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Set valued pexiderized quadratic functional equation Aequat. Math. (IF 0.8) Pub Date : 2024-04-21 Elham Mohammadi, Abbas Najati, Kazimierz Nikodem
Consider a real vector space denoted as X, and let cc(Y) represent the collection of all convex and compact subsets of a real Hausdorff topological vector space Y. This paper investigates set-valued solutions of the Pexiderized quadratic functional equation $$\begin{aligned} f_1(x+y)+f_2(x-y)=f_3(x)+f_4(y), \end{aligned}$$ for unknown functions \(f_1,f_2,f_3,f_4:X\rightarrow cc(Y)\). This functional
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Generalizing the concept of bounded variation Aequat. Math. (IF 0.8) Pub Date : 2024-04-20 Angshuman R. Goswami
Let \([a,b]\subseteq \mathbb {R}\) be a non-empty and non singleton closed interval and \(P=\{a=x_0<\cdots 1\), under minimal assumptions such a function can be treated as an approximately monotone function which can be closely approximated by a nondecreasing majorant. We also prove that for \(0
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About Sobolev spaces on fractals: fractal gradians and Laplacians Aequat. Math. (IF 0.8) Pub Date : 2024-04-16 Alireza Khalili Golmankhaneh, Palle E. T. Jørgensen, Cristina Serpa, Kerri Welch
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Curves that allow the motion of a regular polygon Aequat. Math. (IF 0.8) Pub Date : 2024-04-15 David Rochera
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A characterization of the Euclidean ball via antipodal points Aequat. Math. (IF 0.8) Pub Date : 2024-04-11 Xuguang Lu
Arising from an equilibrium state of a Fermi–Dirac particle system at the lowest temperature, a new characterization of the Euclidean ball is proved: a compact set \(K\subset {{{\mathbb {R}}}^n}\) (having at least two elements) is an n-dimensional Euclidean ball if and only if for every pair \(x, y\in \partial K\) and every \(\sigma \in {{{\mathbb {S}}}^{n-1}}\), either \(\frac{1}{2}(x+y)+\frac{1}{2}|x-y|\sigma
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The cosine addition and subtraction formulas on non-abelian groups Aequat. Math. (IF 0.8) Pub Date : 2024-04-10 Omar Ajebbar, Elhoucien Elqorachi, Henrik Stetkær
Let G be a topological group, and let C(G) denote the algebra of continuous, complex valued functions on G. We determine the solutions \(f,g,h \in C(G)\) of the Levi-Civita equation $$\begin{aligned} g(xy) = g(x)g(y) + f(x)h(y), \ x,y \in G, \end{aligned}$$ that extends the cosine addition law. As a corollary we obtain the solutions \(f,g \in C(G)\) of the cosine subtraction law \(g(xy^*) = g(x)g(y)
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On some classes of multiplicative functions Aequat. Math. (IF 0.8) Pub Date : 2024-04-09 Pentti Haukkanen
An arithmetical function f is multiplicative if \(f(1)=1\) and \(f(mn)=f(m)f(n)\) whenever m and n are coprime. We study connections between certain subclasses of multiplicative functions, such as strongly multiplicative functions, over-multiplicative functions and totients. It appears, among others, that the over-multiplicative functions are exactly same as the totients and the strongly multiplicative
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A note on the Radiant formula and its relations to the sliced Wasserstein distance Aequat. Math. (IF 0.8) Pub Date : 2024-04-06
Abstract In this note, we show that the squared Wasserstein distance can be expressed as the average over the sphere of one dimensional Wasserstein distances. We name this new expression for the Wasserstein Distance Radiant Formula. Using this formula, we are able to highlight new connections between the Wasserstein distances and the Sliced Wasserstein distance, an alternative Wasserstein-like distance
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Global centers of a family of cubic systems Aequat. Math. (IF 0.8) Pub Date : 2024-04-05 Raul Felipe Appis, Jaume Llibre
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Means of Cauchy’s difference type Aequat. Math. (IF 0.8) Pub Date : 2024-03-30 Janusz Matkowski
k-variable means which are the Cauchy differences of additive type generated by a real single variable function f, and denoted by \(C_{f,k}\), are examined. It is shown that \(C_{f,k}\) is an increasing mean in \(\left( 0,\infty \right) \) iff f is a convex solution of the (reflexivity) functional equation \(f\left( kx\right) -kf\left( x\right) =x\), and a construction of a large class of such means
