Abstract: In this article we analyze the notions of amenability and paradoxical decomposition from an algebraic perspective. We consider this dichotomy for locally finite extended metric spaces and for general algebras over fields. In the context of algebras we also study the relation of amenability with proper infiniteness. We apply our general analysis to two important classes of algebras: the unital Leavitt path algebras and the translation algebras on locally finite extended metric spaces. In particular, we show that the amenability of a metric space is equivalent to the algebraic amenability of the corresponding translation algebra. PubDate: 2017-11-09

Abstract: For \(n\in \mathbb {N}\) the nth alternating harmonic number $$\begin{aligned} H_n^*:=\sum _{k=1}^n(-1)^{k-1}\frac{1}{k} \end{aligned}$$ is given in the form $$\begin{aligned} H_n^*=\ln 2 +\frac{(-1)^{n+1}}{4\left\lfloor \frac{n+1}{2}\right\rfloor } +\sum _{i=1}^{q-1}\frac{(4^i-1)B_{2i}}{(2i)\left( 2\left\lfloor \frac{n+1}{2}\right\rfloor \right) ^{2i}}+r_q(n) \end{aligned}$$ where \(q\in \mathbb {N}\) is a parameter controlling the magnitude of the error term \(r_q(n)\) estimated as $$\begin{aligned} 0< (-1)^{q+1}r_q(n)< \frac{ B_{2q} }{2q\cdot \left\lfloor \frac{n+1}{2}\right\rfloor ^{2q}} <2\frac{\exp \left( \frac{1}{24q}\right) }{1-2\cdot 4^{-q}}\sqrt{\frac{\pi }{q}} \left( \frac{q}{e\pi \left\lfloor \frac{n+1}{2}\right\rfloor }\right) ^{2q}. \end{aligned}$$ PubDate: 2017-08-29

Abstract: A nonlinear system with different fractional derivative terms is considered. The existence of positive blowing-up solutions is proved. PubDate: 2017-08-01

Abstract: We study the Nemytskii operators \(u\mapsto u \) and \(u\mapsto u^{\pm }\) in fractional Sobolev spaces \(H^s({\mathbb {R}}^n)\) , \(s>1\) . PubDate: 2017-06-28

Abstract: In this paper, we study semi-slant submanifolds and their warped products in Kenmotsu manifolds. The existence of such warped products in Kenmotsu manifolds is shown by an example and a characterization. A sharp relation is obtained as a lower bound of the squared norm of second fundamental form in terms of the warping function and the slant angle. The equality case is also considered in this paper. Finally, we provide some applications of our derived results. PubDate: 2017-04-10

Abstract: The Wintgen inequality (1979) is a sharp geometric inequality for surfaces in the 4-dimensional Euclidean space involving the Gauss curvature (intrinsic invariant) and the normal curvature and squared mean curvature (extrinsic invariants), respectively. De Smet et al. (Arch. Math. (Brno) 35:115–128, 1999) conjectured a generalized Wintgen inequality for submanifolds of arbitrary dimension and codimension in Riemannian space forms. This conjecture was proved by Lu (J. Funct. Anal. 261:1284–1308, 2011) and by Ge and Tang (Pac. J. Math. 237:87–95, 2008), independently. In the present paper we establish a generalized Wintgen inequality for statistical submanifolds in statistical manifolds of constant curvature. PubDate: 2017-04-01

Abstract: The principal aim of this paper is to derive an abstract form of the third Green identity associated with a proper extension T of a symmetric operator S in a Hilbert space \(\mathfrak {H}\) , employing the technique of quasi boundary triples for T. The general results are illustrated with couplings of Schrödinger operators on Lipschitz domains on smooth, boundaryless, compact Riemannian manifolds. PubDate: 2017-03-30

Abstract: At the turn of this century Durand, and Lagarias and Pleasants established that key features of minimal subshifts (and their higher-dimensional analogues) to be studied are linearly repetitive, repulsive and power free. Since then, generalisations and extensions of these features, namely \(\alpha \) -repetitive, \(\alpha \) -repulsive and \(\alpha \) -finite ( \(\alpha \ge 1\) ), have been introduced and studied. We establish the equivalence of \(\alpha \) -repulsive and \(\alpha \) -finite for general subshifts over finite alphabets. Further, we studied a family of aperiodic minimal subshifts stemming from Grigorchuk’s infinite 2-group G. In particular, we show that these subshifts provide examples that demonstrate \(\alpha \) -repulsive (and hence \(\alpha \) -finite) is not equivalent to \(\alpha \) -repetitive, for \(\alpha > 1\) . We also give necessary and sufficient conditions for these subshifts to be \(\alpha \) -repetitive, and \(\alpha \) -repulsive (and hence \(\alpha \) -finite). Moreover, we obtain an explicit formula for their complexity functions from which we deduce that they are uniquely ergodic. PubDate: 2017-03-29

