Combinatorics of the subshift associated with Grigorchuk’s group

We study combinatorial properties of the subshift induced by the substitution that describes Lysenok’s presentation of Grigorchuk’s group of intermediate growth by generators and relators. This subshift has recently appeared in two different contexts: on the one hand, it allowed embedding Grigorchuk’s group in a topological full group, and on the other hand, it was useful in the spectral theory of Laplacians on the associated Schreier graphs.     

Decomposition of group rings and the converse of Schur’s lemma

In this work, we give necessary and su?cient conditions for a group ring (of finite group) to satisfy the converse of Schur’s lemma for group rings of finite groups. Rings considered here are commutative or noncommutative perfect. Some cases of twisted group rings are studied. Also we introduce the notion of semi-CSL ring.     

Borch’s Theorem from the perspective of comonotonicity

This short note revisits the classical Theorem of Borch on the characterization of Pareto optimal risk exchange treaties under the expected utility paradigm. Our objective is to approach the optimal risk exchange problem by a new method, which is based on a Breeden–Litzenberger type integral representation formula for increasing convex functions and the theory of comonotonicity. Our method allows us to derive Borch’s characterization without using Kuhn–Tucker theory, and also without the need of assuming that all utility functions are continuously differentiable everywhere. We demonstrate that our approach can be used effectively to solve the Pareto optimal risk-sharing problem with a positivity constraint being imposed on the admissible allocations when the aggregate risk is positive.     

Regular subsets of valued fields and Bhargava’s v-orderings

We generalize notions and results obtained by Amice for regular compact subsets S of a local field K and extended by Bhargava to general compact subsets of K. Considering any ultrametric valued field K and subsets S that are regular in a generalized sense (but not necessarily compact), we show that they still have strong properties such as having v-orderings ${\{a_n\}_{n\geq0}}$ which satisfy a generalized Legendre formula, which are very well ordered and well distributed sequences in the sense of Helsmoortel and which remain v-orderings when a finite number of the initial terms of the sequence are deleted.     

Analysis of the hyper-chaos generated from Chen’s system

This letter does some further researches on a newly proposed hyper-chaos found in [Gao TG, Chen ZQ, Chen G. A hyper-chaos Generated from Chen’s system, Int J Mod Phys C 2006;17(4):471–8], and gets some new results with some parameters of the system are definite and the other one varies. The basic dynamical behavior of the system under various conditions is analyzed, and the new findings is not only demonstrated by computer simulations but also verified with bifurcation analysis.     

Green’s relations in some categories of strong graph homomorphisms

The immédiate purpose of this paper is to show how Li’s characterization of Green’s relations on monoids of strong endomorphisms of graphs [8] is related to earlier work of Klasa [5] on catégories. Li’s results are thus extended to some catégories of strong graph homomorphisms with more than one object, and a finiteness requirement is weakened. A further purpose is to extend Klasa’s results in such a way as to better reflect the factorization properties of interesting concrète catégories.     

Dirichlet to Neumann functional and the group of disc’s biholomorphic trasformations

Studying the injectivity of the Dirichlet to Neumann functional on the unit disk in the plane, we arrive at the nonlinear differential equation $S(f)=-\frac{1}{2}([f^\prime ]^2-1)$ , where $S(f)$ is the Schwarzian derivative of $f$ (cf. (1.2)). Let $\mathcal{S}$ be the set of solutions of this equation. In this article, we establish a correspondence between the set of solutions $\mathcal{S}$ and the projective linear transformations on the real projective line. Moreover, we use this to show that the quotient group $\mathbb{H }/\mathcal{T}_{2\pi }$ it is isomorphic to the group of biholomorphic automorphisms of the unit disk in the plane. Here $\mathcal{T}_{2\pi }$ and $\mathbb{H }$ denote the subgroups of $\mathcal{S}$ consisting of translations by integral multiples of $2\pi $ and those solutions with have positive derivative, respectively.     

The Dehn function of Baumslag’s metabelian group

Baumslag??s group is a finitely presented metabelian group with a ${\mathbb Z \wr \mathbb Z}$ subgroup. There is an analogue with an additional torsion relation in which this subgroup becomes ${C_m \wr \mathbb Z}$ . We prove that Baumslag??s group has an exponential Dehn function. This contrasts with the torsion analogues which have quadratic Dehn functions.     

Diversity of firm’s life cycle adapted from the firm’s technology investment decision

The stylized model presented is an optimal control model of technology investment decision of a single product firm. The firm’s technology investment does not have only a long-run positive effect but also a short-run adverse effect on its sales volume. We examine the case of high adverse investment effects where the firm finally leaves the market but we have observed different life cycles till this happens. Depending on the firm’s initial technology stock and sales volume, we compute different firm’s life cycles, which are driven by a trade-off between two strategies: technology versus sales focus strategy. Indifference curves, where managers are indifferent to apply initially technology or sales focus strategies, separate founding conditions of the firm to various classes distinguishable because of the firm’s life cycle.     

A Quasi-isometric embedding into the group of Hamiltonian diffeomorphisms with Hofer’s metric

We construct an embedding Φ of [0, 1]^∞ into Ham(M, ω), the group of Hamiltonian diffeomorphisms of a suitable closed symplectic manifold (M, ω). We then prove that Φ is in fact a quasi-isometry. After imposing further assumptions on (M, ω), we adapt our methods to construct a similar embedding of ? ⊕ [0, 1]^∞ into either Ham(M, ω) or

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, the universal cover of Ham(M, ω). Along the way, we prove results related to the filtered Floer chain complexes of radially symmetric Hamiltonians. Our proofs rely heavily on a continuity result for barcodes (as presented in [28]) associated to filtered Floer homology viewed as a persistence module.

     

Harnack’s inequality for a class of non-divergent equations in the Heisenberg group

We prove an invariant Harnack’s inequality for operators in non-divergence form structured on Heisenberg vector fields when the coe?cient matrix is uniformly positive definite, continuous, and symplectic. The method consists in constructing appropriate barriers to obtain pointwise-to-measure estimates for supersolutions in small balls, and then invoking the axiomatic approach developed by Di Fazio, Gutiérrez, and Lanconelli to obtain Harnack’s inequality.     

Thompson’s conjecture for alternating group of degree 22

For a finite group G, it is denoted by N(G) the set of conjugacy class sizes of G. In 1980s, J. G. Thompson posed the following conjecture: if L is a finite nonabelian simple group, G is a finite group with trivial center, and N(G) = N(L), then L and G are isomorphic. In this paper, it is proved that Thompson’s conjecture is true for the alternating group A ₂₂ with connected prime graph.     

Representation of green’s function for the heat equation on a compact lie group

We obtain an explicit formula which presents the solution of the heat equation on a compact Lie group as the limit of finite-to-one convolutions of Green’s function for the heat equation in Euclidean space. Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 397–413, March, 2000.     

Generators and normal forms of Richard Thompson’s group F and the four-color theorem

We use a result of Kauffman to explore the connection between Richard Thompson’s Group F and the four-color theorem.