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On iterative roots of injective functions Aequat. Math. (IF 0.8) Pub Date : 2024-03-26
Abstract In 1951 Łojasiewicz found a necessary and sufficient condition for the existence of a q-iterative root of an arbitrary bijective function g for any \(q\ge 2\) . In this article we extend this result to the injective case. More precisely, a necessary and sufficient condition for the existence of an iterative root of an injective function is proved, and in the case of existence, the characterization
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The minmin coalition number in graphs Aequat. Math. (IF 0.8) Pub Date : 2024-03-25 Davood Bakhshesh, Michael A. Henning
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Alienation and stability of Jensen’s and other functional equations Aequat. Math. (IF 0.8) Pub Date : 2024-03-21 Mohamed Tial, Driss Zeglami
Let S be a semigroup and \(\mathbb {K}\) be the field of real or complex numbers. We deal with the stability and alienation of Cauchy’s multiplicative (resp. additive) and Jensen’s functional equations, starting from the inequalities $$\begin{aligned} \left| f(xy)+f(x\sigma y)+g(xy)-2f(x)-g(x)g(y)\right|\le & {} \varepsilon ,\ \;x,y\in S, \\ \left| f(xy)+f(x\sigma y)+g(xy)-2f(x)-g(x)-g(y)\right|\le
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A functional equation related to Wigner’s theorem Aequat. Math. (IF 0.8) Pub Date : 2024-03-07
Abstract An open problem posed by G. Maksa and Z. Páles is to find the general solution of the functional equation $$\begin{aligned} \{\Vert f(x)-\beta f(y)\Vert : \beta \in {\mathbb {T}}_n\}=\{\Vert x-\beta y\Vert : \beta \in {\mathbb {T}}_n\} \quad (x,y\in H) \end{aligned}$$ where \(f: H \rightarrow K\) is between two complex normed spaces and \({\mathbb {T}}_n:=\{e^{i\frac{2k\pi }{n}}: k=1, \cdots
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Symmetries in Dyck paths with air pockets Aequat. Math. (IF 0.8) Pub Date : 2024-03-06 Jean-Luc Baril, Rigoberto Flórez, José L. Ramírez
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Odd strength spherical designs attaining the Fazekas–Levenshtein bound for covering and universal minima of potentials Aequat. Math. (IF 0.8) Pub Date : 2024-03-06 Sergiy Borodachov
We characterize the cases of existence of spherical designs of an odd strength attaining the Fazekas–Levenshtein bound for covering and prove some of their properties. We also find all universal minima of the potential of regular spherical configurations in two new cases: the demihypercube on \(S^d\), \(d\ge 4\), and the \(2_{41}\) polytope on \(S^7\) (which is dual to the \(E_8\) lattice).
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Bisector fields and pencils of conics Aequat. Math. (IF 0.8) Pub Date : 2024-03-06
Abstract We introduce the notion of a bisector field, which is a maximal collection of pairs of lines such that for each line in each pair, the midpoint of the points where the line crosses every pair is the same, regardless of choice of pair. We use this to study asymptotic properties of pencils of affine conics over fields and show that pairs of lines in the plane that occur as the asymptotes of
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Quaternion-valued multiplicative functions on semigroups Aequat. Math. (IF 0.8) Pub Date : 2024-02-27 Ayoub Ouhabi, Driss Zeglami, Mohamed Ayoubi
Our aim is to solve a system of functional equations closely related to trigonometric functional equations. This allows us to express quaternion-valued multiplicative functions in terms of complex-valued multiplicative functions. As an application of our results, we give the continuous quaternion-valued solutions of a functional equation on \(({\mathbb {R}},\cdot )\).
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Homi-repair under iteration (II): oscillating discontinuities and pre-discontinuities Aequat. Math. (IF 0.8) Pub Date : 2024-02-25
Abstract It is shown that removable and jumping discontinuities for functions having more than one but finitely many discontinuities have a second order \(C^0\) homi-repair. In this paper we study second order \(C^0\) homi-repair of oscillating discontinuities and pre-discontinuities for those functions and give necessary and sufficient conditions for \(C^0\) repair by second order iteration.
