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# A Characterization of T 1 Spaces via Limit Sets of Nets

Mathematica Pannonica
Author:
Yu-Lin Chou

This article indicates another set-theoretic formula, solely in terms of union and intersection, for the set of the limits of any given sequence (net, in general) in an arbitrary T 1 space; this representation in particular gives a new characterization of a T 1 space.

Open access

# Completely Multiplicative Functions with Special Properties

Mathematica Pannonica
Authors:
Imre Kátai
and
Bui M. Phong

We give all solutions of completely multiplicative functions ƒ , g, for which the equation Ag(n + 1) = Bƒ (n) + C holds for every n ∈ ℕ. We also study the equation G(p + 1) = F(p − 1) + D and we prove some results concerning it.

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# Batch Scheduling with Time Restriction and Clique Search

Mathematica Pannonica
Author:
Sándor Szabó

We consider a graph whose vertices are legally colored using k colors and ask if the graph contains a k-clique. As it turns out this very special type of k-clique problem is in an intimate connection with constructing schedules. The practicality this clique search based construction of schedules is checked by carrying out numerical experiments.

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# Determinant Inequalities for Positive Definite Matrices via Cartwright–Field’s Result for Arithmetic and Geometric Weighted Means

Mathematica Pannonica
Author:
Silvestru Sever Dragomir

Assume that Aj , j ∈ {1, … , m} are positive definite matrices of order n. In this paper we prove among others that, if 0 < l In Aj , j ∈ {1, … , m} in the operator order, for some positive constant l, and In is the unity matrix of order n, then

$o ≤ 1 2 ∑ k = 1 m P k 1 − P k det 2 A j − l I n − 1 / 2 − 2 ∑ 1 ≤ j < k ≤ m P j P k det A j + A k − l I n − 1 / 2 ≤ ∑ j = 1 m P j det A j − 1 / 2 − det ∑ k = 1 m P k A k − 1 / 2 ,$

where Pk ≥ 0 for k ϵ {1, …, m} and $∑ j = 1 m P j = 1$ .

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# Evolutes of Conics in the Pseudo-Euclidean Plane

Mathematica Pannonica
Author:
Ivana Božić Dragun

The evolute of a conic in the pseudo-Euclidean plane is the locus of centers of all its osculating circles. It’s a curve of order six and class four in general case. In this paper we discuss and compute the order and class of evolutes of different types of conics. We will highlight those cases that have no analogy in the Euclidean plane.

Open access

# On Quasi I-Statistical Convergence of Triple Sequences in Cone Metric Spaces

Mathematica Pannonica
Authors:
Işıl Açık Demırcı
,
Ömer Kışı
, and
Mehmet Gürdal

Fast [12] is credited with pioneering the field of statistical convergence. This topic has been researched in many spaces such as topological spaces, cone metric spaces, and so on (see, for example [19, 21]). A cone metric space was proposed by Huang and Zhang [17]. The primary distinction between a cone metric and a metric is that a cone metric is valued in an ordered Banach space. Li et al. [21] investigated the definitions of statistical convergence and statistical boundedness of a sequence in a cone metric space. Recently, Sakaoğlu and Yurdakadim [29] have introduced the concepts of quasi-statistical convergence. The notion of quasi I-statistical convergence for triple and multiple index sequences in cone metric spaces on topological vector spaces is introduced in this study, and we also examine certain theorems connected to quasi I-statistically convergent multiple sequences. Finally, we will provide some findings based on these theorems.

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# The Lower Bipartite Number of a Graph

Mathematica Pannonica
Authors:
Anna Bachstein
and
Wayne Goddard

For a graph G, we define the lower bipartite number LB(G) as the minimum order of a maximal induced bipartite subgraph of G. We study the parameter, and the related parameter bipartite domination, providing bounds both in general graphs and in some graph families. For example, we show that there are arbitrarily large 4-connected planar graphs G with LB(G) = 4 but a 5-connected planar graph has linear LB(G). We also show that if G is a maximal outerplanar graph of order n, then LB(G) lies between (n + 2)/3 and 2 n/3, and these bounds are sharp.

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# Generalized Forms of an Overconstrained Sliding Mechanism Consisting of Two Congruent Tetrahedra

Studia Scientiarum Mathematicarum Hungarica
Authors:
Endre Makai Jr.
and
Tibor Tarnai

The motions of a bar structure consisting of two congruent tetrahedra are investigated, whose edges in their basic position are the face diagonals of a rectangular parallelepiped. The constraint of the motion is the following: the originally intersecting edges have to remain coplanar. All finite motions of our bar structure are determined. This generalizes our earlier work, where we did the same for the case when the rectangular parallelepiped was a cube. At the end of the paper we point out three further possibilities to generalize the question about the cube, and give for them examples of finite motions.

