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Revista Matemática Complutense
Journal Prestige (SJR): 1.04

Citation Impact (citeScore): 1
Number of Followers: 0

Hybrid journal (It can contain Open Access articles)
ISSN (Print) 1139-1138 - ISSN (Online) 1988-2807
Published by Springer-Verlag

[2468 journals]

Correction to: A monotonicity result for the first Steklov–Dirichlet
Laplacian eigenvalue
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  PubDate: 2023-12-02
Non-negative solutions and strong maximum principle for a resonant
quasilinear problem
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  Abstract: Abstract We study the resonant quasilinear problem $$\begin{aligned} -\Delta _p u = \lambda _p u^{p-1} + \lambda g(u) \text { in } \Omega ,\;\; u\ge 0 \text { in } \Omega , \;\; u_{ \partial \Omega }=0, \end{aligned}$$ where $\Omega \subset {\mathbb {R}}^N$ is a smooth, bounded domain, $\lambda _p$ is the first eigenvalue of $-\Delta _p$ in $\Omega $ , and $g:[0,+\infty )\rightarrow {\mathbb {R}}$ is a continuous and subcritical term. By means of variational arguments, we prove the existence of non-negative solutions for any $\lambda >0$ ; positive solutions for sufficiently small $\lambda >0$ . Our results generalize the ones recently obtained by different techniques in the case $p=2$ .
  PubDate: 2023-10-03
Some recent developments on the Steklov eigenvalue problem
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  Abstract: Abstract The Steklov eigenvalue problem, first introduced over 125 years ago, has seen a surge of interest in the past few decades. This article is a tour of some of the recent developments linking the Steklov eigenvalues and eigenfunctions of compact Riemannian manifolds to the geometry of the manifolds. Topics include isoperimetric-type upper and lower bounds on Steklov eigenvalues (first in the case of surfaces and then in higher dimensions), stability and instability of eigenvalues under deformations of the Riemannian metric, optimisation of eigenvalues and connections to free boundary minimal surfaces in balls, inverse problems and isospectrality, discretisation, and the geometry of eigenfunctions. We begin with background material and motivating examples for readers that are new to the subject. Throughout the tour, we frequently compare and contrast the behavior of the Steklov spectrum with that of the Laplace spectrum. We include many open problems in this rapidly expanding area.
  PubDate: 2023-09-28
Correction to: Some q-supercongruences from the Gasper and Rahman
quadratic summation
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  PubDate: 2023-09-01
  DOI: 10.1007/s13163-023-00456-3
Correction: On the Michor'Mumford phenomenon in the infinite
dimensional Heisenberg group
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  PubDate: 2023-09-01
  DOI: 10.1007/s13163-023-00463-4
Deformations of vector bundles over Lie groupoids
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  Abstract: Abstract VB-groupoids are vector bundles in the category of Lie groupoids. They encompass several classical objects, including Lie group representations and 2-vector spaces. Moreover, they provide geometric pictures for 2-term representations up to homotopy of Lie groupoids. We attach to every VB-groupoid a cochain complex controlling its deformations and discuss its fundamental features, such as Morita invariance and a van Est theorem. Several examples and applications are given.
  PubDate: 2023-09-01
  DOI: 10.1007/s13163-022-00441-2
Existence and multiplicity of solutions for a singular anisotropic problem
with a sign-changing term
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  Abstract: Abstract In this paper we investigate the existence and multiplicity of solutions for a class of singular anisotropic problems involving a weight and a term that may change sign. The approach is based on sub-supersolutions and the Mountain Pass Theorem.
  PubDate: 2023-09-01
  DOI: 10.1007/s13163-022-00446-x
Stability results on the Kirchhoff plate equation with delay terms on the
dynamical boundary controls
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  Abstract: Abstract In this paper, we consider the Kirchhoff plate equation with delay terms on the dynamical boundary controls. We prove its well-posedness, strong stability, non-exponential stability, and polynomial stability under a multiplier geometric control condition.
