1,701 research outputs found
On the Complexity of Hilbert Refutations for Partition
Given a set of integers W, the Partition problem determines whether W can be
divided into two disjoint subsets with equal sums. We model the Partition
problem as a system of polynomial equations, and then investigate the
complexity of a Hilbert's Nullstellensatz refutation, or certificate, that a
given set of integers is not partitionable. We provide an explicit construction
of a minimum-degree certificate, and then demonstrate that the Partition
problem is equivalent to the determinant of a carefully constructed matrix
called the partition matrix. In particular, we show that the determinant of the
partition matrix is a polynomial that factors into an iteration over all
possible partitions of W.Comment: Final versio
New Structured Matrix Methods for Real and Complex Polynomial Root-finding
We combine the known methods for univariate polynomial root-finding and for
computations in the Frobenius matrix algebra with our novel techniques to
advance numerical solution of a univariate polynomial equation, and in
particular numerical approximation of the real roots of a polynomial. Our
analysis and experiments show efficiency of the resulting algorithms.Comment: 18 page
Pseudo-Hermiticity for a Class of Nondiagonalizable Hamiltonians
We give two characterization theorems for pseudo-Hermitian (possibly
nondiagonalizable) Hamiltonians with a discrete spectrum that admit a
block-diagonalization with finite-dimensional diagonal blocks. In particular,
we prove that for such an operator H the following statements are equivalent.
1. H is pseudo-Hermitian; 2. The spectrum of H consists of real and/or
complex-conjugate pairs of eigenvalues and the geometric multiplicity and the
dimension of the diagonal blocks for the complex-conjugate eigenvalues are
identical; 3. H is Hermitian with respect to a positive-semidefinite inner
product. We further discuss the relevance of our findings for the merging of a
complex-conjugate pair of eigenvalues of diagonalizable pseudo-Hermitian
Hamiltonians in general, and the PT-symmetric Hamiltonians and the effective
Hamiltonian for a certain closed FRW minisuperspace quantum cosmological model
in particular.Comment: 17 pages, slightly revised version, to appear in J. Math. Phy
Functions of random walks on hyperplane arrangements
Many seemingly disparate Markov chains are unified when viewed as random
walks on the set of chambers of a hyperplane arrangement. These include the
Tsetlin library of theoretical computer science and various shuffling schemes.
If only selected features of the chains are of interest, then the mixing times
may change. We study the behavior of hyperplane walks, viewed on a
subarrangement of a hyperplane arrangement. These include many new examples,
for instance a random walk on the set of acyclic orientations of a graph. All
such walks can be treated in a uniform fashion, yielding diagonalizable
matrices with known eigenvalues, stationary distribution and good rates of
convergence to stationarity.Comment: Final version; Section 4 has been split into two section
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