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Any n × n real symmetric matrix A determines a quadratic form q < sub > A </ sub > in n variables by the formula
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Any and n
* Any ring of characteristic n is a ( Z / nZ )- algebra in the same way.
Any subset of R < sup > n </ sup > ( with its subspace topology ) that is homeomorphic to another open subset of R < sup > n </ sup > is itself open.
Any symmetry group whose elements have a common fixed point, which is true for all finite symmetry groups and also for the symmetry groups of bounded figures, can be represented as a subgroup of orthogonal group O ( n ) by choosing the origin to be a fixed point.
Any n × n permutation matrix can be constructed as a product of no more than n − 1 transpositions.
Any orthogonal matrix of size n × n can be constructed as a product of at most n such reflections.
Any rotation matrix of size n × n can be constructed as a product of at most n ( n − 1 )/ 2 such rotations.
Any n × n matrix A of the form
To show that U ( n ) is connected, recall that any unitary matrix A can be diagonalized by another unitary matrix S. Any diagonal unitary matrix must have complex numbers of absolute value 1 on the main diagonal.
# Any smooth manifold of dimension n ≥ 3 admits a Riemannian metric with negative Ricci curvature.
for some p between 1 and n. Any two such expressions of q ( at the same point m of M ) will have the same number p of positive signs.
* Any positive multiple of the repunit R < sub > n </ sub >< sup >( b )</ sup > contains at least n nonzero digits in base b.
Any covering space of a differentiable manifold may be equipped with a ( natural ) differentiable structure that turns p ( the covering map in question ) into a local diffeomorphism – a map with constant rank n.
Any and ×
Any conjectures are largely based on the date of her father's death ( 1006 ), the approximate date of her betrothal to Cnut ( 1004 × 1016, see below ) and the time by which she had borne him sons, whose ages are themselves difficult to establish.
* Any two objects X and Y of C have a product X × Y in C.
Any one of these decompositions, or Lie algebra renderings, may be necessary for rendering the Lie subalgebra of a 2 × 2 real matrix.
Any 2 × 2 matrix can be written in the form, where I is the identity matrix, σ < sub > i </ sub > are the Pauli matrices and u is a unit vector.
Any and real
# Any real number can be added to both sides.
# Any real number can be subtracted from both sides.
# Any real number can be multiplied to both sides.
# Any non-zero real number can divide both sides.
Any given individual or organization attempting to implement this model in a real situation would be unable to comply with the three requirements.
* Any Lie group G defines an associated real Lie algebra.
* Any closed interval b of real numbers is Lebesgue measurable, and its Lebesgue measure is the length b − a.
Any real mode operating system ( such as Xenix ) could also be made to run using OS / 2's virtual machine capabilities, subject to certain direct hardware access limitations.
Any Dedekind-complete ordered field is isomorphic to the real numbers.
Any poles on the imaginary axis have real strictly positive residues, and similarly at any zeros on the imaginary axis, the function has a real strictly positive derivative.
* Any field, including fields of rational numbers, real numbers, and complex numbers.
Any non-negative real number can be represented as a base-φ numeral using only the digits 0 and 1, and avoiding the digit sequence " 11 "-this is called a standard form.
* Any polynomial ( real or complex ) is an analytic function.
Any Taylor series for this function converges not only for x close enough to x < sub > 0 </ sub > ( as in the definition ) but for all values of x ( real or complex ).
Any real analytic function on some open set on the real line can be extended to a complex analytic function on some open set of the complex plane.
Any destruction by the Occupying Power of real or personal property belonging individually or collectively to private persons, or to the State, or to other public authorities, or to social or cooperative organizations, is prohibited, except where such destruction is rendered absolutely necessary by military operations.
Furthermore, in the opening credits there is a mock disclaimer which reads ( in translation ): “ Any resemblance to real events, to persons dead or living, is not accidental.
Any real advantages to the system were far outweighed by the additional cost of manufacture associated with the device.
Any rational function of a real variable can be written as the sum of a polynomial function and a finite number of algebraic fractions.
Any hope of real progress by the Poles was then lost.
Any entity from the real or virtual world that lends itself to dramatic interpretation may be taken up as a subject.
* Any open subset of a finite-dimensional real, and therefore complex, vector space is a diffeological space.
Any public employee who fell under his power would be replaced by someone in whom he saw real ability instead of a mere useful recipient of a favor.
Any and symmetric
Any square diagonal matrix is also a symmetric matrix.
Any brightness distribution can be written as the sum of a symmetric component and an anti-symmetric component.
Any symmetric matrix A can be transformed into a diagonal matrix
* Any symmetric odd theta divisor on is given by the set points, where w is a Weierstrass point on.
Any payoff symmetric 2x2 game will take one of these three forms.
Any non-singular matrix S of the same size is said to transform A into another symmetric matrix B = SAS < sup > T </ sup >, also of order n, where S < sup > T </ sup > is the transpose of S. If A is the coefficient matrix of some quadratic form of R < sup > n </ sup >, then B is the matrix for the same form after the change of coordinates defined by S.
Any symmetric real matrix A has an eigendecomposition of the form QEQ < sup > T </ sup > where E is a diagonal matrix containing the eigenvalues of A, and Q is an orthonormal square matrix containing the eigenvectors.
Any symmetric space gives an involutory quandle, where is the result of ' reflecting through '.
Any Riemannian symmetric space M is complete and Riemannian homogeneous ( meaning that the isometry group of M acts transitively on M ).
Any compact Riemann surface of genus greater than 1 ( with its usual metric of constant curvature − 1 ) is a locally symmetric space but not a symmetric space.
Any centrally symmetric convex body of surface area A can be squeezed through a noose of length, with the tightest fit achieved by a sphere.
4.910 seconds.