Cauchy–Schwarz inequality

Summary

The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics. The inequality for sums was published by . The corresponding inequality for integrals was published by and . Schwarz gave the modern proof of the integral version. Statement of the inequality The Cauchy–Schwarz inequality states that for all vectors \mathbf{u} and \mathbf{v} of an inner product space where \langle \cdot, \cdot \rangle is the inner product. Examples of inner products include the real and complex dot product; see the examples in inner product. Every inner product gives rise to a Euclidean (l_2) norm, called the or , where the norm of a vector \mathbf{u} is denoted and defined by: |\mathbf{u}|_2 := \sqrt{\langle \mathbf{u}, \mathbf{u} \rangle}

Official source

https://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality

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Regularity in metric spaces

Serguei Timochine

Using arguments developed by De Giorgi in the 1950's, it is possible to prove the regularity of the solutions to a vast class of variational problems in the Euclidean space. The main goal of the present thesis is to extend these results to the more abstract context of metric spaces with a measure. In particular, working in the axiomatic framework of Gol'dshtein – Troyanov, we establish both the interior and the boundary regularity of quasi-minimizers of the p-Dirichlet energy. Our proof works for quite general domains, assuming some natural hypotheses on the (axiomatic) D-structure. Furthermore, we prove analogous results for extremal functions lying in the class of Sobolev functions in the sense of Hajłasz – Koskela, i.e. functions characterized by the single condition that a Poincaré inequality be satisfied. Our strategy to prove these regularity results is first to show that, in a very general setting, the (Hölder) continuity of a function is a consequence of three specific technical hypotheses. This part of the argument is the essence of the De Giorgi method. Then, we verify that for a function u which is a quasi-minimizer in an axiomatic Sobolev space or an extremal Sobolev function in the sense of Hajłasz – Koskela, these technical hypotheses are indeed satisfied and u is thus (Hölder) continuous. In addition to that, we establish the Harnack's inequality for these extremal functions, and we show that the Dirichlet semi-norm of a piecewise-extremal function is equivalent to the sum of the Dirichlet semi-norms of its components.

EPFL2006

Dependence Structure of Uniform Spacings

Eva Maria Restle

This note derives a general, exact formula for the covariance of products of the spacings of a uniform sample in order to determine their order of magnitude. It compares the results with an estimation of the order obtained by applying the Cauchy--Schwarz inequality.

1999

The Minimal Faithful Permutation Degree For A Direct Product Obeying An Inequality Condition

Neil John Saunders

The minimal faithful permutation degree (G) of a finite group G is the least nonnegative integer n such that G embeds in the symmetric group Sym(n). Clearly (G x H) (G) + (H) for all finite groups G and H. In 1975, Wright ([10]) proved that equality occurs when G and H are nilpotent and exhibited an example of strict inequality where G x H embeds in Sym(15). In 2010 Saunders ([7]) produced an infinite family of examples of permutation groups G and H where (G x H) < (G) + (H), including the example of Wright's as a special case. The smallest groups in Saunders' class embed in Sym(10). In this article, we prove that 10 is minimal in the sense that (G x H)=(G) + (H) for all groups G and H such that (G x H)

Taylor & Francis Inc2016