Monte-Carlo sampling of energy-constrained quantum superpositions in high-dimensional Hilbert spaces

Recent studies into the properties of quantum statistical ensembles in high-dimensional Hilbert spaces have encountered difficulties associated with the Monte-Carlo sampling of quantum superpositions constrained by the energy expectation value. A straightforward Monte-Carlo routine would enclose the energy constrained manifold within a larger manifold, which is easy to sample, for example, a hypercube. The efficiency of such a sampling routine decreases exponentially with the increase of the dimension of the Hilbert space, because the volume of the enclosing manifold becomes exponentially larger than the volume of the manifold of interest. The present paper explores the ways to optimise the above routine by varying the shapes of the manifolds enclosing the energy-constrained manifold. The resulting improvement in the sampling efficiency is about a factor of five for a 14-dimensional Hilbert space. The advantage of the above algorithm is that it does not compromise on the rigorous statistical nature of the sampling outcome and hence can be used to test other more sophisticated Monte-Carlo routines. The present attempts to optimise the enclosing manifolds also bring insights into the geometrical properties of the energy constrained manifold itself.Comment: 9 pages, 7 figures, accepted for publication in European Physical Journal

The critical window for the classical Ramsey-Tur\'an problem

The first application of Szemer\'edi's powerful regularity method was the following celebrated Ramsey-Tur\'an result proved by Szemer\'edi in 1972: any K_4-free graph on N vertices with independence number o(N) has at most (1/8 + o(1)) N^2 edges. Four years later, Bollob\'as and Erd\H{o}s gave a surprising geometric construction, utilizing the isoperimetric inequality for the high dimensional sphere, of a K_4-free graph on N vertices with independence number o(N) and (1/8 - o(1)) N^2 edges. Starting with Bollob\'as and Erd\H{o}s in 1976, several problems have been asked on estimating the minimum possible independence number in the critical window, when the number of edges is about N^2 / 8. These problems have received considerable attention and remained one of the main open problems in this area. In this paper, we give nearly best-possible bounds, solving the various open problems concerning this critical window.Comment: 34 page

The history of degenerate (bipartite) extremal graph problems

This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version of our survey presented in Erdos 100. In this version 2 only a citation was complete

Linkages, key sectors and structural change: some new perspectives

Recent exchanges in the literature on the identification and role of key sectors in national and regional economies have highlighted the difficulties of consensus regarding terminology, appropriate measurement as well as economic interpretation. In this paper, some new perspectives are advanced which provide a more comprehensive view of an economy and offer the potential for uncovering alternative perspectives about the role of linkages and multipliers in input-output and expanded social accounting systems. The analysis draws on some pioneering work by Miyazawa in the identification of internal and external multiplier effects. The theoretical techniques are illustrated by reference to a set of input-output tables for the Brazilian economy. The paper thus provides a more comprehensive view than the ones proposed by Baer, Fonseca, and Guilhoto (1987), Hewings, Fonseca, Guilhoto, and Sonis (1989) and the recent contributions of Clements and Rossi (1991, 1992) that draw on some earlier work of Cella (1984)

On the structure of edge graphs II

This note is a sequel to [1]. First let us recall some of the notations. Denote by G(n, in) a graph with n vertices and m edges. Let K d (r,,..., rd) be the complete d-partite graph with r; vertices in its i-th class and put K,(t) = K d (t,..., t), K d = Kd (1). Given integers n> d(> 2), let in d (n) be the minimal integer with the property that every G(n, in), where an> na d (n), contains a K d. The function m d (n) was determined by Turán [5]. It is easily seen that d-2 ntd (n)