PT-symmetric Hamiltonians and transfer matrices arise naturally in statistical mechanics. These classical and quantum models often require the use of complex or negative weights and thus fall outside the conventional equilibrium statistical mechanics of Hermitian systems. PT-symmetric models form a natural class where the partition function is necessarily real, but not necessarily positive. The...

The consequences of diluting the weights of the standard Hopfield architecture associative memory model, trained using perceptron like learning rules, is examined. A proportion of the weights of the network are removed; this can be done in a symmetric and asymmetric way and both methods are investigated. This paper reports experimental investigations into the consequences of dilution in terms o...

We present approximation algorithms for four variations of the maximum latency problem. We consider symmetric graphs and asymmetric graphs and both with general edge weights or weights satisfying the triangle inequality. Moreover, in each variation the starting point of the tour may either be given in the input or be a decision variable. As a tool for our solution, we use a PTAS for the maximum...

Dendrimers are nano-sized, radially symmetric molecules with well-defined, homogeneous, and monodisperse structure that has a typically symmetric core, an inner shell, and an outer shell. Their three traditional macromolecular architectural classes are broadly recognized to generate rather polydisperse products of different molecular weights. A variety of dendrimers exist, and each has biologic...

We describe the cohomology groups of a homogeneous vector bundle E on any Hermitian symmetric variety X = G/P of ADE type as the cohomology of a complex explicitly described. The main tool is the equivalence between the category of homogeneous bundles and the category of representations of a certain quiver QX with relations, whose vertices are the dominant weights of the reductive part of P . T...

An extension of symmetric classical orthogonal polynomials is d-symmetric classical d-orthogonal polynomials, d being a positive number. These polynomials and their derivatives satisfy particular (d+1)-order recurrence relations. Many works deals with these families. Our purpose is to derive some more results for this class of polynomials. In fact, in terms of hypergeometric functions, we expre...

We provide a lower bound for the spectral radius of the universal cover of irregular graphs in the presence of symmetric edge weights. We use this bound to derive an Alon-Boppana type bound for the second eigenvalue of the normalized Laplacian.

We consider the celebrated Cucker-Smale model in finite dimension, modelling interacting collective dynamics and their possible evolution to consensus. The objective of this paper is to study the effect of time delays in the general model. By a Lyapunov functional approach, we provide convergence results to consensus for symmetric as well as nonsymmetric communication weights under some structu...

In a physical neural system, learning rules must be local both in space and time. In order for learning to occur, non-local information must be communicated to the deep synapses through a communication channel, the deep learning channel. We identify several possible architectures for this learning channel (Bidirectional, Conjoined, Twin, Distinct) and six symmetry challenges: (1) symmetry of ar...

We discuss a 4-vertex model on an ensemble of 3-valent (Φ) planar random graphs, which has the effect of coupling the vertex model to 2D quantum gravity. The regular lattice equivalent of the model is the symmetric 8-vertex model on the honeycomb lattice, which can be mapped on to an Ising model in field, as was originally shown by Wu et.al. using generalised weak graph transformation technique...