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: We define a distance function on the bordered punctured disk \(0< z \le 1/e\) in the complex plane, which is comparable with the hyperbolic distance of the punctured unit disk \(0< z <1.\) As an application, we will construct a distance function on an n-times punctured sphere which is comparable with the hyperbolic distance. We also propose a comparable quantity which is not necessarily a distance function on the punctured sphere but easier to compute. PubDate: 2017-11-22

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: 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: 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