Browse Our Mathematics and Statistics Journals

Mathematics and statistics journals publish papers on the theory and application of mathematics, statistics, and probability. Most mathematics journals have a broad scope that encompasses most mathematical fields. These commonly include logic and foundations, algebra and number theory, analysis (including differential equations, functional analysis and operator theory), geometry, topology, combinatorics, probability and statistics, numerical analysis and computation theory, mathematical physics, etc.

# Mathematics and Statistics

## You are looking at 1 - 50 of 11,204 items for

• Refine by Access: All Content
Clear All

## Lattice Cohomology of Partially Ordered Sets

Studia Scientiarum Mathematicarum Hungarica
Authors:
Tamás Ágoston
and
András Némethi

In this paper we introduce a construction for a weighted CW complex (and the associated lattice cohomology) corresponding to partially ordered sets with some additional structure. This is a generalization of the construction seen in [4] where we started from a system of subspaces of a given vector space. We then proceed to prove some basic properties of this construction that are in many ways analogous to those seen in the case of subspaces, but some aspects of the construction result in complexities not present in that scenario.

Open access

## Planar Turán Number of the Θ6

Studia Scientiarum Mathematicarum Hungarica
Authors:
Chuanqi Xiao
,
Debarun Ghosh
,
Ervin Győri
,
, and
Oscar Zamora

Let F be a nonempty family of graphs. A graph 𝐺 is called F -free if it contains no graph from F as a subgraph. For a positive integer 𝑛, the planar Turán number of F, denoted by exp (𝑛, F), is the maximum number of edges in an 𝑛-vertex F -free planar graph.

Let Θ𝑘 be the family of Theta graphs on 𝑘 ≥ 4 vertices, that is, graphs obtained by joining a pair of non-consecutive of a 𝑘-cycle with an edge. Lan, Shi and Song determined an upper bound exp (𝑛, Θ6) ≤ 18𝑛/7−36𝑛/7, but for large 𝑛, they did not verify that the bound is sharp. In this paper, we improve their bound by proving exp (𝑛, Θ6) ≤ 18𝑛/−48𝑛/7 and then we demonstrate the existence of infinitely many positive integer 𝑛 and an 𝑛-vertex Θ6-free planar graph attaining the bound.

Restricted access

## Parallel Covering an Obtuse Triangle with Squares

Studia Scientiarum Mathematicarum Hungarica
Authors:
Zhanjun Su
and
Yufang Wu

Suppose that 𝑇 (𝛼, 𝛽) is an obtuse triangle with base length 1 and with base angles 𝛼 and 𝛽 (where 𝛽 > 90). In this note a tight lower bound of the sum of the areas of squares that can parallel cover 𝑇 (𝛼, 𝛽) is given. This result complements the previous lower bound obtained for the triangles with the interior angles at the base of the measure not greater than 90.

Restricted access

## Positroids are 3-Colorable

Studia Scientiarum Mathematicarum Hungarica
Authors:
Lamar Chidiac
and
Winfried Hochstättler

We show that every positroid of rank 𝑟 ≥ 2 has a good coline. Using the definition of the chromatic number of oriented matroid introduced by J. Nešetřil, R. Nickel, and W. Hochstättler, this shows that every orientation of a positroid of rank at least 2 is 3-colorable.

Restricted access

## Spanning Trees Whose Stems are Caterpillars

Studia Scientiarum Mathematicarum Hungarica
Authors:
Pham Hoang Ha
,
Dang Dinh Hanh
,
Le Dinh Nam
, and
Nguyen Huu Nhan

Let 𝑇 be a tree, a vertex of degree one is called a leaf. The set of all leaves of 𝑇 is denoted by Leaf(𝑇). The subtree 𝑇 − Leaf(𝑇) of 𝑇 is called the stem of 𝑇 and denoted by Stem(𝑇). A tree 𝑇 is called a caterpillar if Stem(𝑇) is a path. In this paper, we give two sufficient conditions for a connected graph to have a spanning tree whose stem is a caterpillar. We also give some examples to show that these conditions are sharp.

Restricted access

## Two-Sided Convexity Testing with Certificates

Studia Scientiarum Mathematicarum Hungarica
Author:

We revisit the problem of property testing for convex position for point sets in ℝ𝑑. Our results draw from previous ideas of Czumaj, Sohler, and Ziegler (2000). First, their testing algorithm is redesigned and its analysis is revised for correctness. Second, its functionality is expanded by (i) exhibiting both negative and positive certificates along with the convexity determination, and (ii) significantly extending the input range for moderate and higher dimensions.

