In mathematical logic , specifically recursion theory computability theory , a range function math f colon mathbb R to mathbb R math is sequentially computable if, for every computable sequence math x i i 1 infty math of real number s, the sequence math f x i i 1 infty math is also computable real number computable . A function math f colon mathbb R to mathbb R math is effectively uniformly continuous if there exists a primitive recursive function recursive function math d colon mathbb N to mathbb N math such that, if math x y 1 over d n math then math f x f y 1 over n math A real function is computable if it is both sequentially computable and effectively uniformly continuous. These definition s can be generalized to functions of more than one Variable mathematics variable or functions only defined on a subset of math mathbb R n. math The generalizations of the latter two need not be restated. A suitable generalization of the first definition is Let math D math be a subset of math mathbb R n. math A function math f colon D to mathbb R math is sequentially computable if, for every math n math tuplet math left x i , 1 i 1 infty, ldots x i , n i 1 infty right math of computable sequences of real numbers such that math forall i quad x i , 1 , ldots x i , n in D qquad , math the sequence math f x i i 1 infty math is also computable. planetmath id 6248 title Computable real function Category Computable analysis math stub ... more details
In mathematics , computable analysis is the study of which parts of real analysis and functional analysis can be carried out in a computability theory computable manner. It is closely related to constructive analysis . Basic results The computable real number s form a real closed field . The equality mathematics equality relation on computable real numbers is not computable, but for unequal computable real numbers the order relation is computable. Computable real function s map computable real numbers to computable real numbers. The function composition composition of computable real functions is again computable. Every computable real function is continuous function continuous . See also Specker sequence References Oliver Aberth 1980 , Computable analysis , McGraw Hill , 1980. Marian Pour El and Ian Richards, Computability in Analysis and Physics , Springer Verlag , 1989. Stephen G. Simpson 1999 , Subsystems of second order arithmetic . Klaus Weihrauch 2000 , Computable analysis , Springer, 2000. mathlogic stub Category Mathematical analysis Category Mathematical constructivism Category Computability theory Category Computable analysis ... more details
Unreferenced stub auto yes date December 2009 In computability theory computer science computability theory two sets A and B are computably isomorphic or recursively isomorphic if there exists a bijective computable function f with f A B . Two numbering computability theory numbering s math nu math and are called computably isomorphic if there exists a bijective computable function math f math so that math nu mu circ f. , math Computably isomorphic numberings induce the same notion of computability on a set. DEFAULTSORT Computable Isomorphism Category Theory of computation Category Computability theory Comp sci stub mathlogic stub ... more details
is not. Formal definition A real number a is said to be computable if it can be approximated by some computablefunction in the following manner given any integer math n ge 1 math , the function produces ... that are equivalent There exists a computablefunction which, given any positive rational error ... There is a computable sequence of rational numbers math q i math converging to math a math such that math q i q i 1 2 i , math for each i . There is another equivalent definition of computable numbers via computable Dedekind cut s. A computable Dedekind cut is a computablefunction math D math which ... to each Turing machine definition. This gives a function from the naturals to the computable reals ... to produce a computable real, a Turing machine must compute a total function , but the corresponding ... numbers it represents when written in binary and viewed as a characteristic function is computable ... also Definable number Semicomputable function References Oliver Aberth 1968, Analysis in the Computable ...In mathematics , particularly theoretical computer science and mathematical logic , the computable numbers , also known as the recursive numbers or the computable reals , are the real numbers that can ... can be given using recursive function s, Turing machines or lambda calculus calculus as the formal representation of algorithms. The computable numbers form a real closed field and can be used ... fractions between 0 and 1 A computable number is one for which there is a Turing machine which ... true math math p 3 3 q 3 Rightarrow D p q mathrm false math A real number is computable if and only if there is a computable Dedekind cut D converging to it. The function D is unique for each irrational computable number although of course two different programs may provide the same function . A complex number is called computable if its real and imaginary parts are computable. Properties Although the set of real numbers is uncountable , the set of computable numbers is countable and thus almost ... more details