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A note on convex solutions to an equation on open intervals Aequat. Math. (IF 0.8) Pub Date : 2024-02-24 Chaitanya Gopalakrishna
The note is concerned with the functional equation $$\begin{aligned} \lambda _1H_1(f(x))+\lambda _2H_2(f^2(x))+\cdots +\lambda _nH_n(f^n(x))=F(x), \end{aligned}$$ which is a generalised form of the so-called polynomial-like iterative equation. We investigate the existence of nondecreasing convex (both usual and higher order) solutions to this equation on open intervals using the Schauder fixed point
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Zonal labelings and Tait colorings from a new perspective Aequat. Math. (IF 0.8) Pub Date : 2024-02-21 Andrew Bowling, Weiguo Xie
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Homi-repair under iteration (I): removable and jumping cases Aequat. Math. (IF 0.8) Pub Date : 2024-02-20 Xiaohua Liu, Liu Liu, Weinian Zhang
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Disprove of a conjecture on the double Roman domination number Aequat. Math. (IF 0.8) Pub Date : 2024-02-16
Abstract A double Roman dominating function (DRDF) on a graph \(G=(V,E)\) is a function \(f:V\rightarrow \{0,1,2,3\}\) having the property that if \(f(v)=0\) , then vertex v must have at least two neighbors assigned 2 under f or one neighbor w with \(f(w)=3\) , and if \(f(v)=1\) , then vertex v must have at least one neighbor w with \(f(w)\ge 2\) . The weight of a DRDF is the sum of its function values
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Exponential semi-polynomials and their characterization on semigroups Aequat. Math. (IF 0.8) Pub Date : 2024-02-13 Bruce Ebanks
Exponential semi-polynomials on semigroups are natural generalizations of exponential polynomials on groups. We show that several of the standard properties of exponential polynomials on groups also hold for exponential semi-polynomials on semigroups. The main result is that for topological commutative monoids S belonging to a certain class, a function in C(S) is an exponential semi-polynomial if and
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Generalized Vincze’s functional equations on any group in connection with the maximum functional equation Aequat. Math. (IF 0.8) Pub Date : 2024-02-03
Abstract In this research paper, we investigate a generalization of Vincze’s type functional equations involving several (up to four) unknown functions in connection with the maximum functional equation as $$\begin{aligned} \max \{\psi (xy), \psi (xy^{-1})\}&= \psi (x)\eta (y)+\psi (y), \\ \max \{\psi (xy), \psi (xy^{-1})\}&= \psi (x)\eta (y)+\chi (y), \\ \max \{\psi (xy), \psi (xy^{-1})\}&= \phi (x)\eta
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Vortex filament flows for curves in a 3-dimensional pseudo-Riemannian manifold Aequat. Math. (IF 0.8) Pub Date : 2024-01-24 Zühal Küçükarslan Yüzbai, Nevin Ertug Gürbüz, Hyun Chul Lee, Dae Won Yoon
In this work, we focus on the evolution of the vortex filament flow \(\frac{\partial \gamma }{\partial t} = \frac{\partial \gamma }{\partial s} \wedge \frac{D}{ds}\frac{\partial \gamma }{\partial s}\) for spacelike and timelike curves in a 3-dimensional pseudo-Riemannian manifold. We study the relations between a partial differential equation and the vortex filament flow for spacelike and timelike
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Notes on the arithmetic–geometric mean inequality Aequat. Math. (IF 0.8) Pub Date : 2024-01-17 Ahmad Al-Natoor, Omar Hirzallah, Fuad Kittaneh
In this paper, we give a matrix version of an equivalent form of the classical arithmetic–geometric mean inequality for two positive scalars. Applications and generalizations of our results are also given.
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$$\varepsilon $$ -isometries in $$l^n_1$$ Aequat. Math. (IF 0.8) Pub Date : 2024-01-17 Igor A. Vestfrid
We show that every \(\varepsilon \)-isometry of the unit ball in \(l^n_1\) can be uniformly approximated by an affine surjective isometry to within \(Cn\varepsilon \) for some absolute constant C.