  PubDate: 2023-09-01
  DOI: 10.1007/s13163-022-00437-y
D’Atri spaces and the total scalar curvature of hemispheres, tubes
and cylinders
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  Abstract: Abstract Csikós and Horváth proved in J Geom Anal 28(4): 3458-3476, (2018) that if a connected Riemannian manifold of dimension at least 4 is harmonic, then the total scalar curvatures of tubes of small radius about an arbitrary regular curve depend only on the length of the curve and the radius of the tube, and conversely, if the latter condition holds for cylinders, i.e., for tubes about geodesic segments, then the manifold is harmonic. In the present paper, we show that in contrast to the higher dimensional case, a connected 3-dimensional Riemannian manifold has the above mentioned property of tubes if and only if the manifold is a D’Atri space, furthermore, if the space has bounded sectional curvature, then it is enough to require the total scalar curvature condition just for cylinders to imply that the space is D’Atri. This result gives a negative answer to a question posed by Gheysens and Vanhecke. To prove these statements, we give a characterization of D’Atri spaces in terms of the total scalar curvature of geodesic hemispheres in any dimension.
  PubDate: 2023-09-01
  DOI: 10.1007/s13163-022-00444-z
Sturm attractors for fully nonlinear parabolic equations
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  Abstract: Abstract We explicitly construct global attractors of fully nonlinear parabolic equations in one spatial dimension. These attractors are decomposed as equilibria (time independent solutions) and heteroclinic orbits (solutions that converge to distinct equilibria backwards and forwards in time). In particular, we state necessary and sufficient conditions for the occurrence of heteroclinics between hyperbolic equilibria, which is accompanied by a method that computes such conditions.
  PubDate: 2023-09-01
  DOI: 10.1007/s13163-022-00435-0
Some q-supercongruences from the Gasper and Rahman quadratic summation
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  Abstract: Abstract We give four families of q-supercongruences modulo the square and cube of a cyclotomic polynomial from Gasper and Rahman’s quadratic summation. As conclusions, we obtain four new supercongruences modulo $p^2$ or $p^3$ , such as: for $d \ge 2, r \ge 1$ with $\gcd (d,r)=1$ and $d+r$ odd, and any prime $p\equiv d+r\pmod {2d}$ with $p\geqslant d+r$ , $$\begin{aligned} \sum _{k=0}^{p-1}(3dk+r)\frac{ (\frac{r}{d})_k (\frac{d-r}{d})_k (\frac{r}{2d})_k^2(\frac{1}{2})_k}{k!^4(\frac{d+2r}{2d})_k}\equiv 0\pmod {p^3}, \end{aligned}$$ where $(x)_n=x(x+1)\cdots (x+n-1)$ is the Pochhammer symbol. We also propose three related conjectures on q-supercongruences.
  PubDate: 2023-09-01
  DOI: 10.1007/s13163-022-00442-1
On the reflexivity of the spaces of variable integrability and summability
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  Abstract: Abstract In this paper, we show that under the condition $ 1<p_-, q_-, p_+, q_+<\infty $ , the space $\ell ^{q(\cdot )} (L^{p(\cdot )})$ is reflexive as long as $\ell ^{q(\cdot )} (L^{p(\cdot )})$ is a Banach space. In this way we give an answer to the open problem posed by Hästö in 2017 about the reflexivity of the variable mixed Lebesgue-sequence spaces $\ell ^{q(\cdot )} (L^{p(\cdot )})$ . What is important here is that the dual space of $\ell ^{q(\cdot )} (L^{p(\cdot )})$ is specified. As its direct corollary, we show that the corresponding Besov space $B^{s(\cdot )}_{p(\cdot )q(\cdot )}$ is reflexive.