wiktionarypar functionFunction may refer to Diatonic function , a term in music theory Function biology , explaining why a feature survived selection Function computer science , or subroutine, a portion of code within a larger program, performs a specific task Function engineering , related to the selected property of a system Function language , in linguistics, a way of achieving an aim using language Function mathematics , an abstract entity that associates an input to a corresponding output according to some rule Function model , a structured representation of the functions, activities or processes Function object , or functor or functionoid, a concept of object oriented programming Function Drinks , a beverage company based in Redondo Beach, California. A formal event such as a party or meeting See also Function hall Functional disambiguation Functionalism disambiguation Functor disambig bs Funkcija vor bg ca Funci desambiguaci cs Funkce da Funktion de Funktion et Funktsioon es Funci n eo Funkcio eu Funtzio argipena fr Fonction ko it Funzione lt Funkcija lmo Funziun nl Functie ja no Funksjon pl Funkcja ujednoznacznienie pt Fun o desambigua o ro Func ie dezambiguizare ru simple Function sk Funkcia sl Funkcija razlo itev sr sh Funkcija razvrstavanje sv Funktion olika betydelser th uk zh ... more details
Image VEST Core4 LowLevel.png thumbnail 320px right VEST 4 T function followed by a transposition layer In cryptography , a T function is a bijection bijective mapping that updates every bit of the state computer science state in a way that can be described as math x i x i f x 0, cdots, x i 1 math , or in simple words an update function in which each bit of the state is updated by a linear combination of the same bit and a function of a subset of its less significant bits. If every single less significant bit is included in the update of every bit in the state, such a T function is called triangular . Thanks to their bijectivity no collisions, therefore no entropy loss regardless of the used Boolean function s and regardless of the selection of inputs as long as they all come from one side of the output bit , T functions are now widely used in cryptography to construct block cipher s, stream cipher s, PRNG s and cryptographic hash function hash functions . T functions were first proposed in 2002 by Alexander Klimov A. Klimov and Adi Shamir A. Shamir in their paper A New Class of Invertible Mappings . Ciphers such as TSC 1 , TSC 3 , TSC 4 , ABC stream cipher ABC , Mir 1 and VEST are built with different types of T functions. Because arithmetic operation s such as addition , subtraction and multiplication are also T functions triangular T functions , software efficient word based T functions can be constructed by combining bitwise logic with arithmetic operations. Another important property of T functions based on arithmetic operations is predictability of their period mathematics period , which is highly attractive to cryptographers. Although triangular T functions are naturally vulnerable to guess and determine attacks, well chosen bitwise transposition mathematics transposition ... bit. Subsequent transposition of the output bits and iteration of the T function also do not affect ... and losing the T function bias of depending only on the less significant bits of the state. References ... more details
Unreferenced date November 2008 In mathematics , computable measure theory is a version of measure theory which deals with computable number s, as opposed to real number s which are used in standard measure theory. Category Measure theory Category Computability theory Mathanalysis stub ... more details
Computable model theory is a branch of model theory which deals with questions of computability as they apply to model theoretical structures. It was developed almost simultaneously by mathematicians in the West, primarily located in the United States and Australia , and Soviet Russia during the middle of the 20th century. Because of the Cold War there was little communication between these two groups and so a number of important results were discovered independently. Computable model theory introduces the ideas of computable and decidable models and theories and one of the basic problems is discovering whether or not computable or decidable models fulfilling certain model theoretic conditions can be shown to exist. References citation last Harizanov first V. S. authorlink Valentina Harizanov contribution Pure Computable Model Theory pages 3 114 title Handbook of Recursive Mathematics, Volume 1 Recursive Model Theory series Studies in Logic and the Foundations of Mathematics volume 138 editor first Iurii Leonidovich editor last Ershov publisher North Holland year 1998 isbn 978 0444500038 id MR 1673621 . Category Mathematical constructivism Category Model theory mathlogic stub ... more details
