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 ... index t ranging over the non negative real numbers. Its quadraticvariation is the process, written ... variables convergence in probability . Note that a process may be of finite quadraticvariation in the sense of the definition given here and its paths be nonetheless a.s. of infinite quadraticvariation for every t 0 in the classical sense of taking the supremum of the sum over all partitions this is in particular the case for Brownian Motion . More generally, the quadratic covariation or quadratic ... 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 ..., all continuously differentiable functions. The quadraticvariation exists for all continuous .... Any c dl g finite variation process X has quadraticvariation equal to the sum of the squares ... X sub t sub X sub t sub . Then, the quadraticvariation is given by math X t sum 0 s le t Delta X s 2. math The proof that continuous finite variation processes have zero quadraticvariation ... The quadraticvariation of a standard Wiener process Brownian motion B exists, and is given by B ... motion. Any such process has quadraticvariation given by math X t int 0 t sigma s 2 ,ds. math ... Martingales All c dl g martingales, and local martingale s have well defined quadraticvariation ... that the quadraticvariation M of a general local martingale M is the unique right continuous and increasing ... positive value of p . An alternative process, the predictable quadraticvariation is sometimes used ... and, for continuous local martingales, it is the same as the quadraticvariation. See also Total ... 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 reducible to 0 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 ... Quadratic time , in referring to algorithms with quadratic time complexity Other Quadratic collection ... more details
wiktionary variation tocright 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 Andrew Lloyd Webber album Variations Andrew Lloyd Webber album Variations Eddie Rabbitt album Variations Eddie Rabbitt album Variations Cage , a series of works by American avant garde composer John Cage Variations Stravinsky , Igor Stravinsky s last orchestral composition, written in 1963 64 Variation on a theme , in art history Ballet Variation ballet , solo dance or dance figure Balanchine s 1947 Theme and Variations ballet Theme and Variations Balanchine s 1966 Variations ballet 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 disambiguation Variations on a Theme disambiguation disambiguation ca Variaci cs Variace de Variation es Variaci n gl Variaci n it Variazione disambigua ka lt Variacija ja no Variasjon pl Wariacja pt Varia o ru sk Vari cia fi Variaatio uk ... more details
In mathematics, a quadratic transformation may be A quadratic transformation in the Cremona group Kummer s quadratic transformation of the hypergeometric function mathdab ... 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 ... quadratic equation can be written in the form math ax 2 bx c 0, , math where x represents a Variable ... 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 ... graphing , Newton s method , and using the quadratic formula given below . 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 coefficients has two ... number real . The roots are given by the quadratic formula math x frac b pm sqrt b 2 4ac 2a , math ... x frac b sqrt b 2 4ac 2a math are solutions of the quadratic equation. Discriminant File Quadratic ... the square root sign is called the discriminant of the quadratic equation, and is often represented ... language Greek word , Diakr nousa , discriminant math Delta b 2 4ac. , math A quadratic ... numbers math frac b sqrt Delta 2a quad text and quad frac b sqrt Delta 2a math For quadratic equations ... are rational number s in other cases they may be quadratic irrational s. If the discriminant is zero ... the quadratic equation by the quadratic coefficient a gives the simplified Monic polynomial monic form ..., A Geometric Algorithm with Solutions to Quadratic Equations in a Sumerian Juridical Document ... century BC quadratic equations of the form ax sup 2 sup c and ax sup 2 sup bx c were explored using ... from circa 200 BC used the method of completing the square to solve quadratic equations ... the quadratic equation. In his work Arithmetica , the Greek mathematician Diophantus solved the quadratic ... more details
externallinks date November 2011 Quadratic programming QP is a special type of mathematical optimization optimization problem problem . It is the problem of optimizing minimizing or maximizing a quadratic ... The quadratic programming problem can be formulated as ref Cite book last1 Nocedal first1 Jorge ... is a convex function In this case the quadratic program has a global minimizer if there exists some ... constrained quadratic program quadratically constrained quadratic programming , can be posed by adding quadratic constraints on the variables. Solution methods For general problems a variety of methods ... quadratic optimization problems. Journal of Convex Analysis , 12, 45 69. ref , Conjugate gradient ... of Professor http www personal.umich.edu murty Katta G. Murty . MR 949214 ref Convex quadratic programming is a special case of the more general field of convex optimization . Equality constraints Quadratic ... simple depending on the structure of math E math . Substituting into the quadratic form gives an unconstrained ... matrix math Z T Q Z math will be positive definite. It s possible to write a variation on the conjugate ... Nocedal first Jorge year 2000 title On the Solution of Equality Constrained Quadratic Programming Problems ... trans title Polynomial solvability of convex quadratic programming journal Doklady Akademii Nauk SSSR ..., even if Q has only one negative eigenvalue , the problem is NP hard . ref Quadratic programming with one ... octave doc interpreter Quadratic Programming.html Octave GPL General purpose GNU Octave solver for Quadratic Programming problems Other Free open source licenses class wikitable Name License Brief ... containing an Interior point method interior point primal dual solver for convex quadratic programming ... scale linear programs, quadratic programs and mixed integer programs. Free for academic use. LINGO ... computing. Quadratic programming in MATLAB requires the Optimization Toolbox in addition to the base ... of the NAG Library includes routines for quadratic programming problems with both sparse and non ... 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 V 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 American Mathematical Society year 2009 volume 361 bibcode 2006math......3475M pages 1129 1172 arxiv math 0603475 class math.RT issue 3 journal Trans. Amer. Math. Soc. doi 10.1090 S0002 9947 08 04539 X p. 1129 1172 Category ... more details
