5,359 research outputs found
Duality for Legendrian contact homology
The main result of this paper is that, off of a `fundamental class' in degree
1, the linearized Legendrian contact homology obeys a version of Poincare
duality between homology groups in degrees k and -k. Not only does the result
itself simplify calculations, but its proof also establishes a framework for
analyzing cohomology operations on the linearized Legendrian contact homology.Comment: This is the version published by Geometry & Topology on 8 December
200
Duality between Lagrangian and Legendrian invariants
Consider a pair , of a Weinstein manifold with an exact Lagrangian
submanifold , with ideal contact boundary , where is a
contact manifold and is a Legendrian submanifold. We
introduce the Chekanov-Eliashberg DG-algebra, , with
coefficients in chains of the based loop space of and study its
relation to the Floer cohomology of . Using the augmentation
induced by , can be expressed as the Adams cobar
construction applied to a Legendrian coalgebra, .
We define a twisting cochain:via holomorphic curve counts, where
denotes the bar construction and the graded linear dual. We show under
simply-connectedness assumptions that the corresponding Koszul complex is
acyclic which then implies that and are Koszul
dual. In particular, induces a quasi-isomorphism between
and the cobar of the Floer homology of , .
We use the duality result to show that under certain connectivity and locally
finiteness assumptions, is quasi-isomorphic to for any Lagrangian filling of . Our constructions have
interpretations in terms of wrapped Floer cohomology after versions of
Lagrangian handle attachments. In particular, we outline a proof that
is quasi-isomorphic to the wrapped Floer cohomology of a
fiber disk in the Weinstein domain obtained by attaching
to along (or, in the
terminology of arXiv:1604.02540 the wrapped Floer cohomology of in with
wrapping stopped by ). Along the way, we give a definition of wrapped
Floer cohomology without Hamiltonian perturbations.Comment: 126 pages, 20 figures. Substantial overall revision based on
referee's comments. The main results remain the same but the exposition has
been improve
Augmented Superfield Approach to Nilpotent Symmetries in the Modified Version of 2D Proca Theory
We derive the complete set of off-shell nilpotent and absolutely
anticommuting Becchi-Rouet-Stora-Tyutin (BRST), anti-BRST and (anti-)co-BRST
symmetry transformations for all the fields of the modified version of two
(1+1)-dimensional (2D) Proca theory by exploiting the "augmented" superfield
formalism where the (dual-)horizontality conditions and (dual-)gauge-invariant
restrictions are exploited together. We capture the (anti-)BRST and
(anti-)co-BRST invariance of the Lagrangian density in the language of
superfield formalism. We also express the nilpotency and absolute
anticommutativity of the (anti-)BRST and (anti-)co-BRST charges within the
framework of augmented superfield formalism. This exercise leads to some novel
observations which have, hitherto, not been pointed out in the literature
within the framework of superfield approach to BRST formalism. For the sake of
completeness, we also mention, very briefly, a unique bosonic symmetry, the
ghost-scale symmetry and discrete symmetries of the theory and show that the
algebra of conserved charges captures the cohomological aspects of differential
geometry. Thus, our present modified 2D Proca theory is a model for the Hodge
Theory.Comment: LaTeX file, 32 pages, journal reference give
Lagrange optimality system for a class of nonsmooth convex optimization
In this paper, we revisit the augmented Lagrangian method for a class of
nonsmooth convex optimization. We present the Lagrange optimality system of the
augmented Lagrangian associated with the problems, and establish its
connections with the standard optimality condition and the saddle point
condition of the augmented Lagrangian, which provides a powerful tool for
developing numerical algorithms. We apply a linear Newton method to the
Lagrange optimality system to obtain a novel algorithm applicable to a variety
of nonsmooth convex optimization problems arising in practical applications.
Under suitable conditions, we prove the nonsingularity of the Newton system and
the local convergence of the algorithm.Comment: 19 page
On barrier and modified barrier multigrid methods for 3d topology optimization
One of the challenges encountered in optimization of mechanical structures,
in particular in what is known as topology optimization, is the size of the
problems, which can easily involve millions of variables. A basic example is
the minimum compliance formulation of the variable thickness sheet (VTS)
problem, which is equivalent to a convex problem. We propose to solve the VTS
problem by the Penalty-Barrier Multiplier (PBM) method, introduced by R.\
Polyak and later studied by Ben-Tal and Zibulevsky and others. The most
computationally expensive part of the algorithm is the solution of linear
systems arising from the Newton method used to minimize a generalized augmented
Lagrangian. We use a special structure of the Hessian of this Lagrangian to
reduce the size of the linear system and to convert it to a form suitable for a
standard multigrid method. This converted system is solved approximately by a
multigrid preconditioned MINRES method. The proposed PBM algorithm is compared
with the optimality criteria (OC) method and an interior point (IP) method,
both using a similar iterative solver setup. We apply all three methods to
different loading scenarios. In our experiments, the PBM method clearly
outperforms the other methods in terms of computation time required to achieve
a certain degree of accuracy
- β¦