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Morphisms between Grassmannians, II Arch. Math. (IF 0.6) Pub Date : 2024-03-26
Abstract Denote by \({\mathbb {G}}(k,n)\) the Grassmannian of linear subspaces of dimension k in \({\mathbb {P}}^n\) . We show that if \(\varphi :{\mathbb {G}}(l,n) \rightarrow {\mathbb {G}}(k,n)\) is a nonconstant morphism and \(l \not =0,n-1\) , then \(l=k\) or \(l=n-k-1\) and \(\varphi \) is an isomorphism.
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Genus and crosscap of solvable conjugacy class graphs of finite groups Arch. Math. (IF 0.6) Pub Date : 2024-03-24
Abstract The solvable conjugacy class graph of a finite group G, denoted by \(\Gamma _{sc}(G)\) , is a simple undirected graph whose vertices are the non-trivial conjugacy classes of G and two distinct conjugacy classes C, D are adjacent if there exist \(x \in C\) and \(y \in D\) such that \(\langle x, y\rangle \) is solvable. In this paper, we discuss certain properties of the genus and crosscap of
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Reproducing kernel Hilbert spaces cannot contain all continuous functions on a compact metric space Arch. Math. (IF 0.6) Pub Date : 2024-03-23 Ingo Steinwart
Given an uncountable, compact metric space X, we show that there exists no reproducing kernel Hilbert space that contains the space of all continuous functions on X.
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Half-factorial real quadratic orders Arch. Math. (IF 0.6) Pub Date : 2024-03-12 Paul Pollack
Recall that D is a half-factorial domain (HFD) when D is atomic and every equation \(\pi _1\cdots \pi _k = \rho _1 \cdots \rho _\ell \), with all \(\pi _i\) and \(\rho _j\) irreducible in D, implies \(k=\ell \). We explain how techniques introduced to attack Artin’s primitive root conjecture can be applied to understand half-factoriality of orders in real quadratic number fields. In particular, we
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A note on blocks of finite groups with TI Sylow p-subgroups Arch. Math. (IF 0.6) Pub Date : 2024-03-07 Deniz Yılmaz
Let \(\mathbb {F}\) be an algebraically closed field of characteristic zero. We prove that functorial equivalence over \(\mathbb {F}\) and perfect isometry between blocks of finite groups do not imply each other.
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Matrices for finite group representations that respect Galois automorphisms Arch. Math. (IF 0.6) Pub Date : 2024-03-02 David J. Benson
We are given a finite group H, an automorphism \(\tau \) of H of order r, a Galois extension L/K of fields of characteristic zero with cyclic Galois group \(\langle \sigma \rangle \) of order r, and an absolutely irreducible representation \(\rho :H\rightarrow \textsf {GL} (n,L)\) such that the action of \(\tau \) on the character of \(\rho \) is the same as the action of \(\sigma \). Then the following
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Cesàro means in local Dirichlet spaces Arch. Math. (IF 0.6) Pub Date : 2024-02-24 J. Mashreghi, M. Nasri, M. Withanachchi
The Cesàro means of Taylor polynomials \(\sigma _n,\) \(n \ge 0,\) are finite rank operators on any Banach space of analytic functions on the open unit disc. They are particularly exploited when the Taylor polynomials do not constitute a valid linear polynomial approximation scheme (LPAS). Notably, in local Dirichlet spaces \({\mathcal {D}}_\zeta ,\) they serve as a proper LPAS. The primary objective
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Some new decay estimates for $$(2+1)$$ -dimensional degenerate oscillatory integral operators Arch. Math. (IF 0.6) Pub Date : 2024-02-20 Yuxin Tan, Shaozhen Xu
In this paper, we consider the \((2+1)\)-dimensional oscillatory integral operators with cubic homogeneous polynomial phases, which are degenerate in the sense of (Forum Math. 18:427–444, 2006). We improve the previously known \(L^2\rightarrow L^2\) decay rate to 3/8 and also establish a sharp \(L^2\rightarrow L^6\) decay estimate based on the fractional integration method.
