Abstract: 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: 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: 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: 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: 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: Abstract We relate the Belavin–Drinfeld cohomologies (twisted and untwisted) that have been introduced in the literature to study certain families of quantum groups and Lie bialgebras over a non algebraically closed field \(\mathbb {K}\) of characteristic 0 to the standard non-abelian Galois cohomology \(H^1(\mathbb {K}, \mathbf{H})\) for a suitable algebraic \(\mathbb {K}\) -group \(\mathbf{H}.\) The approach presented allows us to establish in full generality certain conjectures that were known to hold for the classical types of the split simple Lie algebras. PubDate: 2016-12-09

Abstract: 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: 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: 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: Abstract Realistic models of traffic flow are nonlinear and involve nonlocal effects in balance laws. Flow characteristics of different types of vehicles, such as cars and trucks, need to be described differently. Two alternatives are used here, \(L^p\) -valued Lebesgue measurable density functions and signed Radon measures. The resulting solution spaces are metric spaces that do not have a linear structure, so the usual convenient methods of functional analysis are no longer applicable. Instead ideas from mutational analysis will be used, in particular the method of Euler compactness will be applied to establish the well-posedness of the nonlocal balance laws. This involves the concatenation of solutions of piecewise linear systems on successive time subintervals obtained by freezing the nonlinear nonlocal coefficients to their values at the start of each subinterval. Various compactness criteria lead to a convergent subsequence. Careful estimates of the linear systems are needed to implement this program. PubDate: 2016-08-11

Abstract: Abstract We discuss the asymptotics of the eigenvalue counting function for partial differential operators and related expressions paying the most attention to the sharp asymptotics. We consider Weyl asymptotics, asymptotics with Weyl principal parts and correction terms and asymptotics with non-Weyl principal parts. Semiclassical microlocal analysis, propagation of singularities and related dynamics play crucial role. We start from the general theory, then consider Schrödinger and Dirac operators with the strong magnetic field and, finally, applications to the asymptotics of the ground state energy of heavy atoms and molecules with or without a magnetic field. PubDate: 2016-08-01

Abstract: 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

Abstract: Abstract We prove that the Lie ring associated to the lower central series of a finitely generated residually-p torsion group is graded nil. PubDate: 2016-06-10

Abstract: Abstract This work studies the regularity and the geometric significance of solution of the Cauchy problem for a degenerate parabolic equation \(u_{t}=\Delta {}u^{m}\) . Our main objective is to improve the H \(\ddot{o}\) lder estimate obtained by pioneers and then, to show the geometric characteristic of free boundary of degenerate parabolic equation. To be exact, for the weak solution u(x, t), the present work will show that: The function \(\phi =(u(x,t))^{\beta }\in {}C^{1}(\mathbb {R}^{n})\) for given \(t>0\) if \(\beta \) is large sufficiently; The surface \(\phi =\phi (x,t)\) is tangent to \(\mathbb {R}^{n}\) at the boundary of the positivity set of u(x, t); The function \(\phi (x,t)\) is a classical solution to another degenerate parabolic equation. Moreover, some explicit derivative estimates and expressions about the speed of propagation of u(x, t) and the continuous dependence on the nonlinearity of the equation are obtained. PubDate: 2016-05-17

Abstract: Abstract We prove geometric \(L^p\) versions of Hardy’s inequality for the sub-elliptic Laplacian on convex domains \(\Omega \) in the Heisenberg group \(\mathbb {H}^n\) , where convex is meant in the Euclidean sense. When \(p=2\) and \(\Omega \) is the half-space given by \(\langle \xi , \nu \rangle > d\) this generalizes an inequality previously obtained by Luan and Yang. For such p and \(\Omega \) the inequality is sharp and takes the form $$\begin{aligned} \int _\Omega \nabla _{\mathbb {H}^n}u ^2 \, d\xi \ge \frac{1}{4}\int _{\Omega } \sum _{i=1}^n\frac{\langle X_i(\xi ), \nu \rangle ^2+\langle Y_i(\xi ), \nu \rangle ^2}{{{\mathrm{\text {dist}}}}(\xi , \partial \Omega )^2} u ^2\, d\xi , \end{aligned}$$ where \({{\mathrm{\text {dist}}}}(\, \cdot \,, \partial \Omega )\) denotes the Euclidean distance from \(\partial \Omega \) . PubDate: 2016-04-28

Abstract: Abstract In this note we prove an analogue of the Rayleigh–Faber–Krahn inequality, that is, that the geodesic ball is a maximiser of the first eigenvalue of some convolution type integral operators, on the sphere \(\mathbb {S}^{n}\) and on the real hyperbolic space \(\mathbb {H}^{n}\) . It completes the study of such question for complete, connected, simply connected Riemannian manifolds of constant sectional curvature. We also discuss an extremum problem for the second eigenvalue on \(\mathbb {H}^{n}\) and prove the Hong–Krahn–Szegö type inequality. The main examples of the considered convolution type operators are the Riesz transforms with respect to the geodesic distance of the space. PubDate: 2016-04-08

Abstract: Abstract We study natural linear representations of self-similar groups over finite fields. In particular, we show that if the group is generated by a finite automaton, then obtained matrices are automatic. This shows a new relation between two separate notions of automaticity: groups generated by automata and automatic sequences. We also show that if the group acts on the tree by p-adic automorphisms, then the corresponding linear representation is a representation by infinite triangular matrices. We relate this observation with the notion of height of an automorphism of a rooted tree due to L. Kaloujnine. PubDate: 2015-11-21

Abstract: Abstract In this paper, the author introduces a concept of the super-pseudoconvex domain. He proves that the solution of the Fefferman equation on a smoothly bounded strictly pseudoconvex domain D in \({\mathbb {C}}^n\) is plurisubharmonic in D if and only if D is super-pseudoconvex. As an application, when D is super-pseudoconvex, he gives the sharp lower bound for the bottom of the spectrum of the Laplace-Beltrami operators by using the result of Li and Wang (Int. Math. Res. Not. 4351–4371, 2012). PubDate: 2015-11-14

Abstract: Abstract We aim at reviewing and extending a number of recent results addressing stability of certain geometric and analytic estimates in the Riemannian approximation of subRiemannian structures. In particular we extend the recent work of the the authors with Rea (Math Ann 357(3):1175–1198, 2013) and Manfredini (Anal Geom Metric Spaces 1:255–275, 2013) concerning stability of doubling properties, Poincare’ inequalities, Gaussian estimates on heat kernels and Schauder estimates from the Carnot group setting to the general case of Hörmander vector fields. PubDate: 2015-10-12