Showing papers in "Contemporary mathematics in 2016"
TL;DR: In this article, the authors considered the Helmholtz equation with real analytic coefficients on a bounded domain Ω ⊂ Rd and showed that for any Ω′ b Ω and almost any d + 1 frequencies ωk in [a, b], there exist d+ 1 subdomains Ωk such that Ω ∈ ∪kΩk and ζ(u1ωk,..., u d+1 ω k,∇u 1 ω, ∇uωk, ∇ uω
Abstract: We consider the Helmholtz equation with real analytic coefficients on a bounded domain Ω ⊂ Rd. We take d+ 1 prescribed boundary conditions f i and frequencies ω in a fixed interval [a, b]. We consider a constraint on the solutions uω of the form ζ(uω , . . . , u d+1 ω ,∇uω , . . . ,∇u d+1 ω ) 6= 0, where ζ is analytic, which is satisfied in Ω when ω = 0. We show that for any Ω′ b Ω and almost any d + 1 frequencies ωk in [a, b], there exist d + 1 subdomains Ωk such that Ω′ ⊂ ∪kΩk and ζ(u1ωk , . . . , u d+1 ωk ,∇u 1 ωk , . . . ,∇u ωk ) 6= 0 in Ωk. This question comes from hybrid imaging inverse problems. The method used is not specific to the Helmholtz model and can be applied to other frequency dependent problems.
6 citations
TL;DR: In this paper, the authors present a survey of recent advances in partial differential equations and their applications in contemporary mathematics series of the American Mathematical Society, vol. 666, No. 1.
Abstract: Publicado em "Recent advances in partial differential equations and applications". Contemporary mathematics series of the American Mathematical Society, vol. 666. ISBN 978-1-4704-1521-1
5 citations
TL;DR: In this article, the authors present a review of how ideas inspired by recent developments in number theory find applications in physics in the context of scattering amplitudes and Feynman integrals and show how one can combine (and conjecturally extend) Goncharov's Hopf algebra on multiple polylogarithms by recent results by Brown on motivic multiple zeta values.
Abstract: We present a review of how ideas inspired by recent developments in number theory find applications in physics in the context of scattering amplitudes and Feynman integrals. In particular, we show how one can combine (and conjecturally extend) Goncharov's Hopf algebra on multiple polylogarithms by recent results by Brown on motivic multiple zeta values. These results can be used to derive in an effective way complicated relations among multiple polylogarithms. We conclude by illustrating the use of these concepts in various contexts related to the computation of scattering amplitudes and Feynman integrals.
5 citations
TL;DR: It is proved that weak solutions constructed by approximating the time-derivative by finite differences are suitable for the three dimensional Navier-Stokes equations.
Abstract: We consider the three dimensional Navier-Stokes equations and we prove that weak solutions constructed by approximating the time-derivative by finite differences are suitable. The so-called method of semi-discretization is of fundamental importance in the numerical analysis and it is one of the building blocks for the full discretization of the equations.
4 citations
TL;DR: In this article, a Minkowski analogue of the Euclidean medial axis of a closed and smooth plane curve is introduced and its generic local configurations are studied and the types of shocks that occur on these are also determined.
Abstract: In this paper a Minkowski analogue of the Euclidean medial axis of a closed and smooth plane curve is introduced. Its generic local configurations are studied and the types of shocks that occur on these are also determined.
3 citations
TL;DR: In this paper, for any d ≥ 3, smooth Hamiltonians have an elliptic equilibrium with an arbitrary frequency, that is not accumulated by a positive measure set of invariant tori.
Abstract: We construct on ℝ2d, for any d ≥ 3, smooth Hamiltonians having an elliptic equilibrium with an arbitrary frequency, that is not accumulated by a positive measure set of invariant tori. For d ≥ 4, t ...
2 citations
TL;DR: A fast and numerically robust algorithm based on structured numerical linear algebra technology for the computation of the zeros of an analytic function inside the unit circle in the complex plane is proposed.
Abstract: We propose a fast and numerically robust algorithm based on structured numerical linear algebra technology for the computation of the zeros of an analytic function inside the unit circle in the complex plane. At the core of our method there are two matrix algorithms: (a) a fast reduction of a certain linearization of the zerofinding problem to a matrix eigenvalue computation involving a perturbed CMV–like matrix and (b) a fast variant of the QR eigenvalue algorithm suited to exploit the structural properties of this latter matrix. We illustrate the reliability of the proposed method by several numerical examples
2 citations
Journal Article•
TL;DR: For the two-dimensional isotropic quantum harmonic oscillator, Lewy and Przybytkowski as discussed by the authors constructed an infinite sequence of regular eigenfunctions with as many nodal domains as allowed by Courant's theorem.