  PubDate: 2023-09-01
  DOI: 10.1007/s13163-022-00447-w
On the Michor–Mumford phenomenon in the infinite dimensional
Heisenberg group
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  Abstract: Abstract In the infinite dimensional Heisenberg group, we construct a left invariant weak Riemannian metric that gives a degenerate geodesic distance. The same construction yields a degenerate sub-Riemannian distance. We show how the standard notion of sectional curvature adapts to our framework, but it cannot be defined everywhere and it is unbounded on suitable sequences of planes. The vanishing of the distance precisely occurs along this sequence of planes, so that the degenerate Riemannian distance appears in connection with an unbounded sectional curvature. In the 2005 paper by Michor and Mumford, this phenomenon was first observed in some specific Fréchet manifolds.
  PubDate: 2023-09-01
  DOI: 10.1007/s13163-022-00430-5
Kirby diagrams and 5-colored graphs representing compact 4-manifolds
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  Abstract: Abstract It is well-known that in dimension 4 any framed link (L, c) uniquely represents the PL 4-manifold $M^4(L,c)$ obtained from ${\mathbb {D}}^4$ by adding 2-handles along (L, c). Moreover, if trivial dotted components are also allowed (i.e. in case of a Kirby diagram $(L^{(*)},d)$ ), the associated PL 4-manifold $M^4(L^{(*)},d)$ is obtained from ${\mathbb {D}}^4$ by adding 1-handles along the dotted components and 2-handles along the framed components. In this paper we study the relationships between framed links and/or Kirby diagrams and the representation theory of compact PL manifolds by edge-colored graphs: in particular, we describe how to construct algorithmically a (regular) 5-colored graph representing $M^4(L^{(*)},d)$ , directly “drawn over” a planar diagram of $(L^{(*)},d)$ , or equivalently how to algorithmically obtain a triangulation of $M^4(L^{(*)},d)$ . As a consequence, the procedure yields triangulations for any closed (simply-connected) PL 4-manifold admitting handle decompositions without 3-handles. Furthermore, upper bounds for both the invariants gem-complexity and regular genus of $M^4(L^{(*)},d)$ are obtained, in terms of the combinatorial properties of the Kirby diagram.
  PubDate: 2023-09-01
  DOI: 10.1007/s13163-022-00438-x
On the Crawford number attaining operators
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  Abstract: Abstract We study the denseness of Crawford number attaining operators on Banach spaces. Mainly, we prove that if a Banach space has the Radon–Nikodým property, then the set of Crawford number attaining operators is dense in the space of bounded linear operators. We also see among others that the set of Crawford number attaining operators is dense in the space of all bounded linear operators while they do not coincide, by observing the case of compact operators when the Banach space has a 1-unconditional basis. Furthermore, we show a Bishop–Phelps–Bollobás type property for the Crawford number for certain Banach spaces, and we finally discuss some difficulties and possible problems on the topic.
  PubDate: 2023-09-01
  DOI: 10.1007/s13163-022-00445-y
Liouville theorem for Hénon-Hardy systems in the unit ball
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  Abstract: Abstract Let (u, v) be a nonnegative solution to the system $$\begin{aligned} (-\Delta )^\frac{\alpha }{2} u = x ^a v^p, \quad (-\Delta )^\frac{\beta }{2} v = x ^b u^q \end{aligned}$$ in the unit ball with the Navier boundary condition, where $\alpha ,\beta ,a,b$ are real numbers such that $\alpha ,\beta \in (0,n)$ , $a>-\alpha $ and $b>-\beta $ . By exploiting the method of scaling spheres in integral forms, we prove that if $p\ge \frac{n+\alpha +2a}{n-\beta }$ , $q\ge \frac{n+\beta +2b}{n-\alpha }$ and $(p,q) \ne (\frac{n+\alpha +2a}{n-\beta },\frac{n+\beta +2b}{n-\alpha })$ , then $(u,v)\equiv (0,0)$ .
