse function is preferably called the bijection réciproque).


It affects the choice of comparable data for use in the analysis. It can also affect the method used to value the property. For example, tree value can contribute up to 27% of property value.


In software engineering, cycle time is a software metric which estimates development speed in agile software projects. The cycle time measures how long it takes to process a given job - whether it's a client request, an order, or a defined production process stage. The crucial aspect of measuring the cycle time is considering only the active, operating processing time and discarding any idle, waiting, or service times occurring mid-process. According to the PMBOK (7th edition) by the Project Management Institute (PMI), cycle time is the "total elapsed time from the start of a particular activity or work item to its completion.


In mathematics, a base-orderable matroid is a matroid that has the following additional property, related to the bases of the matroid. For any two bases A {\displaystyle A} and B {\displaystyle B} there exists a feasible exchange bijection, defined as a bijection f {\displaystyle f} from A {\displaystyle A} to B {\displaystyle B} , such that for every a ∈ A ∖ B {\displaystyle a\in A\setminus B} , both ( A ∖ { a } ) ∪ { f ( a ) } {\displaystyle (A\setminus \{a\})\cup \{f(a)\}} and ( B ∖ { f ( a ) } ) ∪ { a } {\displaystyle (B\setminus \{f(a)\})\cup \{a\}} are bases.The property was introduced by Brualdi and Scrimger. A strongly-base-orderable matroid has the following stronger property:For any two bases A {\displaystyle A} and B {\displaystyle B} , there is a strong feasible exchange bijection, defined as a bijection f {\displaystyle f} from A {\displaystyle A} to B {\displaystyle B} , such that for every X ⊆ A {\displaystyle X\subseteq A} , both ( A ∖ X ) ∪ f ( X ) {\displaystyle (A\setminus X)\cup f(X)} and ( B ∖ f ( X ) ) ∪ X {\displaystyle (B\setminus f(X))\cup X} are bases.


In some implementations of software development processes, issues are investigated by quality assurance analysts a system is verified for correctness, and then assigned back to a member of the development team to resolve the identified issue. They can also be identified by system users during the User Acceptance Testing (UAT) phase. Issues can be recorded and communicated using Issue or Defect Tracking Systems. In the absence of a formal Issue or Defect Tracking system, it is commonplace to simply use any form of written communication such as emails or instant messages to communicate the existence of a found issue.


In one version of airline scheduling the goal is to produce a feasible schedule with at most k crews. To solve this problem one uses a variation of the circulation problem called bounded circulation which is the generalization of network flow problems, with the added constraint of a lower bound on edge flows. Let G = (V, E) be a network with s,t ∈ V as the source and the sink nodes.


Differential variational inequalities were first formally introduced by Pang and Stewart, whose definition should not be confused with the differential variational inequality used in Aubin and Cellina (1984). Differential variational inequalities have the form to find u ( t ) ∈ K {\displaystyle u(t)\in K} such that ⟨ v − u ( t ) , F ( t , x ( t ) , u ( t ) ) ⟩ ≥ 0 {\displaystyle \langle v-u(t),F(t,x(t),u(t))\rangle \geq 0} for every v ∈ K {\displaystyle v\in K} and almost all t; K a closed convex set, where d x d t = f ( t , x ( t ) , u ( t ) ) , x ( t 0 ) = x 0 . {\displaystyle {\frac {dx}{dt}}=f(t,x(t),u(t)),\quad x(t_{0})=x_{0}.} Closely associated with DVIs are dynamic/differential complementarity problems: if K is a closed convex cone, then the variational inequality is equivalent to the complementarity problem: K ∋ u ( t ) ⊥ F ( t , x ( t ) , u ( t ) ) ∈ K ∗ . {\displaystyle K\ni u(t)\quad \perp \quad F(t,x(t),u(t))\in K^{*}.}


In mathematical logic (a subtopic within the field of formal logic), two formulae are equisatisfiable if the first formula is satisfiable whenever the second is and vice versa; in other words, either both formulae are satisfiable or both are not. Equisatisfiable formulae may disagree, however, for a particular choice of variables. As a result, equisatisfiability is different from logical equivalence, as two equivalent formulae always have the same models.


In the beginning of the agglomerative clustering process, each element is in a cluster of its own. The clusters are then sequentially combined into larger clusters, until all elements end up being in the same cluster. At each step, the two clusters separated by the shortest distance are combined. The function used to determine the distance between two clusters, known as the linkage function, is what differentiates the agglomerative clustering methods.


This is particularly common for discrete data. When this happens, the test procedure defined above is usually undefined because there is no way to uniquely rank the data. (The sole exception is if there is a single sample X i {\displaystyle X_{i}} which is zero and no other zeros or ties.) Because of this, the test statistic needs to be modified.


( c ∈ C ) . f ( a ) = c {\displaystyle \forall (a\in A).\,\exists ! (c\in C).f(a)=c} .


PCA is used to identify a set of orthogonal basis vectors (basis signals) which capture as much as possible of the energy in the ensemble of observed samples. The vector space spanned by the basis vectors identified by the analysis is then the signal subspace. The underlying assumption is that information in speech signals is almost completely contained in a small linear subspace of the overall space of possible sample vectors, whereas additive noise is typically distributed through the larger space isotropically (for example when it is white noise).


