infinity The reciprocal function y 1 x . For every x except 0, y represents its multiplicativeinverse. In mathematics , a multiplicativeinverse or reciprocal for a number x , denoted by 1 x or x sup &minus 1 sup , is a number which when multiplied by x yields the multiplicative identity , 1. The multiplicativeinverse of a rational number fraction a b is b a . For the multiplicativeinverse of a real ... Henry Billingsley translation of Elements XI, 34. ref In the phrase multiplicativeinverse , the qualifier multiplicative is often omitted and then tacitly understood in contrast to the additive inverse . Multiplicative inverses can be defined over many mathematical domains as well as numbers. In these cases it can happen that ab ba then inverse typically implies that an element is both a left and right inverse element inverse . Practical applications The multiplicativeinverse has innumerable ... to compute k sup 1 sup , the modular multiplicativeinverse of k mod 2 sup w sup , where w ... unit s, math i , are the only complex numbers with additive inverse equal to multiplicativeinverse. For example, additive and multiplicative inverses of math i are &minus math i &minus math ... , the modular multiplicativeinverse of a is also defined it is the number x such that ax 1 mod n . This multiplicativeinverse exists if and only if a and n are coprime ... every nonzero element has a multiplicativeinverse, but which nonetheless has divisors of zero, i.e. ..., an element x with a multiplicativeinverse cannot be a zero divisor meaning for some y , xy ... divisor is not guaranteed to have a multiplicativeinverse. Within Z , all integers except ... x to 1, ax 1, so that x is an inverse for a . The multiplicativeinverse of a fraction math ... x to 1 x , is one of the simplest examples of a function which is self inverse function inverse ... Britannica 1797 to describe two numbers whose product is 1 geometrical quantities in inverse proportion ... more details
Refimprove date March 2007 The modular multiplicativeinverse of an integer a modular arithmetic modulo m is an integer x such that math a 1 equiv x pmod m . math That is, it is the multiplicativeinverse ... 1 pmod m . math The multiplicativeinverse of a modulo m exists Iff if and only if a and m are coprime i.e., if gcd a , m 1 . If the modular multiplicativeinverse of a modulo m exists, the operation of Division mathematics division by a modulo m can be defined as multiplying by the inverse, which ... the inverse exists, it is always unique in math mathbb Z m math where m is the modulus. Therefore, the x that is selected as the modular multiplicativeinverse is generally a member of math mathbb ... 11 math The smallest x that solves this congruence is 4 therefore, the modular multiplicativeinverse ... m is Euler s totient function . This follows from the fact that a belongs to the multiplicative group of integers modulo n multiplicative group Z m Z sup sup if and only if iff a is coprime to m . Therefore the modular multiplicativeinverse can be found directly math a varphi m 1 equiv a 1 pmod m math ... theory Public key cryptography References reflist DEFAULTSORT Modular MultiplicativeInverse Category ... , which is 11, to the found inverse . Computation Extended Euclidean algorithm wikibooks Algorithm Implementation Mathematics Extended Euclidean algorithm Extended Euclidean algorithm The modular multiplicativeinverse of a modulo m can be found with the extended Euclidean algorithm . The algorithm ... y , and gcd a , b are the integers that the algorithm discovers. So, since the modular multiplicativeinverse is the solution to math ax equiv 1 pmod m , math by the definition of congruence ... given, x the inverse, and q an integer multiple that will be discarded. This is the exact form of equation ... 1 is predetermined instead of discovered. Thus, a needs to be coprime to the modulus, or the inverse ... Euclidean algorithm, Euler s theorem may be used to compute modular inverse ref Thomas Koshy ... more details
Multiplicative may refer to Multiplication Multiplicative partition A Multiplicative function For the Multiplicative numerals , once, twice, and thrice, see English numerals disambig Long comment to avoid being listed on short pages ... more details
Wiktionarypar inverse TOCright Inverse may refer to Inverse logic , a type of immediate inference from a conditional sentence Inverse program , a program for solving inverse and optimization problems Inverse multiplexer or demultiplexer , which breaks a single data stream into several streams with lower data rates Invert , a term used in the exotic pet industry to refer to arthropods Inversions novel Inversions novel , a science fiction novel by Iain M. Banks Science, engineering and mathematics Inverse mathematics Inversive geometry , transformation geometry based on inversion in a circle Inverse problem , in science and mathematics, fitting a model to known data Multiplicativeinverse , a set of numbers which when multiplied yield the multiplicative identity, 1 Invert , the bottom of a sewer or tunnel, particularly if masonry lined. It may be flat or form a continuous curve with the tunnel arch. See also Inversion disambiguation Inverter disambiguation Opposite disambiguation Reverse disambiguation disambig cs Inverze da Invers de Inversion es Inversi n desambiguaci n eo Inverso matematiko fr Inverse homonymie io Inversigeso id Invers it Inversione he nl Inversie nn Invers pl Inwersja pt Invers o simple Inverse sk Inverzia fi Inversio sv Invers uk ... more details
