Bernard Nienhuis

Improving the convergence of the back-propagation algorithm

Abstract We propose a modification to the back-propagation method. The modification consists of a simple change in the total error-of-performance function that is to be minimized by the algorithm. The modified algorithm is slightly simpler than the original. As a result, the convergence of the network is accelerated in two ways. During the learning process according to the original back-propagation method, the network goes through stages in which the improvement of the response is extremely slow. These periods of stagnation are much shorter or even absent in our modified method. Furthermore, the final approach to the desired response function, when the network is already nearly correct, is accelerated by an amount that can be predicted analytically. We compare the original and the modified method in simulations of a variety of functions.

A Guide to stochastic Lowner evolution and its applications

This article is meant to serve as a guide to recent developments in the study of the scaling limit of critical models. These new developments were made possible through the definition of the Stochastic Löwner Evolution (SLE) by Oded Schramm. This article opens with a discussion of Löwners method, explaining how this method can be used to describe families of random curves. Then we define SLE and discuss some of its properties. We also explain how the connection can be made between SLE and the discrete models whose scaling limits it describes, or is believed to describe. Finally, we have included a discussion of results that were obtained from SLE computations. Some explicit proofs are presented as typical examples of such computations. To understand SLE sufficient knowledge of conformal mapping theory and stochastic calculus is required. This material is covered in the appendices.

Exact stationary state for an asymmetric exclusion process with fully parallel dynamics

The exact stationary state of an asymmetric exclusion process with fully parallel dynamics is obtained using the matrix product ansatz. We give a simple derivation for the deterministic case by a physical interpretation of the dimension of the matrices. We prove the stationarity via a cancellation mechanism, and by making use of an explicit representation of the matrix algebra we easily find closed expressions for the correlation functions in the general probabilistic case. Asymptotic expressions, obtained by making use of earlier results, allow us to derive the exact phase diagram.

Long-range strain correlations in sheared colloidal glasses

Glasses behave as solids on experimental time scales due to their slow relaxation. Growing dynamic length scales due to cooperative motion of particles are believed to be central to this slow response. For quiescent glasses, however, the size of the cooperatively rearranging regions has never been observed to exceed a few particle diameters, and the observation of long-range correlations has remained elusive. Here, we provide direct experimental evidence of long-range correlations during the deformation of a dense colloidal glass. By imposing an external stress, we force structural rearrangements, and we identify long-range correlations in the fluctuations of microscopic strain and elucidate their scaling and spatial symmetry. The applied shear induces a transition from homogeneous to inhomogeneous flow at a critical shear rate, and we investigate the role of strain correlations in this transition.

Parity effects in the scaling of block entanglement in gapless spin chains.

We consider the Rényi alpha entropies for Luttinger liquids (LL). For large block lengths l, these are known to grow like lnl. We show that there are subleading terms that oscillate with frequency 2k{F} (the Fermi wave number of the LL) and exhibit a universal power-law decay with l. The new critical exponent is equal to K/(2alpha), where K is the LL parameter. We present numerical results for the anisotropic XXZ model and the full analytic solution for the free fermion (XX) point.

Exact Solutions for Loewner Evolutions

In this note, we solve the Loewner equation in the upper half-plane with forcing function ξ(t), for the cases in which ξ(t)has a power-law dependence on time with powers 0, 1/2, and 1. In the first case the trace of singularities is a line perpendicular to the real axis. In the second case the trace of singularities can do three things. If ξ(t)=2√kt, the trace is a straight line set at an angle to the real axis. If ξ(t)=2√k(1-t), as pointed out by Marshall and Rohde,(12) the behavior of the trace as t approaches 1 depends on the coefficient κ. Our calculations give an explicit solution in which for κ<4 the trace spirals into a point in the upper half-plane, while for κ>4 it intersects the real axis. We also show that for κ=9/2 the trace becomes a half-circle. The third case with forcing ξ(t)=t gives a trace that moves outward to infinity, but stays within fixed distance from the real axis. We also solve explicitly a more general version of the evolution equation, in which ξ(t) is a superposition of the values ±1.

Lattice fermion models with supersymmetry

We investigate a family of lattice models with manifest supersymmetry. The models describe fermions on a 1D lattice, subject to the constraint that no more than k consecutive lattice sites may be occupied. We discuss the special properties arising from the supersymmetry, and present Bethe ansatz solutions of the simplest models. We display the connections of the k = 1 model with the spin- antiferromagnetic XXZ chain at Δ = −1/2, and the k = 2 model with both the su(2|1)-symmetric tJ model in the ferromagnetic regime and the integrable spin-1 XXZ chain at . We argue that these models include critical points described by the superconformal minimal models.

Temperley-Lieb stochastic processes

We discuss one-dimensional stochastic processes defined through the Temperley–Lieb algebra related to the Q = 1 Potts model. For various boundary conditions, we formulate a conjecture relating the probability distribution which describes the stationary state, to the enumeration of a symmetry class of alternating sign matrices, objects that have received much attention in combinatorics.

Intersecting Loop Model as a Solvable Super Spin Chain

In this paper, we investigate an integrable loop model and its connection with a supersymmetric spin chain. The Bethe ansatz solution allows us to study some properties of the ground state. When the loop fugacity q lies in the physical regime, we conjecture that the central charge is c › q 2 1 for q integer ,2. Low-lying excitations are examined, supporting a superdiffusive behavior for q › 1 .W e argue that these systems are interesting examples of integrable lattice models realizing c # 0 conformal field theories. [S0031-9007(98)06495-3]

Solvable lattice models labelled by Dynkin diagrams

An equivalence between generalized restricted solid-on-solid models, associated with sets of graphs, and multi-colour loop models is established. As an application the authors consider solvable loop models and, in this way, obtain new solvable families of critical RSOS models. These families can all be classified by the Dynkin diagrams of the simply laced Lie algebras. For one of the RSOS models, labelled by the Lie algebra pair (AL,AL) and related to the C2(1) vertex model, they give an off-critical extension, which breaks the Z2 symmetry of the Dynkin diagrams.