Authors:Bella; Angelo, Spadaro, Santi Pages: 197 - 203 Abstract: We present a result about covers of a Hausdorff space that implies various known cardinal inequalities, including the following two fundamental results in the theory of cardinal invariants in topology: (Arhangel’skiĭ) and (Hajnal–Juhász). This solves a question that goes back to Bell, Ginsburg and Woods’s 1978 paper (M. Bell, J.N. Ginsburg and R.G. Woods, Cardinal inequalities for topological spaces involving the weak Lindelöf number, Pacific J. Math. 79(1978), 37–45) and is mentioned in Hodel’s survey on Arhangel’skiĭ’s Theorem (R. Hodel, Arhangel’skii’s solution to Alexandroff’s problem: A survey, Topology Appl. 153(2006), 2199–2217).In contrast to previous attempts, we do not need any separation axiom beyond . PubDate: 2020-01-06 DOI: 10.4153/S0008439519000420

Authors:Ballico; E. Pages: 1 - 5 Abstract: We prove the existence of a smooth and non-degenerate curve , , with , , , and general moduli for all such that . It was proved by C. Walter that, for , the inequality is a necessary condition for the existence of a curve with . PubDate: 2019-08-30 DOI: 10.4153/S0008439519000146

Authors:Bavula; V. V., Levandovskyy, V. Pages: 6 - 12 Abstract: The Dixmier Conjecture says that every endomorphism of the (first) Weyl algebra (over a field of characteristic zero) is an automorphism, i.e., if for some , then . The Weyl algebra is a -graded algebra. We prove that the Dixmier Conjecture holds if the elements and are sums of no more than two homogeneous elements of (there is no restriction on the total degrees of and ). PubDate: 2019-08-30 DOI: 10.4153/S0008439519000122

Authors:Bramburger; Jason J. Pages: 13 - 30 Abstract: It is now well known that ultracontractive properties of semigroups with infinitesimal generator given by an undirected graph Laplacian operator can be obtained through an understanding of the geometry of the underlying infinite weighted graph. The aim of this work is to extend these results to semigroups with infinitesimal generators given by a directed graph Laplacian operator through an analogous inspection of the geometry of the underlying directed graph. In particular, we introduce appropriate nomenclature to discuss the geometry of an infinite directed graph, as well as provide sufficient conditions to extend ultracontractive properties of undirected graph Laplacians to those of the directed variety. Such directed graph Laplacians can often be observed in the study of coupled oscillators, where recent work made explicit the link between synchronous patterns to systems of identically coupled oscillators and ultracontractive properties of undirected graph semigroups. Therefore, in this work we demonstrate the applicability of our results on directed graph semigroups by extending the aforementioned investigation beyond the idealized case of identically coupled oscillators. PubDate: 2019-10-31 DOI: 10.4153/S0008439519000390

Authors:Chatterjee; Tapas, Dhillon, Sonika Pages: 31 - 45 Abstract: In 1965, A. Livingston conjectured the -linear independence of logarithms of values of the sine function at rational arguments. In 2016, S. Pathak disproved the conjecture. In this article, we give a new proof of Livingston’s conjecture using some fundamental trigonometric identities. Moreover, we show that a stronger version of her theorem is true. In fact, we modify this conjecture by introducing a co-primality condition, and in that case we provide the necessary and sufficient conditions for the conjecture to be true. Finally, we identify a maximal linearly independent subset of the numbers considered in Livingston’s conjecture. PubDate: 2019-12-12 DOI: 10.4153/S0008439519000468

Authors:Colbois; Bruno, Girouard, Alexandre, Métras, Antoine Pages: 46 - 57 Abstract: Given a smooth compact hypersurface with boundary , we prove the existence of a sequence of hypersurfaces with the same boundary as , such that each Steklov eigenvalue tends to zero as tends to infinity. The hypersurfaces are obtained from by a local perturbation near a point of its boundary. Their volumes and diameters are arbitrarily close to those of , while the principal curvatures of the boundary remain unchanged. PubDate: 2019-12-12 DOI: 10.4153/S000843951900050X

Authors:Dillery; D. Scott, LaGrange, John D. Pages: 58 - 65 Abstract: With entries of the adjacency matrix of a simple graph being regarded as elements of , it is proved that a finite commutative ring with is a Boolean ring if and only if either or the eigenvalues (in the algebraic closure of ) corresponding to the zero-divisor graph of are precisely the elements of . This is achieved by observing a way in which algebraic behavior in a Boolean ring is encoded within Pascal’s triangle so that computations can be carried out by appealing to classical results from number theory. PubDate: 2019-11-04 DOI: 10.4153/S0008439519000365

Authors:Healy; Brendan Burns Pages: 66 - 76 Abstract: We make a few observations on the absence of geometric and topological rigidity for acylindrically hyperbolic and relatively hyperbolic groups. In particular, we demonstrate the lack of a well-defined limit set for acylindrical actions on hyperbolic spaces, even under the assumption of universality. We also prove a statement about relatively hyperbolic groups inspired by a remark by Groves, Manning, and Sisto about the quasi-isometry type of combinatorial cusps. Finally, we summarize these results in a table in order to assert a meta-statement about the decay of metric rigidity as the conditions on actions on hyperbolic spaces are loosened. PubDate: 2019-11-18 DOI: 10.4153/S0008439519000377