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A characterization of translation and modulation invariant Hilbert space of tempered distributions Arch. Math. (IF 0.6) Pub Date : 2024-02-17 Shubham R. Bais, Pinlodi Mohan, D. Venku Naidu
Let \(\mathcal {S}(\mathbb {R}^n)\) be the Schwartz space and \(\mathcal {S'}(\mathbb {R}^n)\) be the space of tempered distributions on \(\mathbb {R}^n\). In this article, we prove that if \(\mathcal {H} \subseteq \mathcal {S'}(\mathbb {R}^n)\) is a non-zero Hilbert space of tempered distributions which is translation and modulation invariant such that $$\begin{aligned} |(f,g)| \le C \Vert f\Vert
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On the finiteness of radii of resolving subcategories Arch. Math. (IF 0.6) Pub Date : 2024-02-17 Yuki Mifune
Let R be a commutative Noetherian ring. Denote by \({\text {mod}}R\) the category of finitely generated R-modules. In this paper, we investigate the finiteness of the radii of resolving subcategories of \({\text {mod}}R\) with respect to a fixed semidualizing module. As an application, we give a partial positive answer to a conjecture of Dao and Takahashi: we prove that for a Cohen–Macaulay local ring
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Sectorial Mertens and Mirsky formulae for imaginary quadratic number fields Arch. Math. (IF 0.6) Pub Date : 2024-02-05 Jouni Parkkonen, Frédéric Paulin
We extend formulae of Mertens and Mirsky on the asymptotic behaviour of the usual Euler function to the Euler functions of principal rings of integers of imaginary quadratic number fields, giving versions in angular sectors and with congruences.
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Vectorial analogues of Cauchy’s surface area formula Arch. Math. (IF 0.6) Pub Date : 2024-01-29 Daniel Hug, Rolf Schneider
Cauchy’s surface area formula says that for a convex body K in n-dimensional Euclidean space, the mean value of the \((n-1)\)-dimensional volumes of the orthogonal projections of K to hyperplanes is a constant multiple of the surface area of K. We prove an analogous formula, with the volumes of the projections replaced by their moment vectors. This requires to introduce a new vector-valued valuation
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On a theorem of Knörr Arch. Math. (IF 0.6) Pub Date : 2024-01-25 Burkhard Külshammer
Knörr has constructed an ideal, in the center of the p-modular group algebra of a finite group G, whose dimension is the number of p-blocks of defect zero in G/Q; here p is a prime and Q is a normal p-subgroup of G. We generalize his construction to symmetric algebras.
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Unbounded periodic constant mean curvature graphs on calibrable Cheeger Serrin domains Arch. Math. (IF 0.6) Pub Date : 2024-01-25 Ignace Aristide Minlend
We prove a general result characterizing a specific class of Serrin domains as supports of unbounded and periodic constant mean curvature graphs. We apply this result to prove the existence of a family of unbounded periodic constant mean curvature graphs, each supported by a Serrin domain and intersecting its boundary orthogonally, up to a translation. We also show that the underlying Serrin domains
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On the sine polynomials of Fejér and Lukács Arch. Math. (IF 0.6) Pub Date : 2024-01-24 Horst Alzer, Man Kam Kwong
The sine polynomials of Fejér and Lukács are defined by $$\begin{aligned} F_n(x)=\sum _{k=1}^n\frac{\sin (kx)}{k} \quad \text{ and } \quad L_n(x)=\sum _{k=1}^n (n-k+1)\sin (kx), \end{aligned}$$ respectively. We prove that for all \(n\ge 2\) and \(x\in (0,\pi )\), we have $$\begin{aligned} F_n(x)\le \lambda \, L_n(x) \quad \text{ and } \quad \mu \le \frac{1}{F_n(x)}-\frac{1}{L_n(x)} \end{aligned}$$
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Co-Bassian and generalized co-Bassian abelian groups Arch. Math. (IF 0.6) Pub Date : 2024-01-22 Patrick W. Keef
The abelian group G is co-Bassian if for all subgroups \(N\subseteq G\), if \(\phi : G\rightarrow G/N\) is an injective homomorphism, then \(\phi (G)=G/N\). And G is generalized co-Bassian if for all subgroups \(N\subseteq G\), if \(\phi : G\rightarrow G/N\) is an injective homomorphism, then \(\phi (G)\) is a summand of G/N. The co-Bassian and generalized co-Bassian groups are completely characterized
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Uncountable groups with finitely many normalizers of large subgroups Arch. Math. (IF 0.6) Pub Date : 2024-01-18 M. De Falco, C. Musella, G. Sabatino
In this paper, the structure of uncountable groups with finitely many normalizers of large subgroups is studied and the connections between this property and other natural finiteness conditions on large subgroups of uncountable groups are investigated. In particular, groups in which every large subgroup is close to be normal with the only obstruction of a finite section and groups with finitely many
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A short proof of the elliptical range theorem Arch. Math. (IF 0.6) Pub Date : 2024-01-17 Gyula Lakos
A short proof of the elliptical range theorem concerning the numerical range of \(2\times 2\) complex matrices is given.
