Constructions of complex Hadamard matrices via tiling Abelian groups

Applications in quantum information theory and quantum tomography have raised current interest in complex Hadamard matrices. In this note we investigate the connection between tiling Abelian groups and constructions of complex Hadamard matrices. First, we recover a recent very general construction of complex Hadamard matrices due to Dita via a natural tiling construction. Then we find some necessary conditions for any given complex Hadamard matrix to be equivalent to a Dita-type matrix. Finally, using another tiling construction, due to Szabo, we arrive at new parametric families of complex Hadamard matrices of order 8, 12 and 16, and we use our necessary conditions to prove that these families do not arise with Dita's construction. These new families complement the recent catalogue of complex Hadamard matrices of small order.Comment: 15 page

Tiles with no spectra

We exhibit a subset of a finite Abelian group, which tiles the group by translation, and such that its tiling complements do not have a common spectrum (orthogonal basis for their L-2 space consisting of group characters). This disproves the Universal Spectrum Conjecture of Lagarias and Wang [7]. Further, we construct a set in some finite Abelian group, which tiles the group but has no spectrum. We extend this last example to the groups Z(d) and R-d (for d >= 5) thus disproving one direction of the Spectral Set Conjecture of Fuglede [1]. The other direction was recently disproved by Tao [12]

Complex Hadamard matrices and the spectral set conjecture

By analyzing the connection between complex Hadamard matrices and spectral sets, we prove the direction "spectral double right arrow tile" of the Spectral Set Conjecture, for all sets A of size \A\ <= 5, in any finite Abelian group. This result is then extended to the infinite grid Z(d) for any dimension d, and finally to R-d. It was pointed out recently in [16] that the corresponding statement fails for \A\ = 6 in the group Z(3)(5), and this observation quickly led to the failure of the Spectral Set Conjecture in R-5 [16], and subsequently in R-4 [13]. In the second part of this note we reduce this dimension further, showing that the direction "spectral double right arrow tile" of the Spectral Set Conjecture is false already in dimension 3. In a computational search for counterexamples in lower dimension (one and two) one needs, at the very least, to be able to decide efficiently if a set is a tile (in, say, a cyclic group) and if it is spectral. Such efficient procedures are lacking however and we make a few comments for the computational complexity of some related problems

On particles in equilibrium on the real line

We study equilibrium configurations of infinitely many identical particles on the real line or finitely many particles on the circle, such that the (repelling) force they exert on each other depends only on their distance. The main question is whether each equilibrium configuration needs to be an arithmetic progression. Under very broad assumptions on the force we show this for the particles on the circle. In the case of infinitely many particles on the line we show the same result under the assumption that the maximal (or the minimal) gap between successive points is finite (positive) and assumed at some pair of successive points. Under the assumption of analyticity for the force field (e.g., the Coulomb force) we deduce some extra rigidity for the configuration: knowing an equilibrium configuration of points in a half-line determines it throughout. Various properties of the equlibrium configuration are proved

How Hard is Counting Triangles in the Streaming Model

The problem of (approximately) counting the number of triangles in a graph is one of the basic problems in graph theory. In this paper we study the problem in the streaming model. We study the amount of memory required by a randomized algorithm to solve this problem. In case the algorithm is allowed one pass over the stream, we present a best possible lower bound of $\Omega(m)$ for graphs $G$ with $m$ edges on $n$ vertices. If a constant number of passes is allowed, we show a lower bound of $\Omega(m/T)$, $T$ the number of triangles. We match, in some sense, this lower bound with a 2-pass $O(m/T^{1/3})$-memory algorithm that solves the problem of distinguishing graphs with no triangles from graphs with at least $T$ triangles. We present a new graph parameter $\rho(G)$ -- the triangle density, and conjecture that the space complexity of the triangles problem is $\Omega(m/\rho(G))$. We match this by a second algorithm that solves the distinguishing problem using $O(m/\rho(G))$-memory

On Counting Triangles through Edge Sampling in Large Dynamic Graphs

Traditional frameworks for dynamic graphs have relied on processing only the stream of edges added into or deleted from an evolving graph, but not any additional related information such as the degrees or neighbor lists of nodes incident to the edges. In this paper, we propose a new edge sampling framework for big-graph analytics in dynamic graphs which enhances the traditional model by enabling the use of additional related information. To demonstrate the advantages of this framework, we present a new sampling algorithm, called Edge Sample and Discard (ESD). It generates an unbiased estimate of the total number of triangles, which can be continuously updated in response to both edge additions and deletions. We provide a comparative analysis of the performance of ESD against two current state-of-the-art algorithms in terms of accuracy and complexity. The results of the experiments performed on real graphs show that, with the help of the neighborhood information of the sampled edges, the accuracy achieved by our algorithm is substantially better. We also characterize the impact of properties of the graph on the performance of our algorithm by testing on several Barabasi-Albert graphs.Comment: A short version of this article appeared in Proceedings of the 2017 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining (ASONAM 2017

On the Fourier transform of the characteristic functions of domains with C1C^1 -smooth boundary

We consider domains $D\subseteq\mathbb R^n$ with $C^1$ -smooth boundary and study the following question: when the Fourier transform $\hat{1_D}$ of the characteristic function $1_D$ belongs to $L^p(\mathbb R^n)$?Comment: added two references; added footnotes on pages 6 and 1

Distances sets that are a shift of the integers and Fourier basis for planar convex sets

The aim of this paper is to prove that if a planar set $A$ has a difference set $\Delta(A)$ satisfying $\Delta(A)\subset \Z^++s$ for suitable $s$ than $A$ has at most 3 elements. This result is motivated by the conjecture that the disk has not more than 3 orthogonal exponentials. Further, we prove that if $A$ is a set of exponentials mutually orthogonal with respect to any symmetric convex set $K$ in the plane with a smooth boundary and everywhere non-vanishing curvature, then # (A \cap {[-q,q]}^2) \leq C(K) q where $C(K)$ is a constant depending only on $K$. This extends and clarifies in the plane the result of Iosevich and Rudnev. As a corollary, we obtain the result from \cite{IKP01} and \cite{IKT01} that if $K$ is a centrally symmetric convex body with a smooth boundary and non-vanishing curvature, then $L^2(K)$ does not possess an orthogonal basis of exponentials