An investigation of PT -symmetry breaking in tight-binding chains

We consider non-Hermitian PT -symmetric tight-binding chains where gain/loss optical potentials of equal magnitudes ±iγ are arbitrarily distributed over all sites. The main focus is on the threshold γ c beyond which PT -symmetry is broken. This threshold generically falls off as a power of the chain length, whose exponent depends on the configuration of optical potentials, ranging between 1 (for balanced periodic chains) and 2 (for unbalanced periodic chains, where each half of the chain experiences a non-zero mean potential). For random sequences of optical potentials with zero average and finite variance, the threshold is itself a random variable, whose mean value decays with exponent 3/2 and whose fluctuations have a universal distribution. The chains yielding the most robust PT -symmetric phase, i.e. the highest threshold at fixed chain length, are obtained by exact enumeration up to 48 sites. This optimal threshold exhibits an irregular dependence on the chain length, presumably decaying asymptotically with exponent 1, up to logarithmic corrections.

Impaired large numerosity estimation and intact subitizing in developmental dyscalculia

It is believed that the approximate estimation of large sets and the exact quantification of small sets (subitizing) are supported by two different systems, the Approximate Number System (ANS) and Object Tracking System (OTS), respectively. It is a current matter of debate whether they are both impaired in developmental dyscalculia (DD), a specific learning disability in symbolic number processing and calculation. Here we tackled this question by asking 32 DD children and 32 controls to perform a series of tasks on visually presented sets, including exact enumeration of small sets as well as comparison of large, uncountable sets. In children with DD, we found poor sensitivity in processing large numerosities, but we failed to find impairments in the exact enumeration of sets within the subitizing range. We also observed deficits in visual short-term memory skills in children with dyscalculia that, however, did not account for their low ANS acuity. Taken together, these results point to a dissociation between quantification skills in dyscalculia, they highlight a link between DD and low ANS acuity and provide support for the notion that DD is a multifaceted disability that covers multiple cognitive skills.

Lower Bounds, and Exact Enumeration in Particular Cases, for the Probability of Existence of a Universal Cycle or a Universal Word for a Set of Words

A universal cycle, or u-cycle, for a given set of words is a circular word that contains each word from the set exactly once as a contiguous subword. The celebrated de Bruijn sequences are a particular case of such a u-cycle, where a set in question is the set A n of all words of length n over a k-letter alphabet A. A universal word, or u-word, is a linear, i.e., non-circular, version of the notion of a u-cycle, and it is defined similarly. Removing some words in A n may, or may not, result in a set of words for which u-cycle, or u-word, exists. The goal of this paper is to study the probability of existence of the universal objects in such a situation. We give lower bounds for the probability in general cases, and also derive explicit answers for the case of removing up to two words in A n , or the case when k = 2 and n ≤ 4 .

About clique number estimates and exact enumeration in scale-free networks

In this article, we present several analytical methods for both correlated and uncorrelated scale-free networks to obtain the clique number upper bound estimate. To test these estimates, we developed several tools to construct networks with scale-free degree distribution. To compute an exact clique number of network in short time, we also developed a clique finder algorithm. As the empirical results show, the various estimate methods gives good upper bounding for the clique numbers. Our clique finder proves to be able to solve clique number problem of arbitrary graphs in a good computation time. To show this, we applied our clique finder on benchmark graphs from various known databases.

An Exact Enumeration Method to Find d-MPs in Multistate Networks

The enumeration approaches are important topics in network reliability calculation. For general purposes, the explicit enumeration (EE) is the popular method to apply. However, the low-cost feature of EE sacrifices its efficiency. A great improvement of EE is the invention of fast enumeration (FE), which creates a low-cost way of general purpose enumeration method with very good efficiency for applications. But for the enumeration in network reliability calculation, the optimal number of enumerations can be shown to be [Formula: see text], where [Formula: see text] is the number of minimal paths and [Formula: see text] is the demand of flow. FE still has the complexity far greater than the optimal one. This paper proposes an exact enumeration method for network reliability calculation, which has the complexity no greater than [Formula: see text]. This method greatly improves the enumeration efficiency than FE. So, it is believed to be very valuable to the large real-life applications. Benchmarks are made to show the efficiency of the proposed method.

Double Circulant Self-Dual and LCD Codes Over ℤp2

We study double circulant LCD codes over [Formula: see text] for all odd primes [Formula: see text] and self-dual double circulant codes over [Formula: see text] for primes [Formula: see text]. We derive exact enumeration formulae, and asymptotic lower bounds on the minimum distance of the [Formula: see text]-ary images of these codes by the classical Gray maps.

Passing through a stack k times

We consider the number of passes a permutation needs to take through a stack if we only pop the appropriate output values and start over with the remaining entries in their original order. We define a permutation [Formula: see text] to be [Formula: see text]-pass sortable if [Formula: see text] is sortable using [Formula: see text] passes through the stack. Permutations that are [Formula: see text]-pass sortable are simply the stack sortable permutations as defined by Knuth. We define the permutation class of [Formula: see text]-pass sortable permutations in terms of their basis. We also show all [Formula: see text]-pass sortable classes have finite bases by giving bounds on the length of a basis element of the permutation class for any positive integer [Formula: see text]. Finally, we define the notion of tier of a permutation [Formula: see text] to be the minimum number of passes after the first pass required to sort [Formula: see text]. We then give a bijection between the class of permutations of tier [Formula: see text] and a collection of integer sequences studied by Parker [The combinatorics of functional composition and inversion, PhD thesis, Brandeis University (1993)]. This gives an exact enumeration of tier [Formula: see text] permutations of a given length and thus an exact enumeration for the class of [Formula: see text]-pass sortable permutations. Finally, we give a new derivation for the generating function in [S. Parker, The combinatorics of functional composition and inversion, PhD thesis, Brandeis University (1993)] and an explicit formula for the coefficients.

Lozenge Tilings of a Halved Hexagon with an Array of Triangles Removed from the Boundary, Part II

Proctor's work on staircase plane partitions yields an exact enumeration of lozenge tilings of a halved hexagon on the triangular lattice. Rohatgi later extended this tiling enumeration to a halved hexagon with a triangle cut off from the boundary. In his previous paper, the author proved a common generalization of Proctor's and Rohatgi's results by enumerating lozenge tilings of a halved hexagon in the case an array of an arbitrary number of triangles has been removed from a non-staircase side. In this paper we consider the other case when the array of triangles has been removed from the staircase side of the halved hexagon. Our result also implies an explicit formula for the number of tilings of a hexagon with an array of triangles removed perpendicularly to the symmetry axis.