Logic of Computable Functions LCF is a deductive system for computable functions proposed by Dana Scott in 1969 in an memorandum unpublished until 1993. ref Dana S. Scott. http www.cs.cmu.edu kw scans scott93tcs.pdf A type theoretical alternative to ISWIM, CUCH, OWHY . Theoretical Computer Science , 121 411 440, 1993. Annotated version of the 1969 manuscript. ref It inspired Logic for Computable Functions LCF , theorem proving logic by Robin Milner . ref Robin Milner 1973 . ftp reports.stanford.edu pub cstr reports cs tr 73 332 CS TR 73 332.pdf Models of LCF ref Programming Computable Functions PCF , small theoretical programming language by Gordon Plotkin . ref cite journal first Gordon D. last Plotkin authorlink Gordon Plotkin title LCF considered as a programming language journal Theoretical Computer Science year 1977 pages 223 255 volume 5 doi 10.1016 0304 3975 77 90044 5 url http homepages.inf.ed.ac.uk gdp publications LCF.pdf ref harv ref References references disambiguation Category Programming language theory ... more details
see also Logic of Computable Functions Logic for Computable Functions LCF is an interactive automated theorem prover developed at the universities of University of Edinburgh Edinburgh and Stanford University Stanford by Robin Milner and others in 1972. LCF introduced the general purpose programming language ML programming language ML to allow users to write theorem proving tactics. Theorems in the system are propositions of a special theorem abstract datatype . The ML type system ensures that theorems are derived using only the inference rule s given by the operations of the abstract type. Successors include Higher Order Logic HOL theorem prover HOL and Isabelle theorem prover Isabelle . References Reflist Refbegin cite web last Gordon first Michael J. C. authorlink Michael J. C. Gordon year 1996 title From LCF to HOL a short history url http www.cl.cam.ac.uk mjcg papers HolHistory.html accessdate 2007 10 11 cite manual author Milner, Robin title Logic for Computable Functions description of a machine implementation. publisher Stanford University date May 1972 url ftp reports.stanford.edu pub cstr reports cs tr 72 288 CS TR 72 288.pdf ref lcf Refend Mathlogic stub Category Logic in computer science Category Interactive theorem proving software es LCF ... more details
In computer science , Programming Computable Functions , ref PCF is a programming language for computable functions, based on LCF, Scott s logic of computable functions harv Plotkin 1977 . Programming Computable Functions is used by harv Mitchell 1996 . It is also refered to as Programming with Computable Functions or Programming language for Computable Functions . ref or PCF , is a typed Functional programming functional language introduced by Gordon Plotkin in 1977. It is based on the Logic of Computable Functions LCF by Dana Scott . It can be considered as an extended version of the typed lambda calculus or a simplified version of modern typed functional languages such as ML programming language ML . A fully abstract model for PCF was first given by Robin Milner Milner 1977 . However, since Milner s model was essentially based on the syntax of PCF it was considered less than satisfactory Ong, 1995 . The first two fully abstract models not employing syntax were formulated during the 1990s. These models are based on game semantics Hyland and Ong, 2000 Abramsky, Jagadeesan, and Malacaria, 2000 and Kripke logical relations O Hearn and Riecke, 1995 . For a time it was felt that neither of these models was completely satisfactory, since they were not effectively presentable. However, Ralph Loader demonstrated that no effectively presentable fully abstract model could exist, since the question of program equivalence in the finitary fragment of PCF is not decidable. Syntax The types of PCF are inductively defined as nat is a type For types and , there is a type A context is a list of pairs x , where x is a variable name and is a type, such that no variable name is duplicated. One then defines typing judgments of terms in context in the usual way for the following syntactical constructs Variables if x is part of a context , then x Application of a term of type to a term of type abstraction The Y combinator Y fixed point combinator making terms of type ... more details
