4,052 research outputs found
Tridiagonal PT-symmetric N by N Hamiltonians and a fine-tuning of their observability domains in the strongly non-Hermitian regime
A generic PT-symmetric Hamiltonian is assumed tridiagonalized and truncated
to N dimensions, and its up-down symmetrized special cases with J=[N/2] real
couplings are considered. In the strongly non-Hermitian regime the secular
equation gets partially factorized at all N. This enables us to reveal a
fine-tuned alignment of the dominant couplings implying an asymptotically
sharply spiked shape of the boundary of the J-dimensional quasi-Hermiticity
domain in which all the spectrum of energies remains real and observable.Comment: 28 pp., 4 tables, 1 figur
Parameter identifiability in a class of random graph mixture models
We prove identifiability of parameters for a broad class of random graph
mixture models. These models are characterized by a partition of the set of
graph nodes into latent (unobservable) groups. The connectivities between nodes
are independent random variables when conditioned on the groups of the nodes
being connected. In the binary random graph case, in which edges are either
present or absent, these models are known as stochastic blockmodels and have
been widely used in the social sciences and, more recently, in biology. Their
generalizations to weighted random graphs, either in parametric or
non-parametric form, are also of interest in many areas. Despite a broad range
of applications, the parameter identifiability issue for such models is
involved, and previously has only been touched upon in the literature. We give
here a thorough investigation of this problem. Our work also has consequences
for parameter estimation. In particular, the estimation procedure proposed by
Frank and Harary for binary affiliation models is revisited in this article
High-precision calculation of multi-loop Feynman integrals by difference equations
We describe a new method of calculation of generic multi-loop master
integrals based on the numerical solution of systems of difference equations in
one variable. We show algorithms for the construction of the systems using
integration-by-parts identities and methods of solutions by means of expansions
in factorial series and Laplace's transformation. We also describe new
algorithms for the identification of master integrals and the reduction of
generic Feynman integrals to master integrals, and procedures for generating
and solving systems of differential equations in masses and momenta for master
integrals. We apply our method to the calculation of the master integrals of
massive vacuum and self-energy diagrams up to three loops and of massive vertex
and box diagrams up to two loops. Implementation in a computer program of our
approach is described. Important features of the implementation are: the
ability to deal with hundreds of master integrals and the ability to obtain
very high precision results expanded at will in the number of dimensions.Comment: 55 pages, 5 figures, LaTe
Chebyshev expansion on intervals with branch points with application to the root of Keplerâs equation: A ChebyshevâHermiteâPadĂ© method
AbstractWhen two or more branches of a function merge, the Chebyshev series of u(λ) will converge very poorly with coefficients an of Tn(λ) falling as O(1/nα) for some small positive exponent α. However, as shown in [J.P. Boyd, Chebyshev polynomial expansions for simultaneous approximation of two branches of a function with application to the one-dimensional Bratu equation, Appl. Math. Comput. 143 (2002) 189â200], it is possible to obtain approximations that converge exponentially fast in n. If the roots that merge are denoted as u1(λ) and u2(λ), then both branches can be written without approximation as the roots of (uâu1(λ))(uâu2(λ))=u2+ÎČ(λ)u+Îł(λ). By expanding the nonsingular coefficients of the quadratic, ÎČ(λ) and Îł(λ), as Chebyshev series and then applying the usual roots-of-a-quadratic formula, we can approximate both branches simultaneously with error that decreases proportional to exp(âÏN) for some constant Ï>0 where N is the truncation of the Chebyshev series. This is dubbed the âChebyshevâShaferâ or âChebyshevâHermiteâPadĂ©â method because it substitutes Chebyshev series for power series in the generalized PadĂ© approximants known variously as âShaferâ or âHermiteâPadĂ©â approximants. Here we extend these ideas. First, we explore square roots with branches that are both real-valued and complex-valued in the domain of interest, illustrated by meteorological baroclinic instability. Second, we illustrate triply branched functions via roots of the Kepler equation, f(u;λ,Ï”)âĄuâÏ”sin(u)âλ=0. Only one of the merging roots is real-valued and the root depends on two parameters (λ,Ï”) rather than one. Nonetheless, the ChebyshevâHermiteâPadĂ© scheme is successful over the whole two-dimensional parameter plane. We also discuss how to cope with poles and logarithmic singularities that arise in our examples at the extremes of the expansion domain
Universality of the mean number of real zeros of random trigonometric polynomials under a weak Cramer condition
We investigate the mean number of real zeros over an interval of a
random trigonometric polynomial of the form where the coefficients are i.i.d. random variables. Under mild
assumptions on the law of the entries, we prove that this mean number is
asymptotically equivalent to as goes to
infinity, as in the known case of standard Gaussian coefficients. Our principal
requirement is a new Cramer type condition on the characteristic function of
the entries which does not only hold for all continuous distributions but also
for discrete ones in a generic sense. To our knowledge, this constitutes the
first universality result concerning the mean number of zeros of random
trigonometric polynomials. Besides, this is also the first time that one makes
use of the celebrated Kac-Rice formula not only for continuous random variables
as it was the case so far, but also for discrete ones. Beyond the proof of a
non asymptotic version of Kac-Rice formula, our strategy consists in using
suitable small ball estimates and Edgeworth expansions for the Kolmogorov
metric under our new weak Cramer condition, which both constitute important
byproducts of our approach
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