A utility program (MOVCPM) was provided with system distribution that allowed relocating the object code to different memory areas. The utility program adjusted the addresses in absolute jump and subroutine call instructions to new addresses required by the new location of the operating system in processor memory. This newly patched version could then be saved on a new disk, allowing application programs to access the additional memory made available by moving the system components.


During the construction of the sequence, if a proof π j {\displaystyle \pi _{j}} happens to be too large, π j + 1 {\displaystyle \pi _{j+1}} is set to be the smallest proof in { π 1 , π 2 , … , π j } {\displaystyle \{\pi _{1},\pi _{2},\ldots ,\pi _{j}\}} . For achieving a better compression/time ratio, a heuristic for variable selection is desirable. For this purpose, Cotton defines the "additivity" of a resolution step (with antecedents p {\displaystyle p} and n {\displaystyle n} and resolvent r {\displaystyle r} ): add  ( r ) := max ( | r | − max ( | p | , | n | ) , 0 ) {\displaystyle \operatorname {add} (r):=\max(|r|-\max(|p|,|n|),0)} Then, for each variable v {\displaystyle v} , a score is calculated summing the additivity of all the resolution steps in π {\displaystyle \pi } with pivot v {\displaystyle v} together with the number of these resolution steps.


The binary logarithm has also been written as log n with a prior statement that the default base for the logarithm is 2. Another notation that is often used for the same function (especially in the German scientific literature) is ld n, from Latin logarithmus dualis or logarithmus dyadis. The DIN 1302, ISO 31-11 and ISO 80000-2 standards recommend yet another notation, lb n. According to these standards, lg n should not be used for the binary logarithm, as it is instead reserved for the common logarithm log10 n.


In mathematics, a linear equation is an equation that may be put in the form a 1 x 1 + … + a n x n + b = 0 , {\displaystyle a_{1}x_{1}+\ldots +a_{n}x_{n}+b=0,} where x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} are the variables (or unknowns), and b , a 1 , … , a n {\displaystyle b,a_{1},\ldots ,a_{n}} are the coefficients, which are often real numbers. The coefficients may be considered as parameters of the equation, and may be arbitrary expressions, provided they do not contain any of the variables. To yield a meaningful equation, the coefficients a 1 , … , a n {\displaystyle a_{1},\ldots ,a_{n}} are required to not all be zero.


In mathematics, an asymmetric relation is a binary relation R {\displaystyle R} on a set X {\displaystyle X} where for all a , b ∈ X , {\displaystyle a,b\in X,} if a {\displaystyle a} is related to b {\displaystyle b} then b {\displaystyle b} is not related to a . {\displaystyle a.}


Most schema migration tools aim to minimize the impact of schema changes on any existing data in the database. Despite this, preservation of data in general is not guaranteed because schema changes such as the deletion of a database column can destroy data (i.e. all values stored under that column for all rows in that table are deleted). Instead, the tools help to preserve the meaning of the data or to reorganize existing data to meet new requirements. Since meaning of the data often cannot be encoded, the configuration of the tools usually needs manual intervention.


This knowledge comes from repeated examinations over time that show changes from marketing initiatives (from brand and competition) and the evolution of consumers’ needs. Harold Geneen in his groundbreaking book “Managing” explains the role of an effective CEO: to repeatedly evaluate performance numbers on a continuous basis. Only long term observation brings true insight of unanticipated changes and “red flags” in the data.All measurement systems are prone to misinterpretation and error.


The SSCG sequence begins slower than SCG, SSCG(0) = 2, SSCG(1) = 5, but then grows rapidly. SSCG(2) = 3 × 2(3 × 295) − 8 ≈ 3.241704 × 1035775080127201286522908640065. Its first and last 20 digits are 32417042291246009846...34057047399148290040.


This allows for forward and reverse branches in code. The limited range of the branch instructions meant that, as code grew, the target addresses of some branches would become unreachable.


In some time-critical situations, failure to act may entail a continuously increasing cost over time, or a continuously decreasing probability over time of achieving the desired outcome. In real-time computing systems, this may be represented by time-utility functions.


Minimal-rights: stronger than non-negativity: each claimant should get at least his minimal right, which is what's left if all other agents get their full claims: ∀ i: x i ≥ m i , where m i := max ( 0 , E − ∑ j ≠ i c j ) {\displaystyle \forall i:x_{i}\geq m_{i},{\text{ where }}m_{i}:=\max(0,E-\sum _{j\neq i}c_{j})} . Note that efficiency, non-negativity and claims-boundedness together imply minimal-rights. Equal treatment of equals (ETE): two claimants with identical claims should get identical allocations: c i = c j ⟹ x i = x j {\displaystyle c_{i}=c_{j}\implies x_{i}=x_{j}} .


With some precomputation, only a single multiplication per sector is required (note that addition in a binary finite field is a simple bitwise addition, also known as xor): F ⊗ I = F ⊗ ( I 0 ⊕ δ ) = F ⊗ I 0 ⊕ F ⊗ δ {\displaystyle F\otimes I=F\otimes (I_{0}\oplus \delta )=F\otimes I_{0}\oplus F\otimes \delta } , where F ⊗ δ {\displaystyle F\otimes \delta } are precomputed for all possible values of δ {\displaystyle \delta } . This mode of operation needs only a single encryption per block and protects against all the above attacks except a minor leak: if the user changes a single plaintext block in a sector then only a single ciphertext block changes.