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On the slope stability of the cotangent bundles of Weierstrass fibrations Arch. Math. (IF 0.6) Pub Date : 2024-01-17 Valentin Boboc
We provide a full classification of the slope stability of the cotangent bundles of relatively minimal smooth Weierstrass fibrations. The classification only depends on the topological Euler characteristic of the surface and the genus of the base curve.
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Tame symmetric algebras of period four Arch. Math. (IF 0.6) Pub Date : 2024-01-17 Karin Erdmann, Adam Hajduk, Adam Skowyrski
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Weight $$\mathbf {1/2}$$ multiplier systems for the group $$\mathbf {\Gamma _0^+({\varvec{p}})}$$ and a geometric formulation Arch. Math. (IF 0.6) Pub Date : 2024-01-16
Abstract We construct a weight 1/2 multiplier system for the group \(\Gamma _0^+(p)\) , the normalizer of the congruence subgroup \(\Gamma _0(p)\) where p is an odd prime, and we define an analogue of the eta function and Rademacher symbol and relate it to the geometry of edge paths in a triangulation of the upper half-plane.
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Monogenity of iterates of polynomials Arch. Math. (IF 0.6) Pub Date : 2024-01-16
Abstract In this article, we study the monogenity of a tower of number fields defined by the iterates of a stable polynomial. We give a necessary condition for the monogenity of the number fields defined by the iterates of a stable polynomial. When the stable polynomial is of certain type, we also give a sufficient condition for the monogenity of the fields defined by each of its iterate. As a consequence
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On conciseness of the word in Olshanskii’s example Arch. Math. (IF 0.6) Pub Date : 2024-01-13
Abstract A group-word w is called concise if the verbal subgroup w(G) is finite whenever w takes only finitely many values in a group G. It is known that there are words that are not concise. In particular, Olshanskii gave an example of such a word, which we denote by \(w_o\) . The problem whether every word is concise in the class of residually finite groups remains wide open. In this note, we observe
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Partition of quadratic residues and non-residues in $$\mathbb {Z}_p^*$$ for an odd prime p Arch. Math. (IF 0.6) Pub Date : 2024-01-12 Yathirajsharma M.V., Manjunatha M.R.
Let p be an odd prime. In this article, we investigate the number of ways in which a quadratic residue and a non-residue modulo p can be expressed as sum of two quadratic residues sum of two quadratic non-residues, and sum of a quadratic residue and non-residue in an elementary way using Gauss sums.
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$$L_p$$ Blaschke–Santaló and Petty projection inequalities in Gaussian space Arch. Math. (IF 0.6) Pub Date : 2024-01-12 Junjie Shan, Wenxue Xu, Leiqin Yin
The analogues of the \(L_p\) mixed volumes inequality, the \(L_p\) Brunn–Minkowski inequality, the \(L_p\) Blaschke–Santaló inequality, and the \(L_p\) Petty projection inequality in Gaussian space are established.
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Characters of prime power degree in principal blocks Arch. Math. (IF 0.6) Pub Date : 2024-01-03
Abstract We describe finite groups whose principal block contains only characters of prime power degree.
