38 research outputs found
The rank of a hypergeometric system
The holonomic rank of the A-hypergeometric system M_A(\beta) is the degree of
the toric ideal I_A for generic parameters; in general, this is only a lower
bound. To the semigroup ring of A we attach the ranking arrangement and use
this algebraic invariant and the exceptional arrangement of nongeneric
parameters to construct a combinatorial formula for the rank jump of
M_A(\beta). As consequences, we obtain a refinement of the stratification of
the exceptional arrangement by the rank of M_A(\beta) and show that the Zariski
closure of each of its strata is a union of translates of linear subspaces of
the parameter space. These results hold for generalized A-hypergeometric
systems as well, where the semigroup ring of A is replaced by a nontrivial
weakly toric module M contained in \CC[\ZZ A]. We also provide a direct proof
of the result of M. Saito and W. Traves regarding the isomorphism classes of
M_A(\beta).Comment: 32 pages. To appear in Compositio Mathematica. Revisions have been
made to the exposition, and the notation has been simplifie
Products of Foldable Triangulations
Regular triangulations of products of lattice polytopes are constructed with
the additional property that the dual graphs of the triangulations are
bipartite. The (weighted) size difference of this bipartition is a lower bound
for the number of real roots of certain sparse polynomial systems by recent
results of Soprunova and Sottile [Adv. Math. 204(1):116-151, 2006]. Special
attention is paid to the cube case.Comment: new title; several paragraphs reformulated; 23 page
The role of the Legendre transform in the study of the Floer complex of cotangent bundles
Consider a classical Hamiltonian H on the cotangent bundle T*M of a closed
orientable manifold M, and let L:TM -> R be its Legendre-dual Lagrangian. In a
previous paper we constructed an isomorphism Phi from the Morse complex of the
Lagrangian action functional which is associated to L to the Floer complex
which is determined by H. In this paper we give an explicit construction of a
homotopy inverse Psi of Phi. Contrary to other previously defined maps going in
the same direction, Psi is an isomorphism at the chain level and preserves the
action filtration. Its definition is based on counting Floer trajectories on
the negative half-cylinder which on the boundary satisfy "half" of the Hamilton
equations. Albeit not of Lagrangian type, such a boundary condition defines
Fredholm operators with good compactness properties. We also present a
heuristic argument which, independently on any Fredholm and compactness
analysis, explains why the spaces of maps which are used in the definition of
Phi and Psi are the natural ones. The Legendre transform plays a crucial role
both in our rigorous and in our heuristic arguments. We treat with some detail
the delicate issue of orientations and show that the homology of the Floer
complex is isomorphic to the singular homology of the loop space of M with a
system of local coefficients, which is defined by the pull-back of the second
Stiefel-Whitney class of TM on 2-tori in M
Continuous time random walk and parametric subordination in fractional diffusion
The well-scaled transition to the diffusion limit in the framework of the
theory of continuous-time random walk (CTRW)is presented starting from its
representation as an infinite series that points out the subordinated character
of the CTRW itself. We treat the CTRW as a combination of a random walk on the
axis of physical time with a random walk in space, both walks happening in
discrete operational time. In the continuum limit we obtain a generally
non-Markovian diffusion process governed by a space-time fractional diffusion
equation. The essential assumption is that the probabilities for waiting times
and jump-widths behave asymptotically like powers with negative exponents
related to the orders of the fractional derivatives. By what we call parametric
subordination, applied to a combination of a Markov process with a positively
oriented L\'evy process, we generate and display sample paths for some special
cases.Comment: 28 pages, 18 figures. Workshop 'In Search of a Theory of Complexity'.
Denton, Texas, August 200
Millisecond and Binary Pulsars as Nature's Frequency Standards. III. Fourier Analysis and Spectral Sensitivity of Timing Observations to Low-Frequency Noise
This paper discusses spectral sensitivity of solitary and binary pulsars to a
low-frequency (colored) noise. It is a third paper in a series of papers
devoted to analysis of the low-frequency noise in pulsar timing observations
and its impact on observable fitting parameters of timing model.Comment: 25 pages, 3 figures, accepted to MNRA