Authors:Lancien; Gilles, Petitjean, Colin, Procházka, Antonin Pages: 77 - 93 Abstract: In this note we prove that the Kalton interlaced graphs do not equi-coarsely embed into the James space nor into its dual . It is a particular case of a more general result on the non-equi-coarse embeddability of the Kalton graphs into quasi-reflexive spaces with a special asymptotic structure. This allows us to exhibit a coarse invariant for Banach spaces, namely the non-equi-coarse embeddability of this family of graphs, which is very close to but different from the celebrated property of Kalton. We conclude with a remark on the coarse geometry of the James tree space and of its predual. PubDate: 2019-12-16 DOI: 10.4153/S0008439519000535

Authors:Li; Yueyue, Tian, Yan, Du, Xiankun Pages: 94 - 105 Abstract: We present conditions for a set of matrices satisfying a permutation identity to be simultaneously triangularizable. As applications of our results, we generalize Radjavi’s result on triangularization of matrices with permutable trace and results by Yan and Tang on linear triangularization of polynomial maps. PubDate: 2019-09-18 DOI: 10.4153/S0008439519000250

Authors:Li; Songxiao, Liu, Junming, Yuan, Cheng Pages: 106 - 117 Abstract: We use the Carleson measure-embedding theorem for weighted Bergman spaces to characterize the positive Borel measures on the unit disc such that certain analytic function spaces of Dirichlet type are embedded (compactly embedded) in certain tent spaces associated with a measure . We apply these results to study Volterra operators and multipliers acting on the mentioned spaces of Dirichlet type. PubDate: 2019-07-22 DOI: 10.4153/S0008439519000201

Authors:Lučić; Danka, Pasqualetto, Enrico Pages: 118 - 140 Abstract: The main result of this paper is the following: any weighted Riemannian manifold , i.e., a Riemannian manifold endowed with a generic non-negative Radon measure , is infinitesimally Hilbertian, which means that its associated Sobolev space is a Hilbert space.We actually prove a stronger result: the abstract tangent module (à la Gigli) associated with any weighted reversible Finsler manifold can be isometrically embedded into the space of all measurable sections of the tangent bundle of that are -integrable with respect to .By following the same approach, we also prove that all weighted (sub-Riemannian) Carnot groups are infinitesimally Hilbertian. PubDate: 2019-09-27 DOI: 10.4153/S0008439519000328

Authors:Saito; Hiroki, Tanaka, Hitoshi, Watanabe, Toshikazu Pages: 141 - 156 Abstract: Block decomposition of spaces with weighted Hausdorff content is established for and the Fefferman–Stein type inequalities are shown for fractional integral operators and some variants of maximal operators. PubDate: 2019-12-16 DOI: 10.4153/S000843951900033X

Authors:San Antolín; A. Pages: 157 - 172 Abstract: We give a characterization of all Parseval wavelet frames arising from a given frame multiresolution analysis. As a consequence, we obtain a description of all Parseval wavelet frames associated with a frame multiresolution analysis. These results are based on a version of Oblique Extension Principle with the assumption that the origin is a point of approximate continuity of the Fourier transform of the involved refinable functions. Our results are written for reducing subspaces. PubDate: 2019-12-06 DOI: 10.4153/S0008439519000341

Authors:Scavia; Federico Pages: 173 - 186 Abstract: For any prime number and field , we characterize the -retract rationality of an algebraic -torus in terms of its character lattice. We show that a -torus is retract rational if and only if it is -retract rational for every prime , and that the Noether problem for retract rationality for a group of multiplicative type has an affirmative answer for if and only if the Noether problem for -retract rationality for has a positive answer for all . For every finite set of primes PubDate: 2019-07-18 DOI: 10.4153/S0008439519000079

Authors:Shparlinski; Igor E., Voloch, José Felipe Pages: 187 - 196 Abstract: We obtain a new lower bound on the size of the value set of a sparse polynomial over a finite field of elements when is prime. This bound is uniform with respect to the degree and depends on some natural arithmetic properties of the degrees of the monomial terms of and the number of these terms. Our result is stronger than those that can be extracted from the bounds on multiplicities of individual values in . PubDate: 2019-09-24 DOI: 10.4153/S0008439519000316

Authors:Suh; Young Jin, Kim, Gyu Jong Pages: 204 - 221 Abstract: We introduce the notion of Lie invariant structure Jacobi operators for real hypersurfaces in the complex quadric . The existence of invariant structure Jacobi operators implies that the unit normal vector field becomes -principal or -isotropic. Then, according to each case, we give a complete classification of real hypersurfaces in with Lie invariant structure Jacobi operators. PubDate: 2019-06-10 DOI: 10.4153/S0008439519000080

Authors:Sun; Hongyan Pages: 222 - 241 Abstract: We establish criteria for Orlicz–Besov extension/imbedding domains via (global) -regular domains that generalize the known criteria for Besov extension/imbedding domains. PubDate: 2019-07-01 DOI: 10.4153/S000843951900002X

Authors:Zubelevic; Oleg Pages: 242 - 255 Abstract: A Lagrangian system is considered. The configuration space is a non-compact manifold that depends on time. A set of periodic solutions has been found. PubDate: 2019-12-18 DOI: 10.4153/S0008439519000456