412,895 research outputs found
Baire reductions and good Borel reducibilities
In reference [8] we have considered a wide class of "well-behaved"
reducibilities for sets of reals. In this paper we continue with the study of
Borel reducibilities by proving a dichotomy theorem for the degree-structures
induced by good Borel reducibilities. This extends and improves the results of
[8] allowing to deal with a larger class of notions of reduction (including,
among others, the Baire class functions).Comment: 21 page
Constrained Reductions of 2D dispersionless Toda Hierarchy, Hamiltonian Structure and Interface Dynamics
Finite-dimensional reductions of the 2D dispersionless Toda hierarchy,
constrained by the ``string equation'' are studied. These include solutions
determined by polynomial, rational or logarithmic functions, which are of
interest in relation to the ``Laplacian growth'' problem governing interface
dynamics. The consistency of such reductions is proved, and the Hamiltonian
structure of the reduced dynamics is derived. The Poisson structure of the
rationally reduced dispersionless Toda hierarchies is also derivedComment: 18 pages LaTex, accepted to J.Math.Phys, Significantly updated
version of the previous submissio
Uncovering Hierarchical Structure in Social Networks using Isospectral Reductions
We employ the recently developed theory of isospectral network reductions to
analyze multi-mode social networks. This procedure allows us to uncover the
hierarchical structure of the networks we consider as well as the hierarchical
structure of each mode of the network. Additionally, by performing a dynamical
analysis of these networks we are able to analyze the evolution of their
structure allowing us to find a number of other network features. We apply both
of these approaches to the Southern Women Data Set, one of the most studied
social networks and demonstrate that these techniques provide new information,
which complements previous findings.Comment: 17 pages, 5 figures, 5 table
Gr\"obner Bases of Bihomogeneous Ideals generated by Polynomials of Bidegree (1,1): Algorithms and Complexity
Solving multihomogeneous systems, as a wide range of structured algebraic
systems occurring frequently in practical problems, is of first importance.
Experimentally, solving these systems with Gr\"obner bases algorithms seems to
be easier than solving homogeneous systems of the same degree. Nevertheless,
the reasons of this behaviour are not clear. In this paper, we focus on
bilinear systems (i.e. bihomogeneous systems where all equations have bidegree
(1,1)). Our goal is to provide a theoretical explanation of the aforementionned
experimental behaviour and to propose new techniques to speed up the Gr\"obner
basis computations by using the multihomogeneous structure of those systems.
The contributions are theoretical and practical. First, we adapt the classical
F5 criterion to avoid reductions to zero which occur when the input is a set of
bilinear polynomials. We also prove an explicit form of the Hilbert series of
bihomogeneous ideals generated by generic bilinear polynomials and give a new
upper bound on the degree of regularity of generic affine bilinear systems.
This leads to new complexity bounds for solving bilinear systems. We propose
also a variant of the F5 Algorithm dedicated to multihomogeneous systems which
exploits a structural property of the Macaulay matrix which occurs on such
inputs. Experimental results show that this variant requires less time and
memory than the classical homogeneous F5 Algorithm.Comment: 31 page
Integrable equations of the dispersionless Hirota type and hypersurfaces in the Lagrangian Grassmannian
We investigate integrable second order equations of the form
F(u_{xx}, u_{xy}, u_{yy}, u_{xt}, u_{yt}, u_{tt})=0.
Familiar examples include the Boyer-Finley equation, the potential form of
the dispersionless Kadomtsev-Petviashvili equation, the dispersionless Hirota
equation, etc. The integrability is understood as the existence of infinitely
many hydrodynamic reductions. We demonstrate that the natural equivalence group
of the problem is isomorphic to Sp(6), revealing a remarkable correspondence
between differential equations of the above type and hypersurfaces of the
Lagrangian Grassmannian. We prove that the moduli space of integrable equations
of the dispersionless Hirota type is 21-dimensional, and the action of the
equivalence group Sp(6) on the moduli space has an open orbit.Comment: 32 page
Lam\'e polynomials, hyperelliptic reductions and Lam\'e band structure
The band structure of the Lam\'e equation, viewed as a one-dimensional
Schr\"odinger equation with a periodic potential, is studied. At integer values
of the degree parameter l, the dispersion relation is reduced to the l=1
dispersion relation, and a previously published l=2 dispersion relation is
shown to be partially incorrect. The Hermite-Krichever Ansatz, which expresses
Lam\'e equation solutions in terms of l=1 solutions, is the chief tool. It is
based on a projection from a genus-l hyperelliptic curve, which parametrizes
solutions, to an elliptic curve. A general formula for this covering is
derived, and is used to reduce certain hyperelliptic integrals to elliptic
ones. Degeneracies between band edges, which can occur if the Lam\'e equation
parameters take complex values, are investigated. If the Lam\'e equation is
viewed as a differential equation on an elliptic curve, a formula is
conjectured for the number of points in elliptic moduli space (elliptic curve
parameter space) at which degeneracies occur. Tables of spectral polynomials
and Lam\'e polynomials, i.e., band edge solutions, are given. A table in the
older literature is corrected.Comment: 38 pages, 1 figure; final revision
Beyond Borel-amenability: scales and superamenable reducibilities
We analyze the degree-structure induced by large reducibilities under the
Axiom of Determinacy. This generalizes the analysis of Borel reducibilities
given in references [1], [6] and [5] e.g. to the projective levels.Comment: 13 page
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