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On the optimal effective stability bounds for quasi-periodic tori of finitely differentiable and Gevrey Hamiltonians Arch. Math. (IF 0.6) Pub Date : 2023-12-19 Gerard Farré
It is known that a Diophantine quasi-periodic torus with frequency \(\omega \in \Omega _{\tau }^d\) of a \(C^{l}\) Hamiltonian is effectively stable for a time T(r) that is polynomial on the inverse of the distance to the torus, that we denote by r, with exponent \(1+(l-2)/(\tau +1)\). It is also known that a Diophantine quasi-periodic torus of a Gevrey Hamiltonian \(H\in G^{\alpha ,L}\) is effectively
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Projective closure of Gorenstein monomial curves and the Cohen–Macaulay property Arch. Math. (IF 0.6) Pub Date : 2023-12-11 Anargyros Katsabekis
Let \(C(\textbf{a})\) be a Gorenstein non-complete intersection monomial curve in the 4-dimensional affine space. There is a vector \(\textbf{v} \in {\mathbb {N}}^{4}\) such that for every integer \(m \ge 0\), the monomial curve \(C(\textbf{a}+m\textbf{v})\) is Gorenstein non-complete intersection whenever the entries of \(\textbf{a}+m\textbf{v}\) are relatively prime. In this paper, we study the arithmetically
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On Laurent polynomial identities Arch. Math. (IF 0.6) Pub Date : 2023-12-11 Ramon Códamo
Let F be a field and denote \(\mathcal {F}_L=F\langle X_1^{\pm 1},X_2^{\pm 1},\ldots \rangle \) the group algebra of the free group freely generated by the \(X_i^{\pm 1}\). Its elements are the (non-commutative) Laurent polynomials in several variables. For an associative unitary algebra R, we denote by U(R) the group of its units. An element of \(\mathcal {F}_L\) is a Laurent polynomial identity (LPI)
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Euclidean minima of algebraic number fields Arch. Math. (IF 0.6) Pub Date : 2023-12-11 Artūras Dubickas, Min Sha, Igor E. Shparlinski
In this paper, we use some of our previous results to improve an upper bound of Bayer–Fluckiger, Borello, and Jossen on the Euclidean minima of algebraic number fields. Our bound depends on the degree n of the field, its signature, discriminant, and the Hermite constant in dimension n.
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Nittka’s invariance criterion and Hilbert space valued parabolic equations in $$L_p$$ Arch. Math. (IF 0.6) Pub Date : 2023-11-30 W. Arendt, A. F. M. ter Elst, M. Sauter
Nittka gave an efficient criterion on a form defined on \(L_2(\Omega )\) which implies that the associated semigroup is \(L_p\)-invariant for some given \(p \in (1,\infty )\). We extend this criterion to the Hilbert space valued \(L_2(\Omega ,H)\). As an application, we consider elliptic systems of purely second order. Our main result shows that the induced semigroup is \(L_p\)-contractive for all
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A note on starshaped hypersurfaces with almost constant mean curvature in space forms Arch. Math. (IF 0.6) Pub Date : 2023-11-30 Julien Roth, Abhitosh Upadhyay
We show that closed starshaped hypersurfaces of space forms with almost constant mean curvature or almost constant higher order mean curvature are close to geodesic spheres.
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On the zeta functions of supersingular isogeny graphs and modular curves Arch. Math. (IF 0.6) Pub Date : 2023-11-29 Antonio Lei, Katharina Müller
Let p and q be distinct prime numbers with \(q\equiv 1\pmod {12}\). Let N be a positive integer that is coprime to pq. We prove a formula relating the Hasse–Weil zeta function of the modular curve \(X_0(qN)_{\mathbb {F}_q}\) to the Ihara zeta function of the p-isogeny graph of supersingular elliptic curves defined over \(\overline{\mathbb {F}_q}\) equipped with a \(\Gamma _0(N)\)-level structure. When
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A-ergodicity of probability measures on locally compact groups Arch. Math. (IF 0.6) Pub Date : 2023-11-29 Heybetkulu Mustafayev
Let G be a locally compact group with the left Haar measure \(m_{G}\) and let \(A=\left[ a_{n,k}\right] _{n,k=0}^{\infty }\) be a strongly regular matrix. We show that if \(\mu \) is a power bounded measure on G, then there exists an idempotent measure \(\theta _{\mu }\) such that $$\begin{aligned} \text {w*-}\lim _{n\rightarrow \infty }\sum _{k=0}^{\infty }a_{n,k}\mu ^{k}=\theta _{\mu }. \end{aligned}$$
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Domination of semigroups on standard forms of von Neumann algebras Arch. Math. (IF 0.6) Pub Date : 2023-11-28 Sahiba Arora, Ralph Chill, Sachi Srivastava
Consider \((T_t)_{t\ge 0}\) and \((S_t)_{t\ge 0}\) as real \(C_0\)-semigroups generated by closed and symmetric sesquilinear forms on a standard form of a von Neumann algebra. We provide a characterisation for the domination of the semigroup \((T_t)_{t\ge 0}\) by \((S_t)_{t\ge 0}\), which means that \(-S_t v\le T_t u\le S_t v\) holds for all \(t\ge 0\) and all real u and v that satisfy \(-v\le u\le
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Terms of recurrence sequences in the solution sets of norm form equations Arch. Math. (IF 0.6) Pub Date : 2023-11-23 Lajos Hajdu, Péter Sebestyén
The structure as well as several arithmetic properties of the solution sets of norm form equations are of classical and recent interest. In this paper, we give a finiteness result for terms of linear recurrence sequences appearing in the coordinates of solutions of norm form equations. Our main theorem yields a common generalization of certain recent results from the literature.
