In mathematics , quadraticvariation is used in the analysis of stochastic process es such as Wiener process Brownian motion and Martingale probability theory martingale s. Quadraticvariation is just one kind of Total variationvariation of a process. Definition Suppose that X sub t sub is a real valued ... the non negative real numbers. Its quadraticvariation is the process, written as X sub t sub , defined ... in probability . More generally, the quadratic covariation of two processes X and Y is math X,Y t lim Vert P Vert to 0 sum k 1 n left X t k X t k 1 right left Y t k Y t k 1 right . math The quadratic covariation may be written in terms of the quadraticvariation by the polarization identity math X,Y t frac 1 2 X Y t X t Y t . math Finite variation processes A process X is said to have finite variation if it has bounded variation over every finite time interval with probability 1 . Such processes are very common including, in particular, all continuously differentiable functions. The quadraticvariation exists for all continuous finite variation processes, and is zero. This statement can be generalized to non continuous processes. Any c dl g finite variation process X has quadraticvariation ... processes have zero quadraticvariation follows from the following inequality. Here, P is a partition ... P Vert math goes to zero. It processes The quadraticvariation of a standard Wiener process Brownian ... 0 t mu s ,ds, math where B is a Brownian motion. Any such process has quadraticvariation given by math ... well defined quadraticvariation, which follows from the fact that such processes are examples of semimartingales. It can be shown that the quadraticvariation M of a general local martingale M is the unique ... of a martingale in terms of the quadraticvariation. For a local martingale M starting ..., the predictable quadraticvariation is sometimes used for locally square integrable martingales. This is written ..., it is the same as the quadraticvariation. See also Total variation Bounded variation References Citation ... more details
TOCright In mathematics, the term quadratic describes something that pertains to Square algebra squares ... and abstract Quadratic function , a polynomial function of degree 2 Quadratic equation , a polynomial equation of degree 2 reducable to y ax sup 2 sup bx c Quadratic formula , calculation to solve a quadratric equation for the independent variable x Quadratic polynomial , a polynomial that contains terms of at most second degree Complex quadratic polynomial s are particularly interesting for their sometimes chaotic properties under iteration Quadratic field , an algebraic number field of degree two over the field of rational numbers Quadratic irrational or quadratic surd , an irrational number that is a root of a quadratic polynomial Calculus Quadratic integral , the integral of the reciprocal of a second degree polynomial Statistics and stochastics Quadratic form statistics , scalar quantity for a n dimensional square matrix Quadratic mean , the square root of the mean of the squares of the data Quadraticvariation , in stochastics, useful for the analysis of Brownian motion and martingales Number theory Quadratic reciprocity , a theorem from number theory Quadratic residue , an integer that is a square modulo n Quadratic sieve , a modern integer factorization algorithm Other Quadratic convergence , in which the distance to a convergent sequence s limit is squared at each step Quadratic differential , a form on a Riemann surface that locally looks like the square of an abelian differential Quadratic form , a homogeneous polynomial of degree two in a number of variables Quadratic programming , a special type of mathematical optimization problem. Quadratic growth , an asymptotic growth rate proportional to a quadratic function Periodic points of complex quadratic mappings ... Quadratic probing , a scheme in computer programming for resolving collisions in hash tables. Quadratic ... Other Quadratic collection Quadratic collection , a 1953 collection of science fiction novels by Olaf ... more details
Variation Biodiversity Genetic diversity , differences within a species Physics Magnetic variation , difference between magnetic north and true north, measured as an angle. Variation astronomy , any perturbation of the mean motion or orbit of a planet or satellite, particularly of the moon Mathematics Bounded variation Calculus of variations , a field of mathematics which deals with functions of functions. Coefficient of variation Permutation Arts Variation music , formal technique where material is altered during repetition Variations album , composed by Andrew Lloyd Webber Variation on a theme , in art history Ballet Variation ballet , solo dance or dance figure Balanchine s 1947 Theme and Variations ballet Balanchine s 1966 Variations ballet , an earlier version of Variations for Orchestra Balanchine s 1982 Variations for Orchestra , a subsequent revision of Variations Other Variation game , modifications made to a game by a community of players as opposed to a central authority Variation game tree , particular series of moves. Variation linguistics See also Variability Variations on a Theme disambiguation disambig ca Variaci cs Variace de Variation es Variaci n gl Variaci n lt Variacija ka ja no Variasjon pl Wariacja pt Varia o ru sk Vari cia fi Variaatio uk ... more details
