The well know theorem of Tverberg states that if n ≥ (d+1)(r − 1)+1 then one can partition any set of n points in R to r disjoint subsets whose convex hulls have a common point. The numbers T (d, r) = (d + 1)(r − 1) + 1 are known as Tverberg numbers. Reay asks the following question: if we add an additional parameter k (2 ≤ k ≤ r) what is the minimal number of points we need in order to guarant...

This work is concerned with the numerical solution of large-scale linear matrix equations A1XB T 1 + · · ·+ AKXB K = C. The most straightforward approach computes X ∈ Rm×n from the solution of an mn×mn linear system, typically limiting the feasible values of m,n to a few hundreds at most. Our new approach exploits the fact that X can often be well approximated by a low-rank matrix. It combines ...

Principal components analysis (PCA) is a well-known technique for approximating a data set represented by a matrix by a low rank matrix. Here, we extend the idea of PCA to handle arbitrary data sets consisting of numerical, Boolean, categorical, ordinal, and other data types. This framework encompasses many well known techniques in data analysis, such as nonnegative matrix factorization, matrix...

This work is concerned with the numerical solution of large-scale linear matrix equations A1XB T 1 + · · ·+ AKXB K = C. The most straightforward approach computes X ∈ Rm×n from the solution of an mn×mn linear system, typically limiting the feasible values of m,n to a few hundreds at most. Our new approach exploits the fact that X can often be well approximated by a low-rank matrix. It combines ...

‎let $n$ and $k$ be two positive integers‎, ‎$kleq n$ and $a$ be an $n-$square quaternion matrix‎. ‎in this paper‎, ‎some results on the $k-$numerical range of $a$ are investigated‎. ‎moreover‎, ‎the notions of $k$-numerical radius‎, ‎right $k$-spectral radius and $k$-norm of $a$ are introduced‎, ‎and some of their algebraic properties are studied‎.

The spectral k-support norm enjoys good estimation properties in low rank matrix learning problems, empirically outperforming the trace norm. Its unit ball is the convex hull of rank k matrices with unit Frobenius norm. In this paper we generalize the norm to the spectral (k, p)support norm, whose additional parameter p can be used to tailor the norm to the decay of the spectrum of the underlyi...

Given an m × n matrix A and a positive integer k, we introduce a randomized procedure for the approximation of A with a matrix Z of rank k. The procedure relies on applying an l×m random matrix with special structure to each column of A, where l is an integer near to, but greater than k. The spectral norm ‖A−Z‖ of the discrepancy between A and Z is of the same order as √ lm times the (k + 1)st ...

there are different methods to reduce drag in high speed hulls. one of these methods is a change in the shape of the body by adding longitudinal side tunnels. in this paper it has been attempted to determine the influence of the tunnel height on hydrodynamic characteristics of the hull to achieving an optimum shape for the tunnel. to achieve this purpose, numerical simulation of the problem has...