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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
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# 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.

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# On the Mean Value of Arithmetic Error Terms

Mathematica Pannonica
Author:
János Pintz

In the 1980’s the author proved lower bounds for the mean value of the modulus of the error term of the prime number theorem and other important number theoretic functions whose oscillation is in connection with the zeros of the Riemann zeta function. In the present work a general theorem is shown in a simple way which gives a lower bound for the mentioned mean value as a function of a hypothetical pole of the Mellin transform of the function. The conditions are amply satisfied for the Riemann zeta function. In such a way the results recover the earlier ones (even in a slightly sharper form). The obtained estimates are often optimal apart from a constant factor, at least under reasonable conditions as the Riemann Hypothesis. This is the case, in particular, for the error term of the prime number theorem.

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# Some Inequalities of Ostrowski Type for Double Integral Mean of Absolutely Continuous Functions

Mathematica Pannonica
Author:
Silvestru Sever Dragomir

In this paper we establish some Ostrowski type inequalities for double integral mean of absolutely continuous functions. An application for special means is given as well.

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# Trimmed Least Square Estimators for Stable Ar(1) Processes

Mathematica Pannonica
Authors:
Alina Bazarova
,
István Berkes
, and
Lajos Horváth

We prove the weak consistency of the trimmed least square estimator of the covariance parameter of an AR(1) process with stable errors.

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# Interval Chains and Completeness in Ultrapowers of Ordered Sets

Mathematica Pannonica
Authors:
Zoltán Boros
and
Péter Tóth

The ultrapower T* of an arbitrary ordered set T is introduced as an infinitesimal extension of T. It is obtained as the set of equivalence classes of the sequences in T, where the corresponding relation is generated by a free ultrafilter on the set of natural numbers. It is established that T* always satisfies Cantor’s property, while one can give the necessary and sufficient conditions for T so that T* would be complete or it would fulfill the open completeness property, respectively. Namely, the density of the original set determines the open completeness of the extension, while independently, the completeness of T* is determined by the cardinality of T.

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# Seiberg–Witten Floer Homotopy Contact Invariant

Studia Scientiarum Mathematicarum Hungarica
Authors:
Nobuo Iida
and
Masaki Taniguchi

We introduce a Floer homotopy version of the contact invariant introduced by Kronheimer–Mrowka–Ozsváth–Szabó. Moreover, we prove a gluing formula relating our invariant with the first author’s Bauer–Furuta type invariant, which refines Kronheimer–Mrowka’s invariant for 4-manifolds with contact boundary. As an application, we give a constraint for a certain class of symplectic fillings using equivariant KO-cohomology.

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# Upsilon Invariants from Cyclic Branched Covers

Studia Scientiarum Mathematicarum Hungarica
Authors:
Antonio Alfieri
,
Daniele Celoria
, and
András Stipsicz

We extend the construction of Y-type invariants to null-homologous knots in rational homology three-spheres. By considering m-fold cyclic branched covers with m a prime power, this extension provides new knot concordance invariants $Y m C ( K )$ of knots in S3. We give computations of some of these invariants for alternating knots and reprove independence results in the smooth concordance group.

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# Inequalities for the First and Second Derivatives of Algebraic Polynomials on an Ellipse

Mathematica Pannonica
Author:
Tatiana M. Nikiforova

We prove a theorem on the preservation of inequalities between functions of a special form after differentiation on an ellipse. In particular, we obtain generalizations of the Duffin–Schaeffer inequality and the Vidensky inequality for the first and second derivatives of algebraic polynomials to an ellipse.

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# A Riemann–von Mangoldt-Type Formula for the Distribution of Beurling Primes

Mathematica Pannonica
Author:
Szilárd Gy. Révész

In this paper we work out a Riemann–von Mangoldt type formula for the summatory function $ψ x$ :=  $∑ g ∈ G , g ≤ x Λ G g$ , where $G$ is an arithmetical semigroup (a Beurling generalized system of integers) and $Λ G$ is the corresponding von Mangoldt function attaining with a prime element $p ∈ G$ and zero otherwise. On the way towards this formula, we prove explicit estimates on the Beurling zeta function $ζ G$ , belonging to $G$ , to the number of zeroes of $ζ G$ in various regions, in particular within the critical strip where the analytic continuation exists, and to the magnitude of the logarithmic derivative of $ζ G$ , under the sole additional assumption that Knopfmacher’s Axiom A is satisfied. We also construct a technically useful broken line contour to which the technic of integral transformation can be well applied. The whole work serves as a first step towards a further study of the distribution of zeros of the Beurling zeta function, providing appropriate zero density and zero clustering estimates, to be presented in the continuation of this paper.

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# A Theory of Congruences and Birkhoff’s Theorem for Matroids

Mathematica Pannonica
Author:
Stefan Veldsman

A congruence is defined for a matroid. This leads to suitable versions of the algebraic isomorphism theorems for matroids. As an application of the congruence theory for matroids, a version of Birkhoff’s Theorem for matroids is given which shows that every nontrivial matroid is a subdirect product of subdirectly irreducible matroids.

