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A suitable generalization of the first definition is: Let D {\displaystyle D} be a subset of R n . {\displaystyle \mathbb {R} ^{n}.} | Wikipedia - Computable real function - Summary |
A function f: D → R {\displaystyle f\colon D\to \mathbb {R} } is sequentially computable if, for every n {\displaystyle n} -tuplet ( { x i 1 } i = 1 ∞ , … { x i n } i = 1 ∞ ) {\displaystyle \left(\{x_{i\,1}\}_{i=1}^{\infty },\ldots \{x_{i\,n}\}_{i=1}^{\infty }\right)} of computable sequences of real numbers such that (... | Wikipedia - Computable real function - Summary |
In mathematical logic, specifically in the discipline of model theory, the Fraïssé limit (also called the Fraïssé construction or Fraïssé amalgamation) is a method used to construct (infinite) mathematical structures from their (finite) substructures. It is a special example of the more general concept of a direct limi... | Wikipedia - Fraïssé limit - Summary |
Given a class K {\displaystyle \mathbf {K} } of finite relational structures, if K {\displaystyle \mathbf {K} } satisfies certain properties (described below), then there exists a unique countable structure Flim ( K ) {\displaystyle \operatorname {Flim} (\mathbf {K} )} , called the Fraïssé limit of K {\displaystyle \... | Wikipedia - Fraïssé limit - Summary |
In mathematical logic, stratification is any consistent assignment of numbers to predicate symbols guaranteeing that a unique formal interpretation of a logical theory exists. Specifically, we say that a set of clauses of the form Q 1 ∧ ⋯ ∧ Q n ∧ ¬ Q n + 1 ∧ ⋯ ∧ ¬ Q n + m → P {\displaystyle Q_{1}\wedge \dots \wedge Q_{... | Wikipedia - Stratified formula - In mathematical logic |
In mathematical logic, structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof, a kind of proof whose semantic properties are exposed. When all the theorems of a logic formalised in a structural proof theory have analytic proofs, then the proof the... | Wikipedia - Display logic - Summary |
In mathematical logic, such concepts as primitive recursive functions and μ-recursive functions represent integer-valued functions of several natural variables or, in other words, functions on Nn. Gödel numbering, defined on well-formed formulae of some formal language, is a natural-valued function. Computability theor... | Wikipedia - Integer-valued function - Mathematical logic and computability theory |
In mathematical logic, the Barwise compactness theorem, named after Jon Barwise, is a generalization of the usual compactness theorem for first-order logic to a certain class of infinitary languages. It was stated and proved by Barwise in 1967. | Wikipedia - Barwise compactness theorem - Summary |
In mathematical logic, the Borel hierarchy is a stratification of the Borel algebra generated by the open subsets of a Polish space; elements of this algebra are called Borel sets. Each Borel set is assigned a unique countable ordinal number called the rank of the Borel set. The Borel hierarchy is of particular interes... | Wikipedia - Borel hierarchy - Summary |
In mathematical logic, the Brouwer–Heyting–Kolmogorov interpretation, or BHK interpretation, of intuitionistic logic was proposed by L. E. J. Brouwer and Arend Heyting, and independently by Andrey Kolmogorov. It is also sometimes called the realizability interpretation, because of the connection with the realizability ... | Wikipedia - Brouwer–Heyting–Kolmogorov interpretation - Summary |
In mathematical logic, the Cantor–Dedekind axiom is the thesis that the real numbers are order-isomorphic to the linear continuum of geometry. In other words, the axiom states that there is a one-to-one correspondence between real numbers and points on a line. This axiom became a theorem proved by Emil Artin in his boo... | Wikipedia - Cantor–Dedekind axiom - Summary |
Analytic geometry was developped from the Cartesian coordinate system introduced by René Descartes. It implicitly assumed this axiom by blending the distinct concepts of real numbers and points on a line, sometimes referred to as the real number line. Artin's proof, not only makes this blend explicity, but also that an... | Wikipedia - Cantor–Dedekind axiom - Summary |
In mathematical logic, the De Bruijn index is a tool invented by the Dutch mathematician Nicolaas Govert de Bruijn for representing terms of lambda calculus without naming the bound variables. Terms written using these indices are invariant with respect to α-conversion, so the check for α-equivalence is the same as tha... | Wikipedia - De Bruijn index - Summary |
λy. x, sometimes called the K combinator, is written as λ λ 2 with De Bruijn indices. The binder for the occurrence x is the second λ in scope. | Wikipedia - De Bruijn index - Summary |
The term λx. λy. λz. | Wikipedia - De Bruijn index - Summary |
