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A Marcinkiewicz Type Interpolation Theorem for Orlicz Spaces and Its Application

Mathematica Pannonica
Authors:
Xiaoqiang Xie
,
Xi Fu
, and
Changmin Li

In this paper, we show a Marcinkiewicz type interpolation theorem for Orlicz spaces. As an application, we obtain an existence result for a parabolic equation in divergence form.

Open access

Convex Bodies of Constant Width in Spaces of Constant Curvature and the Extremal Area of Reuleaux Triangles

Studia Scientiarum Mathematicarum Hungarica
Authors:
Károly J. Böröczky
and
Ádám Sagmeister

Extending Blaschke and Lebesgue’s classical result in the Euclidean plane, it has been recently proved in spherical and the hyperbolic cases, as well, that Reuleaux triangles have the minimal area among convex domains of constant width D. We prove an essentially optimal stability version of this statement in each of the three types of surfaces of constant curvature. In addition, we summarize the fundamental properties of convex bodies of constant width in spaces of constant curvature, and provide a characterization in the hyperbolic case in terms of horospheres.

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On Sets of Points in General Position That Lie on a Cubic Curve in the Plane

Studia Scientiarum Mathematicarum Hungarica
Authors:
Mehdi Makhul
and
Rom Pinchasi

Let P be a set of n points in general position in the plane. Let R be a set of points disjoint from P such that for every x, y € P the line through x and y contains a point in R. We show that if is contained in a cubic curve c in the plane, then P has a special property with respect to the natural group structure on c. That is, P is contained in a coset of a subgroup H of c of cardinality at most |R|.

We use the same approach to show a similar result in the case where each of B and G is a set of n points in general position in the plane and every line through a point in B and a point in G passes through a point in R. This provides a partial answer to a problem of Karasev.

The bound $R < 3 2 n$ is best possible at least for part of our results. Our extremal constructions provide a counterexample to an old conjecture attributed to Jamison about point sets that determine few directions.

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A Note on Common Nuclear Köthe Subspaces and Quotients

Mathematica Pannonica
Authors:
Emre Taştüner
and
Murat Hayrettin Yurdakul

Let E, G be Fréchet spaces and F be a complete locally convex space. It is observed that the existence of a continuous linear not almost bounded operator T on E into F factoring through G causes the existence of a common nuclear Köthe subspace of the triple (E, G, F). If, in addition, F has the property (y), then (E, G, F) has a common nuclear Köthe quotient.

Open access

The Average Number of Divisors in Certain Arithmetic Sequences

Mathematica Pannonica
Author:
Liubomir Chiriac

In this paper we study the sum , where $τ ( n )$ denotes the number of divisors of n, and {np } is a sequence of integers indexed by primes. Under certain assumptions we show that the aforementioned sum is . As an application, we consider the case where the sequence is given by the Fourier coefficients of a modular form.

Open access

Some New Inequalities Involving the Generalized Hardy Operator

Mathematica Pannonica
Author:
Kristina Krulić Himmelreich

In this paper we derive new inequalities involving the generalized Hardy operator. The obtained results generalized known inequalities involving the Hardy operator. We also get new inequalities involving the classical Hardy–Hilbert inequality.

Open access

Bipartite Domination in Graphs

Mathematica Pannonica
Authors:
Anna Bachstein
,
Wayne Goddard
, and
Michael A. Henning

The bipartite domination number of a graph is the minimum size of a dominating set that induces a bipartite subgraph. In this paper we initiate the study of this parameter, especially bounds involving the order, the ordinary domination number, and the chromatic number. For example, we show for an isolate-free graph that the bipartite domination number equals the domination number if the graph has maximum degree at most 3; and is at most half the order if the graph is regular, 4-colorable, or has maximum degree at most 5.

Open access

The Gudermannian Generated Family of Distributions with Characterizations, Regression Models and Applications

Studia Scientiarum Mathematicarum Hungarica
Authors:
Emrah Altun
,
,
Haitham M. Yousof
, and
G. G. Hamedani

This study proposes a new family of continuous distributions, called the Gudermannian generated family of distributions, based on the Gudermannian function. The statistical properties, including moments, incomplete moments and generating functions, are studied in detail. Simulation studies are performed to discuss and evaluate the maximum likelihood estimations of the model parameters. The regression model of the proposed family considering the heteroscedastic structure of the scale parameter is defined. Three applications on real data sets are implemented to convince the readers in favour of the proposed models.

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On an Equation with Prime Numbers Close to Squares

Studia Scientiarum Mathematicarum Hungarica
Author:
Stoyan I. Dimitrov

Let [ · ] be the fioor function. In this paper, we show that when 1 < c < 37/36, then every sufficiently large positive integer N can be represented in the form

$N = P 1 c + P 2 c + P 3 c ,$

where p1, p2, p3 are primes close to squares.

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Farey-Subgraphs and Continued Fractions

Studia Scientiarum Mathematicarum Hungarica
Authors:
Seema Kushwaha
and
Ritumoni Sarma

In this article, we study a family of subgraphs of the Farey graph, denoted as N for every N ∈ ℕ. We show that N is connected if and only if N is either equal to one or a prime power. We introduce a class of continued fractions referred to as N -continued fractions for each N > 1. We establish a relation between N -continued fractions and certain paths from infinity in the graph N . Using this correspondence, we discuss the existence and uniqueness of N -continued fraction expansions of real numbers.

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