discriminant

In mathematics, and particularly in analysis, an ordinary differential equation (or ODE) is an equation that involves the derivatives of an unknown function of one variable. A simple example of an ordinary differential equation is , where is an unknown function, and is its derivative. See differential calculus and integral calculus for basic calculus background. Important theorems in the field of ODEs include broad existence and uniqueness theorems and for ODEs in the plane, the Poincaré-Ben ...

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• Ordinary differential equation - Definition
• Ordinary differential equation - General application
• Ordinary differential equation - Existence and nature of solutions
• Ordinary differential equation - Types of differential equations with some history
• Ordinary differential equation - Linear ODEs with constant coefficients
• Ordinary differential equation - Linear ODEs with variable coefficient
• Ordinary differential equation - General solution method for first-order linear ODEs
• Ordinary differential equation - Linear PDEs
• Ordinary differential equation - First-order PDEs
• Ordinary differential equation - Singular solutions
• Ordinary differential equation - Reduction to quadratures
• Ordinary differential equation - The Fuchsian theory
• Ordinary differential equation - Lie's theory
• Ordinary differential equation - Bibliography

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In number theory, a cusp form is a particular kind of modular form, distinguished in the case of modular forms for the modular group by the vanishing in the Fourier series expansion Σanqn of the constant coefficient a0. This Fourier expansion exists as a consequence of the presence in the modular group's action on the upper half- ...

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In PDEs, it is common to write the unknown function as u and its partial derivative with respect to the variable x as ux, that is: Especially in (mathematical) physics, one often prefers use of the nabla operator for spatial derivatives and a dot () for time derivatives, e.g. to write the wave equation (see below) as . See also:

Partial differential equation, Partial differential equation - Notation and examples, Partial differential equation - Laplace's equation, Partial differential equation - Wave equation, Partial differential equation - Heat equation, Partial differential equation - Euler-Tricomi equation, Partial differential equation - Advection equation, Partial differential equation - Ginzburg-Landau equation, Partial differential equation - The Dym equation, Partial differential equation - Other examples, Partial differential equation - Methods to solve PDEs, Partial differential equation - Classification, Partial differential equation - Equations of mixed type

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Clifford algebra - Relation to the exterior algebra. Given a vector space V one can construct the exterior algebra Λ(V), whose definition is independent of any quadratic form on V. It turns out that if F does not have characteristic 2 then there is a natural isomorphism between Λ(V) and Cℓ(V,Q) considered as vector spaces. This is an algebra isomorphism if and only if Q = 0. One can thus consider the Clifford algebra Cℓ(V,Q) ...

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Clifford algebra, Clifford algebra - Introduction and basic properties, Clifford algebra - Universal property and construction, Clifford algebra - Basis and dimension, Clifford algebra - Examples: Real and complex Clifford algebras, Clifford algebra - Properties, Clifford algebra - Relation to the exterior algebra, Clifford algebra - Grading, Clifford algebra - Antiautomorphisms, Clifford algebra - The Clifford scalar product, Clifford algebra - Structure of Clifford algebras, Clifford algebra - The Clifford group Γ, Clifford algebra - Spin and Pin groups, Clifford algebra - Spinors, Clifford algebra - Applications, Clifford algebra - Differential geometry, Clifford algebra - Physics, Clifford algebra - Footnotes

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The quadratic formula explicitly gives the solutions of a quadratic equation in terms of the coefficients , and , which we temporarily assume to be real (but see below for generalizations) with a being non-zero. These solutions are also called the roots of the equation. The formula reads An alternative form sometimes encountered is given by This gives the solutions and . The term is called the discriminant of the quadratic equation, because it discriminat ...

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Quadratic equation, Quadratic equation - Quadratic formula, Quadratic equation - Derivation, Quadratic equation - Generalizations, Quadratic equation - Viète's formulas, Quadratic equation - Solving Equations of a Higher Degree, Quadratic equation - History

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The quadratic formula explicitly gives the solutions of a quadratic equation in terms of the coefficients , and , which we temporarily assume to be real (but see below for generalizations) with a being non-zero. These solutions are also called the roots of the equation. The formula reads An alternative form sometimes encountered is given by This gives the solutions and . The term is called the discriminant of the quadratic equation, because it discriminat ...