x z (y z) (the S combinator), with De Bruijn indices, is λ λ λ 3 1 (2 1). The term λz. (λy. | Wikipedia - De Bruijn index - Summary |
y (λx. x)) (λx. z x) is λ (λ 1 (λ 1)) (λ 2 1). See the following illustration, where the binders are coloured and the references are shown with arrows. De Bruijn indices are commonly used in higher-order reasoning systems such as automated theorem provers and logic programming systems. | Wikipedia - De Bruijn index - Summary |
In mathematical logic, the De Bruijn notation is a syntax for terms in the λ calculus invented by the Dutch mathematician Nicolaas Govert de Bruijn. It can be seen as a reversal of the usual syntax for the λ calculus where the argument in an application is placed next to its corresponding binder in the function instead... | Wikipedia - De Bruijn notation - Summary |
In mathematical logic, the Friedberg–Muchnik theorem is a theorem about Turing reductions that was proven independently by Albert Muchnik and Richard Friedberg in the middle of the 1950s. It is a more general view of the Kleene–Post theorem. The Kleene–Post theorem states that there exist incomparable languages A and B... | Wikipedia - Friedberg–Muchnik theorem - Summary |
In mathematical logic, the Friedman translation is a certain transformation of intuitionistic formulas. Among other things it can be used to show that the Π02-theorems of various first-order theories of classical mathematics are also theorems of intuitionistic mathematics. It is named after its discoverer, Harvey Fried... | Wikipedia - Friedman translation - Summary |
In mathematical logic, the Hilbert–Bernays provability conditions, named after David Hilbert and Paul Bernays, are a set of requirements for formalized provability predicates in formal theories of arithmetic (Smith 2007:224). These conditions are used in many proofs of Kurt Gödel's second incompleteness theorem. They a... | Wikipedia - Hilbert–Bernays provability conditions - Summary |
In mathematical logic, the Kanamori–McAloon theorem, due to Kanamori & McAloon (1987), gives an example of an incompleteness in Peano arithmetic, similar to that of the Paris–Harrington theorem. They showed that a certain finitistic theorem in Ramsey theory is not provable in Peano arithmetic (PA). | Wikipedia - Kanamori–McAloon theorem - Summary |
In mathematical logic, the Lindenbaum–Tarski algebra (or Lindenbaum algebra) of a logical theory T consists of the equivalence classes of sentences of the theory (i.e., the quotient, under the equivalence relation ~ defined such that p ~ q exactly when p and q are provably equivalent in T). That is, two sentences are e... | Wikipedia - Lindenbaum algebra - Summary |
The algebra is named for logicians Adolf Lindenbaum and Alfred Tarski. Starting in the academic year 1926-1927, Lindenbaum pioneered his method in Jan Łukasiewicz's mathematical logic seminar, and the method was popularized and generalized in subsequent decades through work by Tarski. The Lindenbaum–Tarski algebra is c... | Wikipedia - Lindenbaum algebra - Summary |
In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf Skolem. The precise formulation is given below. It implies that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has... | Wikipedia - Löwenheim–Skolem theorem - Summary |
As a consequence, first-order theories are unable to control the cardinality of their infinite models. The (downward) Löwenheim–Skolem theorem is one of the two key properties, along with the compactness theorem, that are used in Lindström's theorem to characterize first-order logic. In general, the Löwenheim–Skolem th... | Wikipedia - Löwenheim–Skolem theorem - Summary |
In mathematical logic, the Mostowski collapse lemma, also known as the Shepherdson–Mostowski collapse, is a theorem of set theory introduced by Andrzej Mostowski (1949, theorem 3) and John Shepherdson (1953). | Wikipedia - Mostowski collapse - Summary |
In mathematical logic, the Paris–Harrington theorem states that a certain combinatorial principle in Ramsey theory, namely the strengthened finite Ramsey theorem, which is expressible in Peano arithmetic, is not provable in this system. The combinatorial principle is however provable in slightly stronger systems. This ... | Wikipedia - Paris–Harrington theorem - Summary |
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th-century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research... | Wikipedia - Peano axiom - Summary |
In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic. In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of them as a collection of axioms in his book The principles of arithmetic presented by a new... | Wikipedia - Peano axiom - Summary |
