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# 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.

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# 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.

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# 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.

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# 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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# Revisiting the universal linear algebraic model for the characteristic two case

Studia Scientiarum Mathematicarum Hungarica
Author: Máté L. Juhász

## Abstract

In , a universal linear algebraic model was proposed for describing homogeneous conformal geometries, such as the spherical, Euclidean, hyperbolic, Minkowski, anti-de Sitter and Galilei planes (). This formalism was independent from the underlying field, providing an extension and general approach to other fields, such as finite fields. Some steps were taken even for the characteristic 2 case.

In this article, we undertake the study of the characteristic 2 case in more detail. In particular, the concept of virtual quadratic spaces is used (), and a similar result is achieved for finite fields of characteristic 2 as for other fields. Some differences from the non-characteristic 2 case are also pointed out.

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# An algorithm to compute a primary decomposition of modules in polynomial rings over the integers

Studia Scientiarum Mathematicarum Hungarica
Authors: Nazeran Idrees, Gerhard Pfister, and Afshan Sadiq

We present an algorithm to compute the primary decomposition of a submodule N of the free module ℤ[x 1,...,x n]m. For this purpose we use algorithms for primary decomposition of ideals in the polynomial ring over the integers. The idea is to compute first the minimal associated primes of N, i.e. the minimal associated primes of the ideal Ann (ℤ[x 1,...,x n]m/N) in ℤ[x 1,...,x n] and then compute the primary components using pseudo-primary decomposition and extraction, following the ideas of Shimoyama-Yokoyama. The algorithms are implemented in Singular.

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# Bivariate Bonferroni-type inequalities based on multivariate Lagrange interpolation

Studia Scientiarum Mathematicarum Hungarica
Authors: Gergely Mádi-Nagy and András Prékopa

Let A 1,...,A N and B 1,...,B M be two sequences of events and let ν N(A) and ν M(B) be the number of those A i and B j, respectively, that occur. Based on multivariate Lagrange interpolation, we give a method that yields linear bounds in terms of S k,t, k+tm on the distribution of the vector (ν N(A), ν M(B)). For the same value of m, several inequalities can be generated and all of them are best bounds for some values of S k,t. Known bivariate Bonferroni-type inequalities are reconstructed and new inequalities are generated, too.

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# A classifier for simple isolated complete intersection singularities

Studia Scientiarum Mathematicarum Hungarica
Authors: Deeba Afzal and Gerhard Pfister

M. Giusti’s classification of the simple complete intersection singularities is characterized in terms of invariants. This is a basis for the implementation of a classifier in the computer algebra system Singular.

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