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L2 to Lp bounds for spectral projectors on the Euclidean two-dimensional torus Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2024-03-15 Ciprian Demeter, Pierre Germain
We consider spectral projectors associated to the Euclidean Laplacian on the two-dimensional torus, in the case where the spectral window is narrow. Bounds for their L2 to Lp operator norm are derived, extending the classical result of Sogge; a new question on the convolution kernel of the projector is introduced. The methods employed include $\ell^2$ decoupling, small cap decoupling and estimates
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Canonical decompositions and algorithmic recognition of spatial graphs Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2024-03-14 Stefan Friedl, Lars Munser, José Pedro Quintanilha, Yuri Santos Rego
We prove that there exists an algorithm for determining whether two piecewise-linear spatial graphs are isomorphic. In its most general form, our theorem applies to spatial graphs furnished with vertex colourings, edge colourings and/or edge orientations. We first show that spatial graphs admit canonical decompositions into blocks, that is, spatial graphs that are non-split and have no cut vertices
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On the moduli of hypersurfaces in toric orbifolds Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2024-03-14 Dominic Bunnett
We construct and study the moduli of stable hypersurfaces in toric orbifolds. Let X be a projective toric orbifold and $\alpha \in \operatorname{Cl}(X)$ an ample class. The moduli space is constructed as a quotient of the linear system $|\alpha|$ by $G = \operatorname{Aut}(X)$ . Since the group G is non-reductive in general, we use new techniques of non-reductive geometric invariant theory. Using the
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A boundary maximum principle for stationary pairs of varifolds with fixed contact angle Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2024-03-08 Xuwen Zhang
In this note, we establish a boundary maximum principle for a class of stationary pairs of varifolds satisfying a fixed contact angle condition in any compact Riemannian manifold with smooth boundary.
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Unbounded Sturm attractors for quasilinear parabolic equations Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2024-03-08 Phillipo Lappicy, Juliana Fernandes
We analyse the asymptotic dynamics of quasilinear parabolic equations when solutions may grow up (i.e. blow up in infinite time). For such models, there is a global attractor which is unbounded and the semiflow induces a nonlinear dynamics at infinity by means of a Poincaré projection. In case the dynamics at infinity is given by a semilinear equation, then it is gradient, consisting of the so-called
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Proof of some conjectural congruences involving Apéry and Apéry-like numbers Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2024-03-07 Guo-shuai Mao, Lilong Wang
In this paper, we mainly prove the following conjectures of Sun [16]: Let p > 3 be a prime. Then \begin{align*} &A_{2p}\equiv A_2-\frac{1648}3p^3B_{p-3}\ ({\rm{mod}}\ p^4),\\ &A_{2p-1}\equiv A_1+\frac{16p^3}3B_{p-3}\ ({\rm{mod}}\ p^4),\\ &A_{3p}\equiv A_3-36738p^3B_{p-3}\ ({\rm{mod}}\ p^4), \end{align*} where $A_n=\sum_{k=0}^n\binom{n}k^2\binom{n+k}{k}^2$ is the nth Apéry number, and Bn is the nth
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Structure of generalized Yamabe solitons and its applications Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2024-03-07 Shun Maeta
We consider the broadest concept of the gradient Yamabe soliton, the conformal gradient soliton. In this paper, we elucidate the structure of complete gradient conformal solitons under some assumption, and provide some applications to gradient Yamabe solitons. These results enhance the understanding gained from previous research. Furthermore, we give an affirmative partial answer to the Yamabe soliton
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Unit sphere fibrations in Euclidean space Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2024-03-07 Daniel Asimov, Florian Frick, Michael Harrison, Wesley Pegden
We show that if an open set in $\mathbb{R}^d$ can be fibered by unit n-spheres, then $d \geq 2n+1$ , and if $d = 2n+1$ , then the spheres must be pairwise linked, and $n \in \left\{0, 1, 3, 7 \right\}$ . For these values of n, we construct unit n-sphere fibrations in $\mathbb{R}^{2n+1}$ .
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The Schwarzian norm estimates for Janowski convex functions Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2024-02-12 Md Firoz Ali, Sanjit Pal
For $-1\leq B \lt A\leq 1$ , let $\mathcal{C}(A,B)$ denote the class of normalized Janowski convex functions defined in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z| \lt 1\}$ that satisfy the subordination relation $1+zf''(z)/f'(z)\prec (1+Az)/(1+Bz)$ . In the present article, we determine the sharp estimate of the Schwarzian norm for functions in the class $\mathcal{C}(A,B)$ . The Dieudonné’s lemma
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Estimating the Hausdorff measure using recurrence Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2024-01-12 Łukasz Pawelec
We show a new method of estimating the Hausdorff measure of a set from below. The method requires computing the subsequent closest return times of a point to itself.