The behavior of the randomized tester on input set 𝑃 ⊂ ℝ𝑑 is as follows: (i) if 𝑃 is in convex position, it accepts; (ii) if 𝑃 is far from convex position, with probability at least 2/3, it rejects and outputs a (𝑑 +2)-point witness of non-convexity as a negative certificate; (iii) if 𝑃 is close to convex position, with probability at least 2/3, it accepts and outputs a subset in convex position that is a suitable approximation of the largest subset in convex position. The algorithm examines a sublinear number of points and runs in subquadratic time for every fixed dimension 𝑑.

Restricted access

## A Note on the Zeros of the Dedekind Zeta Functions

Mathematica Pannonica
Author:
János Pintz

We prove zero density theorems for Dedekind zeta functions in the vicinity of the line Re s = 1, improving an earlier result of W. Staś.

Open access

## Incidence Functions of the Exponential Divisor Poset

Mathematica Pannonica
Author:
Pentti Haukkanen

A positive integer $d = ∏ i = 1 r p i d i$ is said to be an exponential divisor or an e-divisor of $n = ∏ i = 1 r p i n i > 1$ if 𝑑𝑖 ∣ 𝑛𝑖 for all prime divisors 𝑝𝑖 of 𝑛. In addition, 1 is an e-divisor of 1. It is easy to see that ℤ+ is a poset under the e-divisibility relation. Utilizing this observation we show that e-convolution of arithmetical functions is an example of the convolution of incidence functions of posets. We also note that the identity, units and the Möbius function are preserved in this process.

Open access

## On Mixed 𝐵-Concatenations of Pell and Pell–Lucas Numbers which are Pell Numbers

Mathematica Pannonica
Authors:
and
Marija Bliznac Trebješanin

Let (𝑃𝑛)𝑛≥0 and (𝑄𝑛)𝑛≥0 be the Pell and Pell–Lucas sequences. Let 𝑏 be a positive integer such that 𝑏 ≥ 2. In this paper, we prove that the following two Diophantine equations 𝑃𝑛 = 𝑏𝑑𝑃𝑚 + 𝑄𝑘 and 𝑃𝑛 = 𝑏𝑑𝑄𝑚 + 𝑃𝑘 with 𝑑, the number of digits of 𝑃𝑘 or 𝑄𝑘 in base 𝑏, have only finitely many solutions in nonnegative integers (𝑚, 𝑛, 𝑘, 𝑏, 𝑑). Also, we explicitly determine these solutions in cases 2 ≤ 𝑏 ≤ 10.

Open access

## Answer to a 1971 Question of Grätzer and Lakser on Pseudocomplemented Lattices

Mathematica Pannonica
Author:
Jonathan David Farley

Grätzer and Lakser asked in the 1971 Transactions of the American Mathematical Society if the pseudocomplemented distributive lattices in the amalgamation class of the subvariety generated by 𝟐𝑛 ⊕ 𝟏 can be characterized by the property of not having a *-homomorphism onto 𝟐𝑖 ⊕ 𝟏 for 1 < 𝑖 < 𝑛.

Open access

## Notice of Self-Retraction and Replacement: A 1971 Question of Grätzer and Lakser on Pseudocomplemented Lattices

Mathematica Pannonica
Authors:
Jonathan David Farley
and
Dominic van der Zypen
Open access

## *-Rickart Property for Rings with Involution

Mathematica Pannonica
Authors:
,
Usama A. Aburawash
, and

This paper introduces and examines the concept of a *-Rickart *-ring, and proves that every Rickart *-ring is also a *-Rickart *-ring. A necessary and sufficient condition for a *-Rickart *-ring to be a Rickart *-ring is also provided. The relationship between *-Rickart *-rings and *-Baer *-rings is investigated, and several properties of *-Rickart *-rings are presented. The paper demonstrates that the property of *-Rickart extends to both the center and *-corners of a *-ring, and investigates the extension of a *-Rickart *-ring to its polynomial *-ring. Additionally, *-Rickart *-rings with descending chain condition on *-biideals are studied, and all *-Rickart (*-Baer) *-rings with finitely many elements are classified.