In search of "Framework for Modeling Space Systems Architectures" Peter Shames and Joseph Skipper (2006) defined a "nominal set of views", Derived from CCSDS RASDS, RM-ODP, ISO 10746 and compliant with IEEE 1471. This "set of views", as described below, is a listing of possible modeling viewpoints. Not all of these views may be used for any one project and other views may be defined as necessary. Note that for some analyses elements from multiple viewpoints may be combined into a new view, possibly using a layered representation.


They are studied in the eight queens puzzle, where eight non-attacking queens are placed on a standard 8 × 8 {\displaystyle 8\times 8} chessboard. Dominating sets represent arrangements of queens where every square is attacked or occupied by a queen; five queens, but no fewer, can dominate the 8 × 8 {\displaystyle 8\times 8} chessboard. Colourings of the graphs represent ways to colour each square so that a queen cannot move between any two squares of the same colour; at least n colours are needed for an n × n {\displaystyle n\times n} chessboard, but 9 colours are needed for the 8 × 8 {\displaystyle 8\times 8} board.


In actual implementation, that is not two separate steps; the DFT replaces the DTFT. So J.Cooley (pp. 77–78) describes the implementation as discrete finite Fourier transform.or another name for the Fourier series coefficients.or another name for one snapshot of a short-time Fourier transform.


In more technical terms, roaming refers to the ability for a cellular customer to automatically make and receive voice calls, send and receive data, or access other services, including home data services, when travelling outside the geographical coverage area of the home network, by means of using a visited network. For example: should a subscriber travel beyond their cell phone company's transmitter range, their cell phone would automatically hop onto another phone company's service, if available. The process is supported by the Telecommunication processes of mobility management, authentication, authorization and accounting billing procedures (known as AAA or 'triple A').


The RNN hierarchy can be "collapsed" into a single RNN, by "distilling" a higher level "chunker" network into a lower level "automatizer" network. In 1993, a chunker solved a deep learning task whose CAP depth exceeded 1000. Such history compressors can substantially facilitate downstream supervised deep learning.Geoffrey Hinton et al. (2006) proposed learning a high-level internal representation using successive layers of binary or real-valued latent variables with a restricted Boltzmann machine to model each layer.


(b) "Instrument" means a negotiable instrument. (c) An order that meets all of the requirements of subsection (a), except paragraph (1), and otherwise falls within the definition of "check" in subsection (f) is a negotiable instrument and a check. (d) A promise or order other than a check is not an instrument if, at the time it is issued or first comes into possession of a holder, it contains a conspicuous statement, however expressed, to the effect that the promise or order is not negotiable or is not an instrument governed by this Article.


In most of Europe, LPD433 band is allowed for license-free voice communication in addition to PMR446. Wireless network devices use wavebands as follows: IEEE 802.11/Wi-Fi 2450 MHz and 5800 MHz bands Bluetooth 2450 MHz band falls under WPANIEEE 802.15.4, Zigbee and other personal area networks may use the 915 MHz and 2450 MHz ISM bands because of frequency sharing between different allocations. Wireless LANs and cordless phones can also use bands other than those shared with ISM, but such uses require approval on a country by country basis.


The name "uniform norm" derives from the fact that a sequence of functions { f n } {\displaystyle \left\{f_{n}\right\}} converges to f {\displaystyle f} under the metric derived from the uniform norm if and only if f n {\displaystyle f_{n}} converges to f {\displaystyle f} uniformly.If f {\displaystyle f} is a continuous function on a closed and bounded interval, or more generally a compact set, then it is bounded and the supremum in the above definition is attained by the Weierstrass extreme value theorem, so we can replace the supremum by the maximum. In this case, the norm is also called the maximum norm. In particular, if x {\displaystyle x} is some vector such that x = ( x 1 , x 2 , … , x n ) {\displaystyle x=\left(x_{1},x_{2},\ldots ,x_{n}\right)} in finite dimensional coordinate space, it takes the form: ‖ x ‖ ∞ := max ( | x 1 | , … , | x n | ) . {\displaystyle \|x\|_{\infty }:=\max \left(\left|x_{1}\right|,\ldots ,\left|x_{n}\right|\right).}


{\displaystyle \left|S_{n}-\ell \right|<\varepsilon .} If the series is convergent, the (necessarily unique) number ℓ {\displaystyle \ell } is called the sum of the series. The same notation ∑ k = 1 ∞ a k {\displaystyle \sum _{k=1}^{\infty }a_{k}} is used for the series, and, if it is convergent, to its sum. This convention is similar to that which is used for addition: a + b denotes the operation of adding a and b as well as the result of this addition, which is called the sum of a and b. Any series that is not convergent is said to be divergent or to diverge.


In the UK, Ofcom requires that predictive dialers abandon fewer than 3% of answered calls on a daily basis. Ofcom also requires that if an agent is not available within 2 seconds the call is considered "abandoned" and an automated message is played. The automated message must identify the company making the call, the purpose of the call, a free phone or basic rate phone number to call back on and must not contain any form of marketing. A phone call to the return number must not be treated by the company as an opportunity to market, but to be removed from the calling list.