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Commutants and complex symmetry of finite Blaschke product multiplication operator in Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2024-01-11 Arup Chattopadhyay, Soma Das
Consider the multiplication operator MB in $L^2(\mathbb{T})$, where the symbol B is a finite Blaschke product. In this article, we characterize the commutant of MB in $L^2(\mathbb{T})$. As an application of this characterization result, we explicitly determine the class of conjugations commuting with $M_{z^2}$ or making $M_{z^2}$ complex symmetric by introducing a new class of conjugations in $L^2(\mathbb{T})$
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Goldie dimension for C*-algebras Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2024-01-11 Mohammad Rouzbehani, Massoud Amini, Mohammad B. Asadi
In this article, we introduce and study the notion of Goldie dimension for C*-algebras. We prove that a C*-algebra A has Goldie dimension n if and only if the dimension of the center of its local multiplier algebra is n. In this case, A has finite-dimensional center and its primitive spectrum is extremally disconnected. If moreover, A is extending, we show that it decomposes into a direct sum of n
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Infinitesimally Moebius bendable hypersurfaces Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2024-01-11 M.I. Jimenez, R. Tojeiro
Li, Ma and Wang have provided in [13] a partial classification of the so-called Moebius deformable hypersurfaces, that is, the umbilic-free Euclidean hypersurfaces $f\colon M^n\to \mathbb{R}^{n+1}$ that admit non-trivial deformations preserving the Moebius metric. For $n\geq 5$, the classification was completed by the authors in [12]. In this article we obtain an infinitesimal version of that classification
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Multidimensional Frank–Laptev–Weidl improvement of the hardy inequality Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2024-01-11 Prasun Roychowdhury, Michael Ruzhansky, Durvudkhan Suragan
We establish a new improvement of the classical Lp-Hardy inequality on the multidimensional Euclidean space in the supercritical case. Recently, in [14], there has been a new kind of development of the one-dimensional Hardy inequality. Using some radialisation techniques of functions and then exploiting symmetric decreasing rearrangement arguments on the real line, the new multidimensional version
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On traces of Bochner representable operators on the space of bounded measurable functions Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2024-01-11 Marian Nowak, Juliusz Stochmal
Let Σ be a σ-algebra of subsets of a set Ω and $B(\Sigma)$ be the Banach space of all bounded Σ-measurable scalar functions on Ω. Let $\tau(B(\Sigma),ca(\Sigma))$ denote the natural Mackey topology on $B(\Sigma)$. It is shown that a linear operator T from $B(\Sigma)$ to a Banach space E is Bochner representable if and only if T is a nuclear operator between the locally convex space $(B(\Sigma),\tau(B(\Sigma)
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The spherical growth series of Dyer groups Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-12-21 Luis Paris, Olga Varghese
Graph products of cyclic groups and Coxeter groups are two families of groups that are defined by labelled graphs. The family of Dyer groups contains these both families and gives us a framework to study these groups in a unified way. This paper focuses on the spherical growth series of a Dyer group D with respect to the standard generating set. We give a recursive formula for the spherical growth
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Isomorphisms of quadratic quasigroups Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-11-24 Aleš Drápal, Ian M. Wanless
Let $\mathbb F$ be a finite field of odd order and $a,b\in\mathbb F\setminus\{0,1\}$ be such that $\chi(a) = \chi(b)$ and $\chi(1-a)=\chi(1-b)$, where χ is the extended quadratic character on $\mathbb F$. Let $Q_{a,b}$ be the quasigroup over $\mathbb F$ defined by $(x,y)\mapsto x+a(y-x)$ if $\chi(y-x) \geqslant 0$, and $(x,y)\mapsto x+b(y-x)$ if $\chi(y-x) = -1$. We show that $Q_{a,b} \cong Q_{c,d}$
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On the representability of actions of Leibniz algebras and Poisson algebras Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-11-22 Alan S. Cigoli, Manuel Mancini, Giuseppe Metere
In a recent paper, motivated by the study of central extensions of associative algebras, George Janelidze introduces the notion of weakly action representable category. In this paper, we show that the category of Leibniz algebras is weakly action representable and we characterize the class of acting morphisms. Moreover, we study the representability of actions of the category of Poisson algebras and
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Generalized divisor functions in arithmetic progressions: II Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-11-16 D. T. Nguyen
We obtain a new bound on the second moment of modified shifted convolutions of the generalized threefold divisor function and show that, for applications, the modified version is sufficient.
