We present some identities related to the Cauchy-Schwarz inequality in complex inner product spaces. A new proof of the basic result on the subject of strengthened CauchySchwarz inequalities is derived using these identities. Also, an analogous version of this result is given for strengthened Hölder inequalities.

Among all linear operators on Hilbert spaces, the compact ones (defined below) are the simplest, and most imitate the more familiar linear algebra of finite-dimensional operator theory. In addition, these are of considerable practical value and importance. We prove a spectral theorem for self-adjoint operators with minimal fuss. Thus, we do not invoke broader discussions of properties of spectr...

The time dependence of nonclassical correlations is investigated for two fields (1, 2) generated by an ensemble of cold Cesium atoms via the protocol of Duan et al. [Nature 414, 413 (2001)]. The correlation function R(t1, t2) for the ratio of cross to auto-correlations for the (1, 2) fields at times (t1, t2) is found to have a maximum value R max = 292±57, which significantly violates the Cauch...

The analytic hierarchy process (AHP) is a structured technique for dealing with complex decisionmaking and is already used to solve many group decision problems. This paper used the CauchySchwarz inequality to improve the AHP algorithm that was developed by Escobar et al. (2004). They provided an upper bound for the group geometric consistency index in order to prove that the group geometric co...

We consider a large class of residuum based a posteriori eigenvalue/eigenvector estimates and present an abstract framework for proving their asymptotic exactness. Equivalence of the estimator and the error is also established. To demonstrate the strength of our abstract approach we present a detailed study of hierarchical error estimators for Laplace eigenvalue problems in planar polygonal reg...

Many fundamental theorems in extremal graph theory can be expressed as algebraic inequalities between subgraph densities. As is explained below, for dense graphs, it is possible to replace subgraph densities with homomorphism densities. An easy observation shows that one can convert any algebraic inequality between homomorphism densities to a linear inequality. Inspired by the work of Freedman,...

We suggest the use of alternative distance measures for similarity based classification in Learning Vector Quantization. Divergences can be employed whenever the data consists of non-negative normalized features, which is the case for, e.g., spectral data or histograms. As examples, we derive gradient based training algorithms in the framework of Generalized Learning Vector Quantization based o...

Several inequalities are presented which, in part, generalize inequalities by Weinstein and Weiss, giving rise to new lower bounds for the Bayes risk under squared error loss. Consider a Bayesian setting in which y denotes the observed data and θ denotes an unknown scalar parameter. Suppose that E [ (θ − E(θ|y)) |y ] and E [ (θ − E(θ|y)) ] exist. For any estimator δ(y) of θ, E [ (θ − δ(y))|y ] ...

Multithreshold Entropy Linear Classifier (MELC) is a density based model which searches for a linear projection maximizing the CauchySchwarz Divergence of dataset kernel density estimation. Despite its good empirical results, one of its drawbacks is the optimization speed. In this paper we analyze how one can speed it up through solving an approximate problem. We analyze two methods, both simil...