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# The Metrics G = agS + bgH + cgV on the Tangent Bundle of a Weyl Manifold

Mathematica Pannonica
Author:
Murat Altunbaş

Let (M, [g]) be a Weyl manifold and TM be its tangent bundle equipped with Riemannian g−natural metrics which are linear combinations of Sasaki, horizontal and vertical lifts of the base metric with constant coefficients. The aim of this paper is to construct a Weyl structure on TM and to show that TM cannot be Einstein-Weyl even if (M, g) is fiat.

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# Arithmetical Functions Commutable with Sums of Squares II

Mathematica Pannonica
Authors:
Imre Kátai
and
Bui Minh Phong

We give all functions ƒ , E: ℕ → ℂ which satisfy the relation

for every a, b, c ∈ ℕ, where h ≥ 0 is an integers and K is a complex number. If n cannot be written as a2 + b2 + c2 + h for suitable a, b, c ∈ ℕ, then ƒ (n) is not determined. This is more complicated if we assume that ƒ and E are multiplicative functions.

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# Harvest Management Problem with a Fractional Logistic Equation

Mathematica Pannonica
Authors:
Melani Barrios
,
Gabriela Reyero
, and
Mabel Tidball

In this article, we study a fractional control problem that models the maximization of the profit obtained by exploiting a certain resource whose dynamics are governed by the fractional logistic equation. Due to the singularity of this problem, we develop different resolution techniques, both for the classical case and for the fractional case. We perform several numerical simulations to make a comparison between both cases.

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# On the Negativity of the Walsh–Kaczmarz–Riesz Logarithmic Kernels

Mathematica Pannonica
Authors:
György Gát
and
Gábor Lucskai

The main aim of this paper is to prove that the nonnegativity of the Riesz’s logarithmic kernels with respect to the Walsh– Kaczmarz system fails to hold.

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# Another Look at Threshold Phenomena for Random Cones

Studia Scientiarum Mathematicarum Hungarica
Authors:
Daniel Hug
and
Rolf Schneider

In stochastic geometry there are several instances of threshold phenomena in high dimensions: the behavior of a limit of some expectation changes abruptly when some parameter passes through a critical value. This note continues the investigation of the expected face numbers of polyhedral random cones, when the dimension of the ambient space increases to infinity. In the focus are the critical values of the observed threshold phenomena, as well as threshold phenomena for differences instead of quotients.

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# From Binary Groups to Terminal Rings

Mathematica Pannonica
Author:
Stuart D. Scott

Binary groups are a meaningful step up from non-associative rings and nearrings. It makes sense to study them in terms of their nearrings of zero-fixing polynomial maps. As this involves algebras of a more specialized nature these are looked into in sections three and four. One of the main theorems of this paper occurs in section five where it is shown that a binary group V is a P 0(V) ring module if, and only if, it is a rather restricted form of non-associative ring. Properties of these non-associative rings (called terminal rings) are investigated in sections six and seven. The finite case is of special interest since here terminal rings of odd order really are quite restricted. Sections eight to thirteen are taken up with the study of terminal rings of order p n (p an odd prime and n ≥ 1 an integer ≤ 7).

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# On Zero Determinant Matrices that are Full

Mathematica Pannonica
Authors:
Grigore Călugăreanu
and
Horia F. Pop

Column-row products have zero determinant over any commutative ring. In this paper we discuss the converse. For domains, we show that this yields a characterization of pre-Schreier rings, and for rings with zero divisors we show that reduced pre-Schreier rings have this property.

Finally, for the rings of integers modulo n, we determine the 2x2 matrices which are (or not) full and their numbers.

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# Error Bounds Related to Midpoint and Trapezoid Rules for the Monotonic Integral Transform of Positive Operators in Hilbert Spaces

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 followingmonotonic integral transform

where the integral is assumed to exist forT a positive operator on a complex Hilbert spaceH. We show among others that, if β ≥ A, B ≥ α > 0, and 0 < δ ≤ (B − A)2 ≤ Δ for some constants α, β, δ, Δ, then

$0 ≤ 1 24 δ M ″ ( w , μ ) ( β ) ≤ M ( w , μ ) A + B 2 − ∫ 0 1 M ( w , μ ) ( ( 1 − t ) A + t B ) d t ≤ − 1 24 Δ M ″ ( w , μ ) ( α )$

and

$0 ≤ − 1 12 δ M ″ ( w , μ ) ( β ) ≤ ∫ 0 1 M ( w , μ ) ( ( 1 − t ) A + t B ) d t − M ( w , μ ) ( A ) + M ( w , μ ) ( B ) 2 ≤ 1 12 Δ M ″ ( w , μ ) ( α ) ,$

where $M ″ ( w , μ )$ is the second derivative of $M ( w , μ )$ as a real function.

Applications for power function and logarithm are also provided.

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