Triples of arrangements and local systems

For a triple of complex hyperplane arrangements, there is a well-known long exact sequence relating the cohomology of the complements. We observe that this result extends to certain local coefficient systems, and use this extension to study the characteristic varieties of arrangements. We show that the first characteristic variety may contain components that are translated by characters of any order, thereby answering a question of A. Suciu.

     

On the Nature of Buffering

We state a theorem on the instability of self-similar cycles and tori of a certain type in a system which is a quasinormal form of a boundary-value problem for a nonlinear wave equation in the square.     

Unfolding the torus: Oscillator geometry from time delays

Summary We present a simple method of plotting the trajectories of systems of weakly coupled oscillators. Our algorithm uses the time delays between the “firings” of the oscillators. For any system ofn weakly coupled oscillators there is an attracting invariantn-dimensional torus, and the attractor is a subset of this invariant torus. The invariant torus intersects a suitable codimension-1 surface of section at an (n−1)-dimensional torus. The dynamics ofn coupled oscillators can thus be reduced,in principle, to the study of Poincaré maps of the (n−1)-dimensional torus. This paper gives apractical algorithm for measuring then−1 angles on the torus. Since visualization of 3 (or higher) dimensional data is difficult we concentrate onn=3 oscillators. For three oscillators, a standard projection of the Poincaré map onto the plane yields a projection of the 2-torus which is 4-to-1 over most of the torus, making it difficult to observe the structure of the attractor. Our algorithm allows a direct measurement of the 2 angles on the torus, so we can plot a 1-to-1 map from the invariant torus to the “unfolded torus” where opposite edges of a square are identified. In the cases where the attractor is a torus knot, the knot type of the attractor is obvious in our projection.     

Necessary and Sufficient Conditions for Mcshane’s Identity and Variations

Greg McShane introduced a remarkable identity for lengths of simple closed geodesics on the once punctured torus with a complete, finite volume hyperbolic structure. Bowditch later generalized this and gave sufficient conditions for the identity to hold for general type-preserving representations of a free group on two generators Γ to SL(2,C), this was further generalized by the authors to obtain sufficient conditions for a generalized McShane’s identity to hold for arbitrary (not necessarily type-preserving) non-reducible representations in Tan et al. (Submitted). Here we extend the above by giving necessary and sufficient conditions for the generalized McShane identity to hold (Akiyoshi, Miyachi and Sakuma had proved it for type-preserving representations). We also give a version of Bowditch’s variation of McShane’s identity to once-punctured torus bundles, in the case where the monodromy is generated by a reducible element, and provide necessary and sufficient conditions for the variations to hold.     

Homology of Fortuin–Kasteleyn Clusters of Potts Models on the Torus

Topological properties of Fortuin–Kasteleyn clusters are studied on the torus. Namely, the probability that their topology yields a given subgroup of the first homology group of the torus is computed for Q=1, 2, 3 and 4. The expressions generalize those obtained by Pinson for percolation (Q=1). Numerical results are also presented for three tori of different moduli. They agree with the theoretical predictions for Q=1, 2 and 3. For Q=4 agreement is not ruled out but logarithmic corrections are probably present and they make it harder to decide.     

On Completely Integrable Systems with Local Torus Actions

This paper concerns with symplectic topology of compact completely integrable Hamiltonian systems with only stable nondegenerate elliptic singularities. We describe all systems whose universal coverings admit actian-angle coordinates and, in particular, prove that some finite cover of a base space is diffeomorphic to a product of a convex polytope and a solvmanifold. We also construct an obstruction, vanishing of which guarantees splitting of some finite cover of a phase space as a toric variety and a torus fibering over a solvmanifold.     

Exact partition functions and correlation functions of multiple Hamiltonian walks on the Manhattan lattice

This is a general and exact study of multiple Hamiltonian walks (HAW) filling the two-dimensional (2D) Manhattan lattice. We generalize the original exact solution for a single HAW by Kasteleyn to a system ofmultiple closed walks, aimed at modeling a polymer melt. In 2D, two basic nonequivalent topological situations are distinguished. (1) the Hamiltonian loops are allrooted andcontractible to a point:adjacent one to another, and, on a torus,homotopic to zero. (2) the loops can encircle one another and, on a torus, canwind around it. Forcase 1, the grand canonical partition function and multiple correlation functions are calculated exactly as those of multiple rooted spanningtrees or of a massive 2Dfree field, critical at zero mass (zero fugacity). The conformally invariant continuum limit on a Manhattantorus is studied in detail. The melt entropy is calculated exactly. We also consider the relevant effect of free boundary conditions. The number of single HAWs on Manhattan lattices with other perimeter shapes (rectangular, Kagomé, triangular, and arbitrary) is studied and related to the spectral theory of the Dirichlet Laplacian. This allows the calculation of exact shape-dependent configuration exponents y. An exact surface critical exponent is obtained. Forcase 2, nested and winding Hamiltonian circuits are allowed. An exact equivalence to thecritical Q-state Potts model exists, whereQ ^1/2 is the walk fugacity. The Hamiltonian system is then always critical (forQ<-4). The exact critical exponents, in infinite numbers, are universal and identical to those of theO(n=Q ^1/2) model in its low-temperature phase, i.e. are those of dense polymers. The exact critical partition functions on the torus are given from conformai invariance theory. These models 1 and 2 yield the two first exactly solved models of polymer melts.     

An Estimate of Smoothing and Composition with Applications to Conjugation Problems

An estimation of the difference between smoothing a composition of two functions and composing their smoothings is given. The smoothing operator considered here is defined by the convolution operator with a holomorphic kernel. It is also shown, by means of an example, how to apply the estimate given here to obtain finite differentiable versions of theorems on conjugation of maps in non-perturbative settings.      

两类幂零的n-Lie代数

本文提出并构造了两类幂零的n-Lie代数:特征幂零的n-Lie代数与最大秩的幂零的n-Lie代数.证明了n-Lie代数是特征幂零的n-Lie代数的充分必要条件,以及最大秩的幂零的n-Lie代数的结构特征.     

Minimal invariant tori in the resonant regions for nearly integrable Hamiltonian systems

Consider a real analytical Hamiltonian system of KAM type that has degrees of freedom (2$">) and is positive definite in . Let . In this paper we show that for most rotation vectors in , in the sense of ()-dimensional Lebesgue measure, there is at least one ()-dimensional invariant torus. These tori are the support of corresponding minimal measures. The Lebesgue measure estimate on this set is uniformly valid for any perturbation.