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Quadratic equation, Quadratic equation - Quadratic formula, Quadratic equation - Derivation, Quadratic equation - Generalizations, Quadratic equation - Viète's formulas, Quadratic equation - History

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By adding a "point at infinity", we obtain the projective version of this curve. If P and Q are two points on the curve, then we can uniquely describe a third point which is the intersection of the curve with the line through P and Q. If the line is tangent to the curve at a point, then that point is counted twice; and if the line is parallel to the y-axis, we define the third point as the point "at infinity". Exactly one of these conditions then holds for ...

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Elliptic curve, Elliptic curve - Elliptic curves over the real numbers, Elliptic curve - The group law, Elliptic curve - Elliptic curves over the complex numbers, Elliptic curve - Elliptic curves over a general field, Elliptic curve - Connections to number theory, Elliptic curve - Algorithms that use elliptic curves

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Partial fraction - Distinct first-degree factors in the denominator. Suppose it is desired to decompose the rational function into partial fractions. The denominator factors as and so we seek scalars A and B such that One way of finding A and B begins by "clearing fractions", i.e., multiplying both sides by the common denominator (x − 8)(x + 5). This yields Collecting like terms gives Equating coefficien ...

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Partial fraction, Partial fraction - Some examples, Partial fraction - Distinct first-degree factors in the denominator, Partial fraction - An irreducible quadratic factor in the denominator, Partial fraction - A repeated first-degree factor in the denominator, Partial fraction - Repeated factors in the denominator generally, Partial fraction - Basic principles, Partial fraction - Examples

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The two transformations and together generate a group called the modular group, which we may identify with the projective linear group . By a suitable choice of transformation belonging to this group, , with ad − bc = 1, we may reduce τ to a value giving the same value for j, and lying in the fundamental region for j, which consists of values for τ satisfying the conditions See also:

J-invariant, J-invariant - The fundamental region, J-invariant - Class field theory and j, J-invariant - The q-series and moonshine, J-invariant - Algebraic definition, J-invariant - Inverse

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The television show Futurama contains a running joke about the Hardy-Ramanujan number. In one episode, the robot Bender receives a card labeled "SON 1729". Ken Keeler, a writer on the show with a Ph. D. in Applied Math, said that "that 'joke' alone is worth six years of grad school." In another episode, Bender's serial number is one of a pair of elegant taxicab numbers: his number is 9523 + (−951)3 = 2716057, while that of fellow robot Flexo is 1193 + 1193 = 3370318. (This datum is one of ...

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1729 number, 1729 number - References to 1729, 1729 number - Quotation

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The following test can be applied at a non-degenerate critical point x. If the Hessian is positive definite at x, then f attains a local minimum at x. If the Hessian is negative definite at x, then f attains a local maximum at x. If the Hessian has both positive and negative eigenvalues then x is a saddle point for f (this is true even if x is degenerate). Otherwise the test is inconclusive. Note that for positive semidefinite and negative semidefinite Hessians the test is inconclusive. However, ...

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Hessian matrix, Hessian matrix - Mixed derivatives and symmetry of the Hessian, Hessian matrix - Critical points and discriminant, Hessian matrix - Second derivative test, Hessian matrix - Vector-valued functions

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Next we consider such integrals as The quickest way to see that the denominator x2 − 8x + 25 is irreducible is to observe that its discriminant is negative. Alternatively, we can complete the square: and observe that this sum of two squares can never be 0 while x is a real number. In order to make use of the substitution we would need to find x − 4 in the numer ...

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Partial fractions in integration, Partial fractions in integration - A 1st-degree polynomial in the denominator, Partial fractions in integration - A repeated 1st-degree polynomial in the denominator, Partial fractions in integration - An irreducible 2nd-degree polynomial in the denominator, Partial fractions in integration - A repeated irreducible 2nd-degree polynomial in the denominator, Partial fractions in integration - External link

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The definition of the Frobenius for finite fields can be extended to other sorts of field extensions. Given an unramified finite extension L/K of local fields, there is a concept of Frobenius automorphism which induces the Frobenius automorphism in the corresponding extension of residue fields. Suppose L/K is an unramified extension of local fields, with ring of integers OK of K such that the residue field, the integers of K modulo their unique maximal ideal φ, is a finite field of ...

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Frobenius automorphism, Frobenius automorphism - Frobenius for local fields, Frobenius automorphism - Frobenius for global fields, Frobenius automorphism - Examples

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The graph of a quadratic function or is called a parabola. The former is called the general form while the latter is the standard form. In either form, a is non-zero, and If , the parabola opens upward. If , the parabola opens downward. Quadratic function - Vertex. The place where the parabola turns is called the turning point or the vertex of the parabola. If the quadr ...