About quadratic equations and solutions more general information about quadratic functions Quadratic function more information about quadratic polynomials Quadratic polynomial In mathematics , a quadratic ... a , b , and c , are called respectively, the quadratic coefficient , the linear coefficient and the constant term or free term. The term quadratic comes from quadratus , which is the Latin word for Square algebra square . Quadratic equations can be solved by Factorization factoring , completing the square , Graph of a function graphing , Newton s method , and using the quadratic formula given below . One common use of quadratic equations is computing trajectories in projectile motion . Another ... . File Quadratic equation coefficients.png thumb right 300px Plots of real number real valued quadratic function nowrap ax sup 2 sup bx c , varying each coefficient separately Quadratic formula This section is linked from Number A quadratic equation with real number real or complex number complex ... or may not be real number real . The roots are given by the quadratic formula math x frac b ... 4ac 2a math style width 100px align center and math frac b sqrt b 2 4ac 2a math are solutions of the quadratic equation. There is also a shortened version of the quadratic formula which is commonly used ... are given by math x frac k pm sqrt k 2 ac a . math Discriminant File Quadratic equation discriminant.png ... is called the discriminant of the quadratic equation, and is often represented using an upper case ... , discriminant math Delta b 2 4ac. , math A quadratic equation with real coefficients can have either ... width 100px align center and math frac b sqrt Delta 2a math For quadratic equations with integer coefficients ... s in other cases they may be quadratic irrational s. If the discriminant is zero, then there is exactly ... Polynomialdeg2.svg thumb right 200px For the quadratic function br font size 2 f font x x sup 2 sup ... where the graph intersects the x axis, x &minus 1 and x 2, are the solutions of the quadratic equation ... more details
In mathematics , a quadratic algebra is a filtered algebra generated by degree one elements, with defining relations of degree 2. It was pointed out by Yuri Manin that such algebras play an important role in the theory of quantum group s. The most important class of graded quadratic algebras is Koszul algebra s. Definition A graded quadratic algebra A is determined by a vector space of generators V A sub 1 sub and a subspace of homogeneous quadratic relations S V V harv Polishchuk Positselski 2005 p 6 . Thus math A T V langle S rangle math and inherits its grading from the tensor algebra T V . If the subspace of relations may also contain inhomogeneous degree 2 elements, S k &oplus V &oplus V V , this construction results in a filtered quadratic algebra . A graded quadratic algebra A as above admits a quadratic dual the quadratic algebra generated by V sup sup and with quadratic relations forming the orthogonal complement of S in V sup sup V sup sup . Examples Tensor algebra , symmetric algebra and exterior algebra of a finite dimensional vector space are graded quadratic in fact, Koszul algebras. Universal enveloping algebra of a finite dimensional Lie algebra is a filtered quadratic algebra. References Citation last1 Polishchuk first1 Alexander last2 Positselski first2 Leonid title Quadratic algebras url http books.google.com ?id 5 ZrCKU4NpAC&pg PA6 publisher American Mathematical Society location Providence, R.I. series University Lecture Series isbn 978 0 8218 3834 1 id MathSciNet id 2177131 year 2005 volume 37 Citation last1 Mazorchuk first1 Volodymyr last2 Ovsienko first2 Serge last3 Stroppel first3 Catharina title Quadratic duals, Koszul dual functors, and applications url http www.ams.org journals tran 2009 361 03 S0002 9947 08 04539 X home.html publisher Trans. of the American Mathematical Society year 2009 volume 361 p. 1129 1172 algebra stub Category Algebras he ... more details
In mathematics , a quadratic differential on a Riemann surface is a section of the symmetric square of the holomorphic cotangent bundle . If the section is holomorphic, then the quadratic differential is said to be holomorphic. The vector space of holomorphic quadratic differentials on a Riemann surface has a natural interpretation as the cotangent space to the Riemann moduli space or Teichmueller space . Local form Each quadratic differential on a domain math U math in the complex plane may be written as math f z dz otimes dz math where math z math is the complex variable and math f math is a complex valued function on math U math . Such a local quadratic differential is holomorphic if and only if math f math is holomorphic . Given a chart math mu math for a general Riemann surface math R math and a quadratic differential math q math on math R math , the pull back math mu 1 q math defines a quadratic differential on a domain in the complex plane. Relation to abelian differentials If math omega math is an abelian differential on a Riemann surface, then math omega otimes omega math is a quadratic differential. Singular Euclidean structure A holomorphic quadratic differential math q math determines a Riemannian metric math q math on the complement of its zeroes. If math q math is defined on a domain in the complex plane and math q f z dz otimes dz math , then the associated Riemannian metric is math f z dx 2 dy 2 math where math z x i y math . Since math f math is holomorphic, the curvature of this metric is zero. Thus, a holomorphic quadratic differential defines a flat metric on the complement of the set of math z math such that math f z 0 math . References Kurt Strebel, Quadratic differentials . Ergebnisse der Mathematik und ihrer Grenzgebiete 3 , 5. Springer Verlag, Berlin, 1984. xii 184 pp. ISBN 3 540 13035 7 Y. Imayoshi and M. Taniguchi, M. An introduction to Teichm ller spaces . Translated and revised from the Japanese by the authors. Springer Verlag, Tokyo, 1992 ... more details
