Showing papers in "Archiv der Mathematik in 2013"
TL;DR: In this paper, the authors considered a nonlinear problem on R^N in dimension N ≥ 2, where g is a superlinear, subcritical, possibly nonhomogeneous, odd nonlinearity.
Abstract: We consider a nonlinear problem on R^N in dimension N ≥ 2. Here g is a superlinear, subcritical, possibly nonhomogeneous, odd nonlinearity. We deal with the case where the associated functional is not bounded below on the L^2-unit sphere, and we show the existence of infinitely many solutions.
103 citations
TL;DR: In this article, the authors define almost Yamabe solitons as special conformal solutions of the Yamabe flow and obtain some rigidity results concerning Yamabe almost-solitons.
Abstract: The aim of this note is to define almost Yamabe solitons as special conformal solutions of the Yamabe flow. Moreover, we shall obtain some rigidity results concerning Yamabe almost solitons. Finally, we shall give some characterizations for homogeneous gradient Yamabe almost solitons.
81 citations
TL;DR: In this article, it was shown that a gradient almost Ricci soliton with Codazzi tensor has constant sectional curvature and is isometric to a Euclidean sphere and f is a height function.
Abstract: The aim of this paper is to prove that a gradient almost Ricci soliton \({(M^{n}, g,
abla f, \lambda )}\) whose Ricci tensor is Codazzi has constant sectional curvature. In particular, in the compact case, we deduce that (Mn, g) is isometric to a Euclidean sphere and f is a height function. Moreover, we also classify gradient almost Ricci solitons with constant scalar curvature R provided a suitable function achieves a maximum in Mn.
41 citations
TL;DR: In this paper, a new characterization for the boundedness of the product of differentiation and composition operators acting on Bloch-type spaces is given, and an estimate for its essential norm in terms of the sequence is given.
Abstract: In this paper, we give a new characterization for the boundedness of the product of differentiation and composition operators \({C_\varphi D^m}\) acting on Bloch-type spaces and obtain an estimate for its essential norm in terms of the sequence \({\{z^n\}^{\infty}_{n=1}}\), from which the sufficient and necessary condition of compactness of the operator \({C_\varphi D^m}\) follows immediately.
40 citations
TL;DR: In this paper, the largest size m(G) of a minimal generating set of a finite group G can be computed for a given group G and a finite set G of elements.
Abstract: We study how the largest size m(G) of a minimal generating set of a finite group G can be computed.
30 citations
TL;DR: In this article, the maximal circumradius of points on the sphere equals the radius of the sphere, and it is shown that a normed vector space is an inner product space if and only if all spheres are not degenerate.
Abstract: We give a new characterisation of inner product spaces amongst normed vector spaces in terms of the maximal circumradius of spheres. It turns out that a normed vector space is an inner product space if and only if all spheres are not degenerate, i.e., the maximal circumradius of points on the sphere equals the radius of the sphere.
26 citations
TL;DR: In this paper, it was shown that the Omega-invariant subspace A (Omega) of a n-compact homogeneous Riemannian manifold is a compact subgroup of isometries of M.
Abstract: Let M be a non-compact homogeneous Riemannian manifold, and let Omega be a compact subgroup of isometries of M. We show, under general conditions, that the Omega-invariant subspace A (Omega) of a n ...
24 citations
TL;DR: In this paper, the Sato-Tate equidistribution conjecture of Bruinier and Kohnen for the signs of a(n) was studied for integral weight modular forms and an unconditional proof was given for certain subfamilies of coefficients that are accessible via the Shimura lift.
Abstract: Let f be a cusp form of weight k + 1/2 and at most quadratic nebentype character whose Fourier coefficients a(n) are all real. We study an equidistribution conjecture of Bruinier and Kohnen for the signs of a(n). We prove this conjecture for certain subfamilies of coefficients that are accessible via the Shimura lift by using the Sato–Tate equidistribution theorem for integral weight modular forms. Firstly, an unconditional proof is given for the family {a(tp
2)}
p
, where t is a squarefree number and p runs through the primes. In this case, the result is in terms of natural density. To prove it for the family {a(tn
2)}
n
where t is a squarefree number and n runs through all natural numbers, we assume the existence of a suitable error term for the convergence of the Sato–Tate distribution, which is weaker than one conjectured by Akiyama and Tanigawa. In this case, the results are in terms of Dedekind–Dirichlet density.
