Evolution algebras are a special class of nonassociative algebras exhibiting connections with various fields of mathematics. Hilbert evolution algebras generalize the concept in the framework of Hilbert spaces. This allows us to deal with a wide class of infinite-dimensional spaces. We study Hilbert evolution algebras associated to a graph. Inspired by the definitions of evolution algebras we define the Hilbert evolution algebra that is associated to a given graph and the Hilbert evolution algebra that is associated to the symmetric random walk on a graph. For a given graph, we provide the conditions for these structures to be or not to be isomorphic. Our definitions and results extend to the graphs with infinitely many vertices. We also develop a similar theory for the evolution algebras associated to finite graphs.
相似文献We consider a specific method for embedding a countable group that is given by generators and relations into some 2-generated group. This embedding enables us to express the images of generators of the countable group in the 2-generated group and explicitly deduce from the defining relations of the latter those of the former which inherit some special properties. The method can be used to construct the explicit embeddings of recursive groups into finitely presented groups.
相似文献We propose a method for determining parameters in the Schwarz–Christoffel integral. The desired mapping embeds into a one-parametric family of conformal mappings of the upper half-plane onto the family of polygons which was obtained by shifting one or several vertices of some initial polygon with angle preservation. We consider the case when the family of polygons and the initial polygon have the same number of vertices; the case when the family of polygons has two mobile vertices coinciding at the initial moment and not coinciding with other vertices; and the other case that the family of polygons is a polygon with mobile cut. The problem of finding the parameters of a family of mappings is reduced to integrating some system of ordinary differential equations.
相似文献The carpet subgroups admitting a Bruhat decomposition and different from Chevalley groups are exhausted by the groups lying between the Chevalley groups of type \( B_{l} \), \( C_{l} \), \( F_{4} \), or \( G_{2} \) over various imperfect fields of exceptional characteristic 2 or 3, the larger of which is an algebraic extension of the smaller field. Moreover, as regards the types \( B_{l} \) and \( C_{l} \), these subgroups are parametrized by the pairs of additive subgroups one of which may fail to be a field and, for the type \( B_{2} \), even both additive subgroups may fail to be fields. In this paper for the carpet subgroups admitting a Bruhat decomposition we present the relations similar to those well known for Chevalley groups over fields.
相似文献We study the Cauchy problem in the space of continuous functions for some nonlinear differential equation of Sobolev type that simulates longitudinal waves in an infinite viscoelastic rod. Under consideration are the conditions for the existence of the global classical solution and the blow-up of the solution to the Cauchy problem on a finite time interval.
相似文献We study the properties and applications of the directed graph, introduced by Hawkes in 1968, of a finite group \( G \). The vertex set of \( \Gamma_{H}(G) \) coincides with \( \pi(G) \) and \( (p,q) \) is an edge if and only if \( q\in\pi(G/O_{p^{\prime},p}(G)) \). In the language of properties of this graph we obtain commutation conditions for all \( p \)-elements with all \( r \)-elements of \( G \), where \( p \) and \( r \) are distinct primes. We estimate the nilpotence length of a solvable finite group in terms of subgraphs of its Hawkes graph. Given an integer \( n>1 \), we find conditions for reconstructing the Hawkes graph of a finite group \( G \) from the Hawkes graphs of its \( n \) pairwise nonconjugate maximal subgroups. Using these results, we obtain some new tests for the membership of a solvable finite group in the well-known saturated formations.
相似文献Under study is the problem of asymptotic but exponential stability for a class of linear autonomous neutral functional differential equations. We demonstrate that the asymptotic stability of an equation of the class takes place for all integrable initial functions if the roots of the characteristic equation lie on the left of and approach the imaginary axis.
相似文献Given a prime \( p \) and a partition \( \sigma=\{\{p\},\{p\}^{\prime}\} \) of the set of all primes, we describe the structure of the nonnilpotent finite groups whose every Schmidt subgroup is \( \sigma \)-subnormal.
相似文献Number pyramids are common in elementary school mathematics. Trying to express the value of the top block in terms of the values at the base leads to the binomial coefficients. It also seems natural to ask for the maximal number of odd numbers in a number pyramid of a given size. The answer is easy to state, but the proof is nontrivial: A \(k\) step number pyramid can have at most \(\left\lfloor\frac{k(k+1)+1}{3}\right\rfloor\) odd numbers, which equals two thirds of the number of blocks rounded to the nearest integer. All maximal and almost maximal solutions are given explicitly. To this end, we rephrase the question in terms of colored tilings. In the outlook we present relations to other—mostly geometric—subjects and problems.
相似文献Let \( \pi_{x} \) be the set of primes greater than \( x \). We prove that for all \( x\in{??} \) the classes of finite groups \( D_{\pi_{x}} \) and \( E_{\pi_{x}} \) coincide; i.e., a finite group \( G \) possesses a \( \pi_{x} \)-Hall subgroup if and only if \( G \) satisfies the complete analog of the Sylow Theorems for a \( \pi_{x} \)-subgroup.
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