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Vahid Shirbisheh

Publications and source records attributed to Vahid Shirbisheh.

11 recordsLinked to original sources
arXiv2022

Local Graph Embeddings Based on Neighbors Degree Frequency of Nodes

We propose a local-to-global strategy for graph machine learning and network analysis by defining certain local features and vector representations of nodes and then using them to learn globally defined metrics and properties of the nodes by means of deep neural networks. By extending the notion of the degree of a node via Breath-First Search, a general family of {\bf parametric centrality functions} is defined which are able to reveal the importance of nodes. We introduce the {\bf neighbors degree frequency (NDF)}, as a locally defined embedding of nodes of undirected graphs into euclidean spaces. This gives rise to a vectorized labeling of nodes which encodes the structure of local neighborhoods of nodes and can be used for graph isomorphism testing. We add flexibility to our construction so that it can handle dynamic graphs as well. Afterwards, the Breadth-First Search is used to extend NDF vector representations into two different matrix representations of nodes which contain higher order information about the neighborhoods of nodes. Our matrix representations of nodes provide us with a new way of visualizing the shape of the neighborhood of a node. Furthermore, we use these matrix representations to obtain feature vectors, which are suitable for typical deep learning algorithms. To demonstrate these node embeddings actually contain some information about the nodes, in a series of examples, we show that PageRank and closeness centrality can be learned by applying deep learning to these local features. Our constructions are flexible enough to handle evolving graphs. Finally, we explain how to adapt our constructions for directed graphs.

cs.SI↗
arXiv2021

Lectures on C*-algebras

The following topics are presented in these notes: Elements of Banach algebras, Banach algebras of the form $L^1(G)$, where $G$ is a locally compact group, spectrum of elements of Banach algebras, the spectral theory of compact operators on Banach spaces, the holomorphic functional calculus in Banach algebras, the Gelfand transform on commutative Banach algebras and C*-algebras, the continuous functional calculus, the Gelfand duality between commutative C*-algebras and locally compact and Hausdorff topological spaces, positivity in C*-algebras, approximate units, ideals of C*-algebras, hereditary C*-subalgebras, multiplier algebras, Hilbert spaces, the C*-algebra $B(H)$ of bounded operators on a Hilbert space $H$, examples of concrete C*-algebras, the reduced group C*-algebra of a locally compact group $G$, locally convex topologies on the C*-algebra $B(H)$, the Borel functional calculus in $B(H)$, projections in $B(H)$ and the polar decomposition of elements of $B(H)$, C*-algebras of compact operators and the bicommutant theorem.

math.OA↗
arXiv2019

Locally compact Hecke pairs

We introduce an extended setting to study Hecke pairs $(G,H)$ which admit a regular representation on $L^2(H\backslash G)$, and consequently a $C^*$-algebra. As the result, many pairs of locally compact groups which had been studied in noncommutative harmonic analysis, Lie theory and representation theory are included in the theory of Hecke $C^*$-algebras. These Hecke pairs mainly consist of locally compact groups and their compact subgroups, or cocompact subgroups, or open Hecke subgroups. We clarify similarities, differences and relationships of our formulation with the discrete case, and thereby we obtain new results for discrete Hecke pairs too. In the discrete case, using the Schlichting completion and our results, we show that the left regular representations of associated Hecke algebras are bounded homomorphisms. We observe that the relative unimodularity of a discrete Hecke pair amounts to the condition that the left regular representation be a $\ast$-homomorphism. On the other hand, we have recently shown that relative unimodularity is a necessary condition for a Hecke pair to possess property (RD). Motivated by these facts, we also give several criteria for relative unimodularity of discrete Hecke pairs.

math.GR↗
arXiv2014

Length functions and property (RD) for locally compact Hecke pairs

The purpose of this paper is to study property (RD) for locally compact Hecke pairs. We discuss length functions on Hecke pairs and the growth of Hecke pairs. We establish an equivalence between property (RD) of locally compact groups and property (RD) of certain locally compact Hecke pairs. This allows us to transfer several important results concerning property (RD) of locally compact groups into our setting, and consequently to identify many classes of examples of locally compact Hecke pairs with property (RD). We also show that a reduced discrete Hecke pair $(G,H)$ has (RD) if and only if its Schlichting completion $\bar{G}$ has (RD). Then it follows that the relative unimodularity is a necessary condition for a discrete Hecke pair to possess property (RD).

math.OA↗
arXiv2014

Amenability of Hecke pairs and a question of Eymard

In this short paper we show that if $(G,H)$ is an amenable Hecke pair and $Γ$ is a subgroup of $G$ containing $H$, then the Hecke pair $(Γ, H)$ is amenable too. This answers positively a question of Pierre Eymard, asked in 1972, when it is restricted to Hecke pairs

math.GR↗
arXiv2012

Various commensurability relations in Hecke pairs and property (RD)

In this paper we continue our study of property (RD) for Hecke pairs initiated in (V. Shirbisheh, Property (RD) for Hecke pairs. Mathematical Physics, Analysis and Geometry, vol. 15, no. 2, (2012), 173--192). We study the behavior of property (RD) under different commensurability relations of subgroups in Hecke pairs. As an application, we prove that if H is a normal subgroup of a group G and K is a subgroup of G commensurable to H, then the Hecke pair (G,K) has (RD) if and only if the quotient group G/H has (RD). This is used to investigate an infinite number of non-elementary examples of Hecke pairs with property (RD). In particular, we introduce a class of groups whose all subgroups are almost normal and all such Hecke pairs have property (RD). We also discuss property (RD) of certain Hecke pairs arising from group extensions.

math.GR↗
arXiv2012

Property (RD) for Hecke pairs

As the first step towards developing noncommutative geometry over Hecke C*-algebras, we study property (RD) (Rapid Decay) for Hecke pairs. When the subgroup H in a Hecke pair (G,H) is finite, we show that the Hecke pair (G,H) has (RD) if and only if G has (RD). This provides us with a family of examples of Hecke pairs with property (RD). We also adapt Paul Jolissant's works in 1989 to the setting of Hecke C*-algebras and show that when a Hecke pair (G,H) has property (RD), the algebra of rapidly decreasing functions on the set of double cosets is closed under holomorphic functional calculus of the associated (reduced) Hecke C*-algebra. Hence they have the same K_0-groups.

math.OA↗
arXiv2008

Quantum Statistical Mechanics of $\mathbb{Q}$-lattices and noncommutative geometry

After recalling some basic notions of quantum statistical mechanics, we explain the Bost-Connes system that relates the structure of the maximal abelian extension of $\mathbb{Q}$ to the space of \kms states of a \cs-dynamical system. Afterwards, we study briefly the Connes-Marcolli $\text{GL}_2$-system as a generalization of the former system.

math.OA↗
arXiv2008

A relative version of Kummer theory

Let $E/F$ be a cyclic Galois extension of degree $p^l$ with Galois group $G$. It is shown that the Galois module structure of both sides of the Kummer pairing (for Kummer extensions of $E$) are the same. In other words, we show that the Kummer duality holds in the level of finitely generated $G$-modules.

math.NT↗
arXiv2002

K-Theory Tools For Local and Asymptotic Cyclic Cohomology

A generalization of Connes-Thom isomorphism is given for stable, homotopy invariant, and split exact functors on separable $C^*$-algebras. As examples of these functors, we concentrate on asymptotic and local cyclic cohomology and the result is applied to improve some formulas in asymptotic and local cyclic cohomology of $C^*$-algebras. As an other application, it is shown that these cyclic theories are rigid after Rieffel's deformation quantizations.

math.KT↗