"algebra, such polynomials often arise in the form of a quadratic equation formula_5. The solutions to this equation are called the roots of the quadratic polynomial, and may be found through factorization, completing the square, graphing, Newton's method, or through the use of the quadratic formula. Each quadratic polynomial has an associated quadratic function, whose graph is a parabola. Any quadratic polynomial with two variables may be written as where ""x"" and ""y"" are the variables and ""a"", ""b"", ""c"", ""d"", ""e"", and ""f"" are the coefficients. Such polynomials are fundamental to the study of conic sections, which are characterized"
"symmetric bilinear form . However, ""Q""(""x"") can no longer be recovered from this ""B""′ in the same way, since for all ""x"" (and is thus alternating). Alternately, there always exists a bilinear form ""B""″ (not in general either unique or symmetric) such that . The pair consisting of a finite-dimensional vector space ""V"" over ""K"" and a quadratic map ""Q"" from ""V"" to ""K"" is called a quadratic space, and ""B"" as defined here is the associated symmetric bilinear form of ""Q"". The notion of a quadratic space is a coordinate-free version of the notion of quadratic form. Sometimes, ""Q"""
"so in this case all genera contain the same number of classes of forms. Genus of a quadratic form In mathematics, the genus is a classification of quadratic forms and lattices over the ring of integers. An integral quadratic form is a quadratic form on Z, or equivalently a free Z-module of finite rank. Two such forms are in the same ""genus"" if they are equivalent over the local rings Z for each prime ""p"" and also equivalent over R. Equivalent forms are in the same genus, but the converse does not hold. For example, ""x"" + 82""y"" and 2""x"""
"Quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two. For example, formula_1 is a quadratic form in the variables ""x"" and ""y"". This is not to be confused with a quadratic equation which is strictly univariate and includes terms of degree two or less. A quadratic form is one case of the more general concept of homogeneous polynomials. Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal group), differential geometry (Riemannian metric, second fundamental form), differential topology (intersection forms of four-manifolds), and Lie"
"is also called a quadratic form. Two ""n""-dimensional quadratic spaces and are isometric if there exists an invertible linear transformation (isometry) such that The isometry classes of ""n""-dimensional quadratic spaces over ""K"" correspond to the equivalence classes of ""n""-ary quadratic forms over ""K"". Let ""R"" be a commutative ring, ""M"" be an ""R""-module and be an ""R""-bilinear form. A mapping is the ""associated quadratic form"" of ""b"", and is the ""polar form"" of ""Q"". Alternatively, a quadratic form may be characterized as follows: Two elements ""v"" and ""w"" of ""V"" are called orthogonal if . The kernel of a bilinear"
"symmetric matrix ""A"" of ""φ"" and the symmetric matrix ""B"" of ""ψ"" are related as follows: The associated bilinear form of a quadratic form ""q"" is defined by Thus, ""b"" is a symmetric bilinear form over ""K"" with matrix ""A"". Conversely, any symmetric bilinear form ""b"" defines a quadratic form and these two processes are the inverses of one another. As a consequence, over a field of characteristic not equal to 2, the theories of symmetric bilinear forms and of quadratic forms in ""n"" variables are essentially the same. A quadratic form ""q"" in ""n"" variables over ""K"" induces a"
"Let , , be three quadratic spaces over a field ""k"". Assume that Then the quadratic spaces and are isometric: In other words, the direct summand appearing in both sides of an isomorphism between quadratic spaces may be ""cancelled"". Let be a quadratic space over a field ""k"". Then it admits a Witt decomposition: where is the radical of ""q"", is an anisotropic quadratic space and is a split quadratic space. Moreover, the anisotropic summand, termed the core form, and the hyperbolic summand in a Witt decomposition of are determined uniquely up to isomorphism. Quadratic forms with the same core"
"represents all positive integers if and only if it represents all integers up through 15. Quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two. For example, formula_1 is a quadratic form in the variables ""x"" and ""y"". This is not to be confused with a quadratic equation which is strictly univariate and includes terms of degree two or less. A quadratic form is one case of the more general concept of homogeneous polynomials. Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal group),"
"Quadratic Lie algebra A quadratic Lie algebra is a Lie algebra together with a compatible symmetric bilinear form. Compatibility means that it is invariant under the adjoint representation. Examples of such are semisimple Lie algebras, such as su(n) and sl(n,R). A quadratic Lie algebra is a Lie algebra (g,[..]) together with a non-degenerate symmetric bilinear form formula_1 that is invariant under the adjoint action, i.e. where ""X,Y,Z"" are elements of the Lie algebra g. A localization/ generalization is the concept of Courant algebroid where the vector space g is replaced by (sections of) a vector bundle. As a first example,"