Computable general equilibrium CGE models are a class of economic models that use actual economic data to estimate how an economy might react to changes in policy, technology or other external factors. CGE models are also referred to as AGE applied general equilibrium models. Overview A CGE model consists of a equations describing model variables and b a database usually very detailed consistent with the model equations. The equations tend to be Neo classical economics neo classical in spirit, often assuming cost minimizing behaviour by producers, average cost pricing, and household demands based on optimizing behaviour. However, most CGE models conform only loosely to the theoretical general equilibrium paradigm. For example, they may allow for non market clearing, especially for labour unemployment or for commodities inventories imperfect competition eg, monopoly pricing demands not influenced by price eg, government demands a range of taxes externalities, such as pollution A CGE model database consists of tables of transaction values, showing, for example, the value of coal used by the iron industry. Usually the database is presented as an input output analysis input output table or as a social accounting matrix . In either case, it covers the whole economy of a country or even the whole world , and distinguishes a number of sectors, commodities, primary factors and perhaps types of household. elasticities dimensionless parameters that capture behavioural response. For example, export demand elasticities specify by how much export volumes might fall if export prices went up. Other elasticities may belong to the Constant Elasticity of Substitution class. Amongst these are Armington ... Computable General Equilibrium CGE in GAMS, Microcomputers in Policy Research, vol.5, International ... Decision Making in Australia and the Development of Computable General Equilibrium Modelling, CoPS IMPACT ... workingpapers SCEPA 20Working 20Paper 202008 1 20Kahn.pdf Debunking the Myths of Computable ... more details
Orphan date February 2009 Image ICES logo.jpg thumb right ICES Intertemporal Computable Equilibrium System is a recursive dynamic general equilibrium model developed with the purpose to assess the final welfare implication of climate change affects regional and world economies. The model has been developed at the Climate Change Modelling and Policy Research Programme of the Fondazione Eni Enrico Mattei FEEM, a research institution in the field of sustainable development. Overview As in every computable general equilibrium CGE model, its general equilibrium structure in which all markets are interlinked is tailored to capture and highlight the production and consumption substitution processes at play in the social economic system as a response to climate shocks. In doing so, the final economic equilibrium determined, takes into account explicitly the autonomous adaptation of economic systems. The idea behind ICES is to provide a climate change impact assessment tool that can go beyond the simple quantification of direct costs, thus offering an economic evaluation summarising second and higher order effects. In addition to climate change impact assessment, the model can be used to study mitigation and adaptation policies as well as different trade and public policy reforms in the vein of conventional CGE. Model description ICES is a top down recursive growth model with a sequence of static equilibria intertemporally connected by endogenous investment decisions and capital accumulation . On the economic context, the model accounts for intersectoral factor mobility, international trade and also international investment flows allocated by a global financing entity. Within the climate change assessment framework, its general equilibrium characteristics have been suited to assess ... shares are generally fixed, which amounts to saying that the top level utility function has ... Programme Working Papers Series. External links http www.feem web.it ices ICES Intertemporal Computable ... more details
Recursive function may refer to Recursion computer science , a procedure or subroutine, implemented in a programming language, whose implementation references itself A total computablefunction , a function which is defined for all possible inputs See also recursive function , defined from a particular formal model of computable functions using primitive recursion and the operator Recurrence relation , in mathematics, an equation that defines a sequence recursively disambig cs Rekurzivn funkce ... more details
In recursion theory computability theory a subfield of computer science a semicomputable function is a partial function math f mathbb Q rightarrow mathbb R math that can be approximated either from above or from below by a computablefunction . More precisely a partial function math f mathbb Q rightarrow mathbb R math is upper semicomputable , meaning it can be approximated from above, if there exists a computablefunction math phi x,k mathbb Q times mathbb N rightarrow mathbb Q math , where math x math is the desired parameter for math f x math and math k math is the level of approximation, such that math lim k rightarrow infty phi x,k f x math math forall k in mathbb N phi x,k 1 leq phi x,k math Completely analogous a partial function math f mathbb Q rightarrow mathbb R math is lower semicomputable iff math f x math is upper semicomputable or equivalently if there exists a computablefunction math phi x,k math such that math lim k rightarrow infty phi x,k f x math math forall k in mathbb N phi x,k 1 geq phi x,k math If a partial function is both upper and lower semicomputable it is called computable. See also recursion theory computability theory References Ming Li and Paul Vit nyi, An Introduction to Kolmogorov Complexity and Its Applications , pp 37&ndash 38, Springer, 1997. DEFAULTSORT Semicomputable Function Category Mathematical logic mathlogic stub ... more details
