Table of contents

In a paper by M Kac (1966 Am. Math. Mon.73 1–23), Kac asked his famous question 'Can one hear the shape of a drum?', which was answered negatively in Gordon et al (1992 Invent. Math.110 1–22) by construction of planar isospectral pairs. In Buser et al (1994 Int. Math. Res. Not.9), it is observed that all operator groups associated with the known counter examples are isomorphic to one of PSL₃(2), PSL₃(3), PSL₄(2) and PSL₃(4). We show that if (D₁, D₂) is a pair of non-congruent planar isospectral domains constructed from unfolding a polygonal base-tile and with associated operator group PSL_n(q), then (n, q) belongs to this very restricted list.

In this paper we consider vector fields in that are invariant under a suitable symmetry and that possess a 'generalized heteroclinic loop' formed by two singular points (e⁺ and e⁻) and their invariant manifolds: one of dimension 2 (a sphere minus the points e⁺ and e⁻) and one of dimension 1 (the open diameter of the sphere having endpoints e⁺ and e⁻). In particular, we analyse the dynamics of the vector field near the heteroclinic loop by means of a convenient Poincaré map, and we prove the existence of infinitely many symmetric periodic orbits near . We also study two families of vector fields satisfying this dynamics. The first one is a class of quadratic polynomial vector fields in , and the second one is the charged rhomboidal four-body problem.

A general stochastic traffic cellular automaton (CA) model, which includes the slow-to-start effect and driver's perspective, is proposed in this paper. It is shown that this model includes well-known traffic CA models such as the Nagel–Schreckenberg model, the quick-start model and the slow-to-start model as specific cases. Fundamental diagrams of this new model clearly show metastable states around the critical density even when the stochastic effect is present. We also obtain analytic expressions of the phase transition curve in phase diagrams by using approximate flow-density relations at boundaries. These phase transition curves are in excellent agreement with numerical results.

A set of nonlinear differential equations associated with the Eisenstein series of the congruent subgroup Γ₀(2) of the modular group is constructed. These nonlinear equations are analogues of the well-known Ramanujan equations, as well as the Chazy and Darboux–Halphen equations associated with the modular group. The general solutions of these equations can be realized in terms of the Schwarz triangle function S(0, 0, 1/2; z).

A wide variety of different physical systems can be described by a relatively small set of universal equations. For example, small-amplitude nonlinear Schrödinger dark solitons can be described by a Korteweg-de Vries (KdV) equation. Reductive perturbation theory, based on linear boosts and Gallilean transformations, is often employed to establish connections to and between such universal equations. Here, a novel analytical approach reveals that the evolution of small-amplitude Helmholtz–Kerr dark solitons is also governed by a KdV equation. This broadens the class of nonlinear systems that are known to possess KdV soliton solutions, and provides a framework for perturbative analyses when propagation angles are not negligibly small. The derivation of this KdV equation involves an element that appears new to weakly nonlinear analyses, since transformations are required to preserve the rotational symmetry inherent to Helmholtz-type equations.

We discuss a general method to construct correlated binomial distributions by imposing several consistent relations on the joint probability function. We obtain self-consistency relations for the conditional correlations and conditional probabilities. The beta-binomial distribution is derived by a strong symmetric assumption on the conditional correlations. Our derivation clarifies the 'correlation' structure of the beta-binomial distribution. It is also possible to study the correlation structures of other probability distributions of exchangeable (homogeneous) correlated Bernoulli random variables. We study some distribution functions and discuss their behaviours in terms of their correlation structures.

Harmonic maps from or one-connected domain into and U(m) are treated. The GBDT version of the Bäcklund–Darboux transformation is applied to the case of the harmonic maps and a new and simple algebraic procedure to construct new harmonic maps from the initial ones is given, using some methods from system theory. A new general formula on the GBDT transformations of the Sym–Tafel immersions is derived. A new class of the unitary harmonic maps with asymptotics along one line essentially different from the asymptotics in all other directions, similar in certain ways to line solutions, is obtained explicitly and studied.

The super-algebraic structure of a generalized version of the Jaynes–Cummings model is investigated. We find that a Z₂ graded extension of the so(2,1) Lie algebra is the underlying symmetry of this model. It is isomorphic to the four-dimensional super-algebra u(1/1) with two odd and two even elements. Differential matrix operators are taken as realization of the elements of the superalgebra to which the model Hamiltonian belongs. Several examples with various choices of superpotentials are presented. The energy spectrum and corresponding wavefunctions are obtained analytically.

The meaning of superselection rules in Bohm–Bell theories (i.e., quantum theories with particle trajectories) is different from that in orthodox quantum theory. More precisely, there are two concepts of superselection rule, a weak and a strong one. Weak superselection rules exist both in orthodox quantum theory and in Bohm–Bell theories and represent the conventional understanding of superselection rules. We introduce the concept of strong superselection rule, which does not exist in orthodox quantum theory. It relies on the clear ontology of Bohm–Bell theories and is a sharper and, in the Bohm–Bell context, more fundamental notion. A strong superselection rule for the observable G asserts that one can replace every state vector by a suitable statistical mixture of eigenvectors of G without changing the particle trajectories or their probabilities. A weak superselection rule asserts that every state vector is empirically indistinguishable from a suitable statistical mixture of eigenvectors of G. We establish conditions on G for both kinds of superselection. For comparison, we also consider both kinds of superselection in theories of spontaneous wavefunction collapse.

The new approach to the analysis of four-body systems and computation of various four-body integrals is proposed. The approach is based on the use of six perimetric coordinates which can be introduced for an arbitrary four-body system. The proper (i.e. non-conflicting) definition of the four-body perimetric coordinates is given for an arbitrary four-body system. It is shown that these six internal perimetric coordinates describe all possible configurations in an arbitrary four-body system and can be used to simplify computations of many four-body integrals written in the relative coordinates r₁₂, r₁₃, r₂₃, r₁₄, r₂₄ and r₃₄. In addition to this, a number of new, very effective procedures for variational computation of different four-body systems can now be developed.

The generalized vector is defined on an n-dimensional manifold. The interior product and Lie derivative acting on generalized p-forms, −1 ⩽ p ⩽ n are introduced. The generalized commutator of two generalized vectors is defined. Adding a correction term to Cartan's formula, the generalized Lie derivative's action on a generalized vector field is defined. We explore various identities of the generalized Lie derivative with respect to generalized vector fields, and discuss an application.

Loop models have been widely studied in physics and mathematics, in problems ranging from polymers to topological quantum computation to Schramm–Loewner evolution. I present new loop models which have critical points described by conformal field theories. Examples include both fully packed and dilute loop models with critical points described by the superconformal minimal models and the SU(2)₂ WZW models. The dilute loop models are generalized to include SU(2)_k models as well.

The standard relation between the field momentum and the force is generalized for the system with a field singularity: in addition to the regular force, there appears the singular one. This approach is applied to the description of the gyroscopic dynamics of the classical field with topological defects. The collective-variable Lagrangian description is considered for gyroscopical systems taking into account singularities. Using this method, we describe the dynamics of two-dimensional magnetic solitons. We establish a relation between the gyroscopic force and the singular one. An effective Lagrangian description is discussed for the magnetic soliton dynamics.

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