In mathematical finite group theory, a quadratic pair for the odd prime number prime p , introduced by harvtxt Thompson 1971 , is a finite group mathematics group G together with a quadratic module , a faithful representation M on a vector space over the finite field with p elements such that G is generated by elements with minimum polynomial x &minus 1 sup 2 sup . Thompson classified the quadratic pairs for p &ge 5. harvtxt Chermak 2004 classified the quadratic pairs for p 3. With a few exceptions, especially for p 3, groups with a quadratic pair for the prime p tend to be more or less groups of Lie type in characteristic p . References Citation last1 Chermak first1 Andrew title Quadratic pairs doi 10.1016 S0021 8693 03 00334 X id MR 2059620 year 2004 journal Journal of Algebra issn 0021 8693 volume 277 issue 1 pages 36 72 Citation last1 Thompson first1 John G. author1 link John G. Thompson title Actes du Congr s International des Math maticiens Nice, 1970 url http mathunion.org ICM ICM1970.1 publisher Gauthier Villars id MR 0430043 year 1971 volume 1 chapter Quadratic pairs pages 375 376 Category Finite groups ... more details
In algebraic number theory , a quadratic field is an algebraic number field K of degree two over Q . It is easy ... integer s d 0, 1 to the set of all quadratic fields. If d 0 the corresponding quadratic field is called a real quadratic field , and for d 0 an imaginary quadratic field or complex quadratic field , corresponding to whether its Absolute value algebra archimedean embeddings are real or complex. Quadratic fields have been studied in great depth, initially as part of the theory of binary quadratic form s. There remain some unsolved problems. The class number problem is particularly important. Ring of integers Main Quadratic integer Discriminant The Discriminant of an algebraic number field discriminant of the quadratic field Q &radic d is d if d is congruent ... s, given by the complex cube roots of unity . The set of discriminants of quadratic fields is exactly ... rise to an ideal pO sub K sub in the ring of integers O sub K sub of a quadratic field K . In line ... density theorem . ref Samuel, pp. 76 77 ref The law of quadratic reciprocity implies that the splitting behaviour of a prime p in a quadratic field depends only on p modulo D , where D is the field discriminant. Quadratic subfields of cyclotomic fields The quadratic subfield of the prime cyclotomic field A classical example of the construction of a quadratic field is to take the unique quadratic ... of the quadratic field is p for p 4 n 1 and &minus p for p ... the quadratic field discriminant. That rules out the other discriminants &minus 4 p and 4 p in the respective ... groups with extra 2 torsion, and so contain at least three quadratic fields. In general a quadratic .... This expresses the fact that the Conductor algebraic number theory conductor of a quadratic field ... Heegner number Quadratic irrational Quadratic integer Notes reflist References cite book author Duncan Buell title Binary quadratic forms classical theory and modern computations publisher Springer Verlag ... more details
In number theory , an integer q is called a quadratic residue modular arithmetic modulo n if it is Congruence ... x such that math x 2 equiv q text text mod n . math Otherwise, q is called a quadratic nonresidue mod ... as modular arithmetic, quadratic residues are now used in applications ranging from acoustical engineering ... some conjectures ref Lemmermeyer, pp 6&ndash 8, p. 16 ff ref about quadratic residues, but the first ... the terminology quadratic residue and quadratic nonresidue , and states that, if the context makes it clear, the adjective quadratic may be dropped. For a given n a list of the quadratic residues mod ... that high. This can be seen in the table at the end of the article. Thus, the number of quadratic ... of two residues is always a residue. Prime modulus Modulo 2, every integer is a quadratic residue. Modulo ... of quadratic reciprocity is that if p 1 mod 4 then &minus 1 is a quadratic residue mod p , and if p ... & Rosen, p. 50 ref add to the definition that a quadratic residue q must not only be a square ... symbol , also called the Dirichlet character Examples quadratic character , which is defined for all ... many formulas and theorems easier to state. The other related reason is that the quadratic character ... agree. Distribution of quadratic residues Although quadratic residues appear to occur in a rather random pattern mod n , and this has been exploited in such Applications of quadratic residues applications ... in arithmetic progressions, the law of quadratic reciprocity , and the Chinese remainder theorem ... number of binary quadratic form s. ref Davenport, pp 8&ndash 9, 43&ndash 51. These are classical ... , the sum of the quadratic residues minus the sum of the nonresidues in the range 1, 2, , q &minus 1 ... Jacobi Jacobi , proved by Dirichlet ref In contrast, for prime q 1 mod 4 , the sum of the quadratic ... that there are more quadratic residues than nonresidues among the numbers 1, 2, , q &minus 1 ... O notation . Setting math chi n left frac n q right , math this shows that the number of quadratic ... more details