Open access

## Simultaneous Approximation for an Exponential Operator Connected with x4/3

Mathematica Pannonica
Authors:
Anjali
and
Vijay Gupta

Very recently, the authors in [5] proposed the exponential-type operator connected with $x 4 3$ and studied its convergence estimates. In the present research, we extend the study and obtain the general form of its 𝑝-th order moment; 𝑝 ∈ ℕ ∪ {0}. Further, we establish the simultaneous approximation for the operator under consideration.

Open access

## A Combinatorial Construction of Bi-Cyclic 4-Polytopes

Studia Scientiarum Mathematicarum Hungarica
Author:
Tibor Bisztriczky

A bi-cyclic 4-polytope in ℝ4 was introduced by Z. Smilansky as the convex hull of evenly spaced points on a generalized trigonometric moment curve in ℝ4. We present combinatorial geometric conditions that yield the face lattices of a class of such 4-polytopes.

Restricted access

## On a Dowker-Type Problem for Convex Disks with Almost Constant Curvature

Studia Scientiarum Mathematicarum Hungarica
Authors:
Bushra Basit
and
Zsolt Lángi

A classical result of Dowker (Bull. Amer. Math. Soc. 50: 120-122, 1944) states that for any plane convex body 𝐾, the areas of the maximum (resp. minimum) area convex 𝑛-gons inscribed (resp. circumscribed) in 𝐾 is a concave (resp. convex) sequence. It is known that this theorem remains true if we replace area by perimeter, or convex 𝑛-gons by disk-𝑛-gons, obtained as the intersection of 𝑛 closed Euclidean unit disks. It has been proved recently that if 𝐶 is the unit disk of a normed plane, then the same properties hold for the area of 𝐶-𝑛-gons circumscribed about a 𝐶-convex disk 𝐾 and for the perimeters of 𝐶-𝑛-gons inscribed or circumscribed about a 𝐶-convex disk 𝐾, but for a typical origin-symmetric convex disk 𝐶 with respect to Hausdorff distance, there is a 𝐶-convex disk 𝐾 such that the sequence of the areas of the maximum area 𝐶-𝑛-gons inscribed in 𝐾 is not concave. The aim of this paper is to investigate this question if we replace the topology induced by Hausdorff distance with a topology induced by the surface area measure of the boundary of 𝐶.

Open access

## Results on Extremal Graph Theoretic Questions for Q-Ary Vectors

Studia Scientiarum Mathematicarum Hungarica
Authors:
Koppány Encz
,
Márton Marits
,
Benedek Váli
, and
Máté Weisz

A 𝑞-graph with 𝑒 edges and 𝑛 vertices is defined as an 𝑒 × 𝑛 matrix with entries from {0, … , 𝑞}, such that each row of the matrix (called a 𝑞-edge) contains exactly two nonzero entries. If 𝐻 is a 𝑞-graph, then 𝐻 is said to contain an 𝑠-copy of the ordinary graph 𝐹, if a set 𝑆 of 𝑞-edges can be selected from 𝐻 such that their intersection graph is isomorphic to 𝐹, and for any vertex 𝑣 of 𝑆 and any two incident edges 𝑒, 𝑓 ∈ 𝑆 the sum of the entries of 𝑒 and 𝑓 is at least 𝑠. The extremal number ex(𝑛, 𝐹, 𝑞, 𝑠) is defined as the maximal number of edges in an 𝑛-vertex 𝑞-graph such that it does not contain contain an 𝑠-copy of the forbidden graph 𝐹.

In the present paper, we reduce the problem of finding ex(𝑛, 𝐹, 𝑞, 𝑞 + 1) for even 𝑞 to the case 𝑞 = 2, and determine the asymptotics of ex(𝑛, 𝐶2𝑘+1, 𝑞, 𝑞 + 1).

Restricted access

## Set-Coloring Ramsey Numbers via Codes

Studia Scientiarum Mathematicarum Hungarica
Authors:
David Conlon
,
Jacob Fox
,
Xiaoyu He
,
Dhruv Mubayi
,
Andrew Suk
, and
Jacques Verstraëte

For positive integers 𝑛, 𝑟, 𝑠 with 𝑟 > 𝑠, the set-coloring Ramsey number 𝑅(𝑛; 𝑟, 𝑠) is the minimum 𝑁 such that if every edge of the complete graph 𝐾𝑁 receives a set of 𝑠 colors from a palette of 𝑟 colors, then there is guaranteed to be a monochromatic clique on 𝑛 vertices, that is, a subset of 𝑛 vertices where all of the edges between them receive a common color. In particular, the case 𝑠 = 1 corresponds to the classical multicolor Ramsey number. We prove general upper and lower bounds on 𝑅(𝑛; 𝑟, 𝑠) which imply that 𝑅(𝑛; 𝑟, 𝑠) = 2Θ(𝑛𝑟) if 𝑠/𝑟 is bounded away from 0 and 1. The upper bound extends an old result of Erdős and Szemerédi, who treated the case 𝑠 = 𝑟 − 1, while the lower bound exploits a connection to error-correcting codes. We also study the analogous problem for hypergraphs.

