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The universal typical-signal estimators of entropy and cross-entropy based on the asymptotics of recurrence and waiting times play an important role in information theory. Building on their construction, we introduce and study universal typical-signal estimators of entropy production in the context of nonequilibrium statistical mechanics of one-sided shifts over finite alphabets.

In this paper we determine the effects of winding number on the dynamics of vortex torus knots and unknots in the context of classical, ideal fluid mechanics. We prove that the winding number — a topological invariant of torus knots — has a primary effect on vortex motion. This is done by applying the Moore-Saffman desingularization technique to the full Biot-Savart induction law, determining the influence of winding number on the 3 components of the induced velocity. Results have been obtained for 56 knots and unknots up to 51 crossings. In agreement with previous numerical results we prove that in general the propagation speed increases with the number of toroidal coils, but we notice that the number of poloidal coils may greatly modify the motion. Indeed we prove that for increasing aspect ratio and number of poloidal coils vortex motion can be even reversed, in agreement with previous numerical observations. These results demonstrate the importance of three-dimensional features in vortex dynamics and find useful applications to understand helicity and energy transfers across scales in vortical flows.

We consider the billiard dynamics in a non-compact set of R^d that is constructed as a bi-infinite chain of translated copies of the same d-dimensional polytope. A random configuration of semi-dispersing scatterers is placed in each copy. The ensemble of dynamical systems thus defined, one for each global realization of the scatterers, is called `quenched random Lorentz tube'. Under some fairly general conditions, we prove that every system in the ensemble is hyperbolic and almost every system is recurrent, ergodic, and enjoys some higher chaotic properties., Comment: Final version for J. Stat. Phys., 18 pages, 4 figures

We introduce a model of fluid which has four main features: it readily emerges by a general continuum mechanical framework; it is a generalization maintaining most of the physical features of incompressible Newtonian fluids; it can model adherence interactions with one-dimensional structures surrounded by the fluid; the associated initial boundary-value problem is well-posed on three-dimensional domains. © 2010 Springer-Verlag.

Gaudin algebras form a family of maximal commutative subalgebras in the tensor product of $n$ copies of the universal enveloping algebra $U(\g)$ of a semisimple Lie algebra $\g$. This family is parameterized by collections of pairwise distinct complex numbers $z_1,...,z_n$. We obtain some new commutative subalgebras in $U(\g)^{\otimes n}$ as limit cases of Gaudin subalgebras. These commutative subalgebras turn to be related to the hamiltonians of bending flows and to the Gelfand--Tsetlin bases. We use this to prove the simplicity of spectrum in the Gaudin model for some new cases., Comment: 11 pages, references added

By means of a recently introduced bihamiltonian structure for the homogeneous Gaudin models, we find a new set of Separation Coordinates for the sl(r) case., 31 pages. LaTex with amsmath, amssymb

Let M be a complex projective manifold, and L an Hermitian ample line bundle on it. A fundamental theorem of Gang Tian, reproved and strengthened by Zelditch, implies that the Khaeler form of L can be recovered from the asymptotics of the projective embeddings associated to large tensor powers of L. More precisely, with the natural choice of metrics the projective embeddings associated to the full linear series |kL| are asymptotically symplectic, in the appropriate rescaled sense. In this article, we ask whether and how this result extends to the semiclassical setting. Specifically, we relate the Weinstein symplectic structure on a given isodrastic leaf of half-weighted Bohr-Sommerfeld Lagrangian submanifolds of M to the asymptotics of the the pull-back of the Fubini-Study form under the semiclassical projective maps constructed by Borthwick, Paul and Uribe., exposition improved

We study the existence of small amplitude oscillations near elliptic equilibria of autonomous systems, which mix different normal modes. The reference problem is the Fermi-Pasta-Ulam beta-model: a chain of nonlinear oscillators with nearest-neighborhood interaction. We develop a new bifurcation approach that locates secondary bifurcations from the unimodal primary branches. Two sufficient conditions for bifurcation are given: one involves only the arithmetic properties of the eigenvalues of the linearized system (asymptotic resonance), while the other takes into account the nonlinear character of the interaction between normal modes (nonlinear coupling). Both conditions are checked for the Fermi-Pasta-Ulam problem.

In 1883, S. Lie found the general form of all second-order ordinary differential equations transformable to the linear equation by a change of variables and proved that their solution reduces to integration of a linear third-order ordinary differential equation. He showed that the linearizable equations are at most cubic in the first-order derivative and described a general procedure for constructing linearizing transformations by using an over-determined system of four equations. We present here a simple geometric proof of the theorem, known as Lie's linearization test, stating that the compatibility of Lie's four auxiliary equations furnishes a necessary and sufficient condition for linearization.

A simple example shows the unexpected role of Lenard chains in the theory of classical integrable systems.

We tackle the problem of interpreting the Darboux transformation for the KP hierarchy and its relations with the modified KP hierarchy from a geometric point of view. This is achieved by introducing the concept of a Darboux covering. We construct a Darboux covering of the KP equations and obtain a new hierarchy of equations, which we call the Darboux-KP hierarchy (DKP). We employ the DKP equations to discuss the relationships among the KP equations, the modified KP equations, and the discrete KP equations. Our approach also handles the various reductions of the KP hierarchy. We show that the KP hierarchy is a projection of the DKP, the mKP hierarchy is a DKP restriction to a suitable invariant submanifold, and that the discrete KP equations are obtained as iterations of the DKP ones.

The paper aims to suggest a geometric point of view in the theory of soliton equations. The belief is that a deeper understanding of the origin of these equations may provide a better understanding of their remarkable properties. According to the geometric point of view, soliton equations are the outcome of a specific reduction process of a bi-Hamiltonian manifold. The suggestion of the paper is to pay attention also to the ‘unreduced form’ of soliton equations.

We study a certain family of Schrodinger operators whose eigenfunctions phi(x, lambda) satisfy a differential equation in the spectral parameter lambda of the form B(lambda, partial derivative-lambda)phi = THETA(x)phi. We show that the flows of a hierarchy of master symmetries for KdV are tangent to the manifolds that compose the strata of this class of bispectral potentials. This extends and complements a result of Duistermaat and Grunbaum concerning a similar property for the Adler and Moser potentials and the flows of the KdV hierarchy

We recall some deformation theory of Susy-curves and study obstructions to projectedness of supermoduli spaces, both from a general standpoint and by means of the local "coordinate charts" most commonly used in the physical literature. We prove that these give rise to a projected atlas for genus g=2 only. © 1990 Springer-Verlag.

In the language of tensor analysis on differentiable manifolds, we present a reduction method of integrability structures, and apply it to recover some well-known hierarchies of integrable nonlinear evolution equations. © 1985 Springer-Verlag.

By using the methods of nonlinear functional analysis a single general operator symmetry condition is derived. It supplies a simple and systematic procedure for finding symmetry conditions in particular problems. From it the Lie algebra structure associated with symmetries is easily displayed and the classical Lie theory of symmetries of differential equations is obtained without needing the technique of the group extension. © 1976 Società Italiana di Fisica.

The occurrence of non-abelian anomalies in gauge theories and gravitation, first discovered via perturbative techniques, is now completely explained from the mathematical point of view by means of the family index theorem of Atiyah and Singer. Here we make contact between this approach and BRS cohomology, by showing that they yield the same non-abelian anomalies, provided a certain restriction to "local" functionals is not introduced from the very beginning. In particular, this solves the "unicity" problem for this kind of anomalies. Local BRS cohomology is still relevant for the abelian case. © 1985 Springer-Verlag.