"theory (the Killing form). Quadratic forms are homogeneous quadratic polynomials in ""n"" variables. In the cases of one, two, and three variables they are called unary, binary, and ternary and have the following explicit form: where ""a"", ..., ""f"" are the coefficients. Note that quadratic functions, such as in the one variable case, are not quadratic forms, as they are typically not homogeneous (unless ""b"" and ""c"" are both 0). The theory of quadratic forms and methods used in their study depend in a large measure on the nature of the coefficients, which may be real or complex numbers, rational"
"Genus of a quadratic form In mathematics, the genus is a classification of quadratic forms and lattices over the ring of integers. An integral quadratic form is a quadratic form on Z, or equivalently a free Z-module of finite rank. Two such forms are in the same ""genus"" if they are equivalent over the local rings Z for each prime ""p"" and also equivalent over R. Equivalent forms are in the same genus, but the converse does not hold. For example, ""x"" + 82""y"" and 2""x"" + 41""y"" are in the same genus but not equivalent over Z. Forms in"
"Quadratic function In algebra, a quadratic function, a quadratic polynomial, a polynomial of degree 2, or simply a quadratic, is a polynomial function in one or more variables in which the highest-degree term is of the second degree. For example, a quadratic function in three variables ""x"", ""y,"" and ""z"" contains exclusively terms ""x"", ""y"", ""z"", ""xy"", ""xz"", ""yz"", ""x"", ""y"", ""z"", and a constant: with at least one of the coefficients ""a, b, c, d, e,"" or ""f"" of the second-degree terms being non-zero. A ""univariate"" (single-variable) quadratic function has the form in the single variable ""x"". The graph"
"Further, the isometry groups of ""Q"" and −""Q"" are the same (, but the associated Clifford algebras (and hence pin groups) are different. An quadratic form over a field ""K"" is a map formula_12 from a finite dimensional ""K"" vector space to ""K"" such that formula_13 for all formula_14 and the function formula_15 is bilinear. More concretely, an ""n""-ary quadratic form over a field ""K"" is a homogeneous polynomial of degree 2 in ""n"" variables with coefficients in ""K"": This formula may be rewritten using matrices: let ""x"" be the column vector with components ""x"", ..., ""x"" and be the"
"Simultaneous equations model Simultaneous equation models are a type of statistical model in the form of a set of linear simultaneous equations. They are often used in econometrics. One can estimate these models equation by equation; however, estimation methods that exploit the system of equations, such as generalized method of moments (GMM) and instrumental variables estimation (IV) tend to be more efficient. Suppose there are ""m"" regression equations of the form where ""i"" is the equation number, and is the observation index. In these equations ""x"" is the ""k×""1 vector of exogenous variables, ""y"" is the dependent variable, ""y"" is"
"to the current function, e.g. seconds. Note that the term ""in quadrature"" only implies that two sinusoids are orthogonal, not that they are ""components"" of another sinusoid. In an angle modulation application, with carrier frequency φ is also a time-variant function, giving: When all three terms above are multiplied by an optional amplitude function, the left-hand side of the equality is known as the ""amplitude/phase"" form, and the right-hand side is the ""quadrature-carrier"" or ""IQ"" form. Because of the modulation, the components are no longer completely orthogonal functions. But when and are slowly varying functions compared to the assumption of"
"Definite quadratic form In mathematics, a definite quadratic form is a quadratic form over some real vector space that has the same sign (always positive or always negative) for every nonzero vector of . According to that sign, the quadratic form is called positive-definite or negative-definite. A semidefinite (or semi-definite) quadratic form is defined in the same way, except that ""positive"" and ""negative"" are replaced by ""not negative"" and ""not positive"", respectively. An indefinite quadratic form is one that takes on both positive and negative values. More generally, the definition applies to a vector space over an ordered field. Quadratic"
"forms correspond one-to-one to symmetric bilinear forms over the same space. A symmetric bilinear form is also described as definite, semidefinite, etc. according to its associated quadratic form. A quadratic form and its associated symmetric bilinear form are related by the following equations: As an example, let formula_3, and consider the quadratic form where and and are constants. If and , the quadratic form is positive-definite, so ""Q"" evaluates to a positive number whenever formula_5 If one of the constants is positive and the other is 0, then is positive semidefinite and always evaluates to either 0 or a positive"