If the information word consists of n {\displaystyle n} bits, then the Berger code needs k = ⌈ log 2  ( n + 1 ) ⌉ {\displaystyle k=\lceil \log _{2}(n+1)\rceil } "check bits", giving a Berger code of length k+n. (In other words, the k {\displaystyle k} check bits are enough to check up to n = 2 k − 1 {\displaystyle n=2^{k}-1} information bits). Berger codes can detect any number of one-to-zero bit-flip errors, as long as no zero-to-one errors occurred in the same code word.


In some places and at some times, the one- and two-stroke variants have been used in the same contexts to distinguish between the U.S. dollar and other local currency, such as the former Portuguese escudo.However, such usage is not standardized, and furthermore the two versions are generally considered mere graphic variants of the same symbol—a typeface design choice. Computer and typewriter keyboards usually have a single key for that sign, and many character encodings (including ASCII and Unicode) reserve a single numeric code for it. Indeed, dollar signs in the same digital document may be rendered with one or two strokes, if different computer fonts are used, but the underlying codepoint U+0024 (ASCII 3610) remains unchanged.


Electronic news gathering, including live via satellite interviews, reporters' live shots, and sporting events are all examples of radio or television content that is backhauled to a station or network before being made available to the public through that station or network. Cable TV channels, particularly public, educational, and government access (PEG) along with (local origination) channels, may also backhauled to cable headends before making their way to the subscriber. Finished network feeds are not considered backhauls, even if local insertion is used to modify the content prior to final transmission.


In the case p = 2 {\displaystyle \textstyle p=2} , the inequality holds with A 2 = B 2 = 1 {\displaystyle \textstyle A_{2}=B_{2}=1} , and it reduces to the rule for the sum of variances of independent random variables with zero mean, known from elementary statistics: If E ( X i ) = 0 {\displaystyle \textstyle E\left(X_{i}\right)=0} and E ( | X i | 2 ) < + ∞ {\displaystyle \textstyle E\left(\left\vert X_{i}\right\vert ^{2}\right)<+\infty } , then V a r ( ∑ i = 1 n X i ) = E ( | ∑ i = 1 n X i | 2 ) = ∑ i = 1 n ∑ j = 1 n E ( X i X ¯ j ) = ∑ i = 1 n E ( | X i | 2 ) = ∑ i = 1 n V a r ( X i ) . {\displaystyle \mathrm {Var} \left(\sum _{i=1}^{n}X_{i}\right)=E\left(\left\vert \sum _{i=1}^{n}X_{i}\right\vert ^{2}\right)=\sum _{i=1}^{n}\sum _{j=1}^{n}E\left(X_{i}{\overline {X}}_{j}\right)=\sum _{i=1}^{n}E\left(\left\vert X_{i}\right\vert ^{2}\right)=\sum _{i=1}^{n}\mathrm {Var} \left(X_{i}\right).}


In set theory, an honest leftmost branch of a tree T on ω × γ is a branch (maximal chain) ƒ ∈  such that for each branch g ∈ , one has ∀ n ∈ ω: ƒ(n) ≤ g(n). Here,  denotes the set of branches of maximal length of T, ω is the smallest infinite ordinal (represented by the natural numbers N), and γ is some other ordinal.


In some countries of Europe, the European Telecommunications Standards Institute (ETSI) standards 200 778-1 and -2 – replacing 300 778-1 & -2 – allow 3 physical transport layers (Telcordia Technologies (formerly Bellcore), British Telecom (BT) and Cable Communications Association (CCA)), combined with 2 data formats Multiple Data Message Format (MDMF) & Single Data Message Format (SDMF), plus the Dual-tone multi-frequency (DTMF) system and a no-ring mode for meter-reading and the like. It's more of a recognition that the different types exist than an attempt to define a single "standard".


NLRP3, for example, recruits ASC adaptor protein via PYD-PYD interaction. Both pro-caspase-1 and ASC contain a caspase activation and recruitment domain (CARD), and this homotypic CARD-CARD interaction enables autocatalytic cleavage and reassembly of procaspase-1 to form active caspase-1. Alternatively, NLRC4 can directly recruit pro-caspase-1, as it has a CARD instead of a PYD.


Moreover, if the original cake is rectangular then each piece is a rectangle. Several years after this algorithm has been published, it was proved that envy-free partitions with connected pieces cannot be found by finite protocols. Hence, an approximation algorithm is the best that we can hope for in finite time.


In medical diagnosis, test sensitivity is the ability of a test to correctly identify those with the disease (true positive rate), whereas test specificity is the ability of the test to correctly identify those without the disease (true negative rate). If 100 patients known to have a disease were tested, and 43 test positive, then the test has 43% sensitivity. If 100 with no disease are tested and 96 return a completely negative result, then the test has 96% specificity.


In some Linux desktop environments a letter with double dots can be produced by pressing AltGr⇧ Shift:, then the letter. When the system has a compose key, the same procedure as that described at X-Windows (below) may be used.


In telecommunication, a variable length buffer or elastic buffer is a buffer into which data may be entered at one rate and removed at another rate without changing the data sequence. Most first-in first-out (FIFO) storage devices are variable-length buffers in that the input rate may be variable while the output rate is constant or the output rate may be variable while the input rate is constant. Various clocking and control systems are used to allow control of underflow or overflow conditions.