An inverse number may refer to The multiplicativeinverse of a number A type of Grammatical number Inverse number grammatical number disambig ... more details
Modular arithmetic order group theory References MathWorld urlname MultiplicativeOrder title Multiplicative Order DEFAULTSORT Multiplicative Order Category Modular arithmetic es Orden multiplicativo ... more details
Outside number theory, the term multiplicative function is usually used for completely multiplicative function s. This article discusses number theoretic multiplicative functions. In number theory , a multiplicative ... multiplicative function completely multiplicative or totally multiplicative if f 1 1 ... Some multiplicative functions are defined to make formulas easier to write 1 n the constant function, defined by 1 n 1 completely multiplicative math 1 C n math the indicator function of the set math C subset mathbb Z math . This is multiplicative if the set C has the property that if a and b are in C ... multiplicative Id sub k sub n the power functions, defined by Id sub k sub n n sup k sup for any complex number k completely multiplicative . As special cases we have Id sub 0 sub n 1 n and Id ... n completely multiplicative . Other examples of multiplicative functions include many functions ... dividig n . completely multiplicative . math gamma math n , defined by math gamma math n &minus ... primes dividing n . All Dirichlet character s are completely multiplicative functions. For example ... of a non multiplicative function is the arithmetic function r sub 2 sub n the number of representations ... sup and therefore r sub 2 sub 1 4 1. This shows that the function is not multiplicative. However, r sub 2 sub n 4 is multiplicative. In the On Line Encyclopedia of Integer Sequences , http www.research.att.com njas sequences ?q keyword mult sequences of values of a multiplicative function have the keyword mult . See arithmetic function for some other examples of non multiplicative functions. Properties A multiplicative function is completely determined by its values at the powers of prime number ... of multiplicative functions significantly reduces the need for computation, as in the following ... varphi math 2 sup 4 sup math varphi math 3 sup 2 sup 8 6 48 In general, if f n is a multiplicative ... least common multiple lcm a , b . Every completely multiplicative function is a homomorphism of monoid ... more details
Unreferenced date December 2006 orphan date November 2009 In algebraic geometry , math mu math is said to be a multiplicative distance function over a Field mathematics field if it satisfies, math mu AB 1. , math AB is congruence relation congruent to A B iff math mu AB mu A B . , math AB A nowiki nowiki B nowiki nowiki iff math mu AB mu A B . , math math mu AB CD mu AB mu CD . , math See also Algebraic geometry Hyperbolic geometry Poincar disc model Hilbert s arithmetic of ends DEFAULTSORT Multiplicative Distance Category Algebraic geometry Abstract algebra stub geometry stub ... more details
In number theory , a multiplicative partition or unordered factorization of an integer n that is greater than 1 is a way of writing n as a product of integers greater than 1, treating two products as equivalent if they differ only in the ordering of the factors. The number n is itself considered one of these products. Multiplicative partitions closely parallel the study of multipartite partitions , discussed ... sequences of positive integers, with the addition made pointwise . Although the study of multiplicative partitions has been ongoing since at least 1923, the name multiplicative partition appears to have ... used previously. MathWorld uses the term unordered factorization . Examples The number 20 has four multiplicative ... 3 × 9, 3 × 27, 9 × 9, and 81 are the five multiplicative ... has the same number five of multiplicative partitions as 4 does of partition number theory additive partitions . The number 30 has five multiplicative partitions 2 × 3 × ... of multiplicative partitions of a Square free integer squarefree number with i prime factors is the ith Bell number , B sub i sub . Application harvtxt Hughes Shallit 1983 describe an application of multiplicative ... s these forms correspond to the multiplicative partitions 12, 2× 6, 3× 4, and 2× 2× 3 respectively. More generally, for each multiplicative partition math k prod t i math of the integer ... i t i 1 , math where each p sub i sub is a distinct prime. This correspondence follows from the Multiplicative function multiplicative property of the divisor function . Bounds on the number of partitions harvtxt Oppenheim 1926 credits harvtxt McMahon 1923 with the problem of counting the number of multiplicative ... name of factorisatio numerorum . If the number of multiplicative partitions of n is a sub n sub , McMahon ... n sub of multiplicative partitions of some n the number of values less than N which arise in this way ... Shallit title On the number of multiplicative partitions journal American Mathematical Monthly year ... more details
