Abstract: Consider a function F(X, Y) of pairs of positive matrices with values in the positive matrices such that whenever X and Y commute \(F(X,Y)= X^pY^q.\) Our first main result gives conditions on F such that \(\mathrm{Tr}[ X \log (F(Z,Y))] \le \mathrm{Tr}[X(p\log X + q \log Y)]\) for all X, Y, Z such that \(\mathrm{Tr}Z =\mathrm{Tr}X\) . (Note that Z is absent from the right side of the inequality.) We give several examples of functions F to which the theorem applies. Our theorem allows us to give simple proofs of the well known logarithmic inequalities of Hiai and Petz and several new generalizations of them which involve three variables X, Y, Z instead of just X, Y alone. The investigation of these logarithmic inequalities is closely connected with three quantum relative entropy functionals: The standard Umegaki quantum relative entropy \(D(X Y) = \mathrm{Tr}[X(\log X-\log Y])\) , and two others, the Donald relative entropy \(D_D(X Y)\) , and the Belavkin–Stasewski relative entropy \(D_{BS}(X Y)\) . They are known to satisfy \(D_D(X Y) \le D(X Y)\le D_{BS}(X Y)\) . We prove that the Donald relative entropy provides the sharp upper bound, independent of Z on \(\mathrm{Tr}[ X \log (F(Z,Y))]\) in a number of cases in which F(Z, Y) is homogeneous of degree 1 in Z and −1 in Y. We also investigate the Legendre transforms in X of \(D_D(X Y)\) and \(D_{BS}(X Y)\) , and show how our results about these Legendre transforms lead to new refinements of the Golden–Thompson inequality. PubDate: 2018-05-29

Abstract: In this paper, we study global regularity for oblique boundary value problems of augmented Hessian equations for a class of general operators. By assuming a natural convexity condition of the domain together with appropriate convexity conditions on the matrix function in the augmented Hessian, we develop a global theory for classical elliptic solutions by establishing global a priori derivative estimates up to second order. Besides the known applications for Monge–Ampère type operators in optimal transportation and geometric optics, the general theory here embraces Neumann problems arising from prescribed mean curvature problems in conformal geometry as well as general oblique boundary value problems for augmented k-Hessian, Hessian quotient equations and certain degenerate equations. PubDate: 2018-05-21

Abstract: In this paper we prove the existence and uniqueness of solutions of an inverse problem of the simultaneous recovery of the evolution of two coefficients in the Korteweg-de Vries equation. PubDate: 2018-05-18

Abstract: As a class of Lévy type Markov generators, nonlocal Waldenfels operators appear naturally in the context of investigating stochastic dynamics under Lévy fluctuations and constructing Markov processes with boundary conditions (in particular the construction with jumps). This work is devoted to prove the weak and strong maximum principles for ‘parabolic’ equations with nonlocal Waldenfels operators. Applications in stochastic differential equations with \(\alpha \) -stable Lévy processes are presented to illustrate the maximum principles. PubDate: 2018-05-16

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: 2018-04-01

Abstract: We study spectral properties of a class of global infinite order pseudo-differential operators and obtain the asymptotic behaviour of the spectral counting functions of such operators. Unlike their finite order counterparts, their spectral asymptotics are not of power-log-type but of log-type. The ultradistributional setting of such operators of infinite order makes the theory more complex so that the standard finite order global Weyl calculus cannot be used in this context. PubDate: 2018-04-01

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: 2018-04-01

Abstract: We review the work of Tosio Kato on the mathematics of non-relativistic quantum mechanics and some of the research that was motivated by this. Topics in this first part include analytic and asymptotic eigenvalue perturbation theory, Temple–Kato inequality, self-adjointness results, and quadratic forms including monotone convergence theorems. PubDate: 2018-04-01

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: 2018-04-01

Abstract: We consider a class of Schrödinger operators with complex decaying potentials on the lattice. Using some classical results from complex analysis we obtain some trace formulae and use them to estimate zeros of the Fredholm determinant in terms of the potential. PubDate: 2018-03-13

Abstract: We review the work of Tosio Kato on the mathematics of non-relativistic quantum mechanics and some of the research that was motivated by this. Topics in this second part include absence of embedded eigenvalues, trace class scattering, Kato smoothness, the quantum adiabatic theorem and Kato’s ultimate Trotter Product Formula. PubDate: 2018-02-22

Abstract: We study the stationary Stokes system with variable coefficients in the whole space, a half space, and on bounded Lipschitz domains. In the whole and half spaces, we obtain a priori \(\dot{W}^1_q\) -estimates for any \(q\in [2,\infty )\) when the coefficients are merely measurable functions in one fixed direction. For the system on bounded Lipschitz domains with a small Lipschitz constant, we obtain a \(W^1_q\) -estimate and prove the solvability for any \(q\in (1,\infty )\) when the coefficients are merely measurable functions in one direction and have locally small mean oscillations in the orthogonal directions in each small ball, where the direction is allowed to depend on the ball. PubDate: 2018-02-09

