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

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    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

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    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

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    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 Kepler’s equation: A Chebyshev–Hermite–PadĂ© method

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    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) 189–200], 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 (u−u1(λ))(u−u2(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(−σN) for some constant σ>0 where N is the truncation of the Chebyshev series. This is dubbed the “Chebyshev–Shafer” or “Chebyshev–Hermite–PadĂ©â€ method because it substitutes Chebyshev series for power series in the generalized PadĂ© approximants known variously as “Shafer” or “Hermite–PadĂ©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”)≡u−ϔsin(u)−λ=0. Only one of the merging roots is real-valued and the root depends on two parameters (λ,Ï”) rather than one. Nonetheless, the Chebyshev–Hermite–PadĂ© 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

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    We investigate the mean number of real zeros over an interval [a,b][a,b] of a random trigonometric polynomial of the form ∑k=1nakcos⁥(kt)+bksin⁥(kt)\sum_{k=1}^n a_k \cos(kt)+b_k \sin(kt) 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 n(b−a)π3\frac{n(b-a)}{\pi\sqrt{3}} as nn 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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