  PubDate: 2023-09-01
  DOI: 10.1007/s13163-022-00443-0
Critical metrics for quadratic curvature functionals on some solvmanifolds
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  Abstract: Abstract We prove the existence of four-dimensional compact manifolds admitting some non-Einstein Lorentzian metrics, which are critical points for all quadratic curvature functionals. For this purpose, we consider left-invariant semi-direct extensions $G_{\mathcal S}=H \rtimes \exp ({\mathbb {R}}S)$ of the Heisenberg Lie group H, for any $\mathcal S \in {\mathfrak {s}}{\mathfrak {p}}(1,\mathbb R)$ , equipped with a family $g_a$ of left-invariant metrics. After showing the existence of lattices in all these four-dimensional solvable Lie groups, we completely determine when $g_a$ is a critical point for some quadratic curvature functionals. In particular, some four-dimensional solvmanifolds raising from these solvable Lie groups admit non-Einstein Lorentzian metrics, which are critical for all quadratic curvature functionals.
  PubDate: 2023-09-01
  DOI: 10.1007/s13163-022-00448-9
Convexity of ratios of the modified Bessel functions of the first kind
with applications
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  Abstract: Abstract Let $I_{\nu }\left( x\right) $ be the modified Bessel function of the first kind of order $\nu $ . Motivated by a conjecture on the convexity of the ratio $W_{\nu }\left( x\right) =xI_{\nu }\left( x\right) /I_{\nu +1}\left( x\right) $ for $\nu >-2$ , using the monotonicity rules for a ratio of two power series and an elementary technique, we present fully the convexity of the functions $W_{\nu }\left( x\right) $ , $W_{\nu }\left( x\right) -x^{2}/\left( 2\nu +4\right) $ and $W_{\nu }\left( x^{1/\theta }\right) $ for $\theta \ge 2$ on $\left( 0,\infty \right) $ in different value ranges of $\nu $ , which give an answer to the conjecture and extend known results. As consequences, some monotonicity results and new functional inequalities for $W_{\nu }\left( x\right) $ are established. As applications, an open problem and a conjectures are settled. Finally, a conjecture on the complete monotonicity of $W_{\nu }\left( x^{1/\theta }\right) $ for $\theta \ge 2$ is proposed.
  PubDate: 2023-09-01
  DOI: 10.1007/s13163-022-00439-w
Smooth norms in dense subspaces of $$\ell _p(\Gamma )$$ and operator
ranges
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  Abstract: Abstract For $1\le p<\infty $ , we prove that the dense subspace $\mathcal {Y}_p$ of $\ell _p(\Gamma )$ comprising all elements y such that $y \in \ell _q(\Gamma )$ for some $q \in (0,p)$ admits a $C^{\infty }$ -smooth norm which locally depends on finitely many coordinates. Moreover, such a norm can be chosen as to approximate the $\left\ \cdot \right\ _p$ -norm. This provides examples of dense subspaces of $\ell _p(\Gamma )$ with a smooth norm which have the maximal possible linear dimension and are not obtained as the linear span of a biorthogonal system. Moreover, when $p>1$ or $\Gamma $ is countable, such subspaces additionally contain dense operator ranges; on the other hand, no non-separable operator range in $\ell _1(\Gamma )$ admits a $C^1$ -smooth norm.
  PubDate: 2023-08-22
  DOI: 10.1007/s13163-023-00479-w
On the reduced Bernstein-Sato polynomial of Thom-Sebastiani singularities
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  Abstract: Abstract Given two holomorphic functions f and g defined in two respective germs of complex analytic manifolds (X, x) and (Y, y), we know thanks to M. Saito that, as long as one of them is Euler homogeneous, the reduced (or microlocal) Bernstein-Sato polynomial of the Thom-Sebastiani sum $f+g$ can be expressed in terms of those of f and g. In this note we give a purely algebraic proof of a similar relation between the whole functional equations that can be applied to any setting (not necessarily analytic) in which Bernstein-Sato polynomials can be defined.
  PubDate: 2023-08-10
  DOI: 10.1007/s13163-023-00478-x