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# Random Walks on the Two-Dimensional K-Comb Lattice

Mathematica Pannonica
Authors:
Endre Csáki
and
Antónia Földes

We study the path behavior of the symmetric walk on some special comb-type subsets of ℤ2 which are obtained from ℤ2 by generalizing the comb having finitely many horizontal lines instead of one.

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# Splitting Edge Partitions of Graphs

Mathematica Pannonica
Authors:
Balázs Király
and
Sándor Szabó

In a typical maximum clique search algorithm when optimality testing is inconclusive a forking takes place. The instance is divided into smaller ones. This is the branching step of the procedure. In order to ensure a balanced work load for the processors for parallel algorithms it is essential that the resulting smaller problems are do not overly vary in difficulty. The so-called splitting partitions of the nodes of the given graph were introduced earlier to meliorate this problem. The paper proposes a splitting partition of the edges for the same purpose. In the lack of available theoretical tools we assess the practical feasibility of constructing suboptimal splitting edge partitions by carrying out numerical experiments. While working with splitting partitions we have realized that they can be utilized as preconditioning tools preliminary to a large scale clique search. The paper will discuss this new found role of the splitting edge partitions as well.

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# Convex Polygons and Separation of Convex

Studia Scientiarum Mathematicarum Hungarica
Authors:
Eduardo Rivera-Campo
and
Jorge Urrutia

We prove that for any collection F of n ≥ 2 pairwise disjoint compact convex sets in the plane there is a pair of sets A and B in F such that any line that separates A from B separates either A or B from a subcollection of F with at least n/18 sets.

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# Positive Solutions for a Singular System of Nonlinear Fractional Differential Equations

Studia Scientiarum Mathematicarum Hungarica
Author:
Ping Kang

In this paper, we study the existence of positive solutions for a system of nonlinear fractional differential equations. The results are based upon the fixed-point theorem of cone expansion and compression type due to Krasnosel’skill. Moreover, Our results generalize and include some known results.

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# Lifting Diffeomorphisms to Vector Bundles

Studia Scientiarum Mathematicarum Hungarica
Authors:
Jaime Muñoz Masqué
,
, and
Ignacio Sánchez Rodríguez

Criteria for a diffeomorphism of a smooth manifold M to be lifted to a linear automorphism of a given real vector bundle p : V → M, are stated. Examples are included and the metric and complex vector-bundle cases are also considered.

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# A Remark on Nefness of Divisors on Surfaces of General Type

Studia Scientiarum Mathematicarum Hungarica
Authors:
Debojyoti Bhattacharya
and
Joyentanuj Das

Let X be an irreducible complex projective variety of dimension n ≥ 1. Let D be a Cartier divisor on X such that Hi(X, OX (mD)) = 0 for m > 0 and for all i > 0, then it is not true in general that D is a nef divisor (cf. [4]). Also, in general, effective divisors on smooth surfaces are not necessarily nef (they are nef provided they are semiample). In this article, we show that, if X is a smooth surface of general type and C is a smooth hyperplane section of it, then for any non-zero effective divisor D on X satisfying H1(X, OX (mD)) = 0 for all m > C.KX , D is a nef divisor.

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# Gel’fand Γ-Semirings

Mathematica Pannonica
Authors:
Tilak Raj Sharma
and
Hitesh Kumar Ranote

In this paper, we introduce the notion of a Gel’fand Γ-semiring and discuss the various characterization of simple, k-ideal, strong ideal, t-small elements and additively cancellative elements of a Gel’fand Γ-semiring R, and prove that the set of additively cancellative elements, set of all t-small elements of R and set of all maximal ideal of R are strong ideals. Further, let R be a simple Gel’fand Γ-semiring and 1 ≠ tR. Let M be the set of all maximal left (right) ideals of R. Then an element x of R is t-small if and only if it belongs to every maximal one sided left (right)ideal of R containing t.

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# Operator Convexity of an Integral Transform with Applications

Mathematica Pannonica
Author:
Silvestru Sever Dragomir

For a continuous and positive function w(λ), λ > 0 and μ a positive measure on (0, ∞) we consider the following integral transform

$D w , μ t : = ∫ 0 ∞ w λ λ + t − 1 d μ λ ,$

where the integral is assumed to exist for t > 0.

