5,409 research outputs found
On The L{2}-Solutions of Stochastic Fractional Partial Differential Equations; Existence, Uniqueness and Equivalence of Solutions
The aim of this work is to prove existence and uniqueness of
solutions of stochastic fractional partial differential equations in
one spatial dimension. We prove also the equivalence between several notions of
solutions. The Fourier transform is used to give meaning to SFPDEs.
This method is valid also when the diffusion coefficient is random
The Secant Conjecture in the real Schubert calculus
We formulate the Secant Conjecture, which is a generalization of the Shapiro
Conjecture for Grassmannians. It asserts that an intersection of Schubert
varieties in a Grassmannian is transverse with all points real, if the flags
defining the Schubert varieties are secant along disjoint intervals of a
rational normal curve. We present theoretical evidence for it as well as
computational evidence obtained in over one terahertz-year of computing, and we
discuss some phenomena we observed in our data.Comment: 19 page
Visually building Smale flows in S3
A Smale flow is a structurally stable flow with one dimensional invariant
sets. We use information from homology and template theory to construct,
visualize and in some cases, classify, nonsingular Smale flows in the 3-sphere
On the Scaling Limits of Determinantal Point Processes with Kernels Induced by Sturm-Liouville Operators
By applying an idea of Borodin and Olshanski [J. Algebra 313 (2007), 40-60],
we study various scaling limits of determinantal point processes with trace
class projection kernels given by spectral projections of selfadjoint
Sturm-Liouville operators. Instead of studying the convergence of the kernels
as functions, the method directly addresses the strong convergence of the
induced integral operators. We show that, for this notion of convergence, the
Dyson, Airy, and Bessel kernels are universal in the bulk, soft-edge, and
hard-edge scaling limits. This result allows us to give a short and unified
derivation of the known formulae for the scaling limits of the classical random
matrix ensembles with unitary invariance, that is, the Gaussian unitary
ensemble (GUE), the Wishart or Laguerre unitary ensemble (LUE), and the MANOVA
(multivariate analysis of variance) or Jacobi unitary ensemble (JUE)
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