Restricted access

## Approximations of Singular Surfaces with Standard Cuspidal Edges

Studia Scientiarum Mathematicarum Hungarica
Authors:
Kentaro Saji
and
Yoshiki Yamamoto

We consider a function from the Euclidean three space whose zero set is the image of the standard cuspidal edge. The composition of a parametrized singular surface in the three space with this function provides an approximation of the surface by the standard cuspidal edge. Taking a look at singularities of this composition, we study various approximations of singular surfaces like the cross cap, the generalized cuspidal edge and the swallowtail by standard cuspidal edges.

Restricted access

## Maximizing the Area of Polygons via Quasicyclic Polygons

Studia Scientiarum Mathematicarum Hungarica
Authors:
Giuseppina Anatriello
and
Giovanni Vincenzi

Based on Peter’s work from 2003, quadrilaterals can be characterized in the following way: “among all quadrilaterals with given side lengths 𝑎, 𝑏, 𝑐 and 𝑑, those of the largest possible area are exactly the cyclic ones”. In this paper, we will give the corresponding characterization for every polygon, by means of quasicyclic polygons properties.

Restricted access

## The Bloch Spaces with Differentiable Strictly Positive Weights

Mathematica Pannonica
Authors:
Ding Nan
and
Hasi Wulan

In this paper, some basic characterizations of a weighted Bloch space with the differentiable strictly positive weight 𝜔 on the unit disc are given, including the growth, the higher order or free derivative descriptions, and integral characterizations of functions in the space.

Open access

## Beurling-Integers with Lacunarity

Mathematica Pannonica
Author:
Imre Z. Ruzsa

We present examples of multiplicative semigroups of positive reals (Beurling’s generalized integers) with gaps bounded from below.

Open access

## Construction of New Operators by Composition of Integral-Type Operators and Discrete Operators

Mathematica Pannonica
Authors:
Ulrich Abel
and
Vijay Gupta

In this paper, we propose some new positive linear approximation operators, which are obtained from a composition of certain integral type operators with certain discrete operators. It turns out that the new operators can be expressed in discrete form. We provide representations for their coefficients. Furthermore, we study their approximation properties and determine their moment generating functions, which may be useful in finding several other convergence results in different settings.

Open access

## On the Fourier Coefficients for General Product 𝐿-Functions

Mathematica Pannonica
Author:
Guodong Hua

Let 𝑓 be a normalized primitive cusp form of even integral weight for the full modular group Γ = 𝑆𝐿(2, ℤ). In this paper, we investigate upper bounds for the error terms related to the average behavior of Fourier coefficients 𝜆𝑓 ⊗𝑓 ⊗⋯⊗𝑙𝑓 (𝑛) of 𝑙-fold product 𝐿-functions, where 𝑓 ⊗ 𝑓 ⊗ ⋯ ⊗𝑙 𝑓 denotes the 𝑙-fold product of 𝑓. These results improves and generalizes the recent developments of Venkatasubbareddy and Sankaranarayanan [41]. We also provide some other similar results related to the error terms of general product 𝐿-functions.

Open access

## The Extreme Polygons for the Self Chebyshev Radius of the Boundary

Studia Scientiarum Mathematicarum Hungarica
Authors:
Evgeniĭ Vitalievich Nikitenko
and

The paper is devoted to some extremal problems for convex polygons on the Euclidean plane, related to the concept of self Chebyshev radius for the polygon boundary. We consider a general problem of minimization of the perimeter among all 𝑛-gons with a fixed self Chebyshev radius of the boundary. The main result of the paper is the complete solution of the mentioned problem for 𝑛 = 4: We proved that the quadrilateral of minimum perimeter is a so called magic kite, that verified the corresponding conjecture by Rolf Walter.