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On the $${A_{\!\mathbb {C}}}$$ -rank of multidigraphs Aequat. Math. (IF 0.8) Pub Date : 2023-12-26 Sasmita Barik, Sane Umesh Reddy
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On the state of the second part of Hilbert’s fifth problem Aequat. Math. (IF 0.8) Pub Date : 2023-12-22 Antal Járai
In the second part of his fifth problem Hilbert asks for functional equations “In how far are the assertions which we can make in the case of differentiable functions true under proper modifications without this assumption.” In the case of the general functional equation $$\begin{aligned} f(x)=h\Bigl (x,y,\bigl (g_1(x,y)\bigr ),\ldots ,\bigl (g_n(x,y)\bigr )\Bigr ) \end{aligned}$$ for the unknown function
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On a class of functional difference equations: explicit solutions, asymptotic behavior and applications Aequat. Math. (IF 0.8) Pub Date : 2023-12-21 Nataliya Vasylyeva
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Remarks on Wright-convex functions Aequat. Math. (IF 0.8) Pub Date : 2023-12-12 Andrzej Olbryś
In the present paper we prove a generalized version of the famous decomposition theorem of Ng. We also focus on the problem posed by Zsolt Páles concerning the Wright-convex functions.
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Further remarks on local K-boundedness of K-subadditive set-valued maps Aequat. Math. (IF 0.8) Pub Date : 2023-11-21 Eliza Jabłońska, Kazimierz Nikodem
Let X be an abelian metric group with an invariant metric, Y be a real normed space and K be a convex cone in Y. We prove that a K-subadditive (K-superadditive) compact- and convex-valued map \(F:X\rightarrow \mathcal{C}\mathcal{C}(Y)\), for which the functionals \(f_{y^*}(x)=\inf y^*(F(x))\) are lower (upper, resp.) semicontinuous for any real continuous and non-negative on K functional \(y^*\), has
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Quadratic functions fulfilling an additional condition along the hyperbola $$\pmb {xy = 1} $$ Aequat. Math. (IF 0.8) Pub Date : 2023-11-25 Zoltán Boros, Edit Garda-Mátyás
In this paper we give necessary conditions for quadratic functions \( f :\mathbb {R}\rightarrow \mathbb {R}\) that satisfy the additional equation \( y^2 f(x) = x^2 f(y) \) under the condition \( xy = 1 \,\).
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Zero utility principles coinciding on binary risks Aequat. Math. (IF 0.8) Pub Date : 2023-11-20 Jacek Chudziak, Małgorzata Chudziak
It is known that the zero utility principle under Cumulative Prospect Theory can be uniquely extended from the family of all ternary risks. On the other hand, the extension from the family of all binary risks need not be unique. We establish a characterization of those zero utility principles which coincide on the family of binary risks. The characterization is expressed in terms of relations between
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Some extremal problems for polygons in the Euclidean plane Aequat. Math. (IF 0.8) Pub Date : 2023-11-22 Yuriĭ Gennadievich Nikonorov, Ol’ga Yur’evna Nikonorova
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Spectra for upper triangular linear relation matrices through local spectral theory Aequat. Math. (IF 0.8) Pub Date : 2023-11-22 Teresa Álvarez, Sonia Keskes
Let X and Y be Banach spaces. When A and B are linear relations in X and Y, respectively, we denote by \(M_{C}\) the linear relation in \(X\times Y\) of the form \(\left( \begin{array}{cc} A &{} C \\ 0 &{} B \\ \end{array} \right) \), where 0 is the zero operator from X to Y and C is a bounded operator from Y to X. In this paper, by using properties of the SVEP, we study the defect set \((\Sigma (A)\cup
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The equation $$\pmb {f(xy) = f(x)h(y) + g(x)f(y)}$$ and representations on $$\pmb {\mathbb {C}^2}$$ Aequat. Math. (IF 0.8) Pub Date : 2023-11-16 Henrik Stetkær
Let G be a topological group, and let C(G) denote the algebra of continuous, complex valued functions on G. We find the solutions \(f,g,h \in C(G)\) of the Levi-Civita equation $$\begin{aligned} f(xy) = f(x)h(y) + g(x)f(y), \ x,y \in G, \end{aligned}$$ which is an extension of the sine addition law. Representations of G on \(\mathbb {C}^2\) play an important role. As a corollary we get the solutions
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Achromatic arboricity on complete graphs Aequat. Math. (IF 0.8) Pub Date : 2023-11-14 Gabriela Araujo-Pardo, Christian Rubio-Montiel
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Characterizations of Jordan *-isomorphisms of $$C^*$$ -algebras by Schatten p-norm of Kubo–Ando means Aequat. Math. (IF 0.8) Pub Date : 2023-11-12 Kaixuan Li, Lei Li, Zheng Shi, Liguang Wang
We consider three operations on the positive definite cones of \(C^*\)-algebras which are Schatten p-norm of various versions of the quantum Rényi relative entropy. The main results concern how Jordan \(*\)-isomorphisms between \(C^*\)-algebras can be characterized by the preservation of the Schatten p-norms of those operations.