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Quadratic function, Quadratic function - Roots, Quadratic function - Graph, Quadratic function - Vertex, Quadratic function - Number of x-intercepts, Quadratic function - Bivariate quadratic function, Quadratic function - Minimum/Maximum

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Over the field R of real numbers, the orthogonal group O(n,R) and the special orthogonal group SO(n,R) are often simply denoted by O(n) and SO(n) if no confusion is possible. They form real compact Lie groups of dimension n(n-1)/2. O(n,R) has two connected components, with SO(n,R) being the identity component, i.e., the connected component containing the identity matrix. The real orthogonal and real special ...

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Orthogonal group, Orthogonal group - Over the real number field, Orthogonal group - 3D isometries which leave the origin fixed, Orthogonal group - Over the complex number field, Orthogonal group - The Dickson invariant, Orthogonal group - Orthogonal groups of characteristic 2, Orthogonal group - The spinor norm, Orthogonal group - Galois cohomology and orthogonal groups

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The influence of geometry, physics, and astronomy, starting with Newton and Leibniz, and further manifested through the Bernoullis, Riccati, and Clairaut, but chiefly through d'Alembert and Euler, has been very marked, and especially on the theory of linear partial differential equations with constant coefficients. Ordinary differential equation - Homogeneous linear ODEs with constant coefficients. The first method of integrating linear ordinary differential equations with constant coefficients is due to E ...

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Ordinary differential equation, Ordinary differential equation - Definition, Ordinary differential equation - General application, Ordinary differential equation - Existence and nature of solutions, Ordinary differential equation - Types of differential equations with some history, Ordinary differential equation - Homogeneous linear ODEs with constant coefficients, Ordinary differential equation - Linear ODEs with constant coefficients, Ordinary differential equation - Linear ODEs with variable coefficient, Ordinary differential equation - General solution method for first-order linear ODEs, Ordinary differential equation - Linear PDEs, Ordinary differential equation - First-order PDEs, Ordinary differential equation - Singular solutions, Ordinary differential equation - Reduction to quadratures, Ordinary differential equation - The Fuchsian theory, Ordinary differential equation - Lie's theory, Ordinary differential equation - Bibliography

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For indefinite lattices, the classification is easy to describe. Write Rm,n for the m+n dimensional vector space Rm+n with the inner product of (a1,...,am+n) and (b1,...,bm+n) given by a1b1+...+ambm − am+1bm+1 − ... − am+nbm+n. In Rm,n there is one odd unimodular lattice up to isomorphism, d ...

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Unimodular lattice, Unimodular lattice - Definitions, Unimodular lattice - Examples, Unimodular lattice - Classification, Unimodular lattice - Properties, Unimodular lattice - Applications

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These matrices are useful in polynomial interpolation, since solving the system of linear equations Vu = y for u with V the n × n Vandermonde matrix is equivalent to finding the coefficients uj of the polynomial of degree ≤ n−1 which has the values yi at αi. The Vandermonde determinant plays a central role in the Frobenius formula, which gives the character of conjugacy ...

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Vandermonde matrix, Vandermonde matrix - Applications

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Diagonalizable matrix - How to diagonalize a matrix. Consider a matrix This matrix has eigenvalues So A is a 3-by-3 matrix with 3 different eigenvalues, therefore it is diagonalizable. If we want to diagonalize A, we need to compute the corresponding eigenvectors. They are One can easily check that Avk = λkvk. Now, let P be the matrix with these ...

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Diagonalizable matrix, Diagonalizable matrix - Examples, Diagonalizable matrix - How to diagonalize a matrix, Diagonalizable matrix - Matrices that are not diagonalizable, Diagonalizable matrix - An application

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Determinants are used to characterize invertible matrices (namely as those matrices, and only those matrices, with non-zero determinants), and to explicitly describe the solution to a system of linear equations with Cramer's rule. They can be used to find the eigenvalues of the matrix A through the characteristic polynomial where I is the ide ...

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Determinant, Determinant - Determinants of 2-by-2 matrices, Determinant - Applications, Determinant - General definition and computation, Determinant - Example, Determinant - Properties, Determinant - Derivative, Determinant - Generalizations and related functions, Determinant - Algorithmic implementation, Determinant - History

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