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1

One some cancellation algorithms II

Zakarczemny M.

Technical Transactions

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2017

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Vol. 5(114)

97--103

EN

We define bƒ(n) to be the smallest integer (a natural number) d such that numbers ƒ(n1 , n2, ..., nm), where n1+n2+ ... + nm ≤ n are not divisible by d. For the given functions ƒ : ℕm → ℕ, we will obtain the asymptotic characterisation of the sequence of the least non canceled numbers (bƒ(n)) n∈ℕ. In the case ƒ : ℕ2ͽ (k, l)→k3+l3∈ℕ, this characterisation can be rewritten in the terms of the permutations polynomials of finite commutative quotient ring ℤ/mℤ. There are situations in which we cannot expect formula for bƒ(n) to be simple, but we can provide the upper and lower bounds of it.

PL

Definiujemy bƒ(n) jako najmniejszą d∈ℕ, taką że liczby ƒ(n1 , n2, ..., nm), gdzie n1+n2+ ... + nm ≤ n są niepodzielne przez d. Dla wybranych funkcji ƒ : ℕm → ℕ, znajdziemy wartości elementów ciągu (bƒ(n)) n∈ℕ, lub podamy inną charakteryzacje. Dla funkcji ƒ : ℕ2ͽ (k, l)→k3+l3∈ℕ, Charakteryzacja ciągu (bƒ(n)) n∈ℕ może być podana z użyciem wielomianów permutacyjnych skończonego, przemiennego, pierścienia ilorazowego ℤ/mℤ. W szczególnych przypadkach funkcji f podamy dolne i górne ograniczenia na wartości ciągubƒ(n).

2

Congruences of Edge-bipartite Graphs with Applications to Grothendieck Group Recognition. [Part 2], Coxeter Type Study

Mróz A.

Fundamenta Informaticae

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2016

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Vol. 146, nr 2

145--177

EN

In this two parts article with the same main title we study a problem of Coxeter-Gram spectral analysis of edge-bipartite graphs (bigraphs), a class of signed graphs. We ask for a criterion deciding if a given bigraph Δ is weakly or strongly Gram-congruent with a graph. The problem is inspired by recent works of Simson et al. started in [SIAM J. Discr. Math. 27 (2013), 827-854], and by problems related to integral quadratic forms, bilinear lattices, representation theory of algebras, algebraic methods in graph theory and the isotropy groups of bigraphs. In this Part II we develop general combinatorial techniques, with the use of inflation algorithm discussed in Part I, morsifications and the isotropy group of a bigraph, and we provide a constructive solution of the problem for the class of all positive connected loop-free bigraphs. Moreover, we present an application of our results to Grothendieck group recognition problem: deciding if a given bilinear lattice is the Grothendieck group of some category. Our techniques are tested in a series of experiments for so-called Nakayama bigraphs, illustrating the applications in practice and certain related phenomena. The results show that a computer algebra technique and discrete mathematical computing provide important tools in solving theoretical problems of high complexity.

3

Congruences of Edge-bipartite Graphs with Applications to Grothendieck Group Recognition. [Part 1], Inflation Algorithm Revisited

Mróz A.

Fundamenta Informaticae

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2016

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Vol. 146, nr 2

121--144

EN

We study edge-bipartite graphs (bigraphs), a class of signed graphs, by means of the inflation algorithm which relies on performing certain elementary transformations on a given bigraph Δ, or equivalently, on the associated integral quadratic form qΔ: Zn → Z, preserving Gram Z-congruence. The ideas are inspired by classical results of Ovsienko and recent studies of Simson started in [SIAM J. Discr. Math. 27 (2013), 827-854], concerning classifications of integral quadratic and bilinear forms, and their Coxeter spectral analysis. We provide few modifications of the inflation algorithm and new estimations of its complexity for positive and principal loop-free bigraphs. We discuss in a systematic way the behavior and computational aspects of inflation techniques. As one of the consequences we obtain relatively simple proofs of several interesting properties of quadratic forms and their roots, extending known facts. On the other hand, the results are a first step of a solution of a variant of Grothendieck group recognition, a difficult combinatorial problem arising in representation theory of finite dimensional algebras and their derived categories, which we discuss in Part II of this two parts article with the same main title.

4

Automorphisms of Witt rings of local type

Stępień M. R.

Journal of Applied Mathematics and Computational Mechanics

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2015

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Vol. 14, nr 3

109-119

EN

We present a description of the group of strong automorphisms of Witt rings of local type (Witt rings of local fields) using quaternionic structures.

5

Inflation Algorithms for Positive and Principal Edge-bipartite Graphs and Unit Quadratic Forms

Kosakowska J.

Fundamenta Informaticae

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2012

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Vol. 119, nr 2

149-162

EN

We describe combinatorial algorithms that compute the Dynkin type (resp. Euclidean type) of any positive (resp. principal) unit quadratic form q : Z^n →Z and of any positive (resp. principal) edge-bipartite connected graph Δ. The study of the problem is inspired by applications of the algorithms in the representation theory, in solving a class of Diophantine equations, in the study of mesh geometries of roots, in the spectral analysis of graphs, and in the Coxeter-Gram classification of edge-bipartite graphs.

6

Some remarks on quadratic forms in stable random variables

Mijnheer J.

Demonstratio Mathematica

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2001

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Vol. 34, nr 2

327-338

EN

We consider quadratic forms that appear in the least squares estimator for the unknown parameter in the AR(1) model with stable innovations. In three cases we obtain different limit distributions.

7

Absolute continuity, quadratic forms, and causal perturbations of product measures

Kanter M.

Probability and Mathematical Statistics

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1999

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Vol. 19, Fasc. 2

429--439

EN

Necessary and sufficient conditions are given for the absolute continuity of a causally perturbed product measure with respect to the non-perturbed measure. When the perturbation is linear, these conditions involve the convergence of a quadratic form of independent random variables. The convergence of this form is studied when the independent random variables are symmetric or strictly stable.