Garside theory and subsurfaces: Some examples in braid groups

Garside-theoretical solutions to the conjugacy problem in braid groups depend on the determination of a characteristic subset of the conjugacy class of any given braid, e.g. the sliding circuit set. It is conjectured that, among rigid braids with a fixed number of strands, the size of this set is bounded by a polynomial in the length of the braids. In this paper we suggest a more precise bound: for rigid braids with N strands and of Garside length L, the sliding circuit set should have at most {C\cdot L^{N-2}} elements, for some constant C. We construct a family of braids which realise this potential worst case. Our example braids suggest that having a large sliding circuit set is a geometric property of braids, as our examples have multiple subsurfaces with large subsurface projection; thus they are “almost reducible” in multiple ways, and act on the curve graph with small translation distance.

Existence and control of weighted pseudo almost periodic solutions for abstract nonlinear vibration systems

In many vibration problems, it is very important to know precisely the bounds of the stability/instability frequencies and the associated amplitude ranges. This paper investigates the solvability and control of some weighted pseudo almost periodic solutions of abstract nonlinear vibration differential systems. Some sufficient conditions for the solvability and exponential stability of these systems are obtained. Moreover, the precise bound of Lyapunov exponents is estimated.

Filtering Algorithm for Real Eigenvalue Bounds of Interval and Fuzzy Generalized Eigenvalue Problems

This paper deals with an interval and fuzzy generalized eigenvalue problem involving uncertain parameters. Based on a sufficient regularity condition for intervals, an interval filtering eigenvalue procedure for generalized eigenvalue problems with interval parameters is proposed, which iteratively eliminates the parts that do not contain an eigenvalue and thus reduces the initial eigenvalue bound to a precise bound. The same iterative procedure has been proposed for generalized fuzzy eigenvalue problems. In general, the solution of dynamic problems of structures using the finite element method (FEM) leads to a generalized eigenvalue problem. Based on the proposed procedures, various structural examples with an interval and fuzzy parameter such as triangular fuzzy number (TFN) are investigated to show the efficiency of the algorithms stated. Finally, fuzzy filtered eigenvalue bounds are depicted by fuzzy plots using the α-cut.

Ascent Sequences Avoiding Pairs of Patterns

Ascent sequences were introduced by Bousquet-Melou et al. in connection with (2+2)-avoiding posets and their pattern avoidance properties were first considered by Duncan and Steingrímsson. In this paper, we consider ascent sequences of length $n$ avoiding two patterns of length 3, and we determine an exact enumeration for 16 different pairs of patterns. Methods include simple recurrences, bijections to other combinatorial objects (including Dyck paths and pattern-avoiding permutations), and generating trees. We also provide an analogue of the Erdős-Szekeres Theorem to prove that any sufficiently long ascent sequence contains either many copies of the same number or a long increasing subsequence, with a precise bound.

Complex Dynamical Behaviors of the Chaotic Chen's System

In this paper, the complex dynamical behaviors of the chaotic trajectories of Chen's system are analyzed in detail, with its precise bound derived for the first time. In particular, it is rigorously proved that all nontrivial trajectories of the system always travel alternatively through two specific Poincaré projections for infinitely many times. The results provide an insightful understanding of the complex topological structure of Chen's chaotic attractor.

CLIFFORD'S THEOREM AND HIGHER RANK VECTOR BUNDLES

We give here a refinement of the classical Clifford's theorem for the upper bound of the number of independent global sections of a semistable vector bundle on a smooth curve. We also conjecture a new version of this theorem that takes into account the Clifford index of the curve. In the case of a bi-elliptic curve we obtain a very precise bound. Finally we study the case of rank 2 bundles.

Covering groups with subgroups

A group is covered by a collection of subgroups if it is the union of the collection. The intersection of an irredundant cover of n subgroups is known to have index bounded by a function of n, though in general the precise bound is not known. Here we confirm a claim of Tompkinson that the correct bound is 16 when n is 5. The proof depends on determining all the ‘minimal’ groups with an irredundant cover of five maximal subgroups.