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On the difference of two fourth powers Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-11-10 Nguyen Xuan Tho
We investigate the equation $D=x^4-y^4$ in field extensions. As an application, for a prime number p, we find solutions to $p=x^4-y^4$ if $p\equiv 11$ (mod 16) and $p^3=x^4-y^4$ if $p\equiv 3$ (mod 16) in all cubic extensions of $\mathbb{Q}(i)$.
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Multiplicity and stability of normalized solutions to non-autonomous Schrödinger equation with mixed non-linearities Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-11-09 Xinfu Li, Li Xu, Meiling Zhu
This paper first studies the multiplicity of normalized solutions to the non-autonomous Schrödinger equation with mixed nonlinearities\begin{equation*}\begin{cases}-\Delta u=\lambda u+h(\epsilon x)|u|^{q-2}u+\eta |u|^{p-2}u,\quad x\in \mathbb{R}^N, \\\int_{\mathbb{R}^N}|u|^2\,\textrm{d}x=a^2,\end{cases}\end{equation*}where $a, \epsilon, \eta \gt 0$, q is L2-subcritical, p is L2-supercritical, $\lambda\in
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Reversibility of affine transformations Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-11-08 Krishnendu Gongopadhyay, Tejbir Lohan, Chandan Maity
An element g in a group G is called reversible if g is conjugate to g−1 in G. An element g in G is strongly reversible if g is conjugate to g−1 by an involution in G. The group of affine transformations of $\mathbb D^n$ may be identified with the semi-direct product $\mathrm{GL}(n, \mathbb D) \ltimes \mathbb D^n $, where $\mathbb D:=\mathbb R, \mathbb C$ or $ \mathbb H $. This paper classifies reversible
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Bohr radius for Banach spaces on simply connected domains Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-11-03 Vasudevarao Allu, Himadri Halder
Let $H^{\infty}(\Omega,X)$ be the space of bounded analytic functions $f(z)=\sum_{n=0}^{\infty} x_{n}z^{n}$ from a proper simply connected domain Ω containing the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z| \lt 1\}$ into a complex Banach space X with $\left\lVert f\right\rVert_{H^{\infty}(\Omega,X)} \leq 1$. Let $\phi=\{\phi_{n}(r)\}_{n=0}^{\infty}$ with $\phi_{0}(r)\leq 1$ such that $\sum_{n=0}^{\infty}
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Cohomology of a real toric variety and shellability of posets arising from a graph Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-11-03 Boram Park, Seonjeong Park
Given a graph G without loops, the pseudograph associahedron PG is a smooth polytope, so there is a projective smooth toric variety XG corresponding to PG. Taking the real locus of XG, we have the projective smooth real toric variety $X^{\mathbb{R}}_G$. The integral cohomology groups of $X^{\mathbb{R}}_G$ can be computed by studying the topology of certain posets of even subgraphs of G; such a poset
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Rigidity of Ext and Tor via flat–cotorsion theory Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-11-03 Lars Winther Christensen, Luigi Ferraro, Peder Thompson
Let $\mathfrak{p}$ be a prime ideal in a commutative noetherian ring R and denote by $k(\mathfrak{p})$ the residue field of the local ring $R_\mathfrak{p}$. We prove that if an R-module M satisfies $\operatorname{Ext}_R^{n}(k(\mathfrak{p}),M)=0$ for some $n\geqslant\dim R$, then $\operatorname{Ext}_R^i(k(\mathfrak{p}),M)=0$ holds for all $i \geqslant n$. This improves a result of Christensen, Iyengar
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On Liouville theorems of a Hartree–Poisson system Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-10-26 Ling Li, Yutian Lei
In this paper, we are concerned with the non-existence of positive solutions of a Hartree–Poisson system:\begin{equation*}\left\{\begin{aligned}&-\Delta u=\left(\frac{1}{|x|^{n-2}}\ast v^p\right)v^{p-1},\quad u \gt 0\ \text{in} \ \mathbb{R}^{n},\\&-\Delta v=\left(\frac{1}{|x|^{n-2}}\ast u^q\right)u^{q-1},\quad v \gt 0\ \text{in} \ \mathbb{R}^{n},\end{aligned}\right.\end{equation*} where $n \geq3$ and
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Burstein’s permutation conjecture, Hong and Li’s inversion sequence conjecture and restricted Eulerian distributions Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-10-23 Shane Chern, Shishuo Fu, Zhicong Lin
Recently, Hong and Li launched a systematic study of length-four pattern avoidance in inversion sequences, and in particular, they conjectured that the number of 0021-avoiding inversion sequences can be enumerated by the OEIS entry A218225. Meanwhile, Burstein suggested that the same sequence might also count three sets of pattern-restricted permutations. The objective of this paper is not only a confirmation
