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Mahya Ghandehari

Publications and source records attributed to Mahya Ghandehari.

33 records · Page 2Linked to original sources
arXiv2021

Discrete Frames For $L^2({\mathbb R}^{n^2})$ Arising From Tiling Systems On ${\rm GL}_n({\mathbb R})$

A discrete frame for $L^2({\mathbb R}^d)$ is a countable sequence $\{e_j\}_{j\in J}$ in $L^2({\mathbb R}^d)$ together with real constants $0<A\leq B< \infty$ such that $$ A\|f\|_2^2 \leq \sum_{j\in J}|\langle f,e_j \rangle |^2 \leq B\|f\|_2^2,$$ for all $f\in L^2(\mathbb{R}^d)$. We present a method of sampling continuous frames, which arise from square-integrable representations of affine-type groups, to create discrete frames for high-dimensional signals. Our method relies on partitioning the ambient space by using a suitable "tiling system". We provide all relevant details for constructions in the case of ${\rm M}_n({\mathbb R})\rtimes {\rm GL}_n({\mathbb R})$, although the methods discussed here are general and could be adapted to many other settings. Finally, we prove significantly improved frame bounds over the previously known construction for the case of $n=2$.

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arXiv2020

Stability of characters and filters for weighted semilattices

We continue the study of the AMNM property for weighted semilattices that was initiated in [Y. Choi, J. Austral. Math. Soc. 95 (2013), no. 1, 36-67; arXiv 1203.6691]. We reformulate this in terms of stability of filters with respect to a given weight function, and then provide a combinatorial condition which is necessary and sufficient for this "filter stability" property to hold. Examples are given to show that this new condition allows for easier and unified proofs of some results in [Choi, ibid.], and furthermore allows us to verify the AMNM property in situations not covered by the results of that paper. As a final application, we show that for a large class of semilattices, arising naturally as union-closed set systems, one can always construct weights for which the AMNM property fails.

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arXiv2019

An optimization parameter for seriation of noisy data

A square symmetric matrix is a Robinson similarity matrix if entries in its rows and columns are non-decreasing when moving towards the diagonal. A Robinson similarity matrix can be viewed as the affinity matrix between objects arranged in linear order, where objects closer together have higher affinity. We define a new parameter, $Γ_\max$, which measures how badly a given matrix fails to be Robinson similarity. Namely, a matrix is Robinson similarity precisely when its $Γ_\max$ attains zero, and a matrix with small $Γ_\max$ is close (in the normalized $\ell^1$-norm) to a Robinson similarity matrix. Moreover, both $Γ_\max$ and the Robinson similarity approximation can be computed in polynomial time. Thus, our parameter recognizes Robinson similarity matrices which are perturbed by noise, and can therefore be a useful tool in the problem of seriation of noisy data.

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arXiv2018

Derivations on the algebra of Rajchman measures

For a locally compact Abelian group $G$, the algebra of Rajchman measures, denoted by $M_0(G)$, is the set of all bounded regular Borel measures on $G$ whose Fourier transform vanish at infinity. In this paper, we investigate the spectral structure of the algebra of Rajchman measures, and illustrate aspects of the residual analytic structure of its maximal ideal space. In particular, we show that $M_0(G)$ has a nonzero continuous point derivation, whenever $G$ is a non-discrete locally compact Abelian group. We then give the definition of the Rajchman algebra for a general (not necessarily Abelian) locally compact group, and prove that for a non-compact connected SIN group, the Rajchman algebra admits a nonzero continuous point derivation. Moreover, we discuss the analytic behavior of the spectrum of $M_0(G)$. Namely, we show that for every non-discrete metrizable locally compact Abelian group $G$, the maximal ideal space of $M_0(G)$ contains analytic discs.