Image 3d Quadratic.png thumb 350px right Quadratic polynomial function that happens to have two variables x and y . In mathematics, a quadratic polynomial or quadratic is a polynomial of degree of a polynomial degree two, also called second order polynomial. That means the exponents of the polynomial s variables are no larger than 2. For example, nowrap x sup 2 sup &minus 4 x 7 is a quadratic polynomial, while nowrap x sup 3 sup &minus 4 x 7 is not. Coefficients The coefficients of a polynomial are often taken to be Quadratic function real or Complex quadratic polynomial complex number s, but in fact, a polynomial may be defined over any ring mathematics ring . Degree When using the term quadratic ... of degree , e.g. a second order polynomial. Variables A quadratic polynomial may involve ... case Any single variable quadratic polynomial may be written as math ax 2 bx c, , math where ... often arise in the form of a quadratic equation math ax 2 bx c 0 math . The solutions to this equation are called the Root of a function roots of the quadratic polynomial, and may be found through ... the use of the quadratic formula . Each quadratic polynomial has an associated quadratic function ... mathematics variable , it determines a quadratic function in one variable. An example is given by f ... of a function zero es can be found by solving the quadratic equation f x 0. There are three ... quadratic polynomial monic and centered form , math f c x x 2 c , math , used to study complex dynamics . Two variables case Any quadratic polynomial with two variables may be written as math ax 2 bxy .... Such polynomials are fundamental to the study of conic section s. Similarly, quadratic polynomials ... , quadratic polynomials can be generalized to the notion of a quadratic form on a vector space . N variables case In the general case, a quadratic polynomial in n variables x sub 1 sub , ..., x sub ... , and R a constant. See also Periodic points of complex quadratic mappings Quadratic equation Quadratic ... more details
Unreferenced date January 2007 For other uses of the word quadratic in mathematics Quadratic disambiguation In mathematics , a function or sequence is said to exhibit quadratic growth when its values are proportionality mathematics proportional to the squaring square of the function argument or sequence position, in the limit as the argument or sequence position goes to infinity. That is, in Big O notation big Theta notation , math f x Theta x 2 math . Examples of quadratic growth include Any quadratic polynomial . Certain integer sequence s such as the triangular number s. The n th triangular number has value n n 1 2, approximately n sup 2 sup 2. The amount of time taken in the worst case by certain algorithm s, such as insertion sort , as a function of the input length. The numbers of live cells in space filling cellular automaton patterns such as the Breeder CA , as a function of the number of time steps for which the pattern is simulated. Metcalfe s law stating that the value of a communications network grows quadratically as a function of its number of users See also Exponential growth DEFAULTSORT Quadratic Growth Category Asymptotic analysis Mathanalysis stub pt Ordem quadr tica ... more details
In mathematics , a quadratic form is a homogeneous polynomial of Degree of a polynomial degree two in a number of variables. For example, math 4x 2 2xy 3y 2 , math is a quadratic form in the variables x and y . Quadratic forms occupy a central place in various branches of mathematics, including number ... theory the Killing form . Introduction Quadratic forms are homogeneous quadratic polynomials in n variables. In the cases of one, two, and three variables they are called unary , binary quadratic form ... in the literature ref Note that quadratic function s, such as ax sup 2 sup bx c in the one variable case, are not quadratic forms, as they are typically not homogeneous polynomial homogeneous unless b and c are both 0 . The theory of quadratic forms and methods used in their study depend in a large ... of applications of quadratic forms, the coefficients are real or complex numbers. In the algebraic theory of quadratic forms, the coefficients are elements of a certain field algebra field . In the arithmetic theory of quadratic forms, the coefficients belong to a fixed commutative ring , frequently ... Terminology away from 2 , i. e. if 2 is invertible in the ring, quadratic forms are equivalent to symmetric ... is particularly important for quadratic forms over the integers. ref Binary quadratic forms have been extensively studied in number theory , in particular, in the theory of quadratic field s, continued fraction s, and modular forms . The theory of integral quadratic forms in n variables has important applications to algebraic topology . Using homogeneous coordinates , a non zero quadratic ... one may visualize 3 dimensional real quadratic forms as conic sections . A closely related notion with geometric overtones is a quadratic space , which is a pair V , q , with V a vector space over a field k , and q V k a quadratic form on V . An example is given by the three dimensional Euclidean ... of particular quadratic forms, in particular the question of whether a given integer can be the value ... more details
