2,280 research outputs found
Probability of local bifurcation type from a fixed point: A random matrix perspective
Results regarding probable bifurcations from fixed points are presented in
the context of general dynamical systems (real, random matrices), time-delay
dynamical systems (companion matrices), and a set of mappings known for their
properties as universal approximators (neural networks). The eigenvalue spectra
is considered both numerically and analytically using previous work of Edelman
et. al. Based upon the numerical evidence, various conjectures are presented.
The conclusion is that in many circumstances, most bifurcations from fixed
points of large dynamical systems will be due to complex eigenvalues.
Nevertheless, surprising situations are presented for which the aforementioned
conclusion is not general, e.g. real random matrices with Gaussian elements
with a large positive mean and finite variance.Comment: 21 pages, 19 figure
Distributions of flux vacua
We give results for the distribution and number of flux vacua of various
types, supersymmetric and nonsupersymmetric, in IIb string theory compactified
on Calabi-Yau manifolds. We compare this with related problems such as counting
attractor points.Comment: 43 pages, 7 figures. v2: improved discussion of finding vacua with
discrete flux, references adde
Lattice congruences of the weak order
We study the congruence lattice of the poset of regions of a hyperplane
arrangement, with particular emphasis on the weak order on a finite Coxeter
group. Our starting point is a theorem from a previous paper which gives a
geometric description of the poset of join-irreducibles of the congruence
lattice of the poset of regions in terms of certain polyhedral decompositions
of the hyperplanes. For a finite Coxeter system (W,S) and a subset K of S, let
\eta_K:w \mapsto w_K be the projection onto the parabolic subgroup W_K. We show
that the fibers of \eta_K constitute the smallest lattice congruence with
1\equiv s for every s\in(S-K). We give an algorithm for determining the
congruence lattice of the weak order for any finite Coxeter group and for a
finite Coxeter group of type A or B we define a directed graph on subsets or
signed subsets such that the transitive closure of the directed graph is the
poset of join-irreducibles of the congruence lattice of the weak order.Comment: 26 pages, 4 figure
Eigenvalue distributions for some correlated complex sample covariance matrices
The distributions of the smallest and largest eigenvalues for the matrix
product , where is an complex Gaussian matrix
with correlations both along rows and down columns, are expressed as determinants. In the case of correlation along rows, these expressions are
computationally more efficient than those involving sums over partitions and
Schur polynomials reported recently for the same distributions.Comment: 11 page
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