We show among others that D(w, μ) is operator convex on (0, ∞). From this we derive that, if f : [0, ∞) → R is an operator monotone function on [0, ∞), then the function [f(0) -f(t)] t -1 is operator convex on (0, ∞). Also, if f : [0, ∞) → R is an operator convex function on [0, ∞), then the function $f 0 + f + ′ 0 t − f t t − 2$ is operator convex on (0, ∞). Some lower and upper bounds for the Jensen’s difference

$D w , μ A + D w , μ B 2 − D w , μ A + B 2$

under some natural assumptions for the positive operators A and B are given. Examples for power, exponential and logarithmic functions are also provided.

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# Perfect Solutions to Problems on Common Transversals and Submodular Functions from Welsh’s 1976 Text Matroid Theory

Mathematica Pannonica
Author:
Jonathan David Farley

Problem 2 of Welsh’s 1976 text Matroid Theory, asking for criteria telling when two families of sets have a common transversal, is solved.

Another unsolved problem in the text Matroid Theory, on whether the “join” of two non-decreasing submodular functions is submodular, is answered in the negative. This resolves an issue first raised by Pym and Perfect in 1970.

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# Algebra of Data Reconciliation

Studia Scientiarum Mathematicarum Hungarica
Authors:
Elod P. Csirmaz
and
Laszlo Csirmaz

With distributed computing and mobile applications becoming ever more prevalent, synchronizing diverging replicas of the same data is a common problem. Reconciliation – bringing two replicas of the same data structure as close as possible without overriding local changes – is investigated in an algebraic model. Our approach is to consider two sequences of simple commands that describe the changes in the replicas compared to the original structure, and then determine the maximal subsequences of each that can be propagated to the other. The proposed command set is shown to be functionally complete, and an update detection algorithm is presented which produces a command sequence transforming the original data structure into the replica while traversing both simultaneously. Syntactical characterization is provided in terms of a rewriting system for semantically equivalent command sequences. Algebraic properties of sequence pairs that are applicable to the same data structure are investigated. Based on these results the reconciliation problem is shown to have a unique maximal solution. In addition, syntactical properties of the maximal solution allow for an efficient algorithm that produces it.

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# Existence and Stability of Solutions for a Nonlinear Beam Equation with Internal Damping

Mathematica Pannonica
Authors:
Ducival Carvalho Pereira
,
Carlos Alberto Raposo
, and
Huy Hoang Nguyen

This manuscript deals with the global existence and asymptotic behavior of solutions for a Kirchhoff beam equation with internal damping. The existence of solutions is obtained by using the Faedo-Galerkin method. Exponential stability is proved by applying Nakao’s theorem.

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# Hypersphere Having ΔIIx = Ax in E4

Mathematica Pannonica
Authors:
Erhan Güler
and
Kübra Yilmaz

We consider hypersphere x = x(u, v, w) in the four dimensional Euclidean space. We calculate the Gauss map, and the curvatures of it. Moreover, we compute the second Laplace-Beltrami operator the hypersphere satisfying ΔIIx = Ax, where A ϵ Mat (4,4).

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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.

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# 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.

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# 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.

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# 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.

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# 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.

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# 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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# A Note on Visible Islands

Studia Scientiarum Mathematicarum Hungarica
Authors:
Sophie Leuchtner
,
Carlos M. Nicolás
, and
Andrew Suk

Given a finite point set P in the plane, a subset S⊆P is called an island in P if conv(S) ⋂ P = S. We say that S ⊂ P is a visible island if the points in S are pairwise visible and S is an island in P. The famous Big-line Big-clique Conjecture states that for any k ≥ 3 and l ≥ 4, there is an integer n = n(k, l), such that every finite set of at least n points in the plane contains l collinear points or k pairwise visible points. In this paper, we show that this conjecture is false for visible islands, by replacing each point in a Horton set by a triple of collinear points. Hence, there are arbitrarily large finite point sets in the plane with no 4 collinear members and no visible island of size 13.

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# Lefschetz Open Book Embeddings of 4-Manifolds

Studia Scientiarum Mathematicarum Hungarica
Authors:
Abhijeet Ghanwat
,
Suhas Pandit
, and
A. Selvakumar

In this article, we define the notion of a generalized open book of a n-manifold over the k−sphere Sk , k < n. We discuss Lefschetz open book embeddings of Lefschetz open books of closed oriented 4-manifolds into the trivial open book over S2 of the 7−sphere S7 . If X is the double of a bounded achiral Lefschetz fibration over D2 , then X naturally admits a Lefschetz open book given by the bounded achiral Lefschetz fibration. We show that this natural Lefschetz open book of X admits a Lefschetz open book embedding into the trivial open book over S2 of the 7−sphere S7 .