The first axiom asserts the existence of at least one member of the set of natural numbers. The next four are general statements about equality; in modern treatments these are often not taken as part of the Peano axioms, but rather as axioms of the "underlying logic". The next three axioms are first-order statements ab... | Wikipedia - Peano axiom - Summary |
In mathematical logic, the Scott–Curry theorem is a result in lambda calculus stating that if two non-empty sets of lambda terms A and B are closed under beta-convertibility then they are recursively inseparable. | Wikipedia - Scott–Curry theorem - Summary |
In mathematical logic, the ancestral relation (often shortened to ancestral) of a binary relation R is its transitive closure, however defined in a different way, see below. Ancestral relations make their first appearance in Frege's Begriffsschrift. Frege later employed them in his Grundgesetze as part of his definitio... | Wikipedia - Ancestral relation - Summary |
In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej Mostowski) classifies certain sets based on the complexity of formulas that define them. Any set that receives a classification is called arithmetical. The arithmet... | Wikipedia - Arithmetical reducibility - Summary |
In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful (but generally not effective) method for constructing models of any set of sentences that... | Wikipedia - Compactness theorem - Summary |
Likewise, it is analogous to the finite intersection property characterization of compactness in topological spaces: a collection of closed sets in a compact space has a non-empty intersection if every finite subcollection has a non-empty intersection. The compactness theorem is one of the two key properties, along wit... | Wikipedia - Compactness theorem - Summary |
In mathematical logic, the cut rule is an inference rule of sequent calculus. It is a generalisation of the classical modus ponens inference rule. Its meaning is that, if a formula A appears as a conclusion in one proof and a hypothesis in another, then another proof in which the formula A does not appear can be deduce... | Wikipedia - Cut rule - Summary |
In mathematical logic, the diagonal lemma (also known as diagonalization lemma, self-reference lemma or fixed point theorem) establishes the existence of self-referential sentences in certain formal theories of the natural numbers—specifically those theories that are strong enough to represent all computable functions.... | Wikipedia - Diagonal lemma - Summary |
In mathematical logic, the disjunction and existence properties are the "hallmarks" of constructive theories such as Heyting arithmetic and constructive set theories (Rathjen 2005). | Wikipedia - Disjunction property - Summary |
In mathematical logic, the implicational propositional calculus is a version of classical propositional calculus which uses only one connective, called implication or conditional. In formulas, this binary operation is indicated by "implies", "if ..., then ...", "→", " → {\displaystyle \rightarrow } ", etc.. | Wikipedia - Implicational propositional calculus - Summary |
In mathematical logic, the intersection type discipline is a branch of type theory encompassing type systems that use the intersection type constructor ( ∩ ) {\displaystyle (\cap )} to assign multiple types to a single term. In particular, if a term M {\displaystyle M} can be assigned both the type φ 1 {\displaystyle \... | Wikipedia - Intersection type discipline - Summary |
( x x ) {\displaystyle \lambda x.\! (x\;x)} can be assigned the type ( ( α → β ) ∩ α ) → β {\displaystyle ((\alpha \to \beta )\cap \alpha )\to \beta } in most intersection type systems, assuming for the term variable x {\displaystyle x} both the function type α → β {\displaystyle \alpha \to \beta } and the correspondin... | Wikipedia - Intersection type discipline - Summary |
Prominent intersection type systems include the Coppo–Dezani type assignment system, the Barendregt-Coppo–Dezani type assignment system, and the essential intersection type assignment system. Most strikingly, intersection type systems are closely related to (and often exactly characterize) normalization properties of λ... | Wikipedia - Intersection type discipline - Summary |
In mathematical logic, the primitive recursive functionals are a generalization of primitive recursive functions into higher type theory. They consist of a collection of functions in all pure finite types. The primitive recursive functionals are important in proof theory and constructive mathematics. They are a central... | Wikipedia - Primitive recursive functional - Summary |
In mathematical logic, the quantifier rank of a formula is the depth of nesting of its quantifiers. It plays an essential role in model theory. Notice that the quantifier rank is a property of the formula itself (i.e. the expression in a language). Thus two logically equivalent formulae can have different quantifier ra... | Wikipedia - Quantifier rank - Summary |