merge Quadratic function date October 2011 Image 3d Quadratic.png thumb 350px right Quadratic polynomial function of two independent variables x and y . In mathematics, a quadratic polynomial or quadratic ... 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 ... . Degree When using the term quadratic polynomial , authors sometimes mean having degree exactly ... A quadratic polynomial may involve a single Variable mathematics variable x , or multiple variables such as x , y , and z . The one variable case Any single variable quadratic polynomial may be written ... s. In elementary algebra , such polynomials 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 ... graphing , Newton s method , or through the use of the quadratic formula . Each quadratic polynomial has an associated quadratic function , whose graph of a function graph is a parabola . If the polynomial is a polynomial in one Variable mathematics variable , it determines a quadratic function in one ... mathematics line , and its Root of a function zero es can be found by solving the quadratic equation ... Maps discrete dynamics , Complex 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 ... s. Similarly, quadratic polynomials with three or more variables correspond to quadric surfaces and hypersurface s. In linear algebra , 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 ... dimensional Vector geometric vector , and R a constant. See also Periodic points of complex quadratic mappings Quadratic equation Quadratic function References Algebra 1, Glencoe, ISBN 0 07 825083 ... more details
For other uses of the word quadratic in mathematics Quadratic disambiguation Unreferenced date January 2007 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 Note In plain and simple English, quadratic growth is growth where the rate of change changes at a constant rate. For example, if you add 3 the first time, then you add 3.5 the next time, and 4 the time after that, that is quadratic growth. In this case, you added 0.5 to your rate of change each time. See also Exponential growth DEFAULTSORT Quadratic Growth Category Asymptotic analysis Mathanalysis stub pt Ordem quadr tica ... 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 version by the authors. Springer Verlag, Tokyo ... 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 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 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 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 pmod n . math Otherwise, q is called a quadratic nonresidue modulo ... arithmetic, quadratic residues are now used in applications ranging from acoustical engineering ... conjectures ref Lemmermeyer, pp 6&ndash 8, p. 16 ff ref about quadratic residues, but the first systematic ... 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 modulo ... to go 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 ... ref to the law of quadratic reciprocity is that if p 1 mod 4 then &minus 1 is a quadratic residue modulo ... that a quadratic residue q must not only be a square but must also be relatively prime to the modulus ... character Examples quadratic character , which is defined for all integers a and positive odd prime .... The other related reason is that the quadratic character is a homomorphism from the multiplicative ... N m . If m is prime, the Jacobi and Legendre symbols agree. Distribution of quadratic residues Although quadratic residues appear to occur in a rather random pattern modulo n , and this has been exploited in such Applications of quadratic residues applications as Acoustics acoustics and Cryptography ..., the law of quadratic reciprocity , and the Chinese remainder theorem CRT it is easy to see that for any ... For example, if p 1 mod 8 , mod 12 , mod 5 and mod 28 , then by the law of quadratic reciprocity 2, 3 ... number number theory class number of binary quadratic form s. ref Davenport, pp 8&ndash 9, 43&ndash ..., in this case prime q 3 mod 4 , the sum of the quadratic residues minus the sum of the nonresidues ..., for prime q 1 mod 4 , the sum of the quadratic residues minus the sum of the nonresidues in the range ... more details
ref improve date March 2012 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 ... of four manifold s , and Lie 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 binary , and ternary and have the following explicit form math q x ax ... forms. Both conventions occur 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 ... geometry , and in the majority 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 ... Localization of a ring Terminology away from 2 , i. e. if 2 is invertible in the ring, quadratic ... concepts this distinction 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 ... , a non zero quadratic form in n variables defines an n &minus 2 dimensional Quadric projective ... geometry . In this way 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 ... 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 in the coefficients of a quadratic equation can be cleared by multiplying both sides by their common denominator , a quadratic irrational is an irrational root of some quadratic equation whose coefficients are integer s. The quadratic irrationals form the Real number real algebraic number s of degree 2 and can, therefore, 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 rational number s together with all quadratic irrationals with a given c form a Field mathematics field , called a Quadratic field real 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 This field ... Q &radic Overline c . Quadratic irrationals have useful properties, especially in relation to continued fraction s, where we have the result that all quadratic irrationals, and only quadratic ..., 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 those numbers ... calculator for quadratic irrationals http planetmath.org encyclopedia EIsIrrational.html Proof that e is not a quadratic irrational Category Number theory de Quadratisch irrationale Zahl fr Irrationnel ... more details
