33,210 research outputs found
Moment-angle complexes, monomial ideals, and Massey products
Associated to every finite simplicial complex K there is a "moment-angle"
finite CW-complex, Z_K; if K is a triangulation of a sphere, Z_K is a smooth,
compact manifold. Building on work of Buchstaber, Panov, and Baskakov, we study
the cohomology ring, the homotopy groups, and the triple Massey products of a
moment-angle complex, relating these topological invariants to the algebraic
combinatorics of the underlying simplicial complex. Applications to the study
of non-formal manifolds and subspace arrangements are given.Comment: 30 pages. Published versio
On the homotopy Lie algebra of an arrangement
Let A be a graded-commutative, connected k-algebra generated in degree 1. The
homotopy Lie algebra g_A is defined to be the Lie algebra of primitives of the
Yoneda algebra, Ext_A(k,k). Under certain homological assumptions on A and its
quadratic closure, we express g_A as a semi-direct product of the
well-understood holonomy Lie algebra h_A with a certain h_A-module. This allows
us to compute the homotopy Lie algebra associated to the cohomology ring of the
complement of a complex hyperplane arrangement, provided some combinatorial
assumptions are satisfied. As an application, we give examples of hyperplane
arrangements whose complements have the same Poincar\'e polynomial, the same
fundamental group, and the same holonomy Lie algebra, yet different homotopy
Lie algebras.Comment: 20 pages; accepted for publication by the Michigan Math. Journa
Fitness-based network growth with dynamic feedback
This article is a preprint of a paper that is currently under review with Physical Review E.We study a class of network growth models in which the choice of attachment by new nodes is governed by intrinsic attractiveness, or tness, of the existing nodes. The key feature of the models is a feedback mechanism whereby the distribution from which fitnesses of new nodes are drawn is dynamically updated to account for the evolving degree distribution. It is shown that in the case of linear mapping between fitnesses and degrees, the models lead to tunable stationary powerlaw degree distribution, while in the non-linear case the distributions converge to the stretched exponential form
Coarse Brownian Dynamics for Nematic Liquid Crystals: Bifurcation Diagrams via Stochastic Simulation
We demonstrate how time-integration of stochastic differential equations
(i.e. Brownian dynamics simulations) can be combined with continuum numerical
bifurcation analysis techniques to analyze the dynamics of liquid crystalline
polymers (LCPs). Sidestepping the necessity of obtaining explicit closures, the
approach analyzes the (unavailable in closed form) coarse macroscopic
equations, estimating the necessary quantities through appropriately
initialized, short bursts of Brownian dynamics simulation. Through this
approach, both stable and unstable branches of the equilibrium bifurcation
diagram are obtained for the Doi model of LCPs and their coarse stability is
estimated. Additional macroscopic computational tasks enabled through this
approach, such as coarse projective integration and coarse stabilizing
controller design, are also demonstrated
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