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Mathematica Pannonica
Author: Péter Berkics

A linear operator on a Hilbert space $ℍ$, in the classical approach of von Neumann, must be symmetric to guarantee self-adjointness. However, it can be shown that the symmetry could be omitted by using a criterion for the graph of the operator and the adjoint of the graph. Namely, S is shown to be densely defined and closed if and only if $k+l:k,l∈GS∩GS*=ℍ$.

In a more general setup, we can consider relations instead of operators and we prove that in this situation a similar result holds. We give a necessary and sufficient condition for a linear relation to be densely defined and self-adjoint.

Open access

# Conceptions of Topological Transitivity on Symmetric Products

Mathematica Pannonica
Authors: Franco Barragán, Sergio Macías, and Anahí Rojas

Let X be a topological space. For any positive integer n, we consider the n-fold symmetric product of X, ℱn(X), consisting of all nonempty subsets of X with at most n points; and for a given function ƒ : XX, we consider the induced functions ℱn(ƒ): ℱn(X) → ℱn(X). Let M be one of the following classes of functions: exact, transitive, ℤ-transitive, ℤ+-transitive, mixing, weakly mixing, chaotic, turbulent, strongly transitive, totally transitive, orbit-transitive, strictly orbit-transitive, ω-transitive, minimal, I N, T T ++, semi-open and irreducible. In this paper we study the relationship between the following statements: ƒM and ℱn(ƒ) ∈ M.

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# Dual Pairs of Matroids with Coefficients in Finitary Fuzzy Rings of Arbitrary Rank

Mathematica Pannonica
Author: Walter Wenzel

Infinite matroids have been defined by Reinhard Diestel and coauthors in such a way that this class is (together with the finite matroids) closed under dualization and taking minors. On the other hand, Andreas Dress introduced a theory of matroids with coefficients in a fuzzy ring which is – from a combinatorial point of view – less general, because within this theory every circuit has a finite intersection with every cocircuit. Within the present paper, we extend the theory of matroids with coefficients to more general classes of matroids, if the underlying fuzzy ring has certain properties to be specified.

Open access

# Lower Estimate of Clique Size via Edge Coloring

Mathematica Pannonica
Authors: Balázs Király and Sándor Szabó

In many clique search algorithms well coloring of the nodes is employed to find an upper bound of the clique number of the given graph. In an earlier work a non-traditional edge coloring scheme was proposed to get upper bounds that are typically better than the one provided by the well coloring of the nodes. In this paper we will show that the same scheme for well coloring of the edges can be used to find lower bounds for the clique number of the given graph. In order to assess the performance of the procedure we carried out numerical experiments.

Open access

# Pascal’s Hexagram and Desargues Configurations

Mathematica Pannonica
Author: Jaydeep Chipalkatti

This paper solves an enumerative problem which arises naturally in the context of Pascal’s hexagram. We prove that a general Desargues configuration in the plane is associated to six conical sextuples via the theorems of Pascal and Kirkman. Moreover, the Galois group associated to this problem is isomorphic to the symmetric group on six letters.

Open access

# The Principal Fibre Bundle on Lorentzian Almost R-para Contact Structure

Mathematica Pannonica
Authors: Lovejoy S. Das and Mohammad Nazrul Islam Khan

The purpose of this paper is to study the principal fibre bundle (P, M, G, π p ) with Lie group G, where M admits Lorentzian almost paracontact structure (Ø, ξ p, ηp, g) satisfying certain condtions on (1, 1) tensor field J, indeed possesses an almost product structure on the principal fibre bundle. In the later sections, we have defined trilinear frame bundle and have proved that the trilinear frame bundle is the principal bundle and have proved in Theorem 5.1 that the Jacobian map π * is the isomorphism.

Open access

# Some Zero-One Linear Programming Reformulations for the Maximum Clique Problem

Mathematica Pannonica
Authors: Ákos Beke, Sándor Szabó, and Bogdán Zavalnij

Many combinatorial optimization problems can be expressed in terms of zero-one linear programs. For the maximum clique problem the so-called edge reformulation is applied most commonly. Two less frequently used LP equivalents are the independent set and edge covering set reformulations. The number of the constraints (as a function of the number of vertices of the ground graph) is asymptotically quadratic in the edge and the edge covering set LP reformulations and it is exponential in the independent set reformulation, respectively. F. D. Croce and R. Tadei proposed an approach in which the number of the constraints is equal to the number of the vertices. In this paper we are looking for possible tighter variants of these linear programs.

Open access

# Numerical solution of linear differential equations by Walsh polynomials approach

Studia Scientiarum Mathematicarum Hungarica
Authors: György Gát and Rodolfo Toledo

## Abstract

In 1975 C. F. Chen and C. H. Hsiao established a new procedure to solve initial value problems of systems of linear differential equations with constant coefficients by Walsh polynomials approach. However, they did not deal with the analysis of the proposed numerical solution. In a previous article we study this procedure in case of one equation with the techniques that the theory of dyadic harmonic analysis provides us. In this paper we extend these results through the introduction of a new procedure to solve initial value problems of differential equations with not necessarily constant coefficients.

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# Refutation of a claim made by Fejes Tóth on the accuracy of surface meshes

Studia Scientiarum Mathematicarum Hungarica
Authors: Gert Vegter and Mathijs Wintraecken

## Abstract

Fejes Tóth [] studied approximations of smooth surfaces in three-space by piecewise flat triangular meshes with a given number of vertices on the surface that are optimal with respect to Hausdorff distance. He proves that this Hausdorff distance decreases inversely proportional with the number of vertices of the approximating mesh if the surface is convex. He also claims that this Hausdorff distance is inversely proportional to the square of the number of vertices for a specific non-convex surface, namely a one-sheeted hyperboloid of revolution bounded by two congruent circles. We refute this claim, and show that the asymptotic behavior of the Hausdorff distance is linear, that is the same as for convex surfaces.

Open access

# Integral bases of pure fields with square-free parameter

Studia Scientiarum Mathematicarum Hungarica
Author: László Remete

## Abstract

Let m ≠ 0, ±1 and n ≥ 2 be integers. The ring of algebraic integers of the pure fields of type $ℚ(nm)$ is explicitly known for n = 2, 3,4. It is well known that for n = 2, an integral basis of the pure quadratic fields can be given parametrically, by using the remainder of the square-free part of m modulo 4. Such characterisation of an integral basis also exists for cubic and quartic pure fields, but for higher degree pure fields there are only results for special cases.

In this paper we explicitly give an integral basis of the field $ℚ(nm)$, where m ≠ ±1 is square-free. Furthermore, we show that similarly to the quadratic case, an integral basis of $ℚ(nm)$ is repeating periodically in m with period length depending on n.

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