The quadratic sieve algorithm QS is an integer factorization algorithm and, in practice, the second fastest ... 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 an integer ... to find. The quadratic sieve consists of computing a sup 2 sup mod n for several a , then finding ... numbers keeps the vectors and matrices smaller and more tractable. The quadratic sieve searches ... takes its name. The algorithm To summarize, the basic quadratic sieve algorithm has these main ... and extensions of this basic algorithm. How QS optimizes finding congruences The quadratic sieve attempts ... of x, is known as a relation . The quadratic sieve speeds up the process of finding relations by taking ..., such as the Shanks&ndash Tonelli algorithm . This is where the quadratic sieve gets its name y is a quadratic ... starts by setting every entry in a large array A of bytes to zero. For each p , solve the quadratic ... to solve the quadratic equation modulo small powers of p in order to recognise numbers divisible ... of basic sieve This example will demonstrate standard quadratic sieve without logarithm optimizations ... has a square root mod p are 2, 17, 23, and 29 in other words, 15347 is a quadratic residue modulo each ... math Y equiv Z 2 pmod N math , the remainder of the algorithm follows equivalently to any other variation ... being 149. This demonstration should also serve to show that the quadratic sieve is only appropriate ... called MPQS, Multiple Polynomial Quadratic Sieve is ideally suited for parallel algorithm parallelization ... is a fast implementation of the self initialising multiple polynomial quadratic sieve written by William ... by Dario 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 ... the thesis of Thomas Sosnowski. A variant of the quadratic sieve is available in the Magma computer ... more details
In number theory , quadratic integers are a generalization of the rational integers to quadratic field s. These are algebraic integer s of the degree 2 number 2 . Important examples include the Gaussian ..., many open problems remain. Definition Quadratic integers are solutions of equations of the form x ... ref harvnb Bourbaki 1994 , p. 99 ref The set of all quadratic integers is not closed even under addition . But for any fixed mvar D the set of corresponding quadratic integers forms a ring algebra ring , and it is these quadratic integer rings which are usually studied. Medieval Indian mathematics Indian mathematicians had already discovered a multiplication of quadratic integers of the same mvar D , which allows one to solve some cases of Pell s equation . The study of quadratic integers admits an algebraic version the study of quadratic form Integral quadratic forms quadratic forms with integer coefficients . Quadratic integer rings Fixing a square free integer mvar D , the quadratic integer ring math 1 Z mvar a mvar b mvar a , mvar b Z is a subring of the quadratic field math mathbf ... domain . The quadratic integer rings usually form the first class of examples on which one ... field theory, see Class number below . Examples of complex quadratic integer rings Image Punktraster.svg ... real number. Therefore, it is natural to treat a quadratic integer ring as a set of algebraic complex ... of real quadratic integer rings For mvar D 0, is a positive number positive irrational number irrational and the corresponding quadratic integer ring is a set of algebraic real number ... s, naturally leads to these rings for math mvar D &equiv 2, 3 mod 4 . Algebraic study of real quadratic ... of prime ideals in Galois extensions Being 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 ... methods to compute ideal class group s of quadratic integer rings, but many theoretical questions ... more details
Refimprove date October 2010 merge Quadratic polynomial date October 2011 Image Polynomialdeg2.svg thumb right center math x 2 x 2 math center 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 ... bx c math in the definition of a quadratic function is a polynomial of Degree of a polynomial degree 2 or second order, or a 2nd degree polynomial , 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 ... Quadratic equation The root of a function roots zeros of the quadratic function math f x ax 2 bx c , math ... is defined as math Delta b 2 4 a c , . math Forms of a quadratic function A quadratic function ... UTF 8&rlz 1T4GGLJ enBE306BE306&q 22three different forms for a quadratic expression are 22 , http ... forms for a quadratic expression are 22 Search result ref math f x a x 2 b x c , math is called the standard ... x 2 math are the roots of the quadratic equation, it is used in logistic map math f x a x h 2 k .... To convert the standard form to factored form , one needs only the quadratic formula to determine ... of the format, the graph of a quadratic function is a parabola as shown above . If math a 0 , math or is a positive ... opens downward. The coefficient a controls the speed of increase or decrease of the quadratic ... from Quadratic equation The vertex of a parabola is the place where it turns, hence, it s also called the turning point . If the quadratic function is in vertex form, the vertex is math h, k , math ... in the vertex form is math left frac b 2a , frac Delta 4 a right . math If the quadratic function is in factored ... be expressed as math left frac b 2a , frac Delta 4a right . math The square root of a quadratic function The square root of a quadratic function gives rise either to an ellipse or to a hyperbola .If math ... more details
