The harmonic measures of Lucy Garnett

This paper presents an improved approach to the theory of harmonic measures for foliated spaces introduced by Garnett. This approach is based on a method for solving elliptic equations on foliated spaces and on the Hille-Yosida theory. The diffusion semigroup of a general Laplacian and its infinitesimal generator are made explicit. Applications of the path space to the dynamical study of a foliated space are described. In particular, the final section studies cocycles on foliated spaces, a formula for their asymptotic limit, and some analytic and geometric consequences.     

Kazhdan and Haagerup properties from the median viewpoint

We prove the existence of a close connection between spaces with measured walls and median metric spaces.We then relate properties (T) and Haagerup (a-T-menability) to actions on median spaces and on spaces with measured walls. This allows us to explore the relationship between the classical properties (T) and Haagerup and their versions using affine isometric actions on Lp-spaces. It also allows us to answer an open problem on a dynamical characterization of property (T), generalizing results of Robertson and Steger.     

Non-zero component union graph of a finite-dimensional vector space

In this paper, we introduce a graph structure, called non-zero component union graph on finite-dimensional vector spaces. We show that the graph is connected and find its domination number, clique number and chromatic number. It is shown that two non-zero component union graphs are isomorphic if and only if the base vector spaces are isomorphic. In case of finite fields, we study the edge-connectivity and condition under which the graph is Eulerian. Moreover, we provide a lower bound for the independence number of the graph. Finally, we come up with a structural characterization of non-zero component union graph.     

Nonzero Component Graph of a Finite Dimensional Vector Space

In this article, we introduce a graph structure, called a nonzero component graph on finite dimensional vector spaces. We show that the graph is connected and find its domination number and independence number. We also study the inter-relationship between vector space isomorphisms and graph isomorphisms, and it is shown that two graphs are isomorphic if and only if the corresponding vector spaces are so. Finally, we determine the degree of each vertex in case the base field is finite.     

Dynamical systems on graphs through the signless Laplacian matrix

There is a deep and interesting connection between the topological properties of a graph and the behaviour of the dynamical system defined on it. We analyse various kind of graphs, with different contrasting connectivity or degree characteristics, using the signless Laplacian matrix. We expose the theoretical results about the eigenvalue of the matrix and how they are related to the dynamical system. Then, we perform numerical computations on real-like graphs and observe the resulting system. Comparing the theoretical and numerical results, we found a perfect consistency. Furthermore, we define a metric which takes into account the “rigidity” of the graph and enables us to relate all together the topological properties of the graph, the signless Laplacian matrix and the dynamical system.     

The Oscillation of Elliptic Systems

Foliated differential forms were introduced in [7], [9], to study the cohomology on a RIEMANNian foliated manifold with coefficients in the sheaf of germs of foliated differential forms. In this paper the notion of DE RHAM like current of the type (p, q) is defined for a RIEMANNian foliated manifold and some properties of various differential operators acting on the spaces of currents are given. In particular, special DE RHAM like currents are considered namely the foliated ones. It turns out that the space of foliated p-forms is dense in the space of foliated p-currents with the usual topology. We get certain results concerning the cohomology on a RIEMANNian foliated manifold with coefficients in the sheaf of germs of foliated currents.     

Countable connected Hausdorff and Urysohn bunches of arcs in the plane

In this paper, we answer a question by Krasinkiewicz, Reńska and Sobolewski by constructing countable connected Hausdorff and Urysohn spaces as quotient spaces of bunches of arcs in the plane. We also consider a generalization of graphs by allowing vertices to be continua and replacing edges by not necessarily connected sets. We require only that two “vertices” be in the same quasi-component of the “edge” that contains them. We observe that if a graph G cannot be embedded in the plane, then any generalized graph modeled on G is not embeddable in the plane. As a corollary we obtain not planar bunches of arcs with their natural quotients Hausdorff or Urysohn. This answers another question by Krasinkiewicz, Reńska and Sobolewski.     

Continuous random dynamical systems

We study a dynamical system generalizing continuous iterated function systems and stochastic differential equations disturbed by Poisson noise. The main results provide us with sufficient conditions for the existence and uniqueness of an invariant measure for the considered system. Since the dynamical system is defined on an arbitrary Banach space (possibly infinite dimensional), to prove the existence of an invariant measure and its stability we make use of the lower bound technique developed by Lasota and Yorke and extended recently to infinite-dimensional spaces by Szarek. Finally, it is shown that many systems appearing in models of cell division or gene expressions are exactly as those we study. Hence we obtain their stability as well.     

On the Hilbert Evolution Algebras of a Graph

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.

     

A unified framework for harmonic analysis of functions on directed graphs and changing data

We present a general framework for studying harmonic analysis of functions in the settings of various emerging problems in the theory of diffusion geometry. The starting point of the now classical diffusion geometry approach is the construction of a kernel whose discretization leads to an undirected graph structure on an unstructured data set. We study the question of constructing such kernels for directed graph structures, and argue that our construction is essentially the only way to do so using discretizations of kernels. We then use our previous theory to develop harmonic analysis based on the singular value decomposition of the resulting non-self-adjoint operators associated with the directed graph. Next, we consider the question of how functions defined on one space evolve to another space in the paradigm of changing data sets recently introduced by Coifman and Hirn. While the approach of Coifman and Hirn requires that the points on one space should be in a known one-to-one correspondence with the points on the other, our approach allows the identification of only a subset of landmark points. We introduce a new definition of distance between points on two spaces, construct localized kernels based on the two spaces and certain interaction parameters, and study the evolution of smoothness of a function on one space to its lifting to the other space via the landmarks. We develop novel mathematical tools that enable us to study these seemingly different problems in a unified manner.     

