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# Empirical results on distance of two-dimensional samples

Studia Scientiarum Mathematicarum Hungarica
Author: Csaba Noszály

The distance of two-dimensional samples is studied. The distance of two samples is based on the optimal matching method. Simulation results are obtained when the samples are drawn from normal and uniform distributions.

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# On app skew generalized power series rings

Studia Scientiarum Mathematicarum Hungarica
Authors: A. Majidinya and A. Moussavi

By [12], a ring R is left APP if R has the property that “the left annihilator of a principal ideal is pure as a left ideal”. Equivalently, R is a left APP-ring if R modulo the left annihilator of any principal left ideal is flat. Let R be a ring, (S, ≦) a strictly totally ordered commutative monoid and ω: S → End(R) a monoid homomorphism. Following [16], we show that, when R is a (S, ω)-weakly rigid and (S, ω)-Armendariz ring, then the skew generalized power series ring R[[S , ω]] is right APP if and only if r R(A) is S-indexed left s-unital for every S-indexed generated right ideal A of R. We also show that when R is a (S, ω)-strongly Armendariz ring and ω(S) ⫅ Aut(R), then the ring R[[S , ω]] is left APP if and only if R(∑a As S s(a)) is S-indexed right s-unital, for any S-indexed subset A of R. In particular, when R is Armendariz relative to S, then R[[S ]] is right APP if and only if r R(A) is S-indexed left s-unital, for any S-indexed generated right ideal A of R.

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# On the distribution of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\sqrt p$$ \end{document} modulo one involving primes of special type

Studia Scientiarum Mathematicarum Hungarica
Author: Yingchun Cai
Let P r denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper we show that the inequality
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left\{ {\sqrt p } \right\} < p^{ - \tfrac{1} {{15.5}}}$$ \end{document}
has infinitely many solutions in primes p such that p + 2 = P 4.
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# Paths on the doubly covered region of a covering of the plane by unit discs

Studia Scientiarum Mathematicarum Hungarica
Given a covering of the plane by closed unit discs
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{F}$$ \end{document}
and two points A and B in the region doubly covered by
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{F}$$ \end{document}
, what is the length of the shortest path connecting them that stays within the doubly covered region? This is a problem of G. Fejes-Tóth and he conjectured that if the distance between A and B is d, then the length of this path is at most
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\sqrt {2d} + O(1)$$ \end{document}
. In this paper we give a bound of 2.78d + O(1).
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# Shortest path avoiding balls

Studia Scientiarum Mathematicarum Hungarica
Author: Gábor Tóth
It is shown that in a packing of open circular discs with radii not exceeding 1, any two points lying outside the circles at distance d from one another can be connected by a path traveling outside the circles and having length at most
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\tfrac{4} {\pi }d + O\left( {\sqrt d } \right)$$ \end{document}
. Given a packing of open balls with bounded radii in E n and two points outside the balls at distance d from one another, the length of the shortest path connecting the two points and avoiding the balls is d + O(d/n) as d and n approaches infinity.
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# Approximation by complex modified Szász-Mirakjan operators

Studia Scientiarum Mathematicarum Hungarica
Authors: Nursel Çetin and Nurhayat İspir

In this study, we investigate approximation properties and obtain Voronovskaja type results for complex modified Szász-Mirakjan operators. Also, we estimate the exact orders of approximation in compact disks and prove that the complex modified Szász-Mirakjan operators attached to an analytic function preserve the univalence, starlikeness, convexity and spirallikeness in the unit disk.

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# Linearization of the products of the generalized Lauricella polynomials and the multivariate Laguerre polynomials via their integral representations

Studia Scientiarum Mathematicarum Hungarica
Authors: Shuoh-Jung Liu, Shy-Der Lin, Han-Chun Lu, and H. Srivastava

In this paper, the authors investigate the linearization problems associated with two families of generalized Lauricella polynomials of the first and second kinds. By means of their multiple integral representations, it is shown how one can linearize the product of two different members of each of these two families of the generalized Lauricella polynomials. Upon suitable specialization of the main results presented in this paper, the corresponding integral representations are deduced for such familiar classes of multivariable hypergeometric polynomials as (for example) the Lauricella polynomials F A (r) in r variables, the Appell polynomials F 2 in two variables and the multivariable Laguerre polynomials. Each of these integral representations, which are derived as special cases of the main results in this paper, may also be viewed as a linearization relationship for the product of two different members of the associated family of multivariable hypergeometric polynomials.

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# On absolutely nonmeasurable homomorphisms of commutative groups

Studia Scientiarum Mathematicarum Hungarica
Author: A. Kharazishvili

We give a characterization of all those commutative groups which admit at least one absolutely nonmeasurable homomorphism into the real line (or into the one-dimensional torus). These are exactly those commutative groups (G, +) for which the quotient group G/G 0 is uncountable, where G 0 denotes the torsion subgroup of G.

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# On the extensibility of D(−1)-triples {1, b, c} in the ring \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{Z}\left[ {\sqrt { - t} } \right]$$ \end{document}, t > 0

Studia Scientiarum Mathematicarum Hungarica
Author: Ivan Soldo

Let b = 2, 5, 10 or 17 and t > 0. We study the existence of D(−1)-quadruples of the form {1, b, c, d} in the ring \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{Z}\left[ {\sqrt { - t} } \right]$$ \end{document}. We prove that if {1, b, c} is a D(−1)-triple in \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{Z}\left[ {\sqrt { - t} } \right]$$ \end{document}, then c is an integer. As a consequence of this result, we show that for t ∉ {1, 4, 9, 16} there does not exist a subset of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{Z}\left[ {\sqrt { - t} } \right]$$ \end{document} of the form {1, b, c, d} with the property that the product of any two of its distinct elements diminished by 1 is a square of an element in \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{Z}\left[ {\sqrt { - t} } \right]$$ \end{document}.

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# On the minimum cardinality of a planar point set containing two disjoint convex polygons

Studia Scientiarum Mathematicarum Hungarica
Authors: Liping Wu and Wanbing Lu

Let N(k, l) be the smallest positive integer such that any set of N(k, l) points in the plane, no three collinear, contains both a convex k-gon and a convex l-gon with disjoint convex hulls. In this paper, we prove that N(3, 4) = 7, N(4, 4) = 9, N(3, 5) = 10 and N(4, 5) = 11.

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