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The Schur and Steinhaus Theorems for 4-Dimensional Infinite Matrices

Natarajan P. N.

Commentationes Mathematicae

2014

Vol 54, No. 2

159--165

This paper is a sequel to [2]. Throughout this paper, entries of double sequences, double series and 4-dimensional infinite matrices are real or complex numbers. We prove the Schur and Steinhaus theorems for 4-dimensional infinite matrices.

Results of Tadeusz Świątkowski on algebraic sums of sets and their applications in the theory of subadditive functions

Matkowski J.

Journal of Applied Analysis

2012

Vol. 18, nr 1

97-116

A refinement of Steinhaus' theorem on the algebraic sum of subsets of R due to Raikov (1939) was not known to the mathematical community and still is not popular. In 1994, Tadeusz Świątkowski, being not aware of the existence of Raikov's theorem, proved another result of this type. Unfortunately, a few days later he passed away. In this paper we present the theorems of Świątkowski and Raikov and we apply them in the theory of subadditive type inequalities. An improvement of a converse of Minkowski's inequality theorem is presented.

Some characterizations of Schur matrices in ultrametric fields

Natarajan P.N.

Commentationes Mathematicae

2012

Vol. 52, [Z] 2

137--142

In this short paper, K denotes a complete, non-trivially valued, ultra-metric field. Sequences and infinite matrices have entries in K. We prove a few characterizations of Schur matrices in K. We then deduce some non-inclusion theorems modelled on the results of Agnew [1] and Fridy [3] in the classical case.

The Schur and Steinhaus Theorems for 4-Dimensional Matrices in Ultrametric Fields

Natarajan P.N.

Commentationes Mathematicae

2011

Vol. 51, [Z] 2

203-209

Throughout this paper, K denotes a ds-complete, non-trivially valued, ultrametric field. Entries of double sequences, double series and 4-dimensional matrices are in K. We prove the Schur and Steinhaus theorems for 4-dimensional matrices in such fields.