some rearrangement, it can be shown that the resulting separating surface between the classes is a quadratic ... quantities in this formula. Other quadratic classifiers While QDA is the most commonly ... 2, x 1 x 3, x 2 2, x 2x 3, x 3 2 math . Finding a quadratic classifier for the original measurements ..., which corresponds to introducing only the sum of pure quadratic terms math x 1 2 x 2 2 x 3 2 ldots ... PatternRecognitionPrimerII DEFAULTSORT Quadratic Classifier Category Classification algorithms Category ... more details
Original research article date May 2008 Quadratic probing is a scheme in computer programming for resolving collisions in hash table s. It is an open addressing method to handle overflows after a collision takes place in some bucket of a hash table. Quadratic probing operates by taking the original hash value and adding successive values of an arbitrary quadratic polynomial to the starting value. The form of the equation is math f k c 1 k 2 c 2 k c 3 math . The function used might even be math f k c 1 k 2 math if math c 2 math and math c 3 math are taken as zero. In this case, suppose a cell H ... grows, lesser will be the search efficiency for those items. Quadratic probing can be a more efficient ... performance. 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 Function Let math ... to guarantee that insertions will succeed when the load factor is 1 2. Quadratic Probing ... quadratic probing. ref cite book last Horowitz, Sahni, Anderson Freed title Fundamentals of Data ... is full 6. Stop C function for Key Insertion source lang c int quadratic probing insert int hashtable ... the next hash function h k . Consider a hash table initially containing some elements. File Quadratic ... is empty, so key will be inserted here. Quadratic Probing Search Algorithm to Search Element in Hash ... be placed at the next location given by the quadratic function 4.4 Increment j 4.5 Compute new hash ... in the hash table 6. Stop C function for Key Searching source lang c int quadratic probing ... table gets filled. In the case of quadratic probing, the situation is even more drastic. With the exception ... Reflist External links http research.cs.vt.edu AVresearch hashing quadratic.php Tutorial quadratic probing Category Hashing nl Quadratic probing ... 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 , created by combining unbound sheets from the publisher s previous editions of the two volumes. ref Recommended Reading, F&SF , June 1953, p.71 ref 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 Reflist 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 text 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 text ... 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
Total variation denoising Quadraticvariation Notes inline date February 2012 Reflist 2 Bibliography ... axis, shown as a red ball, is the total variation of the function. In mathematics , the total variation ... a , b , its total variation on the interval of definition is a measure of the one ... a , b . Historical note The concept of total variation for functions of one real variable ... series of discontinuous function discontinuous periodic function s whose variation is Bounded variation ... for various reasons. Definitions Total variation for functions of one real variable EquationRef 1 Definition 1.1. The total variation of a real number real valued or more generally complex number ... mathematics interval . Total variation for functions of n 1 real variables EquationRef 2 Definition ... to math L 1 Omega math , the total variation of math f math in is defined as math V f, Omega sup ... function is a bounded set . Total variation in measure theory Following Harvtxt Saks 1937 p ... scriptstyle underline mathrm W mu, cdot math , respectively called upper variation and lower variation , as follows math overline mathrm W mu,E sup left mu A A in Sigma text and A subset E right ... W mu,E qquad forall E in Sigma math EquationRef 3 Definition 1.3. The variation also called absolute variation of the signed measure math mu math is the set function math mu E overline mathrm W mu,E left underline mathrm W mu,E right qquad forall E in Sigma math and its total variation is defined ... decomposition according to his version of this theorem, the upper and lower variation are respectively ... variation measure . If the measure math mu math is Complex number complex valued i.e. is a complex measure , its upper and lower variation cannot be defined and the Hahn&ndash Jordan decomposition ... Rudin 1966 pp 137&ndash 139 and define 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 ... more details
no footnotes 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, which is a change ... is a condition 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 ... . Genetic variation can also be identified by examining variation at the level of enzyme s using ... are less common in vertebrates. Ultimately, genetic variation is caused by variation in the order ... DNA which 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 ... sequence influence 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 ... of genetic variation because they alter the order of bases in the nucleotides of DNA. Mutations ... alleles can be favored by natural selection. Genetic variation can also be produced by the recombination ..., random fertilization also contributes to variation. Variation and recombination can be facilitated ..., SINEs, etc. Maintenance in populations A variety of factors maintain genetic variation in populations ... common zygosity homozygous individuals . Natural selection can also maintain genetic variation in balanced ... more details
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