24 citations
TL;DR: In this paper, it was shown that the tetrablock is a convex domain, and a new class of (\({\mathbb{C}}\)-convex domains is studied.
Abstract: In this short note, we show that the tetrablock is a \({\mathbb{C}}\)-convex domain. In the proof of this fact, a new class of (\({\mathbb{C}}\)-convex) domains is studied. The domains are natural candidates to study on them the behavior of holomorphically invariant functions.
24 citations
TL;DR: In this article, the authors constructed traveling waves of the fractional bistable equation by approximating it with a fractional Laplacian, where the solution is obtained by passing to the limit.
Abstract: We construct traveling waves of the fractional bistable equation by approximating the fractional Laplacian $${(D^{2})^{\alpha}, \alpha \in (0, 1)}$$
, with operators $${J \ast u - (\int_{R} J)u}$$
, where J is nonsingular. Since the resulting approximating equations are known to have traveling waves, the solutions are obtained by passing to the limit. This provides an answer to the statement (about existence and properties) “This construction will be achieved in a future work” before Assumption 2 in Imbert and Souganidis [6]. With a modification of a part of the argument, we also get the existence of traveling waves for the ignition nonlinearity in the case $${\alpha \in (1/2, 1)}$$
.
21 citations
TL;DR: In this article, the Harborth constant of a finite abelian group is defined as the smallest integer such that each subset of the group has a subset of cardinality equal to the exponent of the groups whose elements sum to the neutral element.
Abstract: The Harborth constant of a finite abelian group is the smallest integer $${\ell}$$
such that each subset of G of cardinality $${\ell}$$
has a subset of cardinality equal to the exponent of the group whose elements sum to the neutral element of the group. The plus-minus weighted analogue of this constant is defined in the same way except that instead of considering the sum of all elements of the subset, one can choose to add either the element or its inverse. We determine these constants for certain groups, mainly groups that are the direct sum of a cyclic group and a group of order 2. Moreover, we contrast these results with existing results and conjectures on these problems.
TL;DR: In this paper, it was shown that the spectrum of the Cesaro operator acting in lp, for 1 < p < ∞, consists of the closed disc centered at q/2 and with radius q 2, where q is the conjugate index of p.
Abstract: We present a novel proof of the fact that the spectrum of the Cesaro operator acting in lp, for 1 < p < ∞, consists of the closed disc centered at q/2 and with radius q/2, where q is the conjugate index of p.
TL;DR: In this article, it was shown that the weighted Bergman kernel function is isometrically isomorphic to the subspace H of a proper holomorphic mapping, and that the tetrablock is not a Lu Qi-Keng domain.
Abstract: We consider proper holomorphic maps \({\pi : D\rightarrow G}\), where D and G are domains in \({\mathbb{C}^{n}}\). Let \({\alpha\in \mathcal{C}(G,\mathbb{R}_{ > 0})}\). We show that every π induces some subspace H of \({\mathbb{A}^{2}_{\alpha\circ\pi}(D)}\) such that \({\mathbb{A}^{2}_{\alpha}(G)}\) is isometrically isomorphic to H via some unitary operator Γ. Using this isomorphism we construct the orthogonal projection onto H, and we derive Bell’s transformation formula for the weighted Bergman kernel function under proper holomorphic mappings. As a consequence of the formula, we get that the tetrablock is not a Lu Qi-Keng domain.
TL;DR: The main result of as discussed by the authors is that no Specht module labelled by a hook partition (ab − r, 1r) with r ≥ 1 appears in \({H^{(a^b)}}\).
Abstract: The Foulkes module \({H^{(a^b)}}\) is the permutation module for the symmetric group Sab given by the action of Sab on the collection of set partitions of a set of size ab into b sets each of size a. The main result of this paper is a sufficient condition for a simple \({\mathbb{C} S_{ab}}\) -module to have zero multiplicity in \({H^{(a^b)}}\) . A special case of this result implies that no Specht module labelled by a hook partition (ab − r, 1r) with r ≥ 1 appears in \({H^{(a^b)}}\) .
TL;DR: In this article, the authors revisited the work of Andrews, Jimenez-Urroz, and Ono to construct a natural vector-valued quantum modular form whose components are similarly "strange".