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The extremal problem for weighted combined energy Arch. Math. (IF 0.6) Pub Date : 2023-11-24 Yan Yang, Ruyue Tang, Xiaogao Feng
We study the extremal problem for weighted combined energy between two concentric annuli and obtain that the extremal mapping is a certain radial mapping. This extends the result obtained by Kalaj (J. Differential Equations, 268(2020)) to a non-Euclidean version. Meanwhile, we get a \(\frac{1}{|w|^{2}}\)-Nitsche type inequality, which generalizes the result in Arch. Math., 107(2016). Furthermore, based
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A unified observability result for non-autonomous observation problems Arch. Math. (IF 0.6) Pub Date : 2023-11-24 Fabian Gabel, Albrecht Seelmann
A final-state observability result in the Banach space setting for non-autonomous observation problems is obtained that covers and extends all previously known results in this context, while providing a streamlined proof that follows the established Lebeau-Robbiano strategy.
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Birational invariance of $$H^1({\mathcal {O}})$$ Arch. Math. (IF 0.6) Pub Date : 2023-11-23 Rémi Lodh
We show a vanishing result for the first direct image of a proper birational morphism of normal quasi-excellent schemes with regular codomain. As a consequence, we deduce the birational invariance of \(H^1(X,{\mathcal {O}}_X)\) for regular quasi-excellent schemes X. More generally, this is shown for quasi-excellent X which are \(S_3\) and pseudo-rational in codimension 2.
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$$\Gamma $$ -conjugate weight enumerators and invariant theory Arch. Math. (IF 0.6) Pub Date : 2023-11-18 Gabriele Nebe, Leonie Scheeren
Let K be a field, \(\Gamma \) a finite group of field automorphisms of K, F the \(\Gamma \)-fixed field in K, and \(G\le {{\,\textrm{GL}\,}}_v(K)\) a finite matrix group. Then the action of \(\Gamma \) defines a grading on the symmetric algebra of the F-space \(K^v\) which we use to introduce the notion of homogeneous \(\Gamma \)-conjugate invariants of G. We apply this new grading in invariant theory
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Orthogonal determinants of $${\textrm{SL}\,}_3(q)$$ and $${\textrm{SU}\,}_3(q)$$ Arch. Math. (IF 0.6) Pub Date : 2023-11-19 Linda Hoyer, Gabriele Nebe
We give a full list of the orthogonal determinants of the even degree indicator ‘+’ ordinary irreducible characters of \(\textrm{SL}_3(q)\) and \(\textrm{SU}_3(q)\).
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Generalized torsion elements in groups Arch. Math. (IF 0.6) Pub Date : 2023-11-16 Raimundo Bastos, Csaba Schneider, Danilo Silveira
A group element is called a generalized torsion element if a finite product of its conjugates is equal to the identity. We prove that in a nilpotent or FC-group, the generalized torsion elements are all torsion elements. Moreover, we compute the generalized order of an element in a finite group G using its character table.