Restricted access

## Some Bounds for the Regularity of the Edge Ideals and Their Powers in a Certain Class of Graphs

Studia Scientiarum Mathematicarum Hungarica
Authors:
Tom Ashitha
,
Thangaraj Asir
,
Do Trong Hoang
, and

Let 𝑛 ≥ 2 be an integer. The graph $G n ¯$ is obtained by letting all the elements of {0, … , 𝑛 − 1} to be the vertices and defining distinct vertices 𝑥 and 𝑦 to be adjacent if and only if gcd(𝑥 + 𝑦, 𝑛) ≠ 1. In this paper, we give some bounds for the Castelnuovo–Mumford regularity of the edge ideals and their powers for $G n ¯$ .

Restricted access

## Unavoidable Crossings in Finite Coverings

Studia Scientiarum Mathematicarum Hungarica
Authors:
András Bezdek
and
Włodzimierz Kuperberg

Motivated by the examples of Heppes and Wegner, we present several other examples of the following kind: a bounded convex region 𝐷 and a convex disk 𝐾 in the plane are described, such that every thinnest covering of 𝐷 with congruent copies of 𝐾 contains crossing pairs.

Restricted access

## The Spherical Cap Discrepancy of HEALPix Points

Studia Scientiarum Mathematicarum Hungarica
Authors:
Damir Ferizović
,
, and
Michelle Mastrianni

In this paper we show that the spherical cap discrepancy of the point set given by centers of pixels in the HEALPix tessellation (short for Hierarchical, Equal Area and iso-Latitude Pixelation) of the unit 2-sphere is lower and upper bounded by order square root of the number of points, and compute explicit constants. This adds to the currently known (short) collection of explicitly constructed sets whose discrepancy converges with order 𝑁 −1/2, matching the asymptotic order for i.i.d. random point sets. We describe the HEALPix framework in more detail and give explicit formulas for the boundaries and pixel centers. We then introduce the notion of an 𝑛-convex curve and prove an upper bound on how many fundamental domains are intersected by such curves, and in particular we show that boundaries of spherical caps have this property. Lastly, we mention briefly that a jittered sampling technique works in the HEALPix framework as well.

Restricted access

## A Density Theorem for Dedekind Zeta Functions

Mathematica Pannonica
Author:
János Pintz

We apply a recent general zero density theorem of us (valid for a large class of complex functions) to improve earlier density theorems of Heath-Brown and Paul–Sankaranarayanan for Dedekind zeta functions attached to a number field 𝐾 of degree 𝑛 with 𝑛 > 2.

Open access

## Kahane’s Upper Density and Syndetic Sets in LCA Groups

Mathematica Pannonica
Author:
Szilárd Gy. Révész

Asymptotic uniform upper density, shortened as a.u.u.d., or simply upper density, is a classical notion which was first introduced by Kahane for sequences in the real line.

Syndetic sets were defined by Gottschalk and Hendlund. For a locally compact group 𝐺, a set 𝑆 ⊂ 𝐺 is syndetic, if there exists a compact subset 𝐶 ⋐ 𝐺 such that 𝑆𝐶 = 𝐺. Syndetic sets play an important role in various fields of applications of topological groups and semigroups, ergodic theory and number theory. A lemma in the book of Fürstenberg says that once a subset 𝐴 ⊂ ℤ has positive a.u.u.d., then its difference set 𝐴 − 𝐴 is syndetic.

The construction of a reasonable notion of a.u.u.d. in general locally compact Abelian groups (LCA groups for short) was not known for long, but in the late 2000’s several constructions were worked out to generalize it from the base cases of ℤ𝑑 and ℝ𝑑. With the notion available, several classical results of the Euclidean setting became accessible even in general LCA groups.

Here we work out various versions in a general locally compact Abelian group 𝐺 of the classical statement that if a set 𝑆 ⊂ 𝐺 has positive asymptotic uniform upper density, then the difference set 𝑆 − 𝑆 is syndetic.

Open access

## Remark on a General Zero Density Theorem

Mathematica Pannonica
Author:
János Pintz

Recently [3] we proved a general zero density theorem for a large class of functions which included among others the Riemann zeta function, Dedekind zeta functions, Dirichlet 𝐿-functions. The goal of the present work is a (slight) improvement of this general theorem which might lead to stronger results in some applications.