In mathematical logic, the rules of passage govern how quantifiers distribute over the basic logical connectives of first-order logic. The rules of passage govern the "passage" (translation) from any formula of first-order logic to the equivalent formula in prenex normal form, and vice versa. | Wikipedia - Rules of passage (logic) - Summary |
In mathematical logic, the spectrum of a sentence is the set of natural numbers occurring as the size of a finite model in which a given sentence is true. By a result in descriptive complexity, a set of natural numbers is a spectrum if and only if it can be recognized in non-deterministic exponential time. | Wikipedia - Spectrum of a sentence - Summary |
In mathematical logic, the theory of infinite sets was first developed by Georg Cantor. Although this work has become a thoroughly standard fixture of classical set theory, it has been criticized in several areas by mathematicians and philosophers. Cantor's theorem implies that there are sets having cardinality greater... | Wikipedia - Controversy over Cantor's theory - Summary |
This argument can be improved by using a definition he gave later. The resulting argument uses only five axioms of set theory. | Wikipedia - Controversy over Cantor's theory - Summary |
Cantor's set theory was controversial at the start, but later became largely accepted. Most modern mathematics textbooks implicitly use Cantor's views on mathematical infinity. For example, a line is generally presented as the infinite set of its points, and it is commonly taught that there are more real numbers than r... | Wikipedia - Controversy over Cantor's theory - Summary |
In mathematical logic, there are several formal systems of "fuzzy logic", most of which are in the family of t-norm fuzzy logics. | Wikipedia - Fuzzy logics - Logical analysis |
In mathematical logic, true arithmetic is the set of all true first-order statements about the arithmetic of natural numbers. This is the theory associated with the standard model of the Peano axioms in the language of the first-order Peano axioms. True arithmetic is occasionally called Skolem arithmetic, though this t... | Wikipedia - True arithmetic - Summary |
In mathematical logic, two theories are equiconsistent if the consistency of one theory implies the consistency of the other theory, and vice versa. In this case, they are, roughly speaking, "as consistent as each other". In general, it is not possible to prove the absolute consistency of a theory T. Instead we usually... | Wikipedia - Consistency strength - Summary |
In mathematical logic, various sublanguages of set theory are decidable. These include: Sets with Monotone, Additive, and Multiplicative Functions. Sets with restricted quantifiers. == References == | Wikipedia - Decidable sublanguages of set theory - Summary |
In mathematical logic, weak interpretability is a notion of translation of logical theories, introduced together with interpretability by Alfred Tarski in 1953. Let T and S be formal theories. Slightly simplified, T is said to be weakly interpretable in S if, and only if, the language of T can be translated into the la... | Wikipedia - Weak interpretability - Summary |
In mathematical measure theory, for every positive integer n the ham sandwich theorem states that given n measurable "objects" in n-dimensional Euclidean space, it is possible to divide each one of them in half (with respect to their measure, e.g. volume) with a single (n − 1)-dimensional hyperplane. This is even possi... | Wikipedia - Ham sandwich theorem - Summary |
In mathematical modeling of social networks, link-centric preferential attachment is a node's propensity to re-establish links to nodes it has previously been in contact with in time-varying networks. This preferential attachment model relies on nodes keeping memory of previous neighbors up to the current time. | Wikipedia - Link-centric preferential attachment - Summary |
In mathematical modeling, a guess value is more commonly called a starting value or initial value. These are necessary for most optimization problems which use search algorithms, because those algorithms are mainly deterministic and iterative, and they need to start somewhere. One common type of application is nonlinea... | Wikipedia - Guess value - Summary |
In mathematical modeling, deterministic simulations contain no random variables and no degree of randomness, and consist mostly of equations, for example difference equations. These simulations have known inputs and they result in a unique set of outputs. Contrast stochastic (probability) simulation, which includes ran... | Wikipedia - Deterministic simulation - Summary |