In the Japanese kantenji braille, the standard 8-dot Braille patterns 2568, 12568, 24568, and 124568 are the patterns related to Braille pattern dots-1456, since the two additional dots of kantenji patterns 01456, 14567, and 014567 are placed above the base 6-dot cell, instead of below, as in standard 8-dot braille.


Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cubic curves; see § Elliptic curves over a general field below.) An elliptic curve is an abelian variety – that is, it has a group law defined algebraically, with respect to which it is an abelian group – and O serves as the identity element.


Given any ideal I {\displaystyle I} it may occur that it is properly contained in some larger ideal J {\displaystyle J} with coefficients in the base field of I {\displaystyle I} ; then J {\displaystyle J} is called a divisor of I {\displaystyle I} . In general, a divisor in a ring of partial differential operators need not be principal. The greatest common right divisor (Gcrd) or sum of two ideals I {\displaystyle I} and J {\displaystyle J} is the smallest ideal with the property that both I {\displaystyle I} and J {\displaystyle J} are contained in it.


Noise exposure and its associated disease burden is likely to increase to a level where the disease burden is similar to that of traffic accidents. The rough estimates do not provide a complete picture of the environmental health burden, because data are uncertain, not all environmental-health relationships are known, not all environmental factors have been included, and it was not possible to assess all potential health effects. The effects of a number of these assumptions were evaluated in an uncertainty analysis.


In quantum teleportation, a sender wishes to transmit an arbitrary quantum state of a particle to a possibly distant receiver. Consequently, the teleportation process is a quantum channel. The apparatus for the process itself requires a quantum channel for the transmission of one particle of an entangled-state to the receiver. Teleportation occurs by a joint measurement of the sent particle and the remaining entangled particle. This measurement results in classical information which must be sent to the receiver to complete the teleportation. Importantly, the classical information can be sent after the quantum channel has ceased to exist.


In addition to formalizing mathematics, category theory is also used to formalize many other systems in computer science, such as the semantics of programming languages. Two categories are the same if they have the same collection of objects, the same collection of arrows, and the same associative method of composing any pair of arrows. Two different categories may also be considered "equivalent" for purposes of category theory, even if they do not have precisely the same structure.


In mathematics, the arithmetic–geometric mean of two positive real numbers x and y is the mutual limit of a sequence of arithmetic means and a sequence of geometric means: Begin the sequences with x and y: Then define the two interdependent sequences (an) and (gn) as These two sequences converge to the same number, the arithmetic–geometric mean of x and y; it is denoted by M(x, y), or sometimes by agm(x, y) or AGM(x, y). The arithmetic–geometric mean is used in fast algorithms for exponential and trigonometric functions, as well as some mathematical constants, in particular, computing π. The arithmetic–geometric mean can be extended to complex numbers and when the branches of the square root are allowed to be taken inconsistently, it is, in general, a multivalued function.


In phylogenetics, a single-access key (also called dichotomous key, sequential key, analytical key, or pathway key) is an identification key where the sequence and structure of identification steps is fixed by the author of the key. At each point in the decision process, multiple alternatives are offered, each leading to a result or a further choice. The alternatives are commonly called "leads", and the set of leads at a given point a "couplet". Single access keys are closely related to decision trees or self-balancing binary search trees.


In the 1970s and 1980s, attempts were made to build database systems with integrated hardware and software. The underlying philosophy was that such integration would provide higher performance at a lower cost. Examples were IBM System/38, the early offering of Teradata, and the Britton Lee, Inc. database machine. Another approach to hardware support for database management was ICL's CAFS accelerator, a hardware disk controller with programmable search capabilities.


In the expression named entity, the word named restricts the task to those entities for which one or many strings, such as words or phrases, stands (fairly) consistently for some referent. This is closely related to rigid designators, as defined by Kripke, although in practice NER deals with many names and referents that are not philosophically "rigid". For instance, the automotive company created by Henry Ford in 1903 can be referred to as Ford or Ford Motor Company, although "Ford" can refer to many other entities as well (see Ford). Rigid designators include proper names as well as terms for certain biological species and substances, but exclude pronouns (such as "it"; see coreference resolution), descriptions that pick out a referent by its properties (see also De dicto and de re), and names for kinds of things as opposed to individuals (for example "Bank").


From a mathematical perspective, random graphs are used to answer questions about the properties of typical graphs. Its practical applications are found in all areas in which complex networks need to be modeled – many random graph models are thus known, mirroring the diverse types of complex networks encountered in different areas. In a mathematical context, random graph refers almost exclusively to the Erdős–Rényi random graph model. In other contexts, any graph model may be referred to as a random graph.


Disagreements within ontology are often about whether entities belonging to a certain category exist and, if so, how they are related to other entities.When used as a countable noun, the words ontology and ontologies refer not to the science of being but to theories within the science of being. Ontological theories can be divided into various types according to their theoretical commitments. Monocategorical ontologies hold that there is only one basic category, but polycategorical ontologies rejected this view. Hierarchical ontologies assert that some entities exist on a more fundamental level and that other entities depend on them. Flat ontologies, on the other hand, deny such a privileged status to any entity.