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New constructions of t-norms and t-conorms on bounded lattices Aequat. Math. (IF 0.8) Pub Date : 2023-11-08 Jie Qiong Shi, Bin Zhao
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Generalizations of certain conditions for Drazin inverse expressions of anti-triangular partitioned matrices Aequat. Math. (IF 0.8) Pub Date : 2023-10-31 Daochang Zhang, Yu Jin, Dijana Mosić
The Drazin inverse expressions for two classes of anti-triangular partitioned matrices satisfying certain conditions are obtained. Applying these results, we present a few new expressions for the Drazin inverse of any partitioned matrix. We list the corresponding generalization of conditions and results. To illustrate our result, we give one numerical example.
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Remarks on a linearization of Koopmans recursion Aequat. Math. (IF 0.8) Pub Date : 2023-10-20 Marek Cezary Zdun
Let X be a metric space and \(U:X^{\infty }\rightarrow \mathbb {R}\) be a continuous function satisfying the Koopmans recursion \(U(x_0,x_1,x_2,\ldots )=\varphi (x_0, U(x_1,x_2,\ldots )),\) where \(\varphi :X\times I \rightarrow I\) is a continuous function and I is an interval. Denote by \(\succeq \) a preference relation defined on the product \(X^{\infty }\) represented by a function \(U:X^{\infty
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Functional equations stemming from ‘scientific laws’ Aequat. Math. (IF 0.8) Pub Date : 2023-10-19 Juan C. Candeal
Functional equations involving certain generalized forms of the so-called ‘laws of sciences’ are considered. The resolution of these equations is linked to the concept of comparison meaningfulness that appears in measurement theory and dimension theory. The results obtained are stated without assuming any topological requirement.
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A classification of non-monotonicity height for piecewise monotone functions (I): increasing case Aequat. Math. (IF 0.8) Pub Date : 2023-10-19 Kui Wu, Lin Li, Wei Song
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Extreme area and perimeter bisectors in a triangle Aequat. Math. (IF 0.8) Pub Date : 2023-10-17 Allan Berele, Stefan Catoiu
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Refinements of discrete and integral Jensen inequalities with Jensen’s gap Aequat. Math. (IF 0.8) Pub Date : 2023-10-11 László Horváth
Motivated by a paper of Dragomir, we give new refinements for both discrete and integral Jensen inequalities using the Jensen’s gap. As applications, we give refinements of various inequalities verifiable by Jensen’s inequality. Topics covered: norms, quasi-arithmetic means, Hölder’s inequality and f-divergences in information theory.
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On the functional equation $$\hbox {f}(\hbox {x}+\hbox {y})=\hbox {g}(\hbox {xy})$$ Aequat. Math. (IF 0.8) Pub Date : 2023-10-16 Péter Erdei, Tamás Glavosits, Attila Házy
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On sets related to integer partitions with quasi-required elements and disallowed elements Aequat. Math. (IF 0.8) Pub Date : 2023-10-13 Aureliano M. Robles-Pérez, José Carlos Rosales
Given a set A of non-negative integers and a set B of positive integers, we are interested in computing all sets C (of positive integers) that are minimal in the family of sets K (of positive integers) such that (i) K contains no elements generated by non-negative integer linear combinations of elements in A and (ii) for any partition of an element in B there is at least one summand that belongs to
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Prefixes of bargraph paths Aequat. Math. (IF 0.8) Pub Date : 2023-10-14 Aubrey Blecher, Arnold Knopfmacher
We extend the notion of bargraphs as first quadrant, semi-perimeter defined lattice paths to that of a bargraph prefix where we relax the bargraph defining parameters and allow the bargraph to extend into negative territory. A bargraph prefix is any initial section of a bargraph. Generating functions for the prefixes which separately track the number of up, down and horizontal steps are found. The
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On symbolic computation of C.P. Okeke functional equations using Python programming language Aequat. Math. (IF 0.8) Pub Date : 2023-10-13 Chisom Prince Okeke, Wisdom I. Ogala, Timothy Nadhomi