"Binary quadratic form In mathematics, a binary quadratic form is a quadratic homogeneous polynomial in two variables where ""a"", ""b"", ""c"" are the coefficients. When the coefficients can be arbitrary complex numbers, most results are not specific to the case of two variables, so they are described in quadratic form. A quadratic form with integer coefficients is called an integral binary quadratic form, often abbreviated to ""binary quadratic form"". This article is entirely devoted to integral binary quadratic forms. This choice is motivated by their status as the driving force behind the development of algebraic number theory. Since the late"
"is odd, the even subalgebra are those elements of for which the two factors are identical. Picking either piece then gives an isomorphism with . The real case is significantly more complicated, exhibiting a periodicity of 8 rather than 2, and there is a 2-parameter family of Clifford algebras. Firstly, there are non-isomorphic quadratic forms of a given degree, classified by signature. Every nondegenerate quadratic form on a real vector space is equivalent to the standard diagonal form: where is the dimension of the vector space. The pair of integers (""p"", ""q"") is called the signature of the quadratic form."
"non-trivial *, ε-symmetric forms are equivalent to ε-quadratic forms if there is an element λ ∈ ""R"" such that λ* + λ = 1. If * is trivial, this is equivalent to 2λ = 1 or λ = 1/2, while if * is non-trivial there can be multiple possible λ; for example, over the complex numbers any number with real part 1/2 is such a λ. For instance, in the ring formula_7 (the integral lattice for the quadratic form 2""x"" − 2""x""+1), with complex conjugation, formula_8 are two such elements, though 1/2 ∉ ""R"". In terms of matrices (we take"
"in this case the quadratic form is called positive definite (all 1) or negative definite (all −1). If none of the terms are 0, then the form is called ; this includes positive definite, negative definite, and indefinite (a mix of 1 and −1); equivalently, a nondegenerate quadratic form is one whose associated symmetric form is a nondegenerate ""bilinear"" form. A real vector space with an indefinite nondegenerate quadratic form of index (denoting ""p"" 1s and ""q"" −1s) is often denoted as R particularly in the physical theory of spacetime. The discriminant of a quadratic form, concretely the class of"
"the group of isometries of into itself. If a quadratic space has a product so that ""A"" is an algebra over a field, and satisfies Every quadratic form ""q"" in ""n"" variables over a field of characteristic not equal to 2 is equivalent to a diagonal form Such a diagonal form is often denoted by Classification of all quadratic forms up to equivalence can thus be reduced to the case of diagonal forms. Using Cartesian coordinates in three dimensions, let formula_30, and let formula_31 be a symmetric 3-by-3 matrix. Then the geometric nature of the solution set of the equation"
"of a univariate quadratic function is a parabola whose axis of symmetry is parallel to the -axis, as shown at right. If the quadratic function is set equal to zero, then the result is a quadratic equation. The solutions to the univariate equation are called the roots of the univariate function. The bivariate case in terms of variables ""x"" and ""y"" has the form with at least one of ""a, b, c"" not equal to zero, and an equation setting this function equal to zero gives rise to a conic section (a circle or other ellipse, a parabola, or a"
"another coordinate system moving in the positive ""x""-direction at a constant speed ""v"", relative to the rest system. The general quadratic form ""q(x)"" with coefficients of a symmetric matrix A, the associated bilinear form ""b(x,y)"", and the linear transformations of ""q(x)"" and ""b(x,y)"" into ""q(x′)"" and ""b(x′,y′)"" using the transformation matrix g, can be written as The case ""n=1"" is the binary quadratic form introduced by Lagrange (1773) and Gauss (1798/1801), ""n=2"" is the ternary quadratic form introduced by Gauss (1798/1801), ""n=3"" is the quaternary quadratic form etc. The general Lorentz transformation follows from () by setting A=A′=diag(-1,1...,1) and det"
"Quadratic set In mathematics, a quadratic set is a set of points in a projective space that bears the same essential incidence properties as a quadric (conic section in a projective plane, sphere or cone or hyperboloid in a projective space). Let formula_1 be a projective space. A quadratic set is a non-empty subset formula_2 of formula_3 for which the following two conditions hold: A quadratic set formula_2 is called non-degenerate if for every point formula_16, the set formula_17 is a hyperplane. A Pappian projective space is a projective space in which Pappus's hexagon theorem holds. The following result, due"
