Showing papers in "Mathematical Physics Analysis and Geometry in 2005"
TL;DR: In this paper, the Riemann hypothesis holds if certain conjectured properties of a sequence ηj are valid and the constants of the zeta function enter the Laurent expansion of the logarithmic derivative of the xi function about s = 1 and appear to have remarkable characteristics.
Abstract: We substantially apply the Li criterion for the Riemann hypothesis to hold. Based upon a series representation for the sequence {λk}, which are certain logarithmic derivatives of the Riemann xi function evaluated at unity, we determine new bounds for relevant Riemann zeta function sums and the sequence itself. We find that the Riemann hypothesis holds if certain conjectured properties of a sequence ηj are valid. The constants ηj enter the Laurent expansion of the logarithmic derivative of the zeta function about s=1 and appear to have remarkable characteristics. On our conjecture, not only does the Riemann hypothesis follow, but an inequality governing the values λn and inequalities for the sums of reciprocal powers of the nontrivial zeros of the zeta function.
38 citations
TL;DR: In this article, the radial centre of a convex body A is the maximizer of a suitable generalized dual volume of A. The radial centre maps are continuous with respect to the Hausdorff metric and solve the problem of direct additivity for radial center of order α.
Abstract: The paper concerns a family of selectors for convex bodies in Rn, radial centre maps, defined in the article of M. Moszynska, Looking for selectors of star bodies, Geom. Dedicata 81 (2000), 131–147. A radial centre of a convex body A is the maximizer of a suitable generalized dual volume of A. We give physical interpretations of the notion of radial centre and study its geometric properties. We prove that these selectors are continuous with respect to the Hausdorff metric and solve the problem of direct additivity for radial centre of order α, which corresponds to the dual volume of order α.
17 citations
TL;DR: In this article, the main result of Egorov's theorem for transversally elliptic operators on compact foliated manifolds is applied to describe the noncommutative geodesic flow in noncommuteative geometry of Riemannian foliations.
Abstract: The main result of the paper is Egorov’s theorem for transversally elliptic operators on compact foliated manifolds. This theorem is applied to describe the noncommutative geodesic flow in noncommutative geometry of Riemannian foliations.
14 citations
TL;DR: In this paper, it was shown that rational solutions of the AKNS hierarchy of the form q=σ/τ and r=ρ/τ, where (σ,τ,ρ) are certain Schur functions, naturally yield Dirac operators of strict Huygens' type, i.e., the support of their fundamental solutions is the surface of the light-cone.
Abstract: We prove that rational solutions of the AKNS hierarchy of the form q=σ/τ and r=ρ/τ, where (σ,τ,ρ) are certain Schur functions, naturally yield Dirac operators of strict Huygens' type, i.e., the support of their fundamental solutions is the surface of the light-cone. This strengthens the connection between the theory of completely integrable systems and Huygens' principle by extending to the Dirac operators and the rational solutions of the AKNS hierarchy a classical result of Lagnese and Stellmacher concerning perturbations of wave operators.
3 citations
TL;DR: In this paper, the singularity of Kontsevich's solution of the WDVV equations of associativity was studied and it was shown that it corresponds to a singularity in the change of two coordinates systems of the Frobenius manifold given by the quantum cohomology of CP2.
Abstract: In this paper we study the nature of the singularity of the Kontsevich’s solution of the WDVV equations of associativity. We prove that it corresponds to a singularity in the change of two coordinates systems of the Frobenius manifold given by the quantum cohomology of CP2.
2 citations
Abstract: This paper answers important questions raised by the recent description, by Attal, of a robust and explicit method to approximate basic objects of quantum stochastic calculus on bosonic Fock space by analogues on the state space of quantum spin chains. The existence of that method justifies a detailed investigation of discrete-time quantum stochastic calculus. Here we fully define and study that theory and obtain in particular a discrete-time quantum Ito formula, which one can see as summarizing the commutation relations of Pauli matrices.
An apparent flaw in that approximation method is the difference in the quantum Ito formulas, discrete and continuous, which suggests that the discrete quantum stochastic calculus differs fundamentally from the continuous one and is therefore not a suitable object to approximate subtle phenomena. We show that flaw is only apparent by proving that the continuous-time quantum Ito formula is actually a consequence of its discrete-time counterpart.
TL;DR: In this article, the authors investigated Lifshits-tail behavior of the integrated density of states for a wide class of Schrodinger operators with positive random potentials, including alloy-type and Poissonian random potential potentials.
Abstract: We investigate Lifshits-tail behaviour of the integrated density of states for a wide class of Schrodinger operators with positive random potentials. The setting includes alloy-type and Poissonian random potentials. The considered (single-site) impurity potentials f: ℝd→[0,∞[ decay at infinity in an anisotropic way, for example, $f(x_{1},x_{2})\sim (|x_{1}|^{\alpha_{1}}+|x_{2}|^{\alpha_{2}})^{-1}$
as |(x1,x2)|→∞. As is expected from the isotropic situation, there is a so-called quantum regime with Lifshits exponent d/2 if both α1 and α2 are big enough, and there is a so-called classical regime with Lifshits exponent depending on α1 and α2 if both are small. In addition to this we find two new regimes where the Lifshits exponent exhibits a mixture of quantum and classical behaviour. Moreover, the transition lines between these regimes depend in a nontrivial way on α1 and α2 simultaneously.
TL;DR: In this paper, the boundary conditions that yield a linearly well posed problem for the so-called KdV-KdV system and for the classical Boussinesq system are characterized.
Abstract: We characterise the boundary conditions that yield a linearly well posed problem for the so-called KdV–KdV system and for the classical Boussinesq system. Each of them is a system of two evolution PDEs modelling two-way propagation of water waves. We study these problems with the spatial variable in either the half-line or in a finite interval. The results are obtained by extending a spectral transform approach, recently developed for the analysis of scalar evolution PDEs, to the case of systems of PDEs.
TL;DR: The authors studied ergodic averages for a class of pseudodifferential operators on the flat N-dimensional torus with respect to the Schrodinger evolution and proved that they are translationally invariant operators up to semi-classically negligible corrections.
Abstract: We study ergodic averages for a class of pseudodifferential operators on the flatN-dimensional torus with respect to the Schrodinger evolution. The later can be consider a quantization of the geodesic flow on \(\mathbb{T}^{N}\) . We prove that, up to semi-classically negligible corrections, such ergodic averages are translationally invariant operators.
TL;DR: In this article, a unified approach for construction of symplectic forms for 1D integrable equations with the periodic and rapidly decaying initial data is presented, where the Schrodinger equation is considered.
Abstract: We develop a unified approach for construction of symplectic forms for 1D integrable equations with the periodic and rapidly decaying initial data. As an example we consider the cubic nonlinear Schrodinger equation.