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In this article we formalize a free ℤ-module and its rank. We formally prove that for a free finite rank ℤ-module V , the number of elements in its basis, that is a rank of the ℤ-module, is constant regardless of the selection of its basis. ℤ-module is necessary for lattice problems, LLL(Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic systems with lattice [15]. Some theorems in this article are described by translating theorems in [21] and [8] into theorems of Z-module.

Freie Auflösungen äusserer Potenzen.

Karsten Lebelt (1977)

Manuscripta mathematica

Fröhlich's local quadratic forms.

Irving Kaplansky (1969)

Journal für die reine und angewandte Mathematik

From Eckart and Young approximation to Moreau envelopes and vice versa

Jean-Baptiste Hiriart-Urruty, Hai Yen Le (2013)

RAIRO - Operations Research - Recherche Opérationnelle

In matricial analysis, the theorem of Eckart and Young provides a best approximation of an arbitrary matrix by a matrix of rank at most r. In variational analysis or optimization, the Moreau envelopes are appropriate ways of approximating or regularizing the rank function. We prove here that we can go forwards and backwards between the two procedures, thereby showing that they carry essentially the same information.

From equivalent linear equations to Gauss-Markov theorem.

Stȩpniak, Czesław (2010)

Journal of Inequalities and Applications [electronic only]

From geometry to invertibility preservers

Hans Havlicek, Peter Šemrl (2006)

Studia Mathematica

We characterize bijections on matrix spaces (operator algebras) preserving full rank (invertibility) of differences of matrix (operator) pairs in both directions.

Full rank factorization and the Flanders theorem.

Cantó, Rafael, Ricarte, Beatriz, Urbano, Ana M. (2009)

ELA. The Electronic Journal of Linear Algebra [electronic only]

Further determinant identities related to classical root systems

Wenchang Chu (2017)

Czechoslovak Mathematical Journal

By introducing polynomials in matrix entries, six determinants are evaluated which may be considered extensions of Vandermonde-like determinants related to the classical root systems.

Further results on the Craig-Sakamoto equation.

Maroulas, John (2008)

ELA. The Electronic Journal of Linear Algebra [electronic only]

Further results on the reverse order law for ${1, 3}$ -inverse and ${1, 4}$ -inverse of a matrix product.

Liu, Deqiang, Yang, Hu (2010)

Journal of Inequalities and Applications [electronic only]

Fuzzy transforms in image compression and fusion

Irina Perfilieva (2007)

Acta Mathematica Universitatis Ostraviensis

An overview of direct and inverse fuzzy transforms of three types is given and applications to data processing are considered. The construction and some important properties of fuzzy transforms are presented on the theoretical level. Three applications of $F$ -transform to data processing have been chosen: compressional and reconstruction of data, removing noise and data fusion. All of them successively exploit the filtering property of the inverse fuzzy transform.

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