Basic notions in group theory In mathematics and group theory the term multiplicative group refers to one of the following concepts, depending on the context any group math scriptstyle mathfrak G , math whose binary operation is written in multiplicative notation instead of being written in additive notation as usual for abelian group s , the underlying group under multiplication of the invertible elements of a field mathematics field , ref See Hazewinkel et. al. 2004 , p. 2. ref ring mathematics ring , or other structure having multiplication as one of its operations. In the case of a field F , the group is F 0 , , where 0 refers to the zero element of the F and the binary operation is the field multiplication , the algebraic torus math scriptstyle mathbf GL 1 math . Group scheme of roots of unity The group scheme of math n math th roots of unity is by definition the kernel of the math n math power map on the multiplicative group math scriptstyle mathbf GL 1 math , considered as a group scheme . That is, for any integer math n 1 math we can consider the morphism on the multiplicative group that takes math n math th powers, and take an appropriate fiber product in the sense of scheme theory of it, with the morphism math e math that serves as the identity. The resulting group scheme is written math mu n math . It gives rise to a reduced scheme , when we take it over a field math scriptstyle mathbb K math , if and only if the characteristic field characteristic of math scriptstyle mathbb K math does not divide math n math . This makes it a source of some key examples of non reduced schemes schemes with nilpotent element s in their structure sheaf structure sheaves for example math mu p math over a finite field with math p math elements for any prime number math ..., rings and modules . Volume 1. 2004. Springer, 2004. ISBN 1 4020 2690 0 See also multiplicative group of integers modulo n additive group DEFAULTSORT Multiplicative Group Category Algebraic structures ... more details
In mathematics, a multiplicative cascade ref Meakin P, PRA vol 36 No 6 1987 Diffusion limited aggregation on multifractal lattices ref ref http uk.arxiv.org abs 0803.3212 Cristano G. Sabiu, Luis Teodoro, Martin Hendry, arXiv 0803.3212v1 Resolving the universe with multifractals ref is a fractal multifractal distribution of points produced via an iterative and multiplicative random process . Image 3fractals2.jpg 800px br Model I left plot math lbrace p 1,p 2,p 3,p 4 rbrace lbrace 1,1,1,0 rbrace math Model II middle plot math lbrace p 1,p 2,p 3,p 4 rbrace lbrace 1,0.75,0.75,0.5 rbrace math Model III right plot math lbrace p 1,p 2,p 3,p 4 rbrace lbrace 1,0.5,0.5,0.25 rbrace math The plots above are examples of multiplicative cascade multifractals. To create these distributions there are a few steps to take. Firstly, we must create a lattice of points which will be our underlying probability density field. Then we will populate this lattice with randomly placed points, insisting that the probability that the points be placed are proportional to the cell probability. The fractal is constructed as follows The space is split into four equal parts. Each part is then assigned a probability from the set math lbrace p 1,p 2,p 3,p 4 rbrace math without replacement, where math p i in 0,1 math . Each subspace is then divided again and assigned probabilities randomly from the same set and this is continued to the N th level. At the N th level the probability of a cell being occupied is the product of the cell s p sub i sub and its parents and ancestors up to level 1 i.e. all the cells above it. In constructing this model down to level 8 we produce a 4 sup 8 sup array of cells each with its own probability. To then place particle in the space we invoke a Monte Carlo method Monte Carlo rejection scheme . Choosing x and y coordinates randomly we simply test if a random number between 0 and 1 is less or greater than the cell probability. To produce the plots above we dust the probability ... more details