Refimprove date May 2008 In mathematics , a quadratic irrational , also known as a quadratic irrationality or quadratic surd , is an irrational number that is the solution to some quadratic equation with rational coefficients. ref J rn Steuding, Diophantine Analysis , 2005 , Chapman & Hall, p.72. ref Since fractions can be cleared from a quadratic equation by multiplying both sides by their common denominator , this is the same as saying it is an irrational root of some quadratic equation whose coefficients are integer s. They form the real number subset of the algebraic numbers of degree 2. The quadratic irrationals, therefore, are all those numbers that can be expressed in this form math a b sqrt c over d math for integers a , b , c , d with b and d non zero, and with c > 1 and square free . This implies that the quadratic irrationals have the same cardinality as ordered quadruples of integers, and are therefore countable . The quadratic irrationals with a given c form a field mathematics field , called a quadratic field . In particular, their inverses are of the same form, since math d over a b sqrt c ad bd sqrt c over a 2 b 2c . , math Quadratic irrationals have useful properties, especially in relation to continued fraction s, where we have the result that all quadratic irrationals, and only quadratic irrationals, have periodic continued fraction forms. For example math sqrt 3 1.732 ldots 1 1,2,1,2,1,2, ldots math Square root of non square is irrational The definition of quadratic irrationals requires them to satisfy two conditions they must satisfy a quadratic equation and they must be irrational. The solutions to the quadratic equation ax sup 2 sup bx c 0 are math frac b pm sqrt b 2 4ac 2a . math Thus quadratic irrationals are precisely ... Continued fraction calculator for quadratic irrationals http planetmath.org encyclopedia EIsIrrational.html Proof that e is not a quadratic irrational Category Number theory de Quadratisch irrationale ... more details
The quadratic sieve algorithm QS is a modern integer factorization algorithm and, in practice, the second .... This is roughly the basis of Fermat s factorization method . The quadratic sieve is a modification of Dixon s factorization method . The general running time required for the quadratic sieve to factor ... for a single number a such that a sup 2 sup mod n is a square. But these a are hard to find. The quadratic ... and matrices smaller and more tractable. The quadratic sieve searches for smooth numbers using .... The algorithm To summarize, the basic quadratic sieve algorithm has these main steps Choose a Smooth ... algorithm. How QS optimizes finding congruences The quadratic sieve attempts to find pairs of integers ... . The quadratic sieve speeds up the process of finding relations by taking x close to the square ... algorithms, such as the Shanks&ndash Tonelli algorithm . This is where the quadratic sieve gets its name y is a quadratic polynomial in x , and the sieving process works like the Sieve of Eratosthenes ... the quadratic equation mod p to get two roots and , and then add an approximation to log ... necessary to solve the quadratic equation modulo small powers of p in order to recognise ... sieving. Example of basic sieve This example will demonstrate standard quadratic sieve without ... quadratic residue and will be the basis for sieving. Now we construct our sieve math V X math of math ... equiv Z 2 pmod N math , the remainder of the algorithm follows equivalently to any other variation of Dixon .... This demonstration should also serve to show that the quadratic sieve is only appropriate when n is large ... MPQS, Multiple Polynomial Quadratic Sieve is ideally suited for parallel algorithm parallelization ... implementation of the self initialising multiple polynomial quadratic sieve written by William Hart ... Alpern, that uses the quadratic sieve if certain conditions are met. The PARI GP computer algebra package includes an implementation of the self initialising multiple polynomial quadratic sieve implementing ... more details
A quadratic function , in mathematics , is a polynomial function of the form math f x ax 2 bx c, quad a ne 0. math The graph of a function graph of a quadratic function is a parabola whose axis of symmetry is parallel to the y axis. The expression math ax 2 bx c math in the definition of a quadratic ... , because the highest exponent of x is 2. If the quadratic function is set equal to zero, then the result is a quadratic equation . The solutions to the equation are called the root of a function root s of the equation. Origin of word The adjective quadratic comes from the Latin word quadratum ... a square has four sides. Roots Further Quadratic equation The root of a function roots zeros of the quadratic ... pm sqrt Delta 2 a , math where the discriminant is defined as math Delta b 2 4 a c , . math Forms of a quadratic function A quadratic function can be expressed in three formats ref citation title College ... for a quadratic expression are 22 , http books.google.be books?sourceid navclient&ie UTF 8&rlz 1T4GGLJ enBE306BE306&q 22three different forms for a quadratic expression are 22 Search result ref math ... form , where math x 1 math and math x 2 math are the roots of the quadratic equation, it is used ... form , one needs only the quadratic formula to determine the two roots math x 1 math and math x 2 math ... of a quadratic function is a parabola as shown above . If math a 0 , math or is a positive number .... The coefficient a controls the speed of increase or decrease of the quadratic function from ..., it is the point where the parabola intercept the y axis. Vertex This section is linked from Quadratic ... point . If the quadratic function is in vertex form, the vertex is math h, k , math . By the method ... is math left frac b 2a , frac Delta 4 a right . math If the quadratic function is in factored form ... b 2a , frac Delta 4a right . math The square root of a quadratic function The square root of a quadratic ... and g . And see Complex quadratic polynomial for the chaotic behavior in the general iteration. Bivariate ... more details