Open access

## How to Approach Stability of Bi-Continuous Semigroups?

Mathematica Pannonica
Author:
Christian Budde

This paper serves as a kick-off to address the question of how to define and investigate the stability of bi-continuous semigroups. We will see that the mixed topology is the key concept in this framework.

Open access

## A Classification of 2-Dimensional Endo-Commutative Straight Algebras of Rank 1 over a non-Trivial Field

Mathematica Pannonica
Authors:
Sin-Ei Takahasi
,
Kiyoshi Shirayanagi
, and

An endo-commutative algebra is a nonassociative algebra in which the square mapping preserves multiplication. In this paper, we give a complete classification of 2-dimensional endo-commutative straight algebras of rank one over an arbitrary non-trivial field, where a straight algebra of dimension 2 satisfies the condition that there exists an element x such that x and x 2 are linearly independent. We list all multiplication tables of the algebras up to isomorphism.

Open access

## On the Simultaneous Sign Changes of Coefficients of Rankin–Selberg L-Functions over a Certain Integral Binary Quadratic Form

Mathematica Pannonica
Author:
Guodong Hua

In this paper, we consider the simultaneous sign changes of coefficients of Rankin–Selberg L-functions associated to two distinct Hecke eigenforms supported at positive integers represented by some certain primitive reduced integral binary quadratic form with negative discriminant D. We provide a quantitative result for the number of sign changes of such sequence in the interval (x, 2x] for sufficiently large x.

Open access

## Divisor Problem for the Greatest Common Divisor of Integers in Piatetski-Shapiro and Beatty Sequences

Mathematica Pannonica
Authors:
Sunanta Srisopha
,
Teerapat Srichan
, and
Pinthira Tangsupphathawat

In this paper, we derive several asymptotic formulas for the sum of d(gcd(m,n)), where d(n) is the divisor function and m,n are in Piatetski-Shapiro and Beatty sequences.

Open access

## The Norming Sets of $L$ (2 d *(1, w)2)

Mathematica Pannonica
Author:
Sung Guen Kim

Let 𝑛 ∈ ℕ. An element (x 1, … , x 𝑛) ∈ En is called a norming point of T $L$ ( nE) if ‖x 1‖ = ⋯ = ‖xn ‖ = 1 and |T (x 1, … , xn )| = ‖T‖, where $L$ ( nE) denotes the space of all continuous n-linear forms on E. For T $L$ ( nE), we define

Norm(T) = {(x 1, … , x n) ∈ En ∶ (x 1, … , x n) is a norming point of T}.

Norm(T) is called the norming set of T. We classify Norm(T) for every T $L$ (2 𝑑 (1, w)2), where 𝑑 (1, w)2 = ℝ2 with the octagonal norm of weight 0 < w < 1 endowed with $x , y d * 1 , w = max x , y , x + y 1 + w$ .

Open access

## On k-Strictly Quasi-Fredholm Linear Relations

Mathematica Pannonica
Authors:
Hafsa Bouaniza
,
Imen Issaoui
, and
Maher Mnif

In this paper, we introduce and study the class of k-strictly quasi-Fredholm linear relations on Banach spaces for nonnegative integer k. Then we investigate its robustness through perturbation by finite rank operators.

Open access

## A Continuum Dimensional Algebra of Nowhere Differentiable Functions

Mathematica Pannonica
Author:
Jan-Christoph Schlage-Puchta

We construct an algebra of dimension 2ℵ0 consisting only of functions which in no point possess a finite one-sided derivative. We further show that some well known nowhere differentiable functions generate algebras, which contain functions which are differentiable at some points, but where for all functions in the algebra the set of points of differentiability is quite small.

Open access

## Convexity in (Colored) Affine Semigroups

Studia Scientiarum Mathematicarum Hungarica
Authors:
Jesús A. De Loera
,
Christopher O’Neill
, and
Chengyang Wang

In this paper, we explore affine semigroup versions of the convex geometry theorems of Helly, Tverberg, and Carathéodory. Additionally, we develop a new theory of colored affine semigroups, where the semigroup generators each receive a color and the elements of the semigroup take into account the colors used (the classical theory of affine semigroups coincides with the case in which all generators have the same color). We prove an analog of Tverberg’s theorem and colorful Helly’s theorem for semigroups, as well as a version of colorful Carathéodory’s theorem for cones. We also demonstrate that colored numerical semigroups are particularly rich by introducing a colored version of the Frobenius number.

