arXiv subjects

Chai Wah Wu

Publications and source records attributed to Chai Wah Wu.

45 records · Page 3Linked to original sources

arXiv2017

Counting the number of isosceles triangles in rectangular regular grids

In general graph theory, the only relationship between vertices are expressed via the edges. When the vertices are embedded in an Euclidean space, the geometric relationships between vertices and edges can be interesting objects of study. We look at the number of isosceles triangles where the vertices are points on a regular grid and show that they satisfy a recurrence relation when the grid is large enough. We also derive recurrence relations for the number of acute, obtuse and right isosceles triangles.

math.CO↗

arXiv2016

Meertens number and its variations

In 1998, Bird introduced Meertens numbers as numbers that are invariant under a map similar to the Godel encoding. In base 10, the only known Meertens number is 81312000. We look at some properties of Meertens numbers and consider variations of this concept. In particular, we consider variations where there is a finite time algorithm to decide whether such numbers exist.

math.NT↗

arXiv2015

On a conjecture regarding primality of numbers constructed from prepending and appending identical digits

Consider the operation of adding the same number of identical digits to the left and to the right of a number n. In OEIS sequence A090287, it was conjectured that this operation will not produce a prime if and only if n is a palindrome with an even number of digits. We show that this conjecture is false by showing that this property also holds for n=231, n=420, and an infinite number of other values of n.

math.NT↗

arXiv2014

Graphs whose normalized Laplacian matrices are separable as density matrices in quantum mechanics

Recently normalized Laplacian matrices of graphs are studied as density matrices in quantum mechanics. Separability and entanglement of density matrices are important properties as they determine the nonclassical behavior in quantum systems. In this note we look at the graphs whose normalized Laplacian matrices are separable or entangled. In particular, we show that the number of such graphs is related to the number of 0-1 matrices that are line sum symmetric and to the number of graphs with at least one vertex of degree 1.

quant-ph↗

arXiv2009

On graphs whose Laplacian matrix's multipartite separability is invariant under graph isomorphism

Normalized Laplacian matrices of graphs have recently been studied in the context of quantum mechanics as density matrices of quantum systems. Of particular interest is the relationship between quantum physical properties of the density matrix and the graph theoretical properties of the underlying graph. One important aspect of density matrices is their entanglement properties, which are responsible for many nonintuitive physical phenomena. The entanglement property of normalized Laplacian matrices is in general not invariant under graph isomorphism. In recent papers, graphs were identified whose entanglement and separability properties are invariant under isomorphism. The purpose of this note is to characterize the set of graphs whose separability is invariant under graph isomorphism. In particular, we show that this set consists of $K_{2,2}$, $\overline{K_{2,2}}$ and all complete graphs.

math-ph↗

arXiv2008

Multipartite separability of Laplacian matrices of graphs

Recently, Braunstein et al. [1] introduced normalized Laplacian matrices of graphs as density matrices in quantum mechanics and studied the relationships between quantum physical properties and graph theoretical properties of the underlying graphs. We provide further results on the multipartite separability of Laplacian matrices of graphs. In particular, we identify complete bipartite graphs whose normalized Laplacian matrix is multipartite entangled under any vertex labeling. Furthermore, we give conditions on the vertex degrees such that there is a vertex labeling under which the normalized Laplacian matrix is entangled. These results address an open question raised in [1]. Finally, we extend some of the results in [1,2] to the multipartite case and show that the Laplacian matrix of any product of graphs (strong, Cartesian, tensor, categorical, etc.) is multipartite separable.

quant-ph↗

arXiv2006

On some properties of contracting matrices

The concepts of paracontracting, pseudocontracting and nonexpanding operators have been shown to be useful in proving convergence of asynchronous or parallel iteration algorithms. The purpose of this paper is to give characterizations of these operators when they are linear and finite-dimensional. First we show that pseudocontractivity of stochastic matrices with respect to sup-norm is equivalent to the scrambling property, a concept first introduced in the study of inhomogeneous Markov chains. This unifies results obtained independently using different approaches. Secondly, we generalize the concept of pseudocontractivity to set-contractivity which is a useful generalization with respect to the Euclidean norm. In particular, we demonstrate non-Hermitian matrices that are set-contractive for ||.||_2, but not pseudocontractive for ||.||_2 or sup-norm. For constant row sum matrices we characterize set-contractivity using matrix norms and matrix graphs. Furthermore, we prove convergence results in compositions of set-contractive operators and illustrate the differences between set-contractivity in different norms. Finally, we give an application to the global synchronization in coupled map lattices.

math.DS↗

arXiv2005

Conditions for separability in generalized Laplacian matrices and nonnegative matrices as density matrices

Recently, Laplacian matrices of graphs are studied as density matrices in quantum mechanics. We continue this study and give conditions for separability of generalized Laplacian matrices of weighted graphs with unit trace. In particular, we show that the Peres-Horodecki positive partial transpose separability condition is necessary and sufficient for separability in ${\mathbb C}^2\otimes {\mathbb C}^q$. In addition, we present a sufficient condition for separability of generalized Laplacian matrices and diagonally dominant nonnegative matrices.

quant-ph↗

arXiv2003

Perturbation of coupling matrices and its effect on the synchronizability in arrays of coupled chaotic systems

In a recent paper, wavelet analysis was used to perturb the coupling matrix in an array of identical chaotic systems in order to improve its synchronization. As the synchronization criterion is determined by the second smallest eigenvalue $λ_2$ of the coupling matrix, the problem is equivalent to studying how $λ_2$ of the coupling matrix changes with perturbation. In the aforementioned paper, a small percentage of the wavelet coefficients are modified. However, this result in a perturbed matrix where every element is modified and nonzero. The purpose of this paper is to present some results on the change of $λ_2$ due to perturbation. In particular, we show that as the number of systems $n \to \infty$, perturbations which only add local coupling will not change $λ_2$. On the other hand, we show that there exists perturbations which affect an arbitrarily small percentage of matrix elements, each of which is changed by an arbitrarily small amount and yet can make $λ_2$ arbitrarily large. These results give conditions on what the perturbation should be in order to improve the synchronizability in an array of coupled chaotic systems. This analysis allows us to prove and explain some of the synchronization phenomena observed in a recently studied network where random coupling are added to a locally connected array. Finally we classify various classes of coupling matrices such as small world networks and scale free networks according to their synchronizability in the limit.

nlin.CD↗