In practice, the objective value may be noisy or even non-numerical, and hence its gradient information may be unreliable or unavailable. This is particularly true when the problem is multi-objective. At present, many designs and refinements are mainly made through a manual trial-and-error process with the help of a CAD simulation package. Usually, such a posteriori learning or adjustments need to be repeated many times until a ‘satisfactory’ or ‘optimal’ design emerges.


Quango (quasi-autonomous-non-governmental organisation) is a commonly used acronym to refer to a non-departmental public body. The Scottish public bodies is used to indicate all quangos and other organisations in Scotland.N.B. An entity described by the undefined term public body is not inevitably a statutory corporation.


In statistics, the Gauss–Markov theorem (or simply Gauss theorem for some authors) states that the ordinary least squares (OLS) estimator has the lowest sampling variance within the class of linear unbiased estimators, if the errors in the linear regression model are uncorrelated, have equal variances and expectation value of zero. The errors do not need to be normal, nor do they need to be independent and identically distributed (only uncorrelated with mean zero and homoscedastic with finite variance). The requirement that the estimator be unbiased cannot be dropped, since biased estimators exist with lower variance. See, for example, the James–Stein estimator (which also drops linearity), ridge regression, or simply any degenerate estimator.


The kernel is usually denoted ker f (or a variation). In symbols: ker  f = { a ∈ A: f ( a ) = e B } .


In multiprocessing, the processors can be used to execute a single sequence of instructions in multiple contexts (single instruction, multiple data or SIMD, often used in vector processing), multiple sequences of instructions in a single context (multiple instruction, single data or MISD, used for redundancy in fail-safe systems and sometimes applied to describe pipelined processors or hyper-threading), or multiple sequences of instructions in multiple contexts (multiple instruction, multiple data or MIMD).


It is important to know which notation is being used when working in different software programs. The respective ISO standard defines both the comma and the small dot as decimal markers, but does not explicitly define universal radix marks for bases other than 10. Fractional numbers are rarely displayed in other number bases, but, when they are, a radix character may be used for the same purpose. When used with the binary (base 2) representation, it may be called "binary point".


In terms of instruction set, ARM11 builds on the preceding ARM9 generation. It incorporates all ARM926EJ-S features and adds the ARMv6 instructions for media support (SIMD) and accelerating IRQ response. Microarchitecture improvements in ARM11 cores include: SIMD instructions which can double MPEG-4 and audio digital signal processing algorithm speed Cache is physically addressed, solving many cache aliasing problems and reducing context switch overhead. Unaligned and mixed-endian data access is supported.


The leftmost digit '1' of the result is then discarded. Discarding the leftmost '1' is especially convenient on calculators or computers that use a fixed number of digits: there is nowhere for it to go so it is simply lost during the calculation. The nines' complement plus one is known as the ten's complement. The method of complements can be extended to other number bases (radices); in particular, it is used on most digital computers to perform subtraction, represent negative numbers in base 2 or binary arithmetic and test underflow and overflow in calculation.


Rather the program's behavior is undefined. To make a fuzzer more sensitive to failures other than crashes, sanitizers can be used to inject assertions that crash the program when a failure is detected. There are different sanitizers for different kinds of bugs: to detect memory related errors, such as buffer overflows and use-after-free (using memory debuggers such as AddressSanitizer), to detect race conditions and deadlocks (ThreadSanitizer), to detect undefined behavior (UndefinedBehaviorSanitizer), to detect memory leaks (LeakSanitizer), or to check control-flow integrity (CFISanitizer).Fuzzing can also be used to detect "differential" bugs if a reference implementation is available.


In software engineering, blue–green (also blue/green) deployment is a method of installing changes to a web, app, or database server by swapping alternating production and staging servers.


The class separation in a direction w → {\displaystyle {\vec {w}}} in this case will be given by S = w → T Σ b w → w → T Σ w → {\displaystyle S={\frac {{\vec {w}}^{\mathrm {T} }\Sigma _{b}{\vec {w}}}{{\vec {w}}^{\mathrm {T} }\Sigma {\vec {w}}}}} This means that when w → {\displaystyle {\vec {w}}} is an eigenvector of Σ − 1 Σ b {\displaystyle \Sigma ^{-1}\Sigma _{b}} the separation will be equal to the corresponding eigenvalue. If Σ − 1 Σ b {\displaystyle \Sigma ^{-1}\Sigma _{b}} is diagonalizable, the variability between features will be contained in the subspace spanned by the eigenvectors corresponding to the C − 1 largest eigenvalues (since Σ b {\displaystyle \Sigma _{b}} is of rank C − 1 at most). These eigenvectors are primarily used in feature reduction, as in PCA.


In several mathematical areas, generalized distributivity laws are considered. This may involve the weakening of the above conditions or the extension to infinitary operations. Especially in order theory one finds numerous important variants of distributivity, some of which include infinitary operations, such as the infinite distributive law; others being defined in the presence of only one binary operation, such as the according definitions and their relations are given in the article distributivity (order theory). This also includes the notion of a completely distributive lattice.


A tensor may be expressed as a linear sum of the tensor product of vector and covector basis elements. The resulting tensor components are labelled by indices of the basis. Each index has one possible value per dimension of the underlying vector space.


They are specified within the LMF document. On the contrary, languageCoding, language, partOfSpeech, commonNoun, writtenForm, grammaticalNumber, singular, plural are data categories that are taken from the Data Category Registry. These marks adorn the structure.