The multiplicative case is a grammatical case used for marking a number of something three times . The case is found in the Hungarian language , ref Mentioned in Istv n Kenesei, Anna Fenyvesi, Robert Michael Vago, Hungarian , page xxviii, 1998 472 pages Google book search ref for example nyolc eight , nyolcszor eight times . ref cite book title The sound pattern of Hungarian first Robert Michael last Vago publisher Georgetown University Press year 1980 isbn 0878401776 page 38 ref The case appears also in Finnish language Finnish as an adverbial adverb forming case. Used with a cardinal number it denotes the number of actions for example, viisi five viidesti five times . Used with adjectives it refers to the mean of the action, corresponding the English suffix ly kaunis beautiful kauniisti beautifully . It is also used with a small number of nouns leikki play leikisti just kidding, not really . In addition, it acts as an intensifier when used with a swearword piru pirusti . ref http www.cc.jyu.fi pamakine kieli suomi sijat sijatadverbien.html Finnish Grammar Adverbial cases ref References references ling morph stub Grammatical cases Category Grammatical cases br Multiplikativel Troad ca Cas multiplicatiu nl Multiplicatief ... more details
noinclude NOTE This article is also transcluded in the disambiguation page Inverse In many contexts in mathematics the term inverse indicates the opposite of something. This word and its derivatives are used widely in mathematics , as illustrated below. noinclude Inverse element of an element x with respect to a binary operation with identity element e is an element y such that x y y x e . In particular, the additive inverse of x is x the multiplicativeinverse or reciprocal of x is x sup 1 sup . inverse function &mdash inverse element with respect to function composition a function that reverses ... of a circle to the inside and vice versa. Inverse limit &mdash a notion in abstract algebra. Inverse logic &mdash p q is the inverse of p q . Inverse matrix &mdash inverse element with respect to matrix multiplication. Pseudoinverse , a generalization of the inverse matrix. Inverse proportion ... math y k x. math Inverse problem &mdash the task of identifying model parameters from observed data see for example inverse scattering problem inverse kinematics inverse dynamics . Perspective graphical Inverse perspective &mdash the further the objects, the larger they are drawn. Inversive ring geometry &mdash classical projective geometry extended by ring theory Inverse semigroup Inverse of an element in a semigroup Inverse square law &mdash the magnitude of a force is proportional to the inverse square of the distance. Inverse transform sampling &mdash generate some random numbers according to a given probability distribution. Inverse chain rule method &mdash related to integration ... of elements , a pair of adjacent out of order elements of a permutation viewed as a list . Inverse ... . noinclude Category Mathematical terminology ca Invers da Invers matematik fr Inverse id Invers lt Inversija nl Inverse pl Inwersja sv Invers NOTE This article is also transcluded in the disambiguation page Inverse noinclude ... more details
In abstract algebra , the idea of an inverse element generalises the concept of a additive inverse negation , in relation to addition , and a Multiplicativeinverse reciprocal , in relation to multiplication ... given by math x math . Every nonzero real number math x math has a multiplicativeinverse i.e. ..., 0 number zero has no multiplicativeinverse, but it has a unique quasi inverse, 0 itself. Functions .... While the precise definition of an inverse element varies depending on the algebraic structure ... a b e math , then math a math is called a left inverse of math b math and math b math is called a right inverse of math a math . If an element math x math is both a left inverse and a right inverse of math y math , then math x math is called a two sided inverse , or simply an inverse , of math y math . An element with a two sided inverse in math S math is called invertible in math S math . An element with an inverse element only on one side is left invertible , resp. right invertible . If all ... math is associative then if an element has both a left inverse and a right inverse, they are equal. In other words, in a monoid every element has at most one inverse as defined in this section . In a monoid ... main Regular semigroup The definition in the previous section generalizes the notion of inverse in group ... of an inverse by dropping the identity element but keeping associativity, i.e. in a semigroup ... z in S such that xzx x z is sometimes called a pseudoinverse . An element y is called simply an inverse of x if xyx x and y yxy . Every regular element has at least one inverse if x xzx then it is easy to verify that y zxz is an inverse of x as defined in this section. Another easy to prove fact if y is an inverse of x then e xy and f yx are idempotent s, that is ee e and ff f . Thus, every pair of mutually inverse elements gives rise to two idempotents, and ex xf x , ye fy y , and e acts as a left .... 2.3.3, p. 51 ref An intuitive description of this is fact is that every pair of mutually inverse elements ... more details