Abstract: Weighted Sobolev spaces play a main role in the study of Sobolev orthogonal polynomials. The aim of this paper is to prove several important properties of weighted Sobolev spaces: separability, reflexivity, uniform convexity, duality and Markov-type inequalities. PubDate: 2017-03-25

Abstract: We construct the first examples of algorithmically complex finitely presented residually finite groups and the first examples of finitely presented residually finite groups with arbitrarily large (recursive) Dehn functions, and arbitrarily large depth functions. The groups are solvable of class 3. PubDate: 2017-03-20

Abstract: This is an expository paper on the theory of gradient flows, and in particular of those PDEs which can be interpreted as gradient flows for the Wasserstein metric on the space of probability measures (a distance induced by optimal transport). The starting point is the Euclidean theory, and then its generalization to metric spaces, according to the work of Ambrosio, Gigli and Savaré. Then comes an independent exposition of the Wasserstein theory, with a short introduction to the optimal transport tools that are needed and to the notion of geodesic convexity, followed by a precise description of the Jordan–Kinderlehrer–Otto scheme and a sketch of proof to obtain its convergence in the easiest cases. A discussion of which equations are gradient flows PDEs and of numerical methods based on these ideas is also provided. The paper ends with a new, theoretical, development, due to Ambrosio, Gigli, Savaré, Kuwada and Ohta: the study of the heat flow in metric measure spaces. PubDate: 2017-03-14

Abstract: We study the inverse problems for the second order hyperbolic equations of general form with time-dependent coefficients assuming that the boundary data are given on a part of the boundary. The main result of this paper is the determination of the time-dependent Lorentzian metric by the boundary measurements. This is achieved by the adaptation of a variant of the boundary control method developed by Eskin (Inverse Probl 22(3):815–833, 2006; Inverse Probl 23:2343–2356, 2007). PubDate: 2017-03-06

Abstract: In this paper, we establish a new multiplicative Sobolev inequality. As applications, we refine and extend the results in Kukavica and Ziane (J Math Phys 48:065203, 2007) and Cao (Discrete Contin Dyn Syst 26:1141–1151, 2010) simultaneously. PubDate: 2017-01-04

Abstract: We derive trace formulas of the Buslaev–Faddeev type for quantum star graphs. One of the new ingredients is high energy asymptotics of the perturbation determinant. PubDate: 2016-12-31

Abstract: We use the type theory for rings of operators due to Kaplansky to describe the structure of modules that are invariant under automorphisms of their injective envelopes. Also, we highlight the importance of Boolean rings in the study of such modules. As a consequence of this approach, we are able to further the study initiated by Dickson and Fuller regarding when a module invariant under automorphisms of its injective envelope is invariant under any endomorphism of it. In particular, we find conditions for several classes of noetherian rings which ensure that modules invariant under automorphisms of their injective envelopes are quasi-injective. In the case of a commutative noetherian ring, we show that any automorphism-invariant module is quasi-injective. We also provide multiple examples to show that our conditions are the best possible, in the sense that if we relax them further then there exist automorphism-invariant modules which are not quasi-injective. We finish this paper by dualizing our results to the automorphism-coinvariant case. PubDate: 2016-12-22

Abstract: We study the eigenvalues of the discrete Schrödinger operator with a complex potential. We obtain bounds on the total number of eigenvalues in the case where V decays exponentially at infinity. PubDate: 2016-11-21

Abstract: We consider resonances for Schrödinger operators with compactly supported potentials on the line and the half-line. We estimate the sum of the negative power of all resonances and eigenvalues in terms of the norm of the potential and the diameter of its support. The proof is based on harmonic analysis and Carleson measures arguments. PubDate: 2016-11-18

Abstract: We give a construction of a family of locally finite residually finite groups with just-infinite \(C^*\) -algebra. This answers a question from Grigorchuk et al. (Just-infinite \(C^*\) -algebras. https://arxiv.org/abs/1604.08774, 2016). Additionally, we show that residually finite groups of finite exponent are never just-infinite. PubDate: 2016-10-14

Abstract: We study the semilinear Poisson equation 1 $$\begin{aligned} \Delta u = f(x, u) \quad \text {in} \quad B_1. \end{aligned}$$ Our main results provide conditions on f which ensure that weak solutions of (1) belong to \(C^{1,1}(B_{1/2})\) . In some configurations, the conditions are sharp. PubDate: 2016-07-08