Abstract: Since their definition in 2010 by Zagier, quantum modular forms have been connected to numerous topics such as strongly unimodal sequences, ranks, cranks, and asymptotics for mock theta functions near roots of unity. These are functions that are not necessarily defined on the upper half plane but a priori are defined only on a subset of \({\mathbb{Q}}\), and whose obstruction to modularity is some analytically “nice” function. Motivated by Zagier’s example of the quantum modularity of Kontsevich’s “strange” function F(q), we revisit work of Andrews, Jimenez-Urroz, and Ono to construct a natural vector-valued quantum modular form whose components are similarly “strange”.
TL;DR: In this article, a general definition of exponential monomials on general commutative hypergroups has been given, which covers the known special cases of polynomial and Sturm-Liouville-hypergroups.
Abstract: Polynomials and exponential polynomials play a fundamental role in the theory of spectral analysis and spectral synthesis on commutative groups. Recently several new results have been published in this field [2, 3, 4,6]. Spectral analysis and spectral synthesis has been studied on some types of commutative hypergroups, as well. However, a satisfactory definition of exponential monomials on general commutative hypergroups has not been available so far. In [5,7,8] and [9], the authors use a special concept on polynomial and Sturm–Liouville-hypergroups. Here we give a general definition which covers the known special cases.
TL;DR: In this article, it was shown that for any > 0, the sum of the sums of the following sums for any = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 28, 30, 31, 34, 35, 36, 38
Abstract: Let φ be a primitive Maass cusp form and t
φ
(n) be its nth Fourier coefficient at the cusp infinity. In this short note, we are interested in the estimation of the sums $${\sum_{n \leq x}t_{\varphi}(n)}$$
and $${\sum_{n \leq x}t_{\varphi}(n^2)}$$
. We are able to improve the previous results by showing that for any $${\varepsilon > 0}$$
$$\sum_{n \leq x}t_{\varphi}(n) \ll\, _{\varphi, \varepsilon} x^{\frac{1027}{2827} + \varepsilon}
\quad {and}\quad\sum_{n \leq x}t_{\varphi}(n^2) \ll\,_{\varphi, \varepsilon} x^{\frac{489}{861} + \varepsilon}.$$
TL;DR: In this article, it was shown that every non-central subnormal subgroup of the multiplicative group of a weakly locally finite division ring contains a non-cyclic free subgroup.
Abstract: A division ring D is said to be weakly locally finite if for every finite subset \({S \subset D}\), the division subring of D generated by S is centrally finite. It is known that the class of weakly locally finite division rings strictly contains the class of locally finite division rings. In this note we prove that every non-central subnormal subgroup of the multiplicative group of a weakly locally finite division ring contains a non-cyclic free subgroup. This generalizes the previous result by Goncalves for centrally finite division rings.
TL;DR: In this article, the authors considered a closed Riemannian manifold M with a time-dependent metric evolving by a symmetric two-tensor and proved differential Harnack inequalities for positive solutions of heat equations with potentials on M,g(t).
Abstract: In the paper we consider a closed Riemannian manifold M with a time-dependent Riemannian metric g
ij
(t) evolving by ∂
t
g
ij
= −2S
ij
, where S
ij
is a symmetric two-tensor on (M,g(t)). We prove some differential Harnack inequalities for positive solutions of heat equations with potentials on (M,g(t)). Some applications of these inequalities will be obtained.
TL;DR: In this paper, the authors obtained sharp reverse estimates for the logarithmic Bloch spaces on the unit disk and studied composition operators with values in the space BMOA.
Abstract: We obtain sharp reverse estimates for the logarithmic Bloch spaces on the unit disk. As an application, we study composition operators with values in the space BMOA.
TL;DR: In this paper, the cone Psd(K) (of polynomials nonnegative on K) is included in the closure of a locally convex topology on the countable dimensional polynomial algebra.
Abstract: Let τ be a locally convex topology on the countable dimensional polynomial \({\mathbb{R}}\) -algebra \({\mathbb{R} [\underline{X}] := \mathbb{R} [X_1, \ldots, X_{n}]}\) . Let K be a closed subset of \({\mathbb{R} ^{n}}\) , and let \({M := M_{\{g_1, \ldots, g_s\}}}\) be a finitely generated quadratic module in \({\mathbb{R} [\underline{X}]}\) . We investigate the following question: When is the cone Psd(K) (of polynomials nonnegative on K) included in the closure of M? We give an interpretation of this inclusion with respect to representing continuous linear functionals by measures. We discuss several examples; we compute the closure of \({M = \sum \mathbb{R} [\underline{X}]^{2}}\) with respect to weighted norm-p topologies. We show that this closure coincides with the cone Psd(K) where K is a certain convex compact polyhedron.