Backward Φ-shifts and universality

In this paper we consider spaces of sequences which are valued in a topological space E and study generalized backward shifts associated to certain selfmappings of E. We characterize their universality in terms of dynamical properties of the underlying selfmappings. Applications to hypercyclicity theory are given. In particular, Rolewicz's theorem on hypercyclicity of scalar multiples of the classical backward shift is extended.     

The spum and sum-diameter of graphs: Labelings of sum graphs

A sum graph is a finite simple graph whose vertex set is labeled with distinct positive integers such that two vertices are adjacent if and only if the sum of their labels is itself another label. The spum of a graph G is the minimum difference between the largest and smallest labels in a sum graph consisting of G and the minimum number of additional isolated vertices necessary so that a sum graph labeling exists. We investigate the spum of various families of graphs, namely cycles, paths, and matchings. We introduce the sum-diameter, a modification of the definition of spum that omits the requirement that the number of additional isolated vertices in the sum graph is minimal, which we believe is a more natural quantity to study. We then provide asymptotically tight general bounds on both sides for the sum-diameter, and study its behavior under numerous binary graph operations as well as vertex and edge operations. Finally, we generalize the sum-diameter to hypergraphs.     

Equicontinuous foliated spaces

Riemannian foliations are characterized as those foliations whose holonomy pseudogroup consists of local isometries of a Riemannian manifold. Their main structural features are well understood since the work of Molina. In this paper we analyze the more general concept of equicontinuous pseudogroup of homeomorphisms, which gives rise to the notion of equicontinuous foliated space. We show that equicontinuous foliated spaces have structural properties similar to those known for Riemannian foliations: the universal covers of their leaves are in the same quasi-isometry class, leaf closures are homogeneous spaces, and the holonomy pseudogroup is indeed given by local isometries.     

Algebraic Cayley graphs over finite local rings

In this work, we define and study the algebraic Cayley directed graph over a finite local ring. Its vertex set is the unit group of a finite extension of a finite local ring R and its adjacency condition is that the quotient is a monic primary polynomial. We investigate its connectedness and diameter bound, and we also show that our graph is an expander graph. In addition, if a local ring has nilpotency two, then we obtain a better view of our graph from the lifting of the graph over its residue field.     

Wreath products and representations of p-local finite groups

Given two finite p-local finite groups and a fusion preserving morphism between their Sylow subgroups, we study the question of extending it to a continuous map between their classifying spaces. The results depend on the construction of the wreath product of p-local finite groups which is also used to study p-local permutation representations.     

Approximation by solutions of elliptic equations in semilocal spaces

In the present work, we investigate the approximability of solutions of elliptic partial differential equations in a bounded domain Ω by solutions of the same equations in a larger domain. We construct an abstract framework which allows us to deal with such density questions, simultaneously for various norms. More specifically, we study approximations with respect to the norms of semilocal Banach spaces of distributions. These spaces are required to satisfy certain postulates. We establish density results for elliptic operators with constant coefficients which unify and extend previous results. In our density results Ω may possess holes and it is required to satisfy the segment condition. We observe that analogous density results do not hold in spaces where the infinitely smooth functions are not dense. Finally, we provide applications related to the method of fundamental solutions.     

Generalization of the second Bogolyubov's theorem for non-almost periodic systems

The article is devoted to the generalization of the second Bogolyubov's theorem to non-almost periodic dynamical systems. We prove the analog of the second Bogolyubov's theorem for recurrent or pseudorecurrent dynamical systems in Banach spaces. Namely, we obtain the relation between a recurrent dynamical system and its averaged dynamical system. We also study existence of recurrent and pseudorecurrent motions (including special cases of periodic, quasi-periodic and almost periodic motions) in related nonautonomous systems.     

Minimal entropy rigidity for foliations of compact spaces

We formulate and prove a foliated version of a theorem of Besson, Courtois, and Gallot establishing the minimal entropy rigidity of negatively curved locally symmetric spaces. One corollary is a foliated version of Mostow’s rigidity theorem.     

Zero-divisor graphs of nilpotent-free semigroups

We find strong relationships between the zero-divisor graphs of apparently disparate kinds of nilpotent-free semigroups by introducing the notion of an Armendariz map between such semigroups, which preserves many graph-theoretic invariants. We use it to give relationships between the zero-divisor graph of a ring, a polynomial ring, and the annihilating-ideal graph. Then we give relationships between the zero-divisor graphs of certain topological spaces (so-called pearled spaces), prime spectra, maximal spectra, tensor-product semigroups, and the semigroup of ideals under addition, obtaining surprisingly strong structure theorems relating ring-theoretic and topological properties to graph-theoretic invariants of the corresponding graphs.