In statutory interpretation, it refers to the problem of giving meaning to groups of words where one of the words is ambiguous or inherently unclear. For example, in road traffic law, a statute may require consideration of large vehicles separately from other vehicles. The word large is ambiguous per se, but may be considered heavy.


In mathematics, a diffiety () is a geometrical object which plays the same role in the modern theory of partial differential equations that algebraic varieties play for algebraic equations, that is, to encode the space of solutions in a more conceptual way. The term was coined in 1984 by Alexandre Mikhailovich Vinogradov as portmanteau from differential variety.


The JavaScript (E4X) extension explicitly defines two specific objects (XML and XMLList), which support XML document nodes and XML node lists as distinct objects and use a dot-notation specifying parent-child relationships. These data structures represent XML documents as a tree structure. An XML Tree represented graphically can be as simple as an ASCII chart or a more graphically complex hierarchy.


Equating the derivative of the Lagrangian with respect to the various probabilities to zero yields a functional form for those probabilities which corresponds to those used in logistic regression.As in the above section on multinomial logistic regression, we will consider M + 1 {\displaystyle M+1} explanatory variables denoted x m {\displaystyle x_{m}} and which include x 0 = 1 {\displaystyle x_{0}=1} . There will be a total of K data points, indexed by k = { 1 , 2 , … , K } {\displaystyle k=\{1,2,\dots ,K\}} , and the data points are given by x m k {\displaystyle x_{mk}} and y k {\displaystyle y_{k}} . The xmk will also be represented as an ( M + 1 ) {\displaystyle (M+1)} -dimensional vector x k = { x 0 k , x 1 k , … , x M k } {\displaystyle {\boldsymbol {x}}_{k}=\{x_{0k},x_{1k},\dots ,x_{Mk}\}} .


{\displaystyle X\times Y.} A binary relation is called a homogeneous relation when X = Y. A binary relation is also called a heterogeneous relation when it is not necessary that X = Y. Since relations are sets, they can be manipulated using set operations, including union, intersection, and complementation, and satisfying the laws of an algebra of sets.


In mathematical optimization, the proximal operator is an operator associated with a proper, lower semi-continuous convex function f {\displaystyle f} from a Hilbert space X {\displaystyle {\mathcal {X}}} to  {\displaystyle } , and is defined by: prox f  ( v ) = arg  min x ∈ X ( f ( x ) + 1 2 ‖ x − v ‖ X 2 ) . {\displaystyle \operatorname {prox} _{f}(v)=\arg \min _{x\in {\mathcal {X}}}\left(f(x)+{\frac {1}{2}}\|x-v\|_{\mathcal {X}}^{2}\right).} For any function in this class, the minimizer of the right-hand side above is unique, hence making the proximal operator well-defined. The proximal operator is used in proximal gradient methods, which is frequently used in optimization algorithms associated with non-differentiable optimization problems such as total variation denoising.


In the examples below, bold denotes the primary stress of the measure, and italics denote a secondary stress. Syllables such as "and" are frequently used for pulsing in between numbers. Simple: 34 is a simple triple meter time signature that represents three quarter notes (crotchets), usually perceived as three beats. In this case the subdivision would be the eighth note (quaver). It is felt as 34: one and two and three and ...Compound: Most often, 68 is felt as two beats, each being a dotted quarter note (crotchet), and each containing subdivisions of three eighth notes (quavers). It is felt as 68: one two three four five six ... (or, if counting dotted-quarter beats, one and a two and a)The table below shows the characteristics of the most frequently used time signatures.


In telephony, multi-frequency signaling (MF) is a type of signaling that was introduced by the Bell System after World War II. It uses a combination of audible tones for address (telephone number) transport and supervision signaling on trunk lines between central offices. The signaling is sent in-band over the same channel as the bearer channel used for voice traffic.


In microeconomics, quasiconcave utility functions imply that consumers have convex preferences. Quasiconvex functions are important also in game theory, industrial organization, and general equilibrium theory, particularly for applications of Sion's minimax theorem. Generalizing a minimax theorem of John von Neumann, Sion's theorem is also used in the theory of partial differential equations.


Rather, the appearance of any given colorant is inherent to its chemical and physical properties, the purity of such a substance being unrelated to whether it conforms to our arbitrary conception of an ideal hue. Moreover, the identity of gamut-optimizing primary colors is determined by the physiology underlying human color vision. Although no set of three primary paints can be mixed to obtain the complete color gamut perceived by humans, red, yellow, and blue are a poor choice if high-chroma mixtures are desired.


The PowerPC model uses a single fence instruction called the SYNC instruction. It is similar to the MB instruction, but with a little exception that reads can occur out of program order even if a SYNC is placed between two reads to the same location.


, A n {\displaystyle A_{1},A_{2},...,A_{n}} of sentences in its language, if A 1 , A 2 , . . .


In visual sensor networks (VSN), sensors are cameras which record images and video sequences. In many applications of VSN, a camera can't give a perfect illustration including all details of the scene. This is because of the limited depth of focus of the optical lens of cameras.