Deterministic simulation models are usually designed to capture some underlying mechanism or natural process. They are different from statistical models (for example linear regression) whose aim is to empirically estimate the relationships between variables. The deterministic model is viewed as a useful approximation o... | Wikipedia - Deterministic simulation - Summary |
In mathematical modeling, overfitting is "the production of an analysis that corresponds too closely or exactly to a particular set of data, and may therefore fail to fit to additional data or predict future observations reliably". An overfitted model is a mathematical model that contains more parameters than can be ju... | Wikipedia - Underfitting - Summary |
: 45 Underfitting occurs when a mathematical model cannot adequately capture the underlying structure of the data. An under-fitted model is a model where some parameters or terms that would appear in a correctly specified model are missing. | Wikipedia - Underfitting - Summary |
Under-fitting would occur, for example, when fitting a linear model to non-linear data. Such a model will tend to have poor predictive performance. The possibility of over-fitting exists because the criterion used for selecting the model is not the same as the criterion used to judge the suitability of a model. | Wikipedia - Underfitting - Summary |
For example, a model might be selected by maximizing its performance on some set of training data, and yet its suitability might be determined by its ability to perform well on unseen data; then over-fitting occurs when a model begins to "memorize" training data rather than "learning" to generalize from a trend. As an ... | Wikipedia - Underfitting - Summary |
Such a model, though, will typically fail severely when making predictions. The potential for overfitting depends not only on the number of parameters and data but also the conformability of the model structure with the data shape, and the magnitude of model error compared to the expected level of noise or error in the... | Wikipedia - Underfitting - Summary |
In particular, the value of the coefficient of determination will shrink relative to the original data. To lessen the chance or amount of overfitting, several techniques are available (e.g., model comparison, cross-validation, regularization, early stopping, pruning, Bayesian priors, or dropout). The basis of some tech... | Wikipedia - Underfitting - Summary |
In mathematical modeling, resilience refers to the ability of a dynamical system to recover from perturbations and return to its original stable steady state. It is a measure of the stability and robustness of a system in the face of changes or disturbances. If a system is not resilient enough, it is more susceptible t... | Wikipedia - Resilience (mathematics) - Summary |
A resilient steady state corresponds to a ball in a deep valley, so any push or perturbation will very quickly lead the ball to return to the resting point where it started. On the other hand, a less resilient steady state corresponds to a ball in a shallow valley, so the ball will take a much longer time to return to ... | Wikipedia - Resilience (mathematics) - Summary |
In mathematical modeling, the dependent variable is studied to see if and how much it varies as the independent variables vary. In the simple stochastic linear model yi = a + bxi + ei the term yi is the ith value of the dependent variable and xi is the ith value of the independent variable. The term ei is known as the ... | Wikipedia - Extraneous variables - In modeling and statistics |
This is also called a bivariate dataset, (x1, y1)(x2, y2) ...(xi, yi). The simple linear regression model takes the form of Yi = a + Bxi + Ui, for i = 1, 2, ... , n. In this case, Ui, ... ,Un are independent random variables. This occurs when the measurements do not influence each other. | Wikipedia - Extraneous variables - In modeling and statistics |
Through propagation of independence, the independence of Ui implies independence of Yi, even though each Yi has a different expectation value. Each Ui has an expectation value of 0 and a variance of σ2. Expectation of Yi Proof: E = E = α + β x i + E = α + β x i . | Wikipedia - Extraneous variables - In modeling and statistics |
{\displaystyle E=E=\alpha +\beta x_{i}+E=\alpha +\beta x_{i}.} The line of best fit for the bivariate dataset takes the form y = α + βx and is called the regression line. α and β correspond to the intercept and slope, respectively.In an experiment, the variable manipulated by an experimenter is something that is proven... | Wikipedia - Extraneous variables - In modeling and statistics |
The dependent variable is the event expected to change when the independent variable is manipulated.In data mining tools (for multivariate statistics and machine learning), the dependent variable is assigned a role as target variable (or in some tools as label attribute), while an independent variable may be assigned a... | Wikipedia - Extraneous variables - In modeling and statistics |