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arXiv2016

Uniform linear embeddings of graphons

Let $w:[0,1]^2\rightarrow [0,1]$ be a symmetric function, and consider the random process $G(n,w)$, where vertices are chosen from $[0,1]$ uniformly at random, and $w$ governs the edge formation probability. Such a random graph is said to have a linear embedding, if the probability of linking to a particular vertex $v$ decreases with distance. The rate of decrease, in general, depends on the particular vertex $v$. A linear embedding is called uniform if the probability of a link between two vertices depends only on the distance between them. In this article, we consider the question whether it is possible to "transform" a linear embedding to a uniform one, through replacing the uniform probability space $[0,1]$ with a suitable probability space on ${\mathbb R}$. We give necessary and sufficient conditions for the existence of a uniform linear embedding for random graphs where $w$ attains only a finite number of values. Our findings show that for a general $w$ the answer is negative in most cases.

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arXiv2015

Projections in $L^1(G)$; the unimodular case

We consider the issue of describing all self-adjoint idempotents (projections) in $L^1(G)$ when $G$ is a unimodular locally compact group. The approach is to take advantage of known facts concerning subspaces of the Fourier-Stieltjes and Fourier algebras of $G$ and the topology of the dual space of $G$. We obtain an explicit description of any projection in $L^1(G)$ which happens to also lie in the coefficient space of a finite direct sum of irreducible representations. This leads to a complete description of all projections in $L^1(G)$ for $G$ belonging to a class of groups that includes $SL(2,R)$ and all almost connected nilpotent locally compact groups.

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arXiv2015

Linear embeddings of graphs and graph limits

Consider a random graph process where vertices are chosen from the interval $[0,1]$, and edges are chosen independently at random, but so that, for a given vertex $x$, the probability that there is an edge to a vertex $y$ decreases as the distance between $x$ and $y$ increases. We call this a random graph with a linear embedding. We define a new graph parameter $Γ^*$, which aims to measure the similarity of the graph to an instance of a random graph with a linear embedding. For a graph $G$, $Γ^*(G)=0$ if and only if $G$ is a unit interval graph, and thus a deterministic example of a graph with a linear embedding. We show that the behaviour of $Γ^*$ is consistent with the notion of convergence as defined in the theory of dense graph limits. In this theory, graph sequences converge to a symmetric, measurable function on $[0,1]^2$. We define an operator $Γ$ which applies to graph limits, and which assumes the value zero precisely for graph limits that have a linear embedding. We show that, if a graph sequence $\{ G_n\}$ converges to a function $w$, then $\{ Γ^*(G_n)\}$ converges as well. Moreover, there exists a function $w^*$ arbitrarily close to $w$ under the box distance, so that $\lim_{n\rightarrow \infty}Γ^*(G_n)$ is arbitrarily close to $Γ(w^*)$.

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arXiv2015

Weak amenability for Fourier algebras of 1-connected nilpotent Lie groups

A special case of a conjecture raised by Forrest and Runde (Math. Zeit., 2005) asserts that the Fourier algebra of every non-abelian connected Lie group fails to be weakly amenable; this was aleady known to hold in the non-abelian compact cases, by earlier work of Johnson (JLMS, 1994) and Plymen (unpublished note). In recent work (JFA, 2014) the present authors verified this conjecture for the real ax+b group and hence, by structure theory, for any semisimple Lie group. In this paper we verify the conjecture for all 1-connected, non-abelian nilpotent Lie groups, by reducing the problem to the case of the Heisenberg group. As in our previous paper, an explicit non-zero derivation is constructed on a dense subalgebra, and then shown to be bounded using harmonic analysis. En route, we use the known fusion rules for Schrödinger representations to give a concrete realization of the "dual convolution" for this group as a kind of twisted, operator-valued convolution. We also give some partial results for solvable groups which give further evidence to support the general conjecture.

math.FA↗
arXiv2014

Weak and cyclic amenability for Fourier algebras of connected Lie groups

Using techniques of non-abelian harmonic analysis, we construct an explicit, non-zero cyclic derivation on the Fourier algebra of the real $ax+b$ group. In particular this provides the first proof that this algebra is not weakly amenable. Using the structure theory of Lie groups, we deduce that the Fourier algebras of connected, semisimple Lie groups also support non-zero, cyclic derivations and are likewise not weakly amenable. Our results complement earlier work of Johnson (JLMS, 1994), Plymen (unpublished note) and Forrest--Samei--Spronk (IUMJ 2009). As an additional illustration of our techniques, we construct an explicit, non-zero cyclic derivation on the Fourier algebra of the reduced Heisenberg group, providing the first example of a connected nilpotent group whose Fourier algebra is not weakly amenable.