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Classification of subpencils for hyperplane sections on certain K3 surfaces Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-10-23 Tomokuni Takahashi
We classify the subpencils of complete linear systems for the hyperplane sections on K3 surfaces obtained as the complete intersection of a hyperquadric and a hypercubic. The classification is done from three points of view, namely, the type of a general fibre, the base locus and the Horikawa index of the essential member. This classification shows the distinct phenomenons depending on the rank of
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Normal families and quasiregular mappings Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-10-23 Alastair N. Fletcher, Daniel A. Nicks
Beardon and Minda gave a characterization of normal families of holomorphic and meromorphic functions in terms of a locally uniform Lipschitz condition. Here, we generalize this viewpoint to families of mappings in higher dimensions that are locally uniformly continuous with respect to a given modulus of continuity. Our main application is to the normality of families of quasiregular mappings through
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Stable solutions to double phase problems involving a nonlocal term Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-10-23 Belgacem Rahal, Phuong Le
In this paper, we study weak solutions, possibly unbounded and sign-changing, to the double phase problem\begin{equation*}-\text{div} (|\nabla u|^{p-2} \nabla u + w(x)|\nabla u|^{q-2} \nabla u) = \left(\frac{1}{|x|^{N-\mu}}*f|u|^r\right) f(x)|u|^{r-2}u \quad\text{in}\ \mathbb{R}^N,\end{equation*}where $q\ge p\ge2$, r > q, $0 \lt \mu \lt N$ and $w,f \in L^1_{\rm loc}(\mathbb{R}^N)$ are two non-negative
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Inequalities concerning maximum modulus and zeros of random entire functions Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-10-19 Hui Li, Jun Wang, Xiao Yao, Zhuan Ye
Let $f(z)=\sum\limits_{j=0}^{\infty} a_j z^j$ be a transcendental entire function and let $f_\omega(z)=\sum\limits_{j=0}^{\infty}\chi_j(\omega) a_j z^j$ be a random entire function, where $\chi_j(\omega)$ are independent and identically distributed random variables defined on a probability space $(\Omega, \mathcal{F}, \mu)$. In this paper, we first define a family of random entire functions, which
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On the vanishing of the coefficients of CM eta quotients Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-10-18 Tim Huber, Chang Liu, James McLaughlin, Dongxi Ye, Miaodan Yuan, Sumeng Zhang
This work characterizes the vanishing of the Fourier coefficients of all CM (Complex Multiplication) eta quotients. As consequences, we recover Serre’s characterization about that of $\eta(12z)^{2}$ and recent results of Chang on the pth coefficients of $\eta(4z)^{6}$ and $\eta(6z)^{4}$. Moreover, we generalize the results on the cases of weight 1 to the setting of binary quadratic forms.
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A weakened Markus–Yamabe condition for planar polynomial differential systems of degree Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-10-17 Jaume Llibre, Claudia Valls
For a general autonomous planar polynomial differential system, it is difficult to find conditions that are easy to verify and which guarantee global asymptotic stability, weakening the Markus–Yamabe condition. In this paper, we provide three conditions that guarantee the global asymptotic stability for polynomial differential systems of the form $x^{\prime}=f_1(x,y)$, $y^{\prime}=f_2(x,y)$, where
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On weakly almost square Banach spaces Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-10-05 José Rodríguez, Abraham Rueda Zoca
We prove some results on weakly almost square Banach spaces and their relatives. On the one hand, we discuss weak almost squareness in the setting of Banach function spaces. More precisely, let $(\Omega,\Sigma)$ be a measurable space, let E be a Banach lattice and let $\nu:\Sigma \to E^+$ be a non-atomic countably additive measure having relatively norm compact range. Then the space $L_1(\nu)$ is weakly
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Discrete restriction estimates for forms in many variables Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-09-18 Brian Cook, Kevin Hughes, Eyvindur Palsson
We prove discrete restriction estimates for a broad class of hypersurfaces arising in seminal work of Birch. To do so, we use a variant of Bourgain’s arithmetic version of the Tomas–Stein method and Magyar’s decomposition of the Fourier transform of the indicator function of the integer points on a hypersurface.
