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At least 1,063 records · Page 59Linked to original sources
arXiv2013

On radial stationary solutions to a model of nonequilibrium growth

We present the formal geometric derivation of a nonequilibrium growth model that takes the form of a parabolic partial differential equation. Subsequently, we study its stationary radial solutions by means of variational techniques. Our results depend on the size of a parameter that plays the role of the strength of forcing. For small forcing we prove the existence and multiplicity of solutions to the elliptic problem. We discuss our results in the context of nonequilibrium statistical mechanics.

math.CA↗
arXiv2013

Analytical Approximation for 2-D Nonlinear Periodic Deep Water Waves

A recently developed method has been extended to a nonlocal equation arising in steady water wave propagation in two dimensions. We obtain analyic approximation of steady water wave solution in two dimensions with rigorous error bounds for a set of parameter values that correspond to heights slightly smaller than the critical. The wave shapes are shown to be analytic. The method presented in quite general and does not assume smallness of wave height or steepness and can be readily extended to other interfacial problems involving Laplace's equation.

physics.flu-dyn↗
arXiv2013

Constructing a Proof of the Riemann Hypothesis

This paper compares the distribution of zeros of the Riemann zeta function $ζ(s)$ with those of a symmetric combination of zeta functions, denoted ${\cal T}_+(s)$, known to have all its zeros located on the critical line $\Re(s)=1/2$. Criteria are described for constructing a suitable quotient function of these, with properties advantageous for establishing an accessible proof that $ζ(s)$ must also have all its zeros on the critical line: the celebrated Riemann hypothesis. While the argument put forward is not at the level of rigour required to constitute a full proof of the Riemann hypothesis, it should convince non-specialists that it must hold.

math.NT↗
arXiv2013

Matrix Fourier transform in dynamic theory of elasticity of piecewise homogeneous medium

The analytical solving dynamic problems of elasticity theory for piecewise homogeneous half-space is found. The explicit construction of direct and inverse Fourier's vector transform with discontinuous coefficients is presented. The technique of applying Fourier's vector transform with discontinuous coefficients for solving problems of mathematical physics in the heterogeneous environments is developed on an example of the dynamic problems of the elasticity theory.

math-ph↗
arXiv2013

Flat G-bundles and regular strata for reductive groups

Let LG be an algebraic loop group associated to a reductive group G. A fundamental stratum is a triple consisting of a point x in the Bruhat-Tits building of LG, a nonnegative real number r, and a character of the corresponding depth r Moy-Prasad subgroup that satisfies a non-degeneracy condition. The authors have shown in previous work how to associate a fundamental stratum to a formal flat G-bundle and used this theory to define its slope. In this paper, the authors study fundamental strata that satisfy an additional regular semisimplicity condition. Flat G-bundles that contain regular strata have a natural reduction of structure to a (not necessarily split) maximal torus in LG, and the authors use this property to compute the corresponding moduli spaces. This theory generalizes a natural condition on algebraic connections (the GL_n case), which plays an important role in the global analysis of meromorphic connections and isomonodromic deformations.

math.AG↗
arXiv2013

A Landau-Ginzburg/Calabi-Yau correspondence for the mirror quintic

We prove a version of the Landau-Ginzburg/Calabi-Yau correspondence for the mirror quintic. In particular we calculate the genus-zero FJRW theory for the pair (W, G) where W is the Fermat quintic polynomial and G = SL(W). We identify it with the Gromov-Witten theory of the mirror quintic three-fold via an explicit analytic continuation and symplectic transformation. In the process we prove a mirror theorem for the corresponding Landau-Ginzburg model (W,G).

math.AG↗
arXiv2013

Completeness and Bethe root distribution of the spin-1/2 Heisenberg chain with arbitrary boundary fields

Recently, the XXX spin chain with arbitrary boundary fields was successfully solved [1] via the off-diagonal Bethe ansatz method [2]. The correctness and the completeness of this solution were numerically verified by Nepomechie for one choