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T K is a non-negative symmetric compact operator on L 2 [a,b]; moreover K(x, x) ≥ 0. To show compactness, show that the image of the unit ball of L 2 [ a , b ] under T K is equicontinuous and apply Ascoli's theorem , to show that the image of the unit ball is relatively compact in C([ a , b ]) with the uniform norm and a fortiori in L 2 [ a ...
Prolog implementations usually omit the occurs check for reasons of efficiency, which can lead to circular data structures and looping. By not performing the occurs check, the worst case complexity of unifying a term with term is reduced in many cases from (() + ()) to (((), ())); in the particular, frequent case of variable-term unifications, runtime shrinks to ().
If L is a third subgroup, the convention that [H,K,L] = [[H,K],L] will be followed. If x and y are elements of a group G, the conjugate of x by y will be denoted by . If H is a subgroup of a group G, then the centralizer of H in G will be denoted by C G (H).
If G is a locally compact Hausdorff group, G carries an essentially unique left-invariant countably additive Borel measure μ called a Haar measure.Using the Haar measure, one can define a convolution operation on the space C c (G) of complex-valued continuous functions on G with compact support; C c (G) can then be given any of various norms and the completion will be a group algebra.
In physics, Hooke's law is an empirical law which states that the force (F) needed to extend or compress a spring by some distance (x) scales linearly with respect to that distance—that is, F s = kx, where k is a constant factor characteristic of the spring (i.e., its stiffness), and x is small compared to the total possible deformation of the spring.
The covariance function K X satisfies the definition of a Mercer kernel. By Mercer's theorem, there consequently exists a set λ k, e k (t) of eigenvalues and eigenfunctions of T K X forming an orthonormal basis of L 2 ([a,b]), and K X can be expressed as (,) = = ()
The coefficients found by Fehlberg for Formula 1 (derivation with his parameter α 2 =1/3) are given in the table below, using array indexing of base 1 instead of base 0 to be compatible with most computer languages:
In algebraic K-theory, a field of mathematics, the Steinberg group of a ring is the universal central extension of the commutator subgroup of the stable general linear group of . It is named after Robert Steinberg , and it is connected with lower K {\displaystyle K} -groups , notably K 2 {\displaystyle K_{2}} and K 3 {\displaystyle K_{3}} .