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# Lifting Generic Maps to Embeddings. Triangulation and Smoothing

Studia Scientiarum Mathematicarum Hungarica
Author:
Sergey A. Melikhov

We show that if a non-degenerate PL map f : NM lifts to a topological embedding in $M × ℝ k$ then it lifts to a PL embedding in there. We also show that if a stable smooth map Nn Mm, mn, lifts to a topological embedding in $M × ℝ$ , then it lifts to a smooth embedding in there.

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# Polynomial Interpolation on Sequences

Mathematica Pannonica
Authors:
Francesc Tugores
and
Laia Tugores

This short note deals with polynomial interpolation of complex numbers verifying a Lipschitz condition, performed on consecutive points of a given sequence in the plane. We are interested in those sequences which provide a bound of the error at the first uninterpolated point, depending only on its distance to the last interpolated one.

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# A Property of Lattices of Sublattices Closed Under Taking Relative Complements and Its Connection to 2-Distributivity

Mathematica Pannonica
Author:
Gábor Czédli

For a lattice L of finite length n, let RCSub(L) be the collection consisting of the empty set and those sublattices of L that are closed under taking relative complements. That is, a subset X of L belongs to RCSub(L) if and only if X is join-closed, meet-closed, and whenever {a, x, b} ⊆ S, yL, xy = a, and xy = b, then yS. We prove that (1) the poset RCSub(L) with respect to set inclusion is lattice of length n + 1, (2) if RCSub(L) is a ranked lattice and L is modular, then L is 2-distributive in András P. Huhn’s sense, and (3) if L is distributive, then RCSub(L) is a ranked lattice.

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# On Derivations and Lie Ideals of Semirings

Mathematica Pannonica
Authors:
and
Neelam

In this paper, centralizing (semi-centralizing) and commuting (semi-commuting) derivations of semirings are characterized. The action of these derivations on Lie ideals is also discussed and as a consequence, some significant results are proved. In addition, Posner’s commutativity theorem is generalized for Lie ideals of semirings and this result is also extended to the case of centralizing (semi-centralizing) derivations of prime semirings. Further, we observe that if there exists a skew-commuting (skew-centralizing) derivation D of S, then D = 0. It is also proved that for any two derivations d 1 and d 2 of a prime semiring S with char S ≠ 2 and x d 1 x d 2 = 0, for all xS implies either d 1 = 0 or d 2 = 0.

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# In Memoriam István Győri

Mathematica Pannonica
Authors:
Ferenc Hartung
and
Mihály Pituk
Open access

# Convexity of Distinct Sum Sets

Studia Scientiarum Mathematicarum Hungarica
Author:
Alexander Lemmens

We study a combinatorial notion where given a set S of lattice points one takes the set of all sums of p distinct points in S, and we ask the question: ‘if S is the set of lattice points of a convex lattice polytope, is the resulting set also the set of lattice points of a convex lattice polytope?’ We obtain a positive result in dimension 2 and a negative result in higher dimensions. We apply this to the corner cut polyhedron.

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# Degree Conditions for Claw-Free Graphs to Have Spanning Trees with at Most Five Branch Vertices and Leaves in Total

Studia Scientiarum Mathematicarum Hungarica
Author:
Dang Dinh Hanh

A leaf of a tree is a vertex of degree one and a branch vertex of a tree is a vertex of degree at least three. In this paper, we show a degree condition for a claw-free graph to have spanning trees with at most five branch vertices and leaves in total. Moreover, the degree sum condition is best possible.

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# An Improved Constant Factor for the Unit Distance Problem

Studia Scientiarum Mathematicarum Hungarica
Authors:
Péter Ágoston
and
Dömötör Pálvölgyi

We prove that the number of unit distances among n planar points is at most 1.94 • n 4/3, improving on the previous best bound of 8 n 4/3. We also give better upper and lower bounds for several small values of n. We also prove some variants of the crossing lemma and improve some constant factors.

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# Intersecting Hexagons in 3-Space

Studia Scientiarum Mathematicarum Hungarica
Authors:
József Solymosi
and
Ching Wong

Two hexagons in the space are said to intersect heavily if their intersection consists of at least one common vertex as well as an interior point. We show that the number of hexagons on n points in 3-space without heavy intersections is o(n 2), under the assumption that the hexagons are ‘fat’.