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The Reidemeister spectrum of finite abelian groups Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-09-06 Pieter Senden
For a finite abelian group A, the Reidemeister number of an endomorphism φ is the same as the number of fixed points of φ, and the Reidemeister spectrum of A is completely determined by the Reidemeister spectra of its Sylow p-subgroups. To compute the Reidemeister spectrum of a finite abelian p-group P, we introduce a new number associated to an automorphism ψ of P that captures the number of fixed
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A Chebyshev-type alternation theorem for best approximation by a sum of two algebras Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-09-01 Aida KH. Asgarova, Ali A. Huseynli, Vugar E. Ismailov
Let X be a compact metric space, C(X) be the space of continuous real-valued functions on X and $A_{1},A_{2}$ be two closed subalgebras of C(X) containing constant functions. We consider the problem of approximation of a function $f\in C(X)$ by elements from $A_{1}+A_{2}$. We prove a Chebyshev-type alternation theorem for a function $u_{0} \in A_{1}+A_{2}$ to be a best approximation to f.
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Perron’s capacity of random sets Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-09-01 Anthony Gauvan
We answer in a probabilistic setting two questions raised by Stokolos in a private communication. Precisely, given a sequence of random variables $\left\{X_k : k \geq 1\right\}$ uniformly distributed in $(0,1)$ and independent, we consider the following random sets of directions\begin{equation*}\Omega_{\text{rand},\text{lin}} := \left\{ \frac{\pi X_k}{k}: k \geq 1\right\}\end{equation*}and\begin{e
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Kaehler submanifolds of the real hyperbolic space Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-08-22 Sergio Chion, Marcos Dajczer
The local classification of Kaehler submanifolds $M^{2n}$ of the hyperbolic space $\mathbb{H}^{2n+p}$ with low codimension $2\leq p\leq n-1$ under only intrinsic assumptions remains a wide open problem. The situation is quite different for submanifolds in the round sphere $\mathbb{S}^{2n+p}$, $2\leq p\leq n-1$, since Florit et al. [7] have shown that the codimension has to be $p=n-1$ and then that
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New mock theta functions and formulas for basic hypergeometric series Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-08-22 Olivia X. M. Yao
In recent years, mock theta functions in the modern sense have received great attention to seek examples of q-hypergeometric series and find their alternative representations. In this paper, we discover some new mock theta functions and express them in terms of Hecke-type double sums based on some basic hypergeometric series identities given by Z.G. Liu.
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Proper Ehresmann semigroups Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-08-22 Ganna Kudryavtseva, Valdis Laan
We propose a notion of a proper Ehresmann semigroup based on a three-coordinate description of its generating elements governed by certain labelled directed graphs with additional structure. The generating elements are determined by their domain projection, range projection and σ-class, where σ denotes the minimum congruence that identifies all projections. We prove a structure result on proper Ehresmann
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The spectral eigenmatrix problems of planar self-affine measures with four digits Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-08-22 Jing-Cheng Liu, Min-Wei Tang, Sha Wu
Given a Borel probability measure µ on $\mathbb{R}^n$ and a real matrix $R\in M_n(\mathbb{R})$. We call R a spectral eigenmatrix of the measure µ if there exists a countable set $\Lambda\subset \mathbb{R}^n$ such that the sets $E_\Lambda=\big\{{\rm e}^{2\pi i \langle\lambda,x\rangle}:\lambda\in \Lambda\big\}$ and $E_{R\Lambda}=\big\{{\rm e}^{2\pi i \langle R\lambda,x\rangle}:\lambda\in \Lambda\big\}$
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The topology of compact rank-one ECS manifolds Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-08-17 Andrzej Derdzinski, Ivo Terek
Pseudo-Riemannian manifolds with parallel Weyl tensor that are not conformally flat or locally symmetric, also known as essentially conformally symmetric (ECS) manifolds, have a natural local invariant, the rank, which equals 1 or 2, and is the rank of a certain distinguished null parallel distribution $\mathcal{D}$. All known examples of compact ECS manifolds are of rank one and have dimensions greater
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Every Salem number is a difference of two Pisot numbers Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-08-08 Artūras Dubickas
In this note, we prove that every Salem number is expressible as a difference of two Pisot numbers. More precisely, we show that for each Salem number α of degree d, there are infinitely many positive integers n for which $\alpha^{2n-1}-\alpha^n+\alpha$ and $\alpha^{2n-1}-\alpha^n$ are both Pisot numbers of degree d and that the smallest such n is at most $6^{d/2-1}+1$. We also prove that every real
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Exact formulae and Turán inequalities for Vafa–Witten invariants of surfaces Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-08-08 Daniel R. Johnston, Joshua Males
We consider three different families of Vafa–Witten invariants of $K3$ surfaces. In each case, the partition function that encodes the Vafa–Witten invariants is given by combinations of twisted Dedekind η-functions. By utilizing known properties of these η-functions, we obtain exact formulae for each of the invariants and prove that they asymptotically satisfy all higher-order Turán inequalities.