Abstract: For the spherical Laplacian on the sphere and for the Dirichlet Laplacian in the square}, Antonie Stern claimed in her PhD thesis (1924) the existence of an infinite sequence of eigenvalues whose corresponding eigenspaces contain an eigenfunction with exactly two nodal domains. These results were given complete proofs respectively by Hans Lewy in 1977, and the authors in 2014 (see also Gauthier-Shalom--Przybytkowski, 2006). In this paper, we obtain similar results for the two dimensional isotropic quantum harmonic oscillator. In the opposite direction, we construct an infinite sequence of regular eigenfunctions with as many nodal domains as allowed by Courant's theorem, up to a factor $\frac{1}{4}$. A classical question for a $2$-dimensional bounded domain is to estimate the length of the nodal set of a Dirichlet eigenfunction in terms of the square root of the energy. In the last section, we consider some Schrodinger operators $-\Delta + V$ in $\mathbb{R}^2$ and we provide bounds for the length of the nodal set of an eigenfunction with energy $\lambda$ in the classically permitted region $\{V(x) < \lambda\}$.
2 citations
Journal Article•
TL;DR: In this paper, a generalization of the squared Bessel process with real nonnegative parameter δ by introducing a predictable almost everywhere positive process γ(t, ω) into the drift and diffusion terms is discussed.
Abstract: We discuss a generalization of the well known squared Bessel process with real nonnegative parameter δ by introducing a predictable almost everywhere positive process γ(t, ω) into the drift and diffusion terms. The resulting generalized process is nonnegative with instantaneous reflection at zero when δ is positive. When δ is a positive integer, the process can be constructed from δ-dimensional Brownian motion. In particular, we consider γt = Xt−τ which makes the process a solution of a stochastic delay differential equation with a discrete delay. The solutions of these equations are constructed in successive steps on time intervals of length τ . We prove that if 0 < δ < 2, zero is an accessible boundary and the process is instantaneously reflecting at zero. If δ ≤ 2, lim inft→∞Xt = 0. Zero is inaccessible if δ ≥ 2.
1 citations
TL;DR: In this article, the first eigenvalue of the Laplace operator −∆ with Dirichlet conditions on ∂Ω ∩D and Neumann or Robin conditions on λ ∆ ∆, where D is a given subset of the Euclidean space R. The equivalent variational formulation λ 1(Ω;D) = min {∫
Abstract: We consider spectral optimization problems of the form min { λ1(Ω;D) : Ω ⊂ D, |Ω| = 1 } , where D is a given subset of the Euclidean space R. Here λ1(Ω;D) is the first eigenvalue of the Laplace operator −∆ with Dirichlet conditions on ∂Ω ∩D and Neumann or Robin conditions on ∂Ω ∩ ∂D. The equivalent variational formulation λ1(Ω;D) = min {∫
1 citations
TL;DR: The relationship between two dynamical systems, one of which is obtained from the other by forming the quotient by an action of an involution commuting with the dynamics, is studied in this paper.
Abstract: The relationship between two dynamical systems, one of which is obtained from the other by forming the quotient by an action of an involution commuting with the dynamics, is studied. The constraints and the possible extent of freedom in the relationship between the growth of closed orbits in pairs of systems related in this way is explored.
Journal Article•
TL;DR: In this article, strong solutions to the steady-state, twodimensional exterior problem for a class of shear-thinning liquids, where shear viscosity is a suitable decreasing function of the shear rate, were shown for data of arbitrary size.
Abstract: We show existence of strong solutions to the steady-state, twodimensional exterior problem for a class of shear-thinning liquids –where shear viscosity is a suitable decreasing function of shear rate– for data of arbitrary size. Notice that the analogous problem is, to date, open for liquids governed by the Navier-Stokes equations, where viscosity is constant. Two important features of this work are that, on the one hand and unlike previous contributions by the same authors, the current results do not require non-vanishing of the constant-viscosity part of the stress tensor, and, on the other hand, we allow the shear-thinning contribution to be “arbitrarily small”, and, therefore, the model used here can be as “close” as we please to the classical NavierStokes one.
TL;DR: In this article, the authors consider a constrained optimization problem arising from the study of the Helmholtz equation in unbounded domains, and prove some estimates on the rate of convergence to the exact solution.
Abstract: We consider a constrained optimization problem arising from the study of the Helmholtz equation in unbounded domains. The optimization problem provides an approximation of the solution in a bounded computational domain. In this paper we prove some estimates on the rate of convergence to the exact solution.
TL;DR: In this article, the authors introduce the conjectural symmetries of Kac-Moody groups in supergravity as well as the known evidence for these conjectures, and describe constructions of kac-moody group over? and? using certain choices of fundamental modules that are considered to have physical relevance.
Abstract: Kac–Moody groups G over ? have been conjectured to occur as symmetry groups of supergravity theories dimensionally reduced to dimensions less than 3, and their integral forms G(?) conjecturally encode quantized symmetries. In this review paper, we briefly introduce the conjectural symmetries of Kac–Moody groups in supergravity as well as the known evidence for these conjectures. We describe constructions of Kac–Moody groups over ? and ? using certain choices of fundamental modules that are considered to have physical relevance. Eisenstein series on certain finite dimensional algebraic groups are known to encode quantum corrections in the low energy limit of superstring theories. We describe briefly how the construction of Eisenstein series extends to Kac–Moody groups. The constant terms of Eisenstein series on E9, E10 and E11 are predicted to encode perturbative string theory corrections.