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# Lazarsfeld-Mukai Bundles on K3 Surfaces Associated with a Pencil Computing the Clifford Index

Studia Scientiarum Mathematicarum Hungarica
Author:
Sarbeswar Pal

Let X be a smooth projective K3 surface over the complex numbers and let C be an ample curve on X. In this paper we will study the semistability of the Lazarsfeld-Mukai bundle EC,A associated to a line bundle A on C such that |A| is a pencil on C and computes the Clifford index of C. We give a necessary and sufficient condition for EC,A to be semistable.

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# Realization of a Graph as the Reeb Graph of a Morse–Bott or a Round Function

Studia Scientiarum Mathematicarum Hungarica
Author:
Irina Gelbukh

We prove criteria for a graph to be the Reeb graph of a function of a given class on a closed manifold: Morse–Bott, round, and in general smooth functions whose critical set consists of a finite number of submanifolds. The criteria are given in terms of whether the graph admits an orientation, which we call S-good orientation, with certain conditions on the degree of sources and sinks, similar to the known notion of good orientation in the context of Morse functions. We also study when such a function is the height function associated with an immersion of the manifold. The condition for a graph to admit an S-good orientation can be expressed in terms of the leaf blocks of the graph.

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# The Sl(2, )-Character Variety of a Montesinos Knot

Studia Scientiarum Mathematicarum Hungarica
Author:
Haimiao Chen

For each Montesinos knot K, we propose an efficient method to explicitly determine the irreducible SL(2, )-character variety, and show that it can be decomposed as χ0(K)⊔χ1(K)⊔χ2(K)⊔χ'(K), where χ0(K) consists of trace-free characters χ1(K) consists of characters of “unions” of representations of rational knots (or rational link, which appears at most once), χ2(K) is an algebraic curve, and χ'(K) consists of finitely many points when K satisfies a generic condition.

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# On a Special Gini Mean

Mathematica Pannonica
Author:
József Sándor

We offer new properties of the special Gini mean S(a, b) = aa /( a + b )bb /( a + b ), in connections with other special means of two arguments.

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# Distance Domination in Vertex Partitioned Graphs

Mathematica Pannonica
Authors:
Allan Frendrup
,
Zsolt Tuza
, and
Preben Dahl Vestergaard

We treat a variation of graph domination which involves a partition (V 1, V 2,..., Vk ) of the vertex set of a graph G and domination of each partition class V i over distance d where all vertices and edges of G may be used in the domination process. Strict upper bounds and extremal graphs are presented; the results are collected in three handy tables. Further, we compare a high number of partition classes and the number of dominators needed.

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# Proof of a Conjecture of Proctor and Scoppetta Related to d-Complete Posets

Mathematica Pannonica
Author:
Jonathan David Farley

Proctor and Scoppetta conjectured that

• (1) there exists an infinite locally finite poset that satisfies their conditions VT and NTC but not SIS;

• (2) there exists an infinite locally finite poset satisfying their conditions D3-C and D3MF but not both VT and FT; and

• (3) there exists an infinite locally finite poset satisfying their conditions D3-C and D3MD but not NCC.

In this note, the conjecture of Proctor and Scoppetta, which is related to d-complete posets, is proven.

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# Hamilton Triangle of a Triangle in the Isotropic Plane

Mathematica Pannonica
Authors:
Zdenka Kolar-Begović
and

In this paper we introduce the concept of the Hamilton triangle of a given triangle in an isotropic plane and investigate a number of important properties of this concept. We prove that the Hamilton triangle is homological with the observed triangle and with its contact and complementary triangles. We also consider some interesting statements about the relationships between the Hamilton triangle and some other significant elements of the triangle, like e.g. the Euler and the Feuerbach line, the Steiner ellipse and the tangential triangle.

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# Menon-Type Identities Concerning Subsets of the Set {1, 2,..., n}

Mathematica Pannonica
Author:
László Tóth

We prove certain Menon-type identities associated with the subsets of the set {1, 2,..., n} and related to the functions f, fk , Ф and Ф k , defined and investigated by Nathanson.

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# On the Almost Sure Central Limit Theorem Along Subsequences

Mathematica Pannonica
Authors:
István Berkes
and
Endre Csáki

Generalizing results of Schatte [11] and Atlagh and Weber [2], in this paper we give conditions for a sequence of random variables to satisfy the almost sure central limit theorem along a given sequence of integers.

Open access