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On the density of bounded bases Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-08-07 Jin-Hui Fang
For a nonempty set A of integers and an integer n, let $r_{A}(n)$ be the number of representations of n in the form $n=a+a'$, where $a\leqslant a'$ and $a, a'\in A$, and $d_{A}(n)$ be the number of representations of n in the form $n=a-a'$, where $a, a'\in A$. The binary support of a positive integer n is defined as the subset S(n) of nonnegative integers consisting of the exponents in the binary expansion
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Counting periodic orbits on fractals weighted by their Lyapounov exponents Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-07-03 Ugo Bessi
Several authors have shown that Kusuoka’s measure κ on fractals is a scalar Gibbs measure; in particular, it maximizes a pressure. There is also a different approach, in which one defines a matrix-valued Gibbs measure µ, which induces both Kusuoka’s measure κ and Kusuoka’s bilinear form. In the first part of the paper, we show that one can define a ‘pressure’ for matrix-valued measures; this pressure
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On the Dales–Żelazko conjecture for Beurling algebras on discrete groups Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-07-03 Jared T. White
Let G be a group that is either virtually soluble or virtually free, and let ω be a weight on G. We prove that if G is infinite, then there is some maximal left ideal of finite codimension in the Beurling algebra $\ell^1(G, \omega)$, which fails to be (algebraically) finitely generated. This implies that a conjecture of Dales and Żelazko holds for these Banach algebras. We then go on to give examples
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Trivial source character tables of , part II Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-06-30 Niamh Farrell, Caroline Lassueur
We compute the trivial source character tables (also called species tables of the trivial source ring) of the infinite family of finite groups $\operatorname{SL}_{2}(q)$ for q even over a large enough field of odd characteristics. This article is a continuation of our article Trivial Source Character Tables of $\operatorname{SL}_{2}(q)$, where we considered, in particular, the case in which q is odd
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The Fueter-Sce mapping and the Clifford–Appell polynomials Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-06-28 Antonino De Martino, Kamal Diki, Ali Guzmán Adán
The Fueter-Sce theorem provides a procedure to obtain axially monogenic functions, which are in the kernel of generalized Cauchy–Riemann operator in ${\mathbb{R}}^{n+1}$. This result is obtained by using two operators. The first one is the slice operator, which extends holomorphic functions of one complex variable to slice monogenic functions in $ \mathbb{R}^{n+1}$. The second one is a suitable power
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Groups of small period growth Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-06-27 Jan Moritz Petschick
We construct finitely generated groups of small period growth, i.e. groups where the maximum order of an element of word length n grows very slowly in n. This answers a question of Bradford related to the lawlessness growth of groups and is connected to an approximative version of the restricted Burnside problem.
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A non-periodic indefinite variational problem in ℝN with critical exponent Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-06-26 Gustavo S. do Amaral Costa, Giovany M. Figueiredo, José Carlos de O. Junior
We consider the non-linear Schrödinger equation(Pμ)\begin{equation*}\begin{array}{lc}-\Delta u + V(x) u = \mu f(u) + |u|^{2^*-2}u, &\end{array}\end{equation*}in $\mathbb{R}^N$, $N\geq3$, where V changes sign and $f(s)/s$, s ≠ 0, is bounded, with V non-periodic in x. The existence of a solution is established employing spectral theory, a general linking theorem due to [12] and interaction between translated
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On the algebra of elliptic curves Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-06-26 Tomasz Brzeziński
It is argued that a nonsingular elliptic curve admits a natural or fundamental abelian heap structure uniquely determined by the curve itself. It is shown that the set of complex analytic or rational functions from a nonsingular elliptic curve to itself is a truss arising from endomorphisms of this heap.