separating surface between the classes is a quadratic. Other quadratic classifiers While ... x 1, x 2, x 3, x 1 2, x 1x 2, x 1 x 3, x 2 2, x 2x 3, x 3 2 math . Finding a quadratic classifier ... DEFAULTSORT Quadratic Classifier Category Machine learning Category Classification algorithms ... more details
Original research article date May 2008 Quadratic probing is a scheme in computer programming for resolving collisions in hash table s. Quadratic probing operates by taking the original hash value and adding successive values of an arbitrary quadratic polynomial to the starting value. This algorithm is used in Open addressing open addressed hash table s. Quadratic probing provides good memory caching because it preserves some locality of reference however, linear probing has greater locality and, thus, better cache performance. Quadratic probing better avoids the clustering problem that can occur with linear probing, although it is not immune. Quadratic probing is used in the Berkeley Fast File System to allocate free blocks. The allocation routine chooses a new cylinder group when the current is nearly full using quadratic probing, because of the speed it shows in finding unused cylinder groups. Quadratic Probing Algorithm Let math h k math be a hash function that maps an element math k math to an integer in math 0,m 1 math , where math m math is the size of the table. Let the math i math sup th sup probe position for a value math k math be given by the function math h k,i h k c 1 i c 2 i 2 pmod m math , where math c 2 neq 0 math . If math c 2 0 math , then math h k,i math degrades to a linear probing linear probe . For a given hash table , the values of math c 1 math and math c 2 math remain constant. Example If math h k,i h k i i 2 pmod m math , then the probe sequence will be math h k , h k 2, h k 6, ... math For math m 2 n math , a good choice for the constants are math c 1 c 2 math 1 2, as the values math h k,i math for math i math in math 0,m 1 math are all distinct. ref Proof assume there exist math i,j math such that math i,j math in math 0,m 1 math and math i i ... Vladimir Batagelj The quadratic hash method when the table size is not a prime number. Communications ...?id 360737 Category Hashing nl Quadratic probing ... more details
In number theory , quadratic integers are a generalization of the rational integers to quadratic field s. Important examples include the Gaussian integers and the Eisenstein integers . Though they have been studied for more than a hundred years, many open problems remain. Definition Quadratic integers are solutions of equations of the form x sup 2 sup Bx C 0 for integers B and C . Such solutions have the form a b , where a , b are integers, and where is defined by math omega begin cases sqrt D & mbox if D equiv 2, 3 pmod 4 1 sqrt D over 2 & mbox if D equiv 1 pmod 4 end cases math D is a square free integer . This characterization was first given by Richard Dedekind in 1871. ref harvnb Dedekind 1871 , Supplement X, p. 447 ref ref harvnb Bourbaki 1994 , p. 99 ref Fixing a square free integer D , the quadratic integer ring Z a b a , b Z is a subring of the quadratic field math mathbf Q sqrt D math . Moreover, Z is the integral closure of Z in math mathbf Q sqrt D math . In other words, it is the ring of integers math mathcal O mathbf Q sqrt D math of math mathbf Q sqrt D math and thus a Dedekind domain . Examples A classic example is math mathbf Z sqrt 1 math , the Gaussian integer s, which was introduced by Carl Gauss around 1800 to state his biquadratic reciprocity law. ref Dummit, pg. 229 ref The elements in math mathcal O mathbf Q sqrt 3 mathbf Z left 1 sqrt 3 over 2 right math are called Eisenstein integer s. In contrast, math mathbf Z sqrt 3 math is not even a Dedekind domain. Class number Equipped with the field norm norm math N a b sqrt D a 2 Db 2 math , math mathcal O mathbf Q sqrt D math is an Euclidean domain a fortiori, Unique factorization domain UFD when ... a Dedekind domain, a quadratic integer ring is a UFD if and only if it is a principal ideal domain i.e., its class number is one. However, there are quadratic integer rings that are principal ideal ... ideal class group s of quadratic integer rings, but many theoretical questions about their structure ... more details
Quadratic programming QP is a special type of mathematical optimization mathematics optimization problem. It is the problem of optimizing minimizing or maximizing a quadratic function of several variables subject to linear constraints on these variables. The quadratic programming problem can be formulated as ref Cite book last1 Nocedal first1 Jorge last2 Wright first2 Stephen J. title Numerical Optimization publisher Springer Verlag location Berlin, New York edition 2nd isbn 978 0 387 30303 1 year 2006 page 449 postscript None . ref Assume x belongs to math mathbb R n math space. The n × n matrix math matrix Q is symmetric, and c is any n × 1 vector. Minimize with respect to x math f mathbf x tfrac 1 2 mathbf x T Q mathbf x mathbf c T mathbf x . math Subject to one or more constraints of the form math A mathbf x leq mathbf b math inequality constraint math E mathbf x mathbf d math equality constraint where math mathbf x T math indicates the vector transpose of math mathbf x math . The notation math Ax leq b math means that every entry of the vector Ax is less than or equal to the corresponding ... matrix positive semidefinite matrix , then math f math is a convex function In this case the quadratic ... programming problem, quadratically constrained quadratic program quadratically constrained quadratic programming , can be posed by adding quadratic constraints on the variables. Solution methods ... murty Katta G. Murty . MR 949214 ref Convex quadratic programming is a special case ... Leonid G. Khachiyan year 1979 trans title Polynomial solvability of convex quadratic programming journal ... Quadratic programming with one negative eigenvalue is NP hard, Panos M. Pardalos and Stephen A. Vavasis ... A general purpose and matrix oriented programming language br for numerical computing. Quadratic ... Sequential quadratic programming quadratically constrained quadratic program Quadratically constrained quadratic programming References Notes references Bibliography cite book last1 Cottle first1 ... more details