TL;DR: In this paper, it was shown that on a complete smooth metric measure space with non-negative Bakry-Emery-Ricci curvature, the weighted volume of the manifold is finite and the universal cover splits isometrically as the product of the real line with a hypersurface.
Abstract: In this paper we prove that on a complete smooth metric measure space with non-negative Bakry–Emery–Ricci curvature if the space of weighted L2 harmonic one-forms is non-trivial, then the weighted volume of the manifold is finite and the universal cover of the manifold splits isometrically as the product of the real line with a hypersurface
TL;DR: In this paper, the largest size of a minimal generating set of a finite group G is estimated in the case when N is the unique minimal normal subgroup of G, and the difference m(G) − m (G/N) is the difference between the largest generating set and the smallest generating set.
Abstract: Denote by m(G) the largest size of a minimal generating set of a finite group G. We want to estimate the difference m(G) − m(G/N) in the case when N is the unique minimal normal subgroup of G.
TL;DR: In this article, a new function from the bounded derived category of a finite dimensional algebra over a field to the set of all natural numbers is introduced, which is a generalized version of the Igusa-Todorov function.
Abstract: We introduce a new function from the bounded derived category of a finite dimensional algebra over a field to the set of all natural numbers, which is a generalized version of the Igusa–Todorov function. Then we extend the results corresponding to the Igusa–Todorov function. As an application, we give a new proof of the finiteness of the finitistic dimension of special biserial algebras.
TL;DR: In this paper, the gradient blowup rate for the one-dimensional p-Laplacian parabolic equation with q > p > 2 was studied and lower and upper bounds were established.
Abstract: This paper is concerned with the gradient blowup rate for the one-dimensional p-Laplacian parabolic equation \({u_t=(|u_x|^{p-2} u_x)_x +|u_x|^q}\) with q > p > 2, for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain bounded. We establish the blowup rate estimates of lower and upper bounds and show that in this case the blowup rate does not match the self-similar one.
TL;DR: In this paper, the authors extend an old result of Rogers and Shephard, provide a new simple proof to a result of Reisner, and study a question related to Geometric Tomography, providing a characterization of central symmetry for convex bodies.
Abstract: We study some problems from various aspects of convexity, concerning shadow systems. Namely, we extend an old result of Rogers and Shephard, we provide a new simple proof to a result of Reisner, and we study a question related to Geometric Tomography, providing a characterization of central symmetry for convex bodies.
TL;DR: This article characterized hypercyclic weighted translation operators generated by non-torsion elements of a locally compact group, extending the recent results in Chen and Chu [6], Chen and Chen [7] and Chu and Chu[8].
Abstract: We characterize hypercyclic weighted translation operators generated by non-torsion elements of a locally compact group, extending the recent results in Chen and Chu [6], Chen and Chu [7].
TL;DR: In this article, it was shown that Stanley's conjecture holds for I and S/I if I is a product of monomial prime ideals or S is a high enough power of a polymatroidal or a stable ideal generated in a single degree.
Abstract: Let \({\mathbb{K}}\) be a field and \({S = \mathbb{K}[x_1,\dots,x_n]}\) be the polynomial ring in n variables over the field \({\mathbb{K}}\) In this paper, it is shown that Stanley’s conjecture holds for I and S/I if I is a product of monomial prime ideals or I is a high enough power of a polymatroidal or a stable ideal generated in a single degree
TL;DR: In this article, a uniform formula for the index of reducibility of parameter ideals of a finitely generated R-module was proposed. But it was only for polynomial types of R-modules.
Abstract: Let \({(R, \mathfrak{m})}\) be a Noetherian local ring, M a finitely generated R-module. The aim of this paper is to prove a uniform formula for the index of reducibility of parameter ideals of M provided the polynomial type of M is at most one.
TL;DR: In this article, the Schatten class weighted composition operators on Fock-Sobolev spaces and a large class of weighted Fock spaces are described, where the weights defining such spaces are radial, decay at least as fast as the classical Gaussian weight, and satisfy a mild smoothness condition.
Abstract: We describe the Schatten class weighted composition operators on Fock–Sobolev spaces and a large class of weighted Fock spaces, where the weights defining such spaces are radial, decay at least as fast as the classical Gaussian weight, and satisfy certain mild smoothness condition. To prove our main results, we characterize the Schatten class membership of the Toeplitz operators Tμ induced by nonnegative measures μ on the complex space \({\mathbb{C}^n}\) .