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Surgeries on iterated torus knots bounding rational homology 4-balls Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-06-23 Lisa Lokteva
We show that all large enough positive integral surgeries on algebraic knots bound a 4-manifold with a negative definite plumbing tree, which we describe explicitly. Then we apply the lattice embedding obstruction coming from Donaldson’s Theorem to classify the ones of the form $S^3_n(T(p_1,k_1p_1+1; p_2, k_2p_2\pm 1))$ that also bound rational homology 4-balls.
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The Lp convergence of Fourier series on triangular domains Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-06-19 Ryan L. Acosta Babb
We prove Lp norm convergence for (appropriate truncations of) the Fourier series arising from the Dirichlet Laplacian eigenfunctions on three types of triangular domains in $\mathbb{R}^2$: (i) the 45-90-45 triangle, (ii) the equilateral triangle and (iii) the hemiequilateral triangle (i.e. half an equilateral triangle cut along its height). The limitations of our argument to these three types are discussed
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Baumslag–Solitar groups and residual nilpotence Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-06-16 C.E. Kofinas, V. Metaftsis, A.I. Papistas
Let G be a Baumslag–Solitar group. We calculate the intersection $\gamma_{\omega}(G)$ of all terms of the lower central series of G. Using this, we show that $[\gamma_{\omega}(G),G]=\gamma_{\omega}(G)$, thus answering a question of Bardakov and Neschadim [1]. For any $c \in \mathbb{N}$, with $c \geq 2$, we show, by using Lie algebra methods, that the quotient group $\gamma_{c}(G)/\gamma_{c+1}(G)$ of
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Schwarz lemma for real harmonic functions onto surfaces with non-negative Gaussian curvature Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-06-15 David Kalaj, Miodrag Mateljević, Iosif Pinelis
Assume that f is a real ρ-harmonic function of the unit disk $\mathbb{D}$ onto the interval $(-1,1)$, where $\rho(u,v)=R(u)$ is a metric defined in the infinite strip $(-1,1)\times \mathbb{R}$. Then we prove that $|\nabla f(z)|(1-|z|^2)\le \frac{4}{\pi}(1-f(z)^2)$ for all $z\in\mathbb{D}$, provided that ρ has a non-negative Gaussian curvature. This extends several results in the field and answers to
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A strongly convergent algorithm for solving multiple set split equality equilibrium and fixed point problems in Banach spaces Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-06-15 E.C. Godwin, O.T. Mewomo, T.O. Alakoya
In this article, using an Halpern extragradient method, we study a new iterative scheme for finding a common element of the set of solutions of multiple set split equality equilibrium problems consisting of pseudomonotone bifunctions and the set of fixed points for two finite families of Bregman quasi-nonexpansive mappings in the framework of p-uniformly convex Banach spaces, which are also uniformly
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On one-dimensional local rings and Berger’s conjecture Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-05-23 Cleto B. Miranda-Neto
Let k be a field of characteristic zero and let $\Omega_{A/k}$ be the universally finite differential module of a k-algebra A, which is the local ring of a closed point of an algebraic or algebroid curve over k. A notorious open problem, known as Berger’s Conjecture, predicts that A must be regular if $\Omega_{A/k}$ is torsion-free. In this paper, assuming the hypotheses of the conjecture and observing
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Higher-order evolution inequalities involving convection and Hardy-Leray potential terms in a bounded domain Proc. Edinburgh. Math. Soc. (IF 0.7) Pub Date : 2023-05-05 Huyuan Chen, Mohamed Jleli, Bessem Samet
We consider a class of nonlinear higher-order evolution inequalities posed in $(0,\infty)\times B_1\backslash\{0\}$, subject to inhomogeneous Dirichlet-type boundary conditions, where B1 is the unit ball in $\mathbb{R}^N$. The considered class involves differential operators of the form\begin{equation*}\mathcal{L}_{\mu_1,\mu_2}=-\Delta +\frac{\mu_1}{|x|^2}x\cdot \nabla +\frac{\mu_2}{|x|^2},\qquad x\in