8,254 research outputs found
Counting eigenvalues in domains of the complex field
A procedure for counting the number of eigenvalues of a matrix in a region
surrounded by a closed curve is presented. It is based on the application of
the residual theorem. The quadrature is performed by evaluating the principal
argument of the logarithm of a function. A strategy is proposed for selecting a
path length that insures that the same branch of the logarithm is followed
during the integration. Numerical tests are reported for matrices obtained from
conventional matrix test sets.Comment: 21 page
Divergences in the Moduli Space Integral and Accumulating Handles in the Infinite-Genus Limit
The symmetries associated with the closed bosonic string partition function
are examined so that the integration region in Teichmuller space can be
determined. The conditions on the period matrix defining the fundamental region
can be translated to relations on the parameters of the uniformizing Schottky
group. The growth of the lower bound for the regularized partition function is
derived through integration over a subset of the fundamental region.Comment: 26 pages, DAMTP-R/94/1
Space-time autocoding
Prior treatments of space-time communications in Rayleigh flat fading generally assume that channel coding covers either one fading interval-in which case there is a nonzero âoutage capacityâ-or multiple fading intervals-in which case there is a nonzero Shannon capacity. However, we establish conditions under which channel codes span only one fading interval and yet are arbitrarily reliable. In short, space-time signals are their own channel codes. We call this phenomenon space-time autocoding, and the accompanying capacity the space-time autocapacity. Let an M-transmitter antenna, N-receiver antenna Rayleigh flat fading channel be characterized by an MĂN matrix of independent propagation coefficients, distributed as zero-mean, unit-variance complex Gaussian random variables. This propagation matrix is unknown to the transmitter, it remains constant during a T-symbol coherence interval, and there is a fixed total transmit power. Let the coherence interval and number of transmitter antennas be related as T=ÎČM for some constant ÎČ. A TĂM matrix-valued signal, associated with R·T bits of information for some rate R is transmitted during the T-symbol coherence interval. Then there is a positive space-time autocapacity Ca such that for all R<Ca, the block probability of error goes to zero as the pair (T, M)ââ such that T/M=ÎČ. The autocoding effect occurs whether or not the propagation matrix is known to the receiver, and Ca=Nlog(1+Ï) in either case, independently of ÎČ, where Ï is the expected signal-to-noise ratio (SNR) at each receiver antenna. Lower bounds on the cutoff rate derived from random unitary space-time signals suggest that the autocoding effect manifests itself for relatively small values of T and M. For example, within a single coherence interval of duration T=16, for M=7 transmitter antennas and N=4 receiver antennas, and an 18-dB expected SNR, a total of 80 bits (corresponding to rate R=5) can theoretically be transmitted with a block probability of error less than 10^-9, all without any training or knowledge of the propagation matrix
An Overview of Polynomially Computable Characteristics of Special Interval Matrices
It is well known that many problems in interval computation are intractable,
which restricts our attempts to solve large problems in reasonable time. This
does not mean, however, that all problems are computationally hard. Identifying
polynomially solvable classes thus belongs to important current trends. The
purpose of this paper is to review some of such classes. In particular, we
focus on several special interval matrices and investigate their convenient
properties. We consider tridiagonal matrices, {M,H,P,B}-matrices, inverse
M-matrices, inverse nonnegative matrices, nonnegative matrices, totally
positive matrices and some others. We focus in particular on computing the
range of the determinant, eigenvalues, singular values, and selected norms.
Whenever possible, we state also formulae for determining the inverse matrix
and the hull of the solution set of an interval system of linear equations. We
survey not only the known facts, but we present some new views as well
Random matrices: Universality of local spectral statistics of non-Hermitian matrices
It is a classical result of Ginibre that the normalized bulk -point
correlation functions of a complex Gaussian matrix with independent
entries of mean zero and unit variance are asymptotically given by the
determinantal point process on with kernel
in the limit
. In this paper, we show that this asymptotic law is universal
among all random matrices whose entries are jointly
independent, exponentially decaying, have independent real and imaginary parts
and whose moments match that of the complex Gaussian ensemble to fourth order.
Analogous results at the edge of the spectrum are also obtained. As an
application, we extend a central limit theorem for the number of eigenvalues of
complex Gaussian matrices in a small disk to these more general ensembles.
These results are non-Hermitian analogues of some recent universality results
for Hermitian Wigner matrices. However, a key new difficulty arises in the
non-Hermitian case, due to the instability of the spectrum for such matrices.
To resolve this issue, we the need to work with the log-determinants
rather than with the Stieltjes transform
, in order to exploit Girko's
Hermitization method. Our main tools are a four moment theorem for these
log-determinants, together with a strong concentration result for the
log-determinants in the Gaussian case. The latter is established by studying
the solutions of a certain nonlinear stochastic difference equation. With some
extra consideration, we can extend our arguments to the real case, proving
universality for correlation functions of real matrices which match the real
Gaussian ensemble to the fourth order. As an application, we show that a real
matrix whose entries are jointly independent, exponentially
decaying and whose moments match the real Gaussian ensemble to fourth order has
real eigenvalues asymptotically almost
surely.Comment: Published in at http://dx.doi.org/10.1214/13-AOP876 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Random matrices: The Universality phenomenon for Wigner ensembles
In this paper, we survey some recent progress on rigorously establishing the
universality of various spectral statistics of Wigner Hermitian random matrix
ensembles, focusing on the Four Moment Theorem and its refinements and
applications, including the universality of the sine kernel and the Central
limit theorem of several spectral parameters.
We also take the opportunity here to issue some errata for some of our
previous papers in this area.Comment: 58 page
Bayesian Inference of Log Determinants
The log-determinant of a kernel matrix appears in a variety of machine
learning problems, ranging from determinantal point processes and generalized
Markov random fields, through to the training of Gaussian processes. Exact
calculation of this term is often intractable when the size of the kernel
matrix exceeds a few thousand. In the spirit of probabilistic numerics, we
reinterpret the problem of computing the log-determinant as a Bayesian
inference problem. In particular, we combine prior knowledge in the form of
bounds from matrix theory and evidence derived from stochastic trace estimation
to obtain probabilistic estimates for the log-determinant and its associated
uncertainty within a given computational budget. Beyond its novelty and
theoretic appeal, the performance of our proposal is competitive with
state-of-the-art approaches to approximating the log-determinant, while also
quantifying the uncertainty due to budget-constrained evidence.Comment: 12 pages, 3 figure
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