The term implies that the hidden complexity is at least in principle understandable, in contrast to black magic and deep magic (see Variants), which describe arcane techniques that are deliberately hidden or extremely difficult to understand. However, the term can also be applied endearingly, suggesting a "charm" about the code. The action of such abstractions is described as being done "automagically", a portmanteau of "automatically" and "magically".


In programming and information security, a buffer overflow or buffer overrun is an anomaly whereby a program writes data to a buffer beyond the buffer's allocated memory, overwriting adjacent memory locations. Buffers are areas of memory set aside to hold data, often while moving it from one section of a program to another, or between programs. Buffer overflows can often be triggered by malformed inputs; if one assumes all inputs will be smaller than a certain size and the buffer is created to be that size, then an anomalous transaction that produces more data could cause it to write past the end of the buffer. If this overwrites adjacent data or executable code, this may result in erratic program behavior, including memory access errors, incorrect results, and crashes.


The descriptors for degree of substitution, primary, secondary, tertiary, and quaternary, are translated as 伯 (bó), 仲 (zhòng), 叔 (shū), 季 (jì), which refer to the first, second, third, and fourth male siblings in a family. For instance, tert-butyllithium is translated as 叔丁基锂 (, , , ). Other commonly used isomeric descriptors normal-, iso-, and neo- are translated as 正 (zhèng, 'proper'), 异 (yì, 'different'), and 新 (xīn, 'new'), respectively.


In order to fully test that all the requirements of an application are met, there must be at least two test cases for each requirement: one positive test and one negative test. If a requirement has sub-requirements, each sub-requirement must have at least two test cases. Keeping track of the link between the requirement and the test is frequently done using a traceability matrix.


The problem is generally attributed to Harold Davenport in about 1927, though he did not publish it at the time. Davenport did not claim to be its discoverer "because he could not believe that it had not been stated earlier". The first publication of a version of the birthday problem was by Richard von Mises in 1939.


Then for each γ ∈ S , {\displaystyle \gamma \in S,} γ ∈ C β {\displaystyle \gamma \in C_{\beta }} for all β < γ . {\displaystyle \beta <\gamma .} Since each C β {\displaystyle C_{\beta }} is closed, α ∈ C β {\displaystyle \alpha \in C_{\beta }} for all β < α , {\displaystyle \beta <\alpha ,} so α ∈ C .


In positional voting, voters complete a ranked ballot by expressing their preferences in rank order. The rank position of each voter preference is allotted a specific fixed weighting. Typically, the higher the rank of the preference, the more points it is worth. Occasionally, it may share the same weighting as a lower-ranked preference but it is never worth fewer points.


In terms of the relational model of databases, a table can be considered a convenient representation of a relation, but the two are not strictly equivalent. For instance, a SQL table can potentially contain duplicate rows, whereas a true relation cannot contain duplicate rows that we call tuples. Similarly, representation as a table implies a particular ordering to the rows and columns, whereas a relation is explicitly unordered.


The cumulants of the sum of the grouped variable and the uniform variable are the sums of the cumulants. As odd cumulants of a uniform distribution are zero; only even moments are affected. The second and fourth cumulants of the uniform distribution on (−0.5c, 0.5c) are respectively, c2/12 and −c4/120. The correction to moments can be derived from the relation between cumulants and moments.


Project planning resource leveling is the process of resolving these conflicts. It can also be used to balance the workload of primary resources over the course of the project, usually at the expense of one of the traditional triple constraints (time, cost, scope). When using specially designed project software, leveling typically means resolving conflicts or over allocations in the project plan by allowing the software to calculate delays and update tasks automatically.


This implies that each element q i ∈ { 0 , 1 , . .


In order to fall within the obligation for disclosure, the matter must meet three conditions: it must be legal authority, it must be directly adverse, and it must be from a controlling jurisdiction. Legal authority The matter for which disclosure is compelled must be a matter of decided law, rather than a mere opinion rendered by an academic. For example, where an attorney is arguing that a certain transfer of assets should be permitted in a bankruptcy proceeding, that attorney need not disclose a law review article, casebook, or op-ed piece in which the author argues that exactly such a transfer of assets should never be permitted. Such a writing need not be disclosed even if the author of the article is a leading expert on the area of law, and the article contains strong and well-written arguments in support of the position.


In mathematics, a Minkowski plane (named after Hermann Minkowski) is one of the Benz planes (the others being Möbius plane and Laguerre plane).


In mathematics, a pandigital number is an integer that in a given base has among its significant digits each digit used in the base at least once. For example, 1234567890 (one billion two hundred thirty four million five hundred sixty seven thousand eight hundred ninety) is a pandigital number in base 10. The first few pandigital base 10 numbers are given by (sequence A171102 in the OEIS): 1023456789, 1023456798, 1023456879, 1023456897, 1023456978, 1023456987, 1023457689The smallest pandigital number in a given base b is an integer of the form b b − 1 + ∑ d = 2 b − 1 d b b − 1 − d = b b − b ( b − 1 ) 2 + ( b − 1 ) × b b − 2 − 1 {\displaystyle b^{b-1}+\sum _{d=2}^{b-1}db^{b-1-d}={\frac {b^{b}-b}{(b-1)^{2}}}+(b-1)\times b^{b-2}-1} The following table lists the smallest pandigital numbers of a few selected bases. OEIS: A049363 gives the base 10 values for the first 18 bases.