In mathematical modelling of infectious disease, the dynamics of spreading is usually described through a set of non-linear ordinary differential equations (ODE). So there is always n {\displaystyle n} coupled equations of form C i ˙ = d C i d t = f ( C 1 , C 2 , . . . | Wikipedia - Effective reproduction number - Next-generation matrix for compartmental models |
, C n ) {\displaystyle {\dot {C_{i}}}={\operatorname {d} \!C_{i} \over \operatorname {d} \!t}=f(C_{1},C_{2},...,C_{n})} which shows how the number of people in compartment C i {\displaystyle C_{i}} changes over time. For example, in a SIR model, C 1 = S {\displaystyle C_{1}=S} , C 2 = I {\displaystyle C_{2}=I} , and C ... | Wikipedia - Effective reproduction number - Next-generation matrix for compartmental models |
In other words, as a rule, there is an infection-free steady state. This solution, also usually ensures that the disease-free equilibrium is also an equilibrium of the system. There is another fixed point known as an Endemic Equilibrium (EE) where the disease is not totally eradicated and remains in the population. | Wikipedia - Effective reproduction number - Next-generation matrix for compartmental models |
Mathematically, R 0 {\displaystyle R_{0}} is a threshold for stability of a disease-free equilibrium such that: R 0 ≤ 1 ⇒ lim t → ∞ ( C 1 ( t ) , C 2 ( t ) , ⋯ , C n ( t ) ) = DFE {\displaystyle R_{0}\leq 1\Rightarrow \lim _{t\to \infty }(C_{1}(t),C_{2}(t),\cdots ,C_{n}(t))={\textrm {DFE}}} R 0 > 1 , I ( 0 ) > 0 ⇒ lim ... | Wikipedia - Effective reproduction number - Next-generation matrix for compartmental models |
Epidemiologically, the linearisation reflects that R 0 {\displaystyle R_{0}} characterizes the potential for initial spread of an infectious person in a naive population, assuming the change in the susceptible population is negligible during the initial spread. A linear system of ODEs can always be described by a matri... | Wikipedia - Effective reproduction number - Next-generation matrix for compartmental models |
Note that this operator (matrix) is responsible for the number of infected people, not all the compartments. Iteration of this operator describes the initial progression of infection within the heterogeneous population. So comparing the spectral radius of this operator to unity determines whether the generations of inf... | Wikipedia - Effective reproduction number - Next-generation matrix for compartmental models |
In mathematical morphology and digital image processing, a morphological gradient is the difference between the dilation and the erosion of a given image. It is an image where each pixel value (typically non-negative) indicates the contrast intensity in the close neighborhood of that pixel. It is useful for edge detect... | Wikipedia - Morphological Gradient - Summary |
In mathematical morphology and digital image processing, a top-hat transform is an operation that extracts small elements and details from given images. There exist two types of top-hat transform: the white top-hat transform is defined as the difference between the input image and its opening by some structuring elemen... | Wikipedia - Top-hat transform - Summary |
In mathematical morphology, a structuring element is a shape, used to probe or interact with a given image, with the purpose of drawing conclusions on how this shape fits or misses the shapes in the image. It is typically used in morphological operations, such as dilation, erosion, opening, and closing, as well as the ... | Wikipedia - Structuring element - Summary |
There are two main characteristics that are directly related to structuring elements: Shape. For example, the structuring element can be a "ball" or a line; convex or a ring, etc. By choosing a particular structuring element, one sets a way of differentiating some objects (or parts of objects) from others, according to... | Wikipedia - Structuring element - Summary |
In mathematical morphology, hit-or-miss transform is an operation that detects a given configuration (or pattern) in a binary image, using the morphological erosion operator and a pair of disjoint structuring elements. The result of the hit-or-miss transform is the set of positions where the first structuring element f... | Wikipedia - Hit-or-miss transform - Summary |
In mathematical morphology, the closing of a set (binary image) A by a structuring element B is the erosion of the dilation of that set, A ∙ B = ( A ⊕ B ) ⊖ B , {\displaystyle A\bullet B=(A\oplus B)\ominus B,\,} where ⊕ {\displaystyle \oplus } and ⊖ {\displaystyle \ominus } denote the dilation and erosion, respectively... | Wikipedia - Closing (morphology) - Summary |