infobox Book See Wikipedia WikiProject Novels or Wikipedia WikiProject Books name Quadratic title orig translator image Image Quadratic.jpg 200px image caption Dust jacket from the first edition author Olaf Stapledon and Murray Leinster illustrator cover artist Walter country United States language English language English series genre Science fiction novels publisher Fantasy Publishing Company, Inc. release date 1953 in literature 1953 english release date media type Print Hardcover Hardback pages 580 pp isbn NA oclc 36939070 preceded by followed by Quadratic is a collection of four science fiction works by Olaf Stapledon and Murray Leinster . It was edited by William L. Crawford and published in 1953 in literature 1953 by Fantasy Publishing Company, Inc. in an edition of 300 copies. The book is an omnibus of Stapledon s Worlds of Wonder collection Worlds of Wonder and Leinster s Murder Madness . Contents Death into Life , by Olaf Stapledon The Flames A Fantasy The Flames , by Olaf Stapledon Old Man in New World Old Man in a New World , by Olaf Stapledon Murder Madness , by Murray Leinster References cite book last Chalker first Jack L. authorlink Jack L. Chalker coauthors Mark Owings title The Science Fantasy Publishers A Bibliographic History, 1923 1998 location Westminster, MD and Baltimore publisher Mirage Press, Ltd. pages 272 date 1998 cite web last Contento first William G. authorlink coauthors title Index to Science Fiction Anthologies and Collections work publisher date url http www.philsp.com homeville ISFAC t30.htm A662 format doi accessdate 2008 05 03 cite book last Tuck first Donald H. authorlink Donald H. Tuck title The Encyclopedia of Science Fiction and Fantasy location Chicago publisher Advent publisher Advent pages 120 date 1974 id ISBN 0 911682 20 1 Category 1953 books Category Science fiction anthologies ... more details
, see Ethic of reciprocity . In number theory , the law of quadratic reciprocity is a theorem about modular arithmetic which gives conditions for the solvability of quadratic equation s modulo prime ... . Quadratic reciprocity br Let q sup sup q where the sign is plus if q 1 mod 4 and minus if q ... whether any quadratic equation modulo a prime number has a solution, it does not provide any help at all for actually finding the solution. The article on quadratic residue s discusses algorithms for this. The theorem ... first proved quadratic reciprocity . See http books.google.com books?id NM36hgqmOLkC&pg PA30&lpg PA30 ... the formulations of quadratic reciprocity found by Legendre and Gauss. The Legendre Jacobi ... of the theorem A quadratic residue quadratic residue mod n is any number congruent to a square mod n . A quadratic nonresidue mod n is any number which is not congruent to a square mod n . The adjective quadratic can be dropped if the context makes it clear that it is implied. When working modulo ... statements become true Modulo a prime, there are an equal number of quadratic residues and nonresidues. Modulo a prime, the product of two quadratic residues is a residue, the product of a residue and a nonresidue is a nonresidue, and the product of two nonresidues is a residue. Table of quadratic ... less than 50. To check whether a number n is a quadratic residue mod one of these primes p , find ..., it is a nonresidue mod p . The quadratic reciprocity law is the statement that certain patterns found ... prime numbers. &ndash 1 and the first supplement First of all, for which prime numbers is &ndash 1 a quadratic ... 3 mod 4 . This leads to The first supplement to quadratic reciprocity math mbox The congruence x 2 ... prime numbers is 2 a quadratic residue? Examining the table, we find 2 in rows 7, 17, 23, 31 ... are all 3 mod 8 . This leads to The second supplement to quadratic reciprocity math mbox ... are 1 mod 3 and the latter 2 mod 3 . Since the only residue mod 3 is 1, we see that &ndash 3 is a quadratic ... more details
In mathematics , a quadratic integral is an integral of the form math int frac dx a bx cx 2 . math It can be evaluated by completing the square in the denominator . math int frac dx a bx cx 2 frac 1 c int frac dx left x frac b 2c right 2 left frac a c frac b 2 4c 2 right . math Positive discriminant case Assume that the discriminant q b sup 2 sup &minus 4 ac is positive. In that case, define u and A by math u x frac b 2c math , and math A 2 frac a c frac b 2 4c 2 frac 1 4c 2 left 4ac b 2 right . math The quadratic integral can now be written as math int frac dx a bx cx 2 frac1c int frac du u 2 A 2 frac1c int frac du u A u A . math The partial fraction decomposition math frac 1 u A u A frac 1 2A left frac 1 u A frac 1 u A right math allows us to evaluate the integral math frac1c int frac du u A u A frac 1 2Ac ln left frac u A u A right text constant . math The final result for the original integral, under the assumption that q 0, is math int frac dx a bx cx 2 frac 1 sqrt q ln left frac 2cx b sqrt q 2cx b sqrt q right text constant, where q b 2 4ac. math Negative