Zero-Temperature Dynamics of Ising Models on the Triangular Lattice

We consider the nature of spin flips of zero-temperature dynamics for ferromagnetic Ising models on the triangular lattice with nearest-neighbor interactions and an initial configuration chosen from a symmetric Bernoulli distribution. We prove that all spins flip infinitely many times for almost every realization of the dynamics and initial configuration.     

Haplotype assembly from aligned weighted SNP fragments

Given an assembled genome of a diploid organism the haplotype assembly problem can be formulated as retrieval of a pair of haplotypes from a set of aligned weighted SNP fragments. Known computational formulations (models) of this problem are minimum letter flips (MLF) and the weighted minimum letter flips (WMLF; Greenberg et al. (INFORMS J. Comput. 2004, 14, 211-213)). In this paper we show that the general WMLF model is NP-hard even for the gapless case. However the algorithmic solutions for selected variants of WMFL can exist and we propose a heuristic algorithm based on a dynamic clustering technique. We also introduce a new formulation of the haplotype assembly problem that we call COMPLETE WMLF (CWMLF). This model and algorithms for its implementation take into account a simultaneous presence of multiple kinds of data errors. Extensive computational experiments indicate that the algorithmic implementations of the CWMLF model achieve higher accuracy of haplotype reconstruction than the WMLF-based algorithms, which in turn appear to be more accurate than those based on MLF.     

Codimension 2 reversible heteroclinic bifurcations with inclination flips

In this paper, the heteroclinic bifurcation problem with real eigenvalues and two inclination-flips is investigated in a four-dimensional reversible system. We perform a detailed study of this case by using the method originally established in the papers “Problems in Homoclinic Bifurcation with Higher Dimensions” and “Bifurcation of Heteroclinic Loops,” and obtain fruitful results, such as the existence and coexistence of R-symmetric homoclinic orbit and R-symmetric heteroclinic loops, R-symmetric homoclinic orbit and R-symmetric periodic orbit. The double R-symmetric homoclinic bifurcation (i.e., two-fold R-symmetric homoclinic bifurcation) for reversible heteroclinic loops is found, and the existence of infinitely many R-symmetric periodic orbits accumulating onto a homoclinic orbit is demonstrated. The relevant bifurcation surfaces and the existence regions are also located. This work was supported by National Natural Science Foundation of China (Grant No. 10671069)     

Bifurcation Analysis of the Multiple Flips Homoclinic Orbit

A high-codimension homoclinic bifurcation is considered with one orbit flip and two inclination flips accompanied by resonant principal eigenvalues. A local active coordinate system in a small neighborhood of homoclinic orbit is introduced. By analysis of the bifurcation equation, the authors obtain the conditions when the original flip homoclinic orbit is kept or broken. The existence and the existence regions of several double periodic orbits and one triple periodic orbit bifurcations are proved. Moreover, the complicated homoclinic-doubling bifurcations are found and expressed approximately.     

Some Results on Secant Varieties Leading to a Geometric Flip Construction

We study the relationship between the equations defining a projective variety and properties of its secant varieties. In particular, we use information about the syzygies among the defining equations to derive smoothness and normality statements about SecX and also to obtain information about linear systems on the blow up of projective space along a variety X. We use these results to geometrically construct, for varieties of arbitrary dimension, a flip first described in the case of curves by M. Thaddeus via Geometric Invariant Theory.     

Diagonal flips in pseudo-triangulations on closed surfaces

Negami has already shown that there is a natural number N(F²) for any closed surface F² such that two triangulations on F² with n vertices can be transformed into each other by a sequence of diagonal flips if nN(F²). We investigate the same theorem for pseudo-triangulations with or without loops, estimating the length of a sequence of diagonal flips. Our arguments will be applied to simple triangulations to obtain a linear upper bound for N(F²) with respect to the genus of F².