The Rayleigh-Ritz method finds the stationary values, called Ritz values, of the Rayleigh quotient on a given trial subspace as approximations to eigenvalues of a Hermitian operator A. If the trial subspace is A-invariant, the Ritz values are exactly some of the eigenvalues of A. Given two subspaces X and Y of the same finite dimension, such that X is A-invariant, the absolute changes in the Ri...

The resemblance between the Horn-Thompson theorem and a recent theorem by Dacorogna-Marcellini-Tanteri indicates that Schur convexity and the majorization relation are relevant for applications in the calculus of variations and its related notions of convexity, such as rank-one convexity or quasiconvexity. We give in theorem 6.6 simple necessary and sufficient conditions for an isotropic object...

Distances in the well known fuzzy c-means algorithm of Bezdek (1973) are measured by the squared Euclidean distance. Other distances have been used as well in fuzzy clustering. For example, Jajuga (1991) proposed to use the L1-distance and Bobrowski and Bezdek (1991) also used the L∞-distance. For the more general case of Minkowski distance and the case of using a root of the squared Minkowski ...

This paper generalizes the axiomatic approach to the design of income inequality measures to the multiattribute context. While the extension of most axioms considered desirable for inequality indices is straightforward, it is not entirely clear when a situation is more unequal than another when each person is characterised by a vector of attributes of wellbeing. We explore two majorization crit...

In this paper we obtain some majorization results for some classes of multivalent functions defined by certain differential operator. Mathematics subject classification (2010): 30C45.

We show that, in general, inequalities between integrands with respect to Brownian motion do not lead to majorization in the convex order for the corresponding stochastic integrals. Particular examples and counter-examples are discussed.

Motivated by Carlson-Shaffer linear operator, we define here a new generalized linear operator. Using this operator, we define a class of analytic functions in the unit disk U. For this class, a majorization problem of analytic functions is discussed.

Write |w| = m, the length of w, and |w|1 = card({1 ≤ i ≤ m : wi = 1}) its 1-length. Define the cyclic shift σ : {0, 1}m → {0, 1}m by σ(w1 . . . wm) = w2 . . . wmw1. A cyclic subword of w is any length-q prefix of some σi−1(w), 1 ≤ i, q ≤ m. To any word w = w1 . . . wm we associate its orbit O(w), the vector O(w) = (O1(w), . . . ,Om(w)) consisting of the iterated cyclic shifts w, σ(w), . . . , σ...

Non-convex optimization is ubiquitous in machine learning. The MajorizationMinimization (MM) procedure systematically optimizes non-convex functions through an iterative construction and optimization of upper bounds on the objective function. The bound at each iteration is required to touch the objective function at the optimizer of the previous bound. We show that this touching constraint is u...

Measures for sparse best–basis selection are analyzed and shown to fit into a general framework based on majorization, Schur-concavity, and concavity. This framework facilitates the analysis of algorithm performance and clarifies the relationships between existing proposed concentration measures useful for sparse basis selection. It also allows one to define new concentration measures, and seve...