Restricted access

## The Endomorphism Conjecture for Graded Posets with Whitney Numbers at most 4

Studia Scientiarum Mathematicarum Hungarica
Authors:
Miklós Bóna
and
Ryan R. Martin

We prove the endomorphism conjecture for graded posets with largest Whitney number at most 4.

Restricted access

## Equalities for the 𝑟3-Crank of 3-Regular Overpartitions

Studia Scientiarum Mathematicarum Hungarica
Authors:
Robert X. J. Hao
and
Erin Y. Y. Shen

Lovejoy introduced the partition function $A l ¯ n$ as the number of 𝑙-regular overpartitions of 𝑛. Andrews defined (𝑘, 𝑖)-singular overpartitions counted by the partition function $C ¯ k , i n$ , and pointed out that $C ¯ 3 , 1 n = A 3 ¯ n$ . Meanwhile, Andrews derived an interesting divisibility property that $C ¯ 3 , 1 9 n + 3 ≡ C ¯ 3 , 1 9 n + 6 ≡ 0$ (mod 3). Recently, we constructed the partition statistic 𝑟𝑙-crank of 𝑙-regular overpartitions and give combinatorial interpretations for some congruences of $A l ¯ n$ as well as the congruences of Andrews. In this paper, we aim to prove some equalities for the 𝑟3-crank of 3-regular overpartitions.

Restricted access

## Hilbert Metric in the Unit Ball

Studia Scientiarum Mathematicarum Hungarica
Authors:
Oona Rainio
and
Matti Vuorinen

The Hilbert metric between two points 𝑥, 𝑦 in a bounded convex domain 𝐺 is defined as the logarithm of the cross-ratio 𝑥, 𝑦 and the intersection points of the Euclidean line passing through the points 𝑥, 𝑦 and the boundary of the domain. Here, we study this metric in the case of the unit ball 𝔹𝑛. We present an identity between the Hilbert metric and the hyperbolic metric, give several inequalities for the Hilbert metric, and results related to the inclusion properties of the balls defined in the Hilbert metric. Furthermore, we study the distortion of the Hilbert metric under conformal and quasiregular mappings.

Restricted access

## Integral Closure of Powers of Generalized Edge Ideals

Studia Scientiarum Mathematicarum Hungarica
Author:
Sirajul Haque

This article studies a new class of monomial ideals associated with a simple graph 𝐺, called generalized edge ideal, denoted by 𝐼𝑔(𝐺). Assuming that all the vertices 𝑥 have an exponent greater than 1 in 𝐼𝑔(𝐺), we completely characterize the graph 𝐺 for which 𝐼𝑔(𝐺) is integrally closed, and show that this is equivalent to 𝐼𝑔(𝐺) being normal i.e., all integral powers of 𝐼𝑔(𝐺) are integrally clased. We also give a necessary and sufficient condition for $I g G = I g G ¯$ when 𝐺 is the star-shaped graph. Finally, we give a necessary and sufficient condition when the generalized edge ideal of a complete graph is integrally closed.

Restricted access

## “Less” Strong Chromatic Indices and the (7, 4)-Conjecture

Studia Scientiarum Mathematicarum Hungarica
Authors:
András Gyárfás
and
Gábor N. Sárközy

A proper edge coloring of a graph 𝐺 is strong if the union of any two color classes does not contain a path with three edges (i.e. the color classes are induced matchings). The strong chromatic index 𝑞(𝐺) is the smallest number of colors needed for a strong coloring of 𝐺. One form of the famous (6, 3)-theorem of Ruzsa and Szemerédi (solving the (6, 3)-conjecture of Brown–Erdős–Sós) states that 𝑞(𝐺) cannot be linear in 𝑛 for a graph 𝐺 with 𝑛 vertices and 𝑐𝑛2 edges. Here we study two refinements of 𝑞(𝐺) arising from the analogous (7, 4)-conjecture. The first is 𝑞𝐴(𝐺), the smallest number of colors needed for a proper edge coloring of 𝐺 such that the union of any two color classes does not contain a path or cycle with four edges, we call it an A-coloring. The second is 𝑞𝐵(𝐺), the smallest number of colors needed for a proper edge coloring of 𝐺 such that all four-cycles are colored with four different colors, we call it a B-coloring. These notions lead to two stronger and one equivalent form of the (7, 4)-conjecture in terms of 𝑞𝐴(𝐺), 𝑞𝐵(𝐺) where 𝐺 is a balanced bipartite graph. Since these are questions about graphs, perhaps they will be easier to handle than the original special(7, 4)-conjecture. In order to understand the behavior of 𝑞𝐴(𝐺) and 𝑞𝐵(𝐺), we study these parameters for some graphs.