"the terms in ""x"", and completing the square. Note the following about the complex constant factors attached to some of the terms: A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length ""k"", and A and B are symmetric, invertible matrices of size formula_303, then where Note that the form x′ A x is called a quadratic form and is a scalar: In other words, it sums up all possible combinations of products of pairs of elements from x, with a separate coefficient for each. In addition, since formula_307,"
"implies that the regular continued fraction expansions of equivalent quadratic irrationalities are eventually the same, that is, their sequences of partial quotients have the same tail. Thus, all numbers in an equivalence class have continued fraction expansions that are eventually periodic with the same tail. There are finitely many equivalence classes of quadratic irrationalities in formula_10. The standard proof of this involves considering the map formula_26 from binary quadratic forms of discriminant formula_6 to formula_10 given by A computation shows that formula_26 is a bijection between that respects the matrix action on each set. The equivalence classes of quadratic irrationalities"
"real variable, quadratic growth is equivalent to the second derivative being constant (i.e., the third derivative being zero), and thus functions with quadratic growth are exactly the quadratic polynomials, as these are the kernel of the third derivative operator ""D"". Similarly, for a sequence (a real function of an integer or natural number variable), quadratic growth is equivalent to the second finite difference being constant (the third finite difference being zero), and thus a sequence with quadratic growth is also a quadratic polynomial. Indeed, an integer-valued sequence with quadratic growth is a polynomial in the zeroth, first, and second binomial"
"that shares three converging edges of a parallelepiped has a volume equal to one sixth of the volume of that parallelepiped (see proof). The surface area of a parallelepiped is the sum the areas of the bounding parallelograms: For parallelepipeds with a symmetry plane there are two cases: See also monoclinic. A rectangular cuboid, also called a ""rectangular parallelepiped"" or sometimes simply a ""cuboid"", is a parallelepiped of which all faces are rectangular; a cube is a cuboid with square faces. A rhombohedron is a parallelepiped with all rhombic faces; a trigonal trapezohedron is a rhombohedron with congruent rhombic faces."
"continuous local martingales, it is the same as the quadratic variation. Quadratic variation In mathematics, quadratic variation is used in the analysis of stochastic processes such as Brownian motion and other martingales. Quadratic variation is just one kind of variation of a process. Suppose that ""X"" is a real-valued stochastic process defined on a probability space formula_1 and with time index ""t"" ranging over the non-negative real numbers. Its quadratic variation is the process, written as [""X""], defined as where ""P"" ranges over partitions of the interval [0,""t""] and the norm of the partition ""P"" is the mesh. This limit,"
"quadratic in the measurements, so ""y"" will be decided based on In the special case where each observation consists of two measurements, this means that the surfaces separating the classes will be conic sections (""i.e."" either a line, a circle or ellipse, a parabola or a hyperbola). In this sense we can state that a quadratic model is a generalization of the linear model, and its use is justified by the desire to extend the classifier's ability to represent more complex separating surfaces. Quadratic discriminant analysis (QDA) is closely related to linear discriminant analysis (LDA), where it is assumed that"
"and perpetual impact of X and Y on each other. This requires a theory that causal effects are simultaneous in time, or so complex that they appear to behave simultaneously; a common example are the moods of roommates. To estimate simultaneous feedback models a theory of equilibrium is also necessary – that X and Y are in relatively steady states or are part of a system (society, market, classroom) that is in a relatively stable state. Simultaneous equations model Simultaneous equation models are a type of statistical model in the form of a set of linear simultaneous equations. They are"
"Simultaneity (music) In music, a simultaneity is more than one complete musical texture occurring at the same time, rather than in succession. This first appeared in the music of Charles Ives, and is common in the music of Conlon Nancarrow and others. In music theory, a pitch simultaneity is more than one pitch or pitch class all of which occur at the same time, or simultaneously: ""A set of notes sounded together."" ""Simultaneity"" is a more specific and more general term than chord: many but not all chords or harmonies are simultaneities, though not all but some simultaneities are chords."