11,713 research outputs found
Generic Regular Decompositions for Parametric Polynomial Systems
This paper presents a generalization of our earlier work in [19]. In this
paper, the two concepts, generic regular decomposition (GRD) and
regular-decomposition-unstable (RDU) variety introduced in [19] for generic
zero-dimensional systems, are extended to the case where the parametric systems
are not necessarily zero-dimensional. An algorithm is provided to compute GRDs
and the associated RDU varieties of parametric systems simultaneously on the
basis of the algorithm for generic zero-dimensional systems proposed in [19].
Then the solutions of any parametric system can be represented by the solutions
of finitely many regular systems and the decomposition is stable at any
parameter value in the complement of the associated RDU variety of the
parameter space. The related definitions and the results presented in [19] are
also generalized and a further discussion on RDU varieties is given from an
experimental point of view. The new algorithm has been implemented on the basis
of DISCOVERER with Maple 16 and experimented with a number of benchmarks from
the literature.Comment: It is the latest version. arXiv admin note: text overlap with
arXiv:1208.611
Coadjoint orbits of symplectic diffeomorphisms of surfaces and ideal hydrodynamics
We give a classification of generic coadjoint orbits for the groups of
symplectomorphisms and Hamiltonian diffeomorphisms of a closed symplectic
surface. We also classify simple Morse functions on symplectic surfaces with
respect to actions of those groups. This gives an answer to V.Arnold's problem
on describing all invariants of generic isovorticed fields for the 2D ideal
fluids. For this we introduce a notion of anti-derivatives on a measured Reeb
graph and describe their properties.Comment: 38 pages, 11 figures; to appear in Annales de l'Institut Fourie
Symmetry adapted Assur decompositions
Assur graphs are a tool originally developed by mechanical engineers to
decompose mechanisms for simpler analysis and synthesis. Recent work has
connected these graphs to strongly directed graphs, and decompositions of the
pinned rigidity matrix. Many mechanisms have initial configurations which are
symmetric, and other recent work has exploited the orbit matrix as a symmetry
adapted form of the rigidity matrix. This paper explores how the decomposition
and analysis of symmetric frameworks and their symmetric motions can be
supported by the new symmetry adapted tools.Comment: 40 pages, 22 figure
Fibrations in complete intersections of quadrics, Clifford algebras, derived categories, and rationality problems
Let X -> Y be a fibration whose fibers are complete intersections of two
quadrics. We develop new categorical and algebraic tools---a theory of relative
homological projective duality and the Morita invariance of the even Clifford
algebra under quadric reduction by hyperbolic splitting---to study
semiorthogonal decompositions of the bounded derived category of X. Together
with new results in the theory of quadratic forms, we apply these tools in the
case where X -> Y has relative dimension 1, 2, or 3, in which case the fibers
are curves of genus 1, Del Pezzo surfaces of degree 4, or Fano threefolds,
respectively. In the latter two cases, if Y is the projective line over an
algebraically closed field of characteristic zero, we relate rationality
questions to categorical representability of X.Comment: 43 pages, changes made and some material added and corrected in
sections 1, 4, and 5; this is the final version accepted for publication at
Journal de Math\'ematiques Pures et Appliqu\'ee
Generating Polynomials and Symmetric Tensor Decompositions
This paper studies symmetric tensor decompositions. For symmetric tensors,
there exist linear relations of recursive patterns among their entries. Such a
relation can be represented by a polynomial, which is called a generating
polynomial. The homogenization of a generating polynomial belongs to the apolar
ideal of the tensor. A symmetric tensor decomposition can be determined by a
set of generating polynomials, which can be represented by a matrix. We call it
a generating matrix. Generally, a symmetric tensor decomposition can be
determined by a generating matrix satisfying certain conditions. We
characterize the sets of such generating matrices and investigate their
properties (e.g., the existence, dimensions, nondefectiveness). Using these
properties, we propose methods for computing symmetric tensor decompositions.
Extensive examples are shown to demonstrate the efficiency of proposed methods.Comment: 35 page
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