In mathematical morphology, the h-maxima transform is a morphological operation used to filter local maxima of an image based on local contrast information. First, all local maxima are defined as connected pixels in a given neighborhood with intensity level greater than pixels outside the neighborhood. Second, all loca... | Wikipedia - H-maxima transform - Summary |
In mathematical notation for numbers, a signed-digit representation is a positional numeral system with a set of signed digits used to encode the integers. Signed-digit representation can be used to accomplish fast addition of integers because it can eliminate chains of dependent carries. In the binary numeral system, ... | Wikipedia - Signed-digit representation - Summary |
In mathematical notation the inverse square law can be expressed as an intensity (I) varying as a function of distance (d) from some centre. The intensity is proportional (see ∝) to the reciprocal of the square of the distance thus: It can also be mathematically expressed as: or as the formulation of a constant quantit... | Wikipedia - Inverse-square law - Formula |
In mathematical notation, ordered set operators indicate whether an object precedes or succeeds another. These relationship operators are denoted by the unicode symbols U+227A-F, along with symbols located unicode blocks U+228x through U+22Ex. | Wikipedia - Ordered set operators - Summary |
In mathematical numeral systems the radix r is usually the number of unique digits, including zero, that a positional numeral system uses to represent numbers. In some cases, such as with a negative base, the radix is the absolute value r = | b | {\displaystyle r=|b|} of the base b. For example, for the decimal system ... | Wikipedia - Place value system - Base of the numeral system |
The highest symbol of a positional numeral system usually has the value one less than the value of the radix of that numeral system. The standard positional numeral systems differ from one another only in the base they use. The radix is an integer that is greater than 1, since a radix of zero would not have any digits,... | Wikipedia - Place value system - Base of the numeral system |
Negative bases are rarely used. In a system with more than | b | {\displaystyle |b|} unique digits, numbers may have many different possible representations. | Wikipedia - Place value system - Base of the numeral system |
It is important that the radix is finite, from which follows that the number of digits is quite low. Otherwise, the length of a numeral would not necessarily be logarithmic in its size. (In certain non-standard positional numeral systems, including bijective numeration, the definition of the base or the allowed digits ... | Wikipedia - Place value system - Base of the numeral system |
In mathematical optimization and computer science, heuristic (from Greek εὑρίσκω "I find, discover") is a technique designed for problem solving more quickly when classic methods are too slow for finding an exact or approximate solution, or when classic methods fail to find any exact solution. This is achieved by tradi... | Wikipedia - Heuristic search - Summary |
In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event. An optimization problem seeks to minimize a ... | Wikipedia - Zero-one loss function - Summary |
The loss function could include terms from several levels of the hierarchy. In statistics, typically a loss function is used for parameter estimation, and the event in question is some function of the difference between estimated and true values for an instance of data. The concept, as old as Laplace, was reintroduced ... | Wikipedia - Zero-one loss function - Summary |
In the context of economics, for example, this is usually economic cost or regret. In classification, it is the penalty for an incorrect classification of an example. In actuarial science, it is used in an insurance context to model benefits paid over premiums, particularly since the works of Harald Cramér in the 1920s... | Wikipedia - Zero-one loss function - Summary |
In mathematical optimization and related fields, relaxation is a modeling strategy. A relaxation is an approximation of a difficult problem by a nearby problem that is easier to solve. A solution of the relaxed problem provides information about the original problem. For example, a linear programming relaxation of an i... | Wikipedia - Relaxation (approximation) - Summary |
A Lagrangian relaxation of a complicated problem in combinatorial optimization penalizes violations of some constraints, allowing an easier relaxed problem to be solved. Relaxation techniques complement or supplement branch and bound algorithms of combinatorial optimization; linear programming and Lagrangian relaxation... | Wikipedia - Relaxation (approximation) - Summary |
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... | Wikipedia - Lagrange duality - Summary |
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