Abstract: We give an overview of existing enhancement techniques for derived and trianguated categories based on the notion of a stable model category, and show how it can be applied to the problem of gluing triangulated categories. The article is mostly expository, but we do prove some new results concerning existence of model structures. PubDate: 2018-02-07

Abstract: We study the bilinear Weyl product acting on quasi-Banach modulation spaces. We find sufficient conditions for continuity of the Weyl product and we derive necessary conditions. The results extend known results for Banach modulation spaces. PubDate: 2018-01-24

Abstract: We extend the resolvent estimate on the sphere to exponents off the line \(\frac{1}{r}-\frac{1}{s}=\frac{2}{n}\) . Since the condition \(\frac{1}{r}-\frac{1}{s}=\frac{2}{n}\) on the exponents is necessary for a uniform bound, one cannot expect estimates off this line to be uniform still. The essential ingredient in our proof is an \((L^{r}, L^{s})\) norm estimate on the operator \(H_{k}\) that projects onto the space of spherical harmonics of degree k. In showing this estimate, we apply an interpolation technique first introduced by Bourgain (C R Acad Sci Paris Ser I Math 301(10):499–502, 1985). The rest of our proof parallels that in Huang–Sogge (J Funct Anal 267(12):4635–4666, 2014). PubDate: 2018-01-22

Abstract: Decompositions of linear ordinary differential equations (ode’s) into components of lower order have successfully been employed for determining their solutions. Here this approach is generalized to nonlinear ode’s. It is not based on the existence of Lie symmetries, in that it is a genuine extension of the usual solution algorithms. If an equation allows a Lie symmetry, the proposed decompositions are usually more efficient and often lead to simpler expressions for the solution. For the vast majority of equations without a Lie symmetry decomposition is the only available systematic solution procedure. Criteria for the existence of diverse decomposition types and algorithms for applying them are discussed in detail and many examples are given. The collection of Kamke of solved equations, and a tremendeous compilation of random equations are applied as a benchmark test for comparison of various solution procedures. Extensions of these proceedings for more general types of ode’s and also partial differential equations are suggested. PubDate: 2017-12-01

Abstract: This is a survey on Nichols algebras of diagonal type with finite dimension, or more generally with arithmetic root system. The knowledge of these algebras is the cornerstone of the classification program of pointed Hopf algebras with finite dimension, or finite Gelfand–Kirillov dimension; and their structure should be indispensable for the understanding of the representation theory, the computation of the various cohomologies, and many other aspects of finite dimensional pointed Hopf algebras. These Nichols algebras were classified in Heckenberger (Adv Math 220:59–124, 2009) as a notable application of the notions of Weyl groupoid and generalized root system (Heckenberger in Invent Math 164:175–188, 2006; Heckenberger and Yamane in Math Z 259:255–276, 2008). In the first part of this monograph, we give an overview of the theory of Nichols algebras of diagonal type. This includes a discussion of the notion of generalized root system and its appearance in the contexts of Nichols algebras of diagonal type and (modular) Lie superalgebras. In the second and third part, we describe for each Nichols algebra in the list of Heckenberger (2009) the following basic information: the generalized root system; its label in terms of Lie theory; the defining relations found in Angiono (J Eur Math Soc 17:2643–2671, 2015; J Reine Angew Math 683:189–251, 2013); the PBW-basis; the dimension or the Gelfand–Kirillov dimension; the associated Lie algebra as in Andruskiewitsch et al. (Bull Belg Math Soc Simon Stevin 24(1):15–34, 2017). Indeed the second part deals with Nichols algebras related to Lie algebras and superalgebras in arbitrary characteristic, while the third contains the information on Nichols algebras related to Lie algebras and superalgebras only in small characteristic, and the few examples yet unidentified in terms of Lie theory. PubDate: 2017-12-01

Abstract: Let \(\pi \) be a set of primes. According to H. Wielandt, a subgroup H of a finite group X is called a \(\pi \) -submaximal subgroup if there is a monomorphism \(\phi :X\rightarrow Y\) into a finite group Y such that \(X^\phi \) is subnormal in Y and \(H^\phi =K\cap X^\phi \) for a \(\pi \) -maximal subgroup K of Y. In his talk at the celebrated conference on finite groups in Santa-Cruz (USA) in 1979, Wielandt posed a series of open questions and among them the following problem: to describe the \(\pi \) -submaximal subgroup of the minimal nonsolvable groups and to study properties of such subgroups: the pronormality, the intravariancy, the conjugacy in the automorphism group etc. In the article, for every set \(\pi \) of primes, we obtain a description of the \(\pi \) -submaximal subgroup in minimal nonsolvable groups and investigate their properties, so we give a solution of Wielandt’s problem. PubDate: 2017-11-30

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