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Weighted Ingham-type inequalities via the positivity of quadratic polynomials Aequat. Math. (IF 0.8) Pub Date : 2023-10-11 Ionel Rovenţa, Laurenţiu Emanuel Temereancă, Mihai Adrian Tudor
We consider nonharmonic Fourier series defined in terms of arbitrarily close exponentials. Our aim is to use the positivity of quadratic polynomials in order to get new Ingham-type weighted inequalities. The proof relies on an Ingham proof technique inspired by Jaffard et al. (J Fourier Anal Appl 3:577–582, 1997). As applications, we consider families of frequencies with relevance in control approximation
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The Artin-Hasse series and Laguerre polynomials modulo a prime Aequat. Math. (IF 0.8) Pub Date : 2023-10-10 Marina Avitabile, Sandro Mattarei
For an odd prime p, let \({{\,\textrm{E}\,}}_{p}(X)=\sum _{n=0}^{\infty } a_{n}X^{n}\in {\mathbb {F}}_p[[X]]\) denote the reduction modulo p of the Artin-Hasse exponential series. It is known that there exists a series \(G(X^p)\in {\mathbb {F}}_{p}[[X]]\), such that \(L_{p-1}^{(-T(X))}(X)={{\,\textrm{E}\,}}_{p}(X)\cdot G(X^p)\), where \(T(X)=\sum _{i=1}^{\infty }X^{p^{i}}\) and \(L_{p-1}^{(\alpha )}(X)\)
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Convex functions and approximate Birkhoff–James orthogonality Aequat. Math. (IF 0.8) Pub Date : 2023-10-09 Jacek Chmieliński, Karol Gryszka, Paweł Wójcik
For the well developed notion of approximate Birkhoff-James orthogonality, in a real or complex normed linear space, we formulate a new characterization. It can be derived from other, already known, characterizations as well as obtained in a more elementary and direct way, on the basis of some simple inequalities for real convex functions.
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On q-generalized (r, s)-Stirling numbers Aequat. Math. (IF 0.8) Pub Date : 2023-10-07 Takao Komatsu
In this paper by using q-numbers and r-shift, the (q, r)-Stirling numbers with level s are studied. One of the main aims is to give several identities in their transforms. We also give some applications to the values of a certain kind of q-multiple zeta functions.
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Paired-domination game played on paths Aequat. Math. (IF 0.8) Pub Date : 2023-10-05 Aaron D. Gray, Michael A. Henning
In this paper, we continue the study of a version of the paired-domination game recently introduced by the authors that embraces both the domination and the matching flavor of the game. The game is played on a graph G by two players, named Dominator and Staller. The players take turns choosing a pair of adjacent vertices of G such that neither vertex in the pair has yet been chosen and the vertices
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Recent progress on minimal hypersurfaces with cylindrical tangent cones Aequat. Math. (IF 0.8) Pub Date : 2023-09-29 Gábor Székelyhidi
We survey some recent results on minimal hypersurfaces in \(\mathbb {R}^{n+1}\) with cylindrical tangent cones. We discuss the question of the uniqueness of tangent cones, the behavior of certain minimal hypersurfaces with cylindrical tangent cones, and a Liouville type theorem for entire minimal hypersurfaces.
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Angular structure of equiangular cones in Euclidean spaces Aequat. Math. (IF 0.8) Pub Date : 2023-09-29 Daniel Gourion, Alberto Seeger
Let K be an n-dimensional equiangular cone in a Euclidean vector space. Let \(0<\phi <\pi \) be the common angle between any pair of vectors taken among the generators of the cone. We obtain an explicit formula for the maximum angle of K in terms of n and \(\phi \). We explain also how to identify a pair \(\{u,v\}\) of unit vectors in K achieving the maximum angle. Finally, we analyze a counterintuitive
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Some study of the approximate Birkhoff orthogonality and orthogonality of bounded linear operators Aequat. Math. (IF 0.8) Pub Date : 2023-09-27 Huayou Xie, Chuanjiang Zhou, Yongjin Li
In this note, we are mainly interested in orthogonality, including Birkhoff orthogonality, isosceles orthogonality, Pythagorean orthogonality and approximate Birkhoff orthogonality\(\perp _{B\epsilon }\). First, we show the relation between linear functionals and hyperplanes by means of approximate Birkhoff orthogonality. Second, we establish the sufficient conditions for Birkhoff orthogonality and