discriminant case This hastily written section may need attention. In case the discriminant q b sup 2 sup &minus 4 ac is negative, the second term in the denominator in math int frac dx a bx cx 2 frac 1 c int frac dx left x frac b 2c right 2 left frac a c frac b 2 4c 2 right . math is positive. Then the integral becomes math begin align & qquad frac 1 c int frac du u 2 A 2 9pt & frac 1 cA int frac du A u A 2 1 9pt & frac 1 cA int frac dw w 2 1 9pt & frac 1 cA arctan w mathrm constant 9pt & frac 1 cA arctan left frac u A right text constant 9pt & frac 1 c sqrt frac a c frac b 2 4c 2 arctan left frac x frac b 2c sqrt frac a c frac b 2 4c 2 right text constant 9pt & frac 2 sqrt 4ac b 2 , arctan left frac 2cx b sqrt 4ac b 2 right text constant . end align math References Weisstein, Eric W. http mathworld.wolfram.com QuadraticIntegral.html Quadratic Integral . From MathWorld A Wolfram Web Resource ... more details
diminishing Total variation denoising Quadraticvariation Notes Reflist 2 Bibliography Citation ... variation of the function. In mathematics , the total variation identifies several slightly different ... , defined on an interval mathematics interval a , b , its total variation on the interval ... variation for functions of one real variable was first introduced by Camille Jordan in the paper ... function s whose variation is Bounded variation bounded . The extension of the concept to functions of more than one variable however is not simple for some reasons. Definitions Total variation for functions of one real variable EquationRef 1 Definition 1.1. The total variation of a real number ... of a,b right math of the given interval mathematics interval . Total variation for functions ... n sup . Given a function math f math belonging to math L 1 Omega math , the total variation of math ... domain math Omega math sup n sup of the given function is a bounded set . Total variation in measure ... called upper variation and lower variation , as follows math overline mathrm W mu,E sup left ... 1.3. The variation also called absolute variation of the signed measure math mu math is the set ... math and its total variation is defined as the value of this measure on the whole space of definition ... and lower variation are respectively a non negative and a non positive Measure mathematics measure ... the last one sometimes being called, by abuse of notation , total variation measure . When the measure math mu math is Complex number complex valued i.e. is a complex measure , its upper and lower variation ... the total variation of the complex valued measure math mu math as follows EquationRef 4 Definition 1.4. The variation of the complex valued measure math mu math is the set function math mu E sup ... math pi math of a measurable set math E math into a finite number of disjoint measurable subsets. The variation ... defined by EquationNote 3 1.3 when math mu math is a signed measure its total variation is defined ... more details
Unreferenced date February 2007 Variation or Classical Variation sometimes referred to as a Pas seul , meaning to Dance Alone in ballet is a solo dance. As with an Aria in opera , which allows the singer to demonstrate his or her interpretive skills, the variation in ballet has the same function. Typically, variations in ballet have traditional choreography, with every dancer dancing the same steps as in opera, a singer singing the same words . Ballet Category Ballet terminology Variation ballet eo Variado baleto sv Variation balett ... more details
nofootnotes date August 2010 Image wheat.gif right thumb Genetic variation , variation in allele s of gene s, occurs both within and among population s. Genetic variation is important because it provides the raw material for natural selection . Genetic variation is brought about by mutation, a change ... wherein organisms have three or more sets of genetic variation 3n or more . Among individuals within a population Genetic variation among individuals within a population can be identified at a variety of levels. It is possible to identify genetic variation from observations of phenotype phenotypic variation in either quantitative traits traits that vary continuously and are coded for by many ... variation can also be identified by examining variation at the level of enzyme s using the process ... common in vertebrates. Ultimately, genetic variation is caused by variation in the order of bases ... has identified even more genetic variation than was previously detected by protein electrophoresis. Examination of DNA has shown genetic variation in both coding regions and in the non coding intron region of genes. Genetic variation will result in phenotypic variation if variation in the order of nucleotides ... the shape, and thus the function of the enzyme. Between populations Geographic variation in genes often occurs among populations living in different locations. Geographic variation may be due to differences in selective pressures or to genetic drift . Measurement Genetic variation within a population ... variation because they alter the order of bases in the nucleotides of DNA. Mutations are likely ... be favored by natural selection. Genetic variation can also be produced by the recombination of chromosome ... also contributes to variation. Variation and recombination can be fascilitated by transposable ... see variation inducing genetic element s. Maintenance in populations A variety of factors maintain genetic variation in populations. Potentially harmful recessive alleles can be hidden from selection ... more details
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