2,487 research outputs found

    Rigorous numerics for nonlinear operators with tridiagonal dominant linear part

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    We present a method designed for computing solutions of infinite dimensional non linear operators f(x)=0f(x) = 0 with a tridiagonal dominant linear part. We recast the operator equation into an equivalent Newton-like equation x=T(x)=x−Af(x)x = T(x) = x - Af(x), where AA is an approximate inverse of the derivative Df(x‾)Df(\overline x) at an approximate solution x‾\overline x. We present rigorous computer-assisted calculations showing that TT is a contraction near x‾\overline x, thus yielding the existence of a solution. Since Df(x‾)Df(\overline x) does not have an asymptotically diagonal dominant structure, the computation of AA is not straightforward. This paper provides ideas for computing AA, and proposes a new rigorous method for proving existence of solutions of nonlinear operators with tridiagonal dominant linear part.Comment: 27 pages, 3 figures, to be published in DCDS-A (Vol. 35, No. 10) October 2015 issu

    On The S-Matrix of Ising Field Theory in Two Dimensions

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    We explore the analytic structure of the non-perturbative S-matrix in arguably the simplest family of massive non-integrable quantum field theories: the Ising field theory (IFT) in two dimensions, which may be viewed as the Ising CFT deformed by its two relevant operators, or equivalently, the scaling limit of the Ising model in a magnetic field. Our strategy is that of collider physics: we employ Hamiltonian truncation method (TFFSA) to extract the scattering phase of the lightest particles in the elastic regime, and combine it with S-matrix bootstrap methods based on unitarity and analyticity assumptions to determine the analytic continuation of the 2 to 2 S-matrix element to the complex s-plane. Focusing primarily on the "high temperature" regime in which the IFT interpolates between that of a weakly coupled massive fermion and the E8 affine Toda theory, we will numerically determine 3-particle amplitudes, follow the evolution of poles and certain resonances of the S-matrix, and exclude the possibility of unknown wide resonances up to reasonably high energies.Comment: typos corrected, references added, additional comparison with perturbation theory added. 35 pages, 21 figure

    Non-Asymptotic Analysis of Tangent Space Perturbation

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    Constructing an efficient parameterization of a large, noisy data set of points lying close to a smooth manifold in high dimension remains a fundamental problem. One approach consists in recovering a local parameterization using the local tangent plane. Principal component analysis (PCA) is often the tool of choice, as it returns an optimal basis in the case of noise-free samples from a linear subspace. To process noisy data samples from a nonlinear manifold, PCA must be applied locally, at a scale small enough such that the manifold is approximately linear, but at a scale large enough such that structure may be discerned from noise. Using eigenspace perturbation theory and non-asymptotic random matrix theory, we study the stability of the subspace estimated by PCA as a function of scale, and bound (with high probability) the angle it forms with the true tangent space. By adaptively selecting the scale that minimizes this bound, our analysis reveals an appropriate scale for local tangent plane recovery. We also introduce a geometric uncertainty principle quantifying the limits of noise-curvature perturbation for stable recovery. With the purpose of providing perturbation bounds that can be used in practice, we propose plug-in estimates that make it possible to directly apply the theoretical results to real data sets.Comment: 53 pages. Revised manuscript with new content addressing application of results to real data set

    Existence and instability of steady states for a triangular cross-diffusion system: a computer-assisted proof

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    In this paper, we present and apply a computer-assisted method to study steady states of a triangular cross-diffusion system. Our approach consist in an a posteriori validation procedure, that is based on using a fxed point argument around a numerically computed solution, in the spirit of the Newton-Kantorovich theorem. It allows us to prove the existence of various non homogeneous steady states for different parameter values. In some situations, we get as many as 13 coexisting steady states. We also apply the a posteriori validation procedure to study the linear stability of the obtained steady states, proving that many of them are in fact unstable

    The role of asymptotic functions in network optimization and feasibility studies

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    Solutions to network optimization problems have greatly benefited from developments in nonlinear analysis, and, in particular, from developments in convex optimization. A key concept that has made convex and nonconvex analysis an important tool in science and engineering is the notion of asymptotic function, which is often hidden in many influential studies on nonlinear analysis and related fields. Therefore, we can also expect that asymptotic functions are deeply connected to many results in the wireless domain, even though they are rarely mentioned in the wireless literature. In this study, we show connections of this type. By doing so, we explain many properties of centralized and distributed solutions to wireless resource allocation problems within a unified framework, and we also generalize and unify existing approaches to feasibility analysis of network designs. In particular, we show sufficient and necessary conditions for mappings widely used in wireless communication problems (more precisely, the class of standard interference mappings) to have a fixed point. Furthermore, we derive fundamental bounds on the utility and the energy efficiency that can be achieved by solving a large family of max-min utility optimization problems in wireless networks.Comment: GlobalSIP 2017 (to appear
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