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(In)decomposability of finite solutions of the Yang-Baxter equation Arch. Math. (IF 0.6) Pub Date : 2023-11-10 Arpan Kanrar
We extend a result of Ramirez and Vendramin on the decomposability of a finite non-degenerate involutive solution of the Yang-Baxter equation. We exhibit examples showing that the conditions for (in)decomposability in our obtained results are not necessary. Furthermore, we answer a question of Ramirez and Vendramin affirmatively in case the permutation group is nilpotent.
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Almost sure summability of the maximal normed partial sums of m-dependent random elements in Banach spaces Arch. Math. (IF 0.6) Pub Date : 2023-11-03 Lê Vǎn Thành
This paper provides sharp sufficient conditions for almost sure summability of the maximal normed partial sums of m-dependent random elements in stable type p Banach spaces, complementing recent results of Li, Qi, and Rosalsky (Trans Amer Math Soc 368(1):539–561, 2016) and Thành (Math Nachr 296(1):402–423, 2023). The main theorems are new even when the underlying random elements are independent. Our
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A criterion for nilpotency in finite groups Arch. Math. (IF 0.6) Pub Date : 2023-11-02 Binbin Li, Jiakuan Lu, Linna Pang, Boru Zhang
For a positive integer n, we denote by \(\pi (n)\) the set of prime divisors of n. For a group G and \(a \in G\), we denote by o(a) the order of the element a. We prove that a finite group G is nilpotent if and only if \(\pi (o(ab))=\pi (o(a)o(b))\) for all a, \(b\in G\) of coprime orders.
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On initial boundary value problems for the compressible Navier–Stokes system with temperature dependent heat conductivity Arch. Math. (IF 0.6) Pub Date : 2023-10-31 Wenchao Dong
This paper studies initial boundary value problems, including boundary damping, for the equations of a viscous, heat-conducting, one-dimensional ideal polytropic gas. The existence of a global strong (or classical) solution for the compressible Navier–Stokes system with temperature dependent heat conductivity is established. It can be regarded as a natural generalization of Nagasawa (J Differ Equ 65(1):49–67
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Celebrating Loday’s associahedron Arch. Math. (IF 0.6) Pub Date : 2023-10-31 Vincent Pilaud, Francisco Santos, Günter M. Ziegler
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Linear and quasilinear evolution equations in the context of weighted $$L_p$$ -spaces Arch. Math. (IF 0.6) Pub Date : 2023-11-01 Mathias Wilke
In 2004, the article Maximal regularity for evolution equations in weighted \(L_p\)-spaces by Prüss and Simonett (Arch Math 82:415–431, 2004) has been published in Archiv der Mathematik. We provide a survey of the main results of that article and outline some applications to semilinear and quasilinear parabolic evolution equations which illustrate their power.
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Characterization by approximation of homogeneous Besov and Triebel–Lizorkin type spaces Arch. Math. (IF 0.6) Pub Date : 2023-10-13 Madani Moussai
By approximation with sequences in a certain set of distributions modulo polynomials, we characterize the homogeneous Besov and Triebel–Lizorkin type spaces, then we obtain equivalent quasi-norms defined by the infimum on such a set. The same result holds for the classical homogeneous Besov and Triebel–Lizorkin spaces.
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Ricci curvature and the size of initial data for the Navier–Stokes equations on Einstein manifolds Arch. Math. (IF 0.6) Pub Date : 2023-10-11 Thieu Huy Nguyen, Thi Ngoc Ha Vu
Consider a noncompact Einstein manifold (M, g) with negative Ricci curvature tensor (\({\textrm{Ric}}_{ij}=rg_{ij}\) for a curvature constant \(r<0\)). Denoting by \(\Gamma (TM)\) the set of all vector fields on M, we study the Navier–Stokes equations $$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u + \nabla _u u + {\text {grad}}\pi = {\text {div}}(\nabla u + \nabla u^t)^{\sharp },\, {\text
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Locally conformally symplectic deformation of Gromov non-squeezing Arch. Math. (IF 0.6) Pub Date : 2023-10-09 Yasha Savelyev
We prove one deformation theoretic extension of the Gromov non-squeezing phenomenon to \({{\,\textrm{lcs}\,}}\) structures, or locally conformally symplectic structures, which suitably generalize both symplectic and contact structures. We also conjecture an analogue in \({{\,\textrm{lcs}\,}}\) geometry of contact non-squeezing of Eliashberg–Polterovich and discuss other related questions.