integers a , so this is not a very strong restriction. Examples The easiest example of a multiplicative ... f a a sup n sup . The Liouville function is a non trivial example of a completely multiplicative function as are Dirichlet character s. Properties A completely multiplicative function is completely ... n f p sup a sup f q sup b sup ... While the Dirichlet convolution of two multiplicative functions is multiplicative, the Dirichlet convolution of two completely multiplicative functions need not be completely multiplicative. There are a variety of statements about a function which are equivalent to it being completely multiplicative. For example, if a function f multiplicative then is completely multiplicative if and only if the Dirichlet inverse is math mu f math where math mu math is the Mobius function . ref Apostol, p. 36 ref Completely multiplicative functions also satisfy a pseudo associative law. If f is completely multiplicative then math f cdot g h f cdot g f cdot h math where represents ... pg. 49 ref . One consequence of this is that for any completely multiplicative function f one has math ... n d right g d h left frac n d right math since f is completely multiplicative math f cdot left g ... n f cdot g f cdot h . math See also References references Category Multiplicative functions ... more details
Refimprove date January 2010 In mathematics, the additive inverse , or opposite , of a number mvar a is the number that, when addition added to mvar a , yields 0 number zero . The additive inverse of mvar a is denoted by unary operation unary minus sign minus mvar a . This can be seen as a shorthand for a common subtraction notation math 1 mvar a 0 mvar a with 0 omitted, although in a correct typography there should be no space punctuation space after unary . For example, the additive inverse of 7 is 7, because 7 7 0, and the additive inverse of 0.3 is 0.3, because 0.3 0.3 0 . In other words, the additive inverse of a number is the number s negative. For example, the additive inverse of 8 is &minus 8, the additive inverse of 10002 is &minus 10002 and the additive inverse of mvar x is &minus mvar x . The additive inverse of a number is defined as its inverse element under the binary operation of addition. It can be calculated using multiplication by 1 that is, math 1 mvar n &minus 1 mvar n . Integer s, rational number s, real number s, and complex number all have additive inverses, as they contain negative as well as positive numbers ... x&prime mvar x&prime mvar x o , then mvar x&prime is called an additive inverse of mvar x . If is associativity associative math 1 x y z x y z for all mvar x , mvar y , mvar z , then an additive inverse ... numbers is associative, each real number has a unique additive inverse. Other examples All the following examples are in fact abelian group s addition of real valued functions here, the additive inverse ... , the modular additive inverse of mvar x is also defined it is the number mvar a such that math 1 mvar a mvar x 0 mod mvar n . This additive inverse does always exist. For example, the inverse of 3 modulo 11 is 8 because it is the solution to math 1 3 mvar x 0 mod 11 . See also Multiplicativeinverse Additive identity References MathWorld title Additive Inverse urlname AdditiveInverse author Margherita ... more details
says that the inverse of a function is analogous to a multiplicativeinverse . This explains ... sin sup 1 sup x usually does not represent the multiplicativeinverse to sin x , but the inverse ... nowrap sin x sup 1 sup is the multiplicativeinverse to the sine, and is called the cosecant ... x for the multiplicativeinverse of sinh x . Properties Uniqueness If an inverse function exists ...Technical date May 2011 Image Inverse Function.png thumb right A function and its inverse sup 1 sup . Because maps a to 3, the inverse sup 1 sup maps 3 back to a . In mathematics , an inverse function ... produces an output y , then putting y into the inverse function g produces the output x , and vice ... with x leaves x unchanged. A function that has an inverse is called invertible the inverse function is then uniquely determined by and is denoted by sup 1 sup read f inverse , not to be confused with exponentiation . A relation can be determined to have an inverse if it is a one to one function. Definitions The word inverse is related to the word invert meaning to reverse, turn upside down, to do the opposite. Wiktionary inverse Image Inverse Functions Domain and Range.png thumb right ... be at most one function g satisfying this property. That function g is then called the inverse of , denoted by sup 1 sup . Stated otherwise, a function is invertible if and only if its inverse relation is a function on the range Y , in which case the inverse relation is the inverse function. Not all functions have an inverse. For this rule to be applicable, each element y Y must ..., or information preserving, or an injective function injection . Example inverse operations that lead to inverse functions Inverse operations are the opposite of direct variation functions. Direct variation ... and an inverse variation function is y k x. Example squaring and square root functions The function ... square root function is the inverse of x sup 2 sup because the first is not Single valued function ... more details
The inverse method can refer to The inverse transform sampling method . The inverse method automated reasoning inverse method in automated reasoning . disambig ... more details