The ramp function is an elementary function elementary unary function unary real function , easily computable as the arithmetic mean mean of its independent variable and its absolute value . This function is applied in engineering e.g., in the theory of Digital signal processing DSP . The name ramp function can be derived by the look of its graph. Definitions Image Ramp function.svg Graph of a function Graph of the ramp function thumb 260px right The ramp function math R x mathbb R rightarrow mathbb R math may be defined analytically in several ways. Possible definitions are center math R x begin cases x, & x ge 0 0, & x 0 end cases math center The mean of a straight line with unity gradient and its modulus center math R x frac x x 2 math center this can be derived by noting the following definition of math max a,b math , math max a,b frac a b a b 2 math , for which math a x math and math b 0 math The Heaviside step function multiplied by a straight line with unity gradient center math R left x right xH left x right math center The convolution of the Heaviside step function with itself center math R left x right H left x right H left x right math center The integral of the Heaviside step function center math R x int infty x H xi mathrm d xi math center Analytic properties Non negativity In the whole domain of a function domain the function is non negative, so its absolute value is itself, i.e. math forall x in mathbb R R x geqslant 0 math and math left R left x right right R left x right math Proof by the mean of definition 2 it is non negative in the I. quarter, and zero in the II. so everywhere it is non negative. Derivative Its derivative is the Heaviside function math R x H x mathrm if x ne 0 math Ugyanis ha x 0, akkor R x 0 konstans, teh t ezen a tartom nyon sup sup R x 0 konstans deriv ltja 0 ami megegyezik a Heaviside f ggv nnyel. ha x 0, akkor R x x, teh t ezen ... s 2 . math center Algebraic properties Iteration invariance Every iterated function of the ramp mapping ... more details
In computability theory , the Ackermann function , named after Wilhelm Ackermann , is one of the simplest and earliest discovered examples of a total function total computablefunction that is not Primitive recursive function primitive recursive . All primitive recursive functions are total and computable, but the Ackermann function illustrates that not all total computable functions are primitive ... 0315 0860 79 90024 7 ref with discovering total function total computablefunction s termed simply recursive in some references that are not primitive recursive function primitive recursive . Sudan published the lesser known Sudan function , then shortly afterwards and independently, in 1928, Ackermann published his function math phi , math . Ackermann s three argument function, math phi m ... its historic role as a total computable but not primitive recursive function, Ackermann s original function is seen to extend the basic arithmetic operations beyond exponentiation, although not as seamlessly as do variants of Ackermann s function that are specifically designed for that purpose such as Reuben ... such as a Turing machine and so is a computablefunction , grows faster than any primitive recursive ... 133 doi 10.1007 BF01459088 ref of his function which had three nonnegative integer arguments , many authors modified it to suit various purposes, so that today the Ackermann function may refer to any of numerous variants of the original function. One common version, the two argument Ackermann P ter function , is defined as follows for nonnegative integers m and n math A m, n begin cases n 1 & mbox ..., Solomon Marcus and Ionel Tevy journal Historia Math. title The first example of a recursive function ... that the Ackermann function was not primitive recursive, but it was Ackermann, Hilbert s personal ... a two variable version of the Ackermann function that became preferred by many authors. ref cite ... 09121 2 ref Definition and properties Ackermann s original three argument function math phi m, n, p ... more details
In mathematics, omega function or &omega function may mean Pearson&ndash Cunningham function Lambert W function Wright Omega function mathdab ... more details
Barnes G function , related to the Gamma function Meijer G function , a generalization of the hypergeometric function Siegel G function , a class of functions in transcendence theory mathdab ... more details