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On the uniqueness of eigenfunctions for the vectorial p-Laplacian Arch. Math. (IF 0.6) Pub Date : 2023-10-09 Ryan Hynd, Bernd Kawohl, Peter Lindqvist
We study a nonlinear eigenvalue problem for vector-valued eigenfunctions and give a succinct uniqueness proof for minimizers of the associated Rayleigh quotient.
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On Wolfgang Lusky’s paper “The Gurarij spaces are unique” Arch. Math. (IF 0.6) Pub Date : 2023-10-09 Dirk Werner
This note surveys Wolfgang Lusky’s proof of uniqueness of the Gurariy spaces and mentions further developments.
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Bounds on the higher degree Erdős–Ginzburg–Ziv constants over $${\mathbb {F}}_q^n$$ Arch. Math. (IF 0.6) Pub Date : 2023-10-07 Simone Costa, Stefano Della Fiore
The classical Erdős–Ginzburg–Ziv constant of a group G denotes the smallest positive integer \(\ell \) such that any sequence S of length at least \(\ell \) contains a zero-sum subsequence of length \(\exp (G).\) In the recent paper (Integers 22: Paper No. A102, 17 pp., 2022), Caro and Schmitt generalized this concept, using the m-th degree symmetric polynomial \(e_m(S)\) instead of the sum of the
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On non-surjective word maps on $$\textrm{PSL}_{2}(\mathbb {F}_{q})$$ Arch. Math. (IF 0.6) Pub Date : 2023-10-06 Arindam Biswas, Jyoti Prakash Saha
Jambor–Liebeck–O’Brien showed that there exist non-proper-power word maps which are not surjective on \(\textrm{PSL}_{2}(\mathbb {F}_{q})\) for infinitely many q. This provided the first counterexamples to a conjecture of Shalev which stated that if a two-variable word is not a proper power of a non-trivial word, then the corresponding word map is surjective on \(\textrm{PSL}_2(\mathbb {F}_{q})\) for
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Baire category and the relative growth rate for partial quotients in continued fractions Arch. Math. (IF 0.6) Pub Date : 2023-09-26 Xinyi Chang, Yihan Dong, Mengchen Liu, Lei Shang
Let \([a_1(x),a_2(x),\ldots ,a_n(x),\ldots ]\) be the continued fraction expansion of an irrational number \(x\in (0,1)\), and \(q_n(x)\) be the denominator of its n-th convergent. In this note, the Baire category of the set $$\begin{aligned} E(\alpha ,\beta ):= & {} \left\{ x\in (0,1)\backslash \mathbb {Q}:\liminf _{n \rightarrow \infty }\frac{\log a_{n+1}(x)}{\log q_n(x)} =\alpha ,\right. \\ {}{}
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Linear independence of certain numbers in the base-b number system Arch. Math. (IF 0.6) Pub Date : 2023-09-19 Shintaro Murakami, Yohei Tachiya
Let \((i,j)\in {\mathbb {N}}\times {\mathbb {N}}_{\ge 2}\) and \(S_{i,j}\) be an infinite subset of positive integers including all prime numbers in some arithmetic progression. In this paper, we prove the linear independence over \({\mathbb {Q}}\) of the numbers $$\begin{aligned} 1, \quad \sum _{n\in S_{i,j}}^{}\frac{a_{i,j}(n)}{b^{in^j}},\quad (i,j)\in {\mathbb {N}}\times {\mathbb {N}}_{\ge 2}, \end{aligned}$$
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Correction to: On the finite index subgroups of Houghton’s groups Arch. Math. (IF 0.6) Pub Date : 2023-09-19 Charles Garnet Cox
This note resolves an issue raised by Prof. Derek Holt for the paper “On the finite index subgroups of Houghton’s groups”.
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Boundedness and compactness of weighted composition operators and monomial operators Arch. Math. (IF 0.6) Pub Date : 2023-09-15 I. Chalendar, J. R. Partington
This paper characterises the boundedness and compactness of Agler–McCarthy monomial operators by reducing them to weighted composition operators and deriving explicit Carleson measure criteria on the half-plane. The results are illustrated by examples.