29 research outputs found
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
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 -smooth boundary
We consider domains with -smooth boundary and
study the following question: when the Fourier transform of the
characteristic function belongs to ?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 has a difference
set satisfying for suitable than
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
is a set of exponentials mutually orthogonal with respect to any symmetric
convex set in the plane with a smooth boundary and everywhere non-vanishing
curvature, then # (A \cap {[-q,q]}^2) \leq C(K) q where is a constant
depending only on . 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 is a centrally symmetric convex body with a smooth
boundary and non-vanishing curvature, then does not possess an
orthogonal basis of exponentials
On quaternary complex Hadamard matrices of small orders
One of the main goals of design theory is to classify, characterize and count
various combinatorial objects with some prescribed properties. In most cases,
however, one quickly encounters a combinatorial explosion and even if the
complete enumeration of the objects is possible, there is no apparent way how
to study them in details, store them efficiently, or generate a particular one
rapidly. In this paper we propose a novel method to deal with these
difficulties, and illustrate it by presenting the classification of quaternary
complex Hadamard matrices up to order 8. The obtained matrices are members of
only a handful of parametric families, and each inequivalent matrix, up to
transposition, can be identified through its fingerprint.Comment: 7 page
Triangle Counting in Dynamic Graph Streams
Estimating the number of triangles in graph streams using a limited amount of
memory has become a popular topic in the last decade. Different variations of
the problem have been studied, depending on whether the graph edges are
provided in an arbitrary order or as incidence lists. However, with a few
exceptions, the algorithms have considered {\em insert-only} streams. We
present a new algorithm estimating the number of triangles in {\em dynamic}
graph streams where edges can be both inserted and deleted. We show that our
algorithm achieves better time and space complexity than previous solutions for
various graph classes, for example sparse graphs with a relatively small number
of triangles. Also, for graphs with constant transitivity coefficient, a common
situation in real graphs, this is the first algorithm achieving constant
processing time per edge. The result is achieved by a novel approach combining
sampling of vertex triples and sparsification of the input graph. In the course
of the analysis of the algorithm we present a lower bound on the number of
pairwise independent 2-paths in general graphs which might be of independent
interest. At the end of the paper we discuss lower bounds on the space
complexity of triangle counting algorithms that make no assumptions on the
structure of the graph.Comment: New version of a SWAT 2014 paper with improved result