presentation has computable Dehn function Dehn n , then the word problem for G is solvable with non ...In the mathematical subject of geometric group theory , a Dehn function , named after Max Dehn , is an optimal function associated to a finitely presented group finite group presentation which bounds the area ... name Gersten . The growth type of the Dehn function is a quasi isometry quasi isometry invariant of a finitely presented group . The Dehn function of a finitely presented group is also closely connected ... for groups word problem if and only if the Dehn function for a finitely presented group finite presentation of this group is recursive function recursive see Theorem 2.1 in ref name Gersten . The notion of a Dehn function is motivated by isoperimetric problems in geometry, such as the classic isoperimetric inequality for the Euclidean plane and, more generally, the notion of a filling area function ... curve of that surface. History The idea of an isoperimetric function for a finitely presented ... consequence of this fact is that for this presentation the Dehn function satisfies Dehn n ... notion of an isoperimetric function and a Dehn function as it is used today appeared in late 1980s ... if the Dehn function of this group is equivalent to the function f n n . Gromov s proof was in large ... over with boundary cycle labelled by w . Isoperimetric function An isoperimetric function for a finite presentation is a monotone non decreasing function math f mathbb N to 0, infty math such that whenever ... &le f w , where w is the length of the word w . Dehn function Then the Dehn function of a finite ... reduced . math Equivalently, Dehn n is the smallest isoperimetric function for , that is, Dehn n is an isoperimetric function for and for any other isoperimetric function f n we have Dehn n &le ... admits an isopeprimetric function f n that is equivalent to a linear respectively, quadratic, cubic, polynomial, exponential, etc. function in n , the presentation is said to satisfy a linear respectively ... more details
Riemann function may refer to one of the several function mathematics functions named after the mathematician Bernhard Riemann , including Riemann zeta function Thomae s function Riemann theta function . dab fr Fonction de Riemann ... more details
In mathematics , a zeta function is usually a function mathematics function analogous to the original example the Riemann zeta function math zeta s sum n 1 infty frac 1 n s . math Zeta functions include Airy zeta function , related to the zeros of the Airy function Artin Mazur zeta function Artin Mazur zeta function of a dynamical system Barnes zeta function Beurling zeta function of Beurling generalized primes Dedekind zeta function Dedekind zeta function of a number field Real analytic Eisenstein series Epstein zeta function Epstein zeta function of a quadratic form. Goss zeta function of a function field Hasse Weil zeta function Hasse Weil zeta function of a variety Hurwitz zeta function Hurwitz zeta function A generalization of the Riemann zeta function Ihara zeta function Ihara zeta function of a graph Igusa zeta function Igusa zeta function Jacobi zeta function This is related to elliptic functions and is not analogous to the Riemann zeta function. L function , a twisted zeta function. Lefschetz zeta function Lefschetz zeta function of a morphism Lerch zeta function Lerch zeta function A generalization of the Riemann zeta function Local zeta function of a characteristic p variety Matsumoto zeta function Minakshisundaram Pleijel zeta function of a Laplacian Motivic zeta function of a motive Mordell Tornheim zeta function of several variables Multiple zeta function Prime zeta function Like the Riemann zeta function, but only summed over primes. Riemann zeta function The archetypal example. Selberg zeta function Selberg zeta function of a Riemann surface Shintani zeta function Weierstrass zeta function This is related to elliptic functions and is not analogous to the Riemann zeta function. Witten zeta function of a Lie group Zeta function operator Zeta function of an operator See also Artin conjecture L functions Artin conjecture Birch and Swinnerton Dyer conjecture Riemann hypothesis and the generalized Riemann hypothesis . Selberg class S External links http www.maths.ex.ac.uk ... more details
In mathematics, by sigma function one can mean one of the following The Divisor function sum of divisors function sub a sub n , an arithmetic function . Weierstrass sigma function , related to elliptic functions Rado s sigma function, see busy beaver . mathdab de Teilersumme fr Fonction sigma ... more details
In mathematics , two different function mathematics functions are known as the pi or Pi function math pi x , math pi function &ndash the prime counting function math Pi x , math Pi function &ndash the Gamma function when offset to coincide with the factorial disambig th ... more details
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