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arXiv2014

Some Beurling-Fourier algebras on compact groups are operator algebras

Let $G$ be a compact connected Lie group. The question of when a weighted Fourier algebra on $G$ is completely isomorphic to an operator algebra will be investigated in this paper. We will demonstrate that the dimension of the group plays an important role in the question. More precisely, we will get a positive answer to the question when we consider a polynomial type weight coming from a length function on $G$ with the order of growth strictly bigger than the half of the dimension of the group. The case of SU(n) will be examined, focusing more on the details including negative results. The proof for the positive directions depends on a non-commutative version of Littlewood multiplier theory, which we will develop in this paper, and the negative directions will be taken care of by restricting to a maximal torus.

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arXiv2011

Amenability properties of Rajchman algebras

Rajchman measures of locally compact Abelian groups are studied for almost a century now, and they play an important role in the study of trigonometric series. Eymard's influential work allowed generalizing these measures to the case of \emph{non-Abelian} locally compact groups $G$. The Rajchman algebra of $G$, which we denote by $B_0(G)$, is the set of all elements of the Fourier-Stieltjes algebra that vanish at infinity. In the present article, we characterize the locally compact groups that have amenable Rajchman algebras. We show that $B_0(G)$ is amenable if and only if $G$ is compact and almost Abelian. On the other extreme, we present many examples of locally compact groups, such as non-compact Abelian groups and infinite solvable groups, for which $B_0(G)$ fails to even have an approximate identity.

math.FA↗
arXiv2007

Amenability constants for semilattice algebras

For any finite unital commutative idempotent semigroup S, a unital semilattice, we show how to compute the amenability constant of its semigroup algebra l^1(S), which is always of the form 4n+1. We then show that these give lower bounds to amenability constants of certain Banach algebras graded over semilattices. We show that there is no commutative semilattice with amenability constant between 5 and 9.

math.FA↗
arXiv2006

On the size of the minimum critical set of a Latin square

A critical set in an $n \times n$ array is a set $C$ of given entries, such that there exists a unique extension of $C$ to an $n\times n$ Latin square and no proper subset of $C$ has this property. For a Latin square $L$, $\scs{L}$ denotes the size of the smallest critical set of $L$, and $\scs{n}$ is the minimum of $\scs{L}$ over all Latin squares $L$ of order $n$. We find an upper bound for the number of partial Latin squares of size $k$ and prove that $$n^2-(e+o(1))n^{10/6} \le \max \scs{L} \le n^2-\frac{\sqrtπ}{2}n^{9/6}.$$ % This improves a result of N. Cavenagh (Ph.D. thesis, The University of Queensland, 2003) and disproves one of his conjectures. Also it improves the previously known lower bound for the size of the largest critical set of any Latin square of order $n$.

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arXiv2006

Circular chromatic index of graphs of maximum degree 3

This paper proves that if $G$ is a graph (parallel edges allowed) of maximum degree 3, then $χ_c'(G) \leq 11/3$ provided that $G$ does not contain $H_1$ or $H_2$ as a subgraph, where $H_1$ and $H_2$ are obtained by subdividing one edge of $K_2^3$ (the graph with three parallel edges between two vertices) and $K_4$, respectively. As $χ_c'(H_1) = χ_c'(H_2) = 4$, our result implies that there is no graph $G$ with $11/3 < χ_c'(G) < 4$. It also implies that if $G$ is a 2-edge connected cubic graph, then $χ'(G) \le 11/3$.

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