A right inverse in mathematics may refer to A right inverse element with respect to a binary operation on a set A right inverse function Left and right inverses inverse function for a mapping between sets mathdab ... more details
A left inverse in mathematics may refer to A left inverse element with respect to a binary operation on a set A left inverse function Left and right inverses inverse function for a mapping between sets A kind of generalized inverse mathdab ... more details
File Simple inverse relationship chart.svg thumb Linear inverse relationship An inverse or negative relationship is a mathematical relationship in which one Variable mathematics variable , say y , decreases as another, say x , increases. For a linear straight line relation, this can be expressed as y a bx , where b is a constant value less than zero and a is a constant. For example, there is an inverse relationship between education and unemployment that is, as education increases, the rate of unemployment decreases. Inverse relationships and their counterpart, direct relationship s , are widely used in the physical science s to describe the relationship between two variables in an equation. In economic graph ing, two variables are said to have an inverse relationship if the graph line slopes downward to the right. See also Direct relationship Inverse relation Proportionality mathematics Category Elementary mathematics Inverse relationship math stub stat stub ar fr Relation inverse ... more details
unreferenced date October 2010 In the concept oriented model dimensions are used to link subconcepts with their superconcepts. Thus dimension is a named position of superconcept within one subconcept. Inverse dimension is produced from dimension by inverting its direction. Thus inverse dimensions identify subconcepts for a superconcept. Inverse dimension takes values from the corresponding subconcept as a domain. However, in contrast to dimensions they are multi valued. Inverse dimension can be represented graphically as a downward edge arrow the source concept to some its subconcept. Normally inverse dimensions do not have their own names. Instead we take a normal dimension and inverse its direction by enclosing into curly brackets. For example, concept tt Orders tt has two inverse dimensions tt OrderParts.order tt and tt OrderOperations.order tt we normally prefix the path by the source concept to avoid ambiguity . Notice that for each order we have many order parts and many order operations. See also Concept oriented model OLAP External links http conceptoriented.com The Concept Oriented Portal Category Data modeling ... more details
In mathematics , the term weak inverse is used with several meanings. Theory of semigroups In the theory of semigroup s, a weak inverse of an element x in a semigroup nowrap S , is an element y such that nowrap 1 y x y y . An element x of S for which there is an element y of S such that nowrap 1 x y x x is called regular. A regular semigroup is a semigroup in which every element is regular. If every element x in S has a unique inverse y in S in the sense that nowrap 1 x y x x and nowrap 1 y x y y then S is called an inverse semigroup . Category theory In category theory , a weak inverse of an object category theory object A in a monoidal category C with monoidal product and unit object I is an object B such that both A B and B A are isomorphism isomorphic to the unit object I of C . A monoidal category in which every morphism is invertible and every object has a weak inverse is called a 2 group . See also Generalized inverse Von Neumann regular ring Category Monoidal categories Category Semigroup theory cattheory stub Abstract algebra stub ... more details
notability date March 2008 Inverse is a general purpose computer program for solving inverse problem inverse and optimization mathematics optimization problems. The program has been designed in particular for solving problems where complex numerical simulation s e.g. by the finite element method are involved in the definition of the objective and constraint functions. The original motivation for design of the program came from the area of metal forming where industrial simulation programs were mainly designed without having in mind application to the solution of optimization problems, while increasing needs were arising especially for application of numerical simulations in inverse identification of material parameters. See also Investigative Optimization Library IOptLib optimization mathematics External links http www2.arnes.si ljc3m2 inverse doc man frame.html Inverse manuals http www2.arnes.si 7Eljc3m2 igor ioptlib index.html IOptLib http www.c3m.si inverse doc phd A General Purpose Computational Shell for Solving Inverse and Optimisation Problems A Ph.D. thesis on Inverse http www unix.mcs.anl.gov otc Guide faq nonlinear programming faq.html Nonlinear programming FAQ http glossary.computing.society.informs.org Mathematical Programming Glossary Category Mathematical optimization software sl Inverse ... more details
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