We note that 𝑞𝐴(𝐺) has already been extensively studied from various motivations. However, as far as we know the behavior of 𝑞𝐵(𝐺) is studied here for the first time.

Open access

## Refined Ehrhart Series and Bigraded Rings

Studia Scientiarum Mathematicarum Hungarica
Authors:
and
Balázs Szendrői

We study a natural set of refinements of the Ehrhart series of a closed polytope, first considered by Chapoton. We compute the refined series in full generality for a simplex of dimension 𝑑, a cross-polytope of dimension 𝑑, respectively a hypercube of dimension 𝑑 ≤ 3, using commutative algebra. We deduce summation formulae for products of 𝑞-integers with different arguments, generalizing a classical identity due to MacMahon and Carlitz. We also present a characterisation of a certain refined Eulerian polynomial in algebraic terms.

Restricted access

## Similarity for 2 × 2 Matrices Obtained by Clockwise “Rotation”

Mathematica Pannonica
Author:
Grigore Călugăreanu

Over integral domains of characteristics different from 2, we determine all the matrices $a b c d$ which are similar to $c a d b$ .

Open access

## A Note on the Pinelis Extension of Stolarsky’s Inequality

Mathematica Pannonica
Author:
Sanja Varošanec

We present generalizations of the Pinelis extension of Stolarsky’s inequality and its reverse. In particular, a new Stolarsky-type inequality is obtained. We study the properties of the linear functional related to the new Stolarsky-type inequality, and finally apply these new results in the theory of fractional integrals.

Open access

## On the Feuerbach Point and Feuerbach Line in the Isotropic Plane

Mathematica Pannonica
Authors:
Ružica Kolar-Šuper
and

In this paper, we consider the Feuerbach point and the Feuerbach line of a triangle in the isotropic plane, and investigate some properties of these concepts and their relationships with other elements of a triangle in the isotropic plane. We also compare these relationships in Euclidean and isotropic cases.

Open access

## Confluent Kampé de Fériet Series Arising in the Solutions of Cauchy Problem for the Degenerate Hyperbolic Equation of the Second Kind with the Spectral Parameter

Mathematica Pannonica
Authors:
,
Tuhtasin G. Ergashev
,
Dildora A. Ergasheva
, and
Anvarjon Hasanov

We define the order of the double hypergeometric series, investigate the properties of the new confluent Kampé de Fériet series, and build systems of partial differential equations that satisfy the new Kampé de Fériet series. We solve the Cauchy problem for a degenerate hyperbolic equation of the second kind with a spectral parameter using the high-order Kampé de Fériet series. Thanks to the properties of the introduced Kampé de Fériet series, it is possible to obtain a solution to the problem in explicit forms.

Open access

## On Blaschke–Santaló-Type Inequalities for r-Ball Bodies

Mathematica Pannonica
Author:
Károly Bezdek

Let 𝔼 𝑑 denote the 𝑑-dimensional Euclidean space. The 𝑟-ball body generated by a given set in 𝔼 𝑑 is the intersection of balls of radius 𝑟 centered at the points of the given set. The author [Discrete Optimization 44/1 (2022), Paper No. 100539] proved the following Blaschke–Santaló-type inequalities for 𝑟-ball bodies: for all 0 < 𝑘 < 𝑑 and for any set of given 𝑑-dimensional volume in 𝔼 𝑑 the 𝑘-th intrinsic volume of the 𝑟-ball body generated by the set becomes maximal if the set is a ball. In this note we give a new proof showing also the uniqueness of the maximizer. Some applications and related questions are mentioned as well.

Open access

## On Graphical Shapes of χ 2 Statistics of Leading Digits of Irrational Rotations

Mathematica Pannonica
Authors:
Yoshiyuki Mori
and
Keizo Takashima

We discuss the outline of the shapes of graphs of χ 2 statistics for distributions of leading digits of irrational rotations under some conditions on mth convergent. We give some estimates of important coefficients Lk ’s, which determine the graphical shapes of χ2 statistics. This means that the denominator qm of mth convergent and the large partial quotient am +1 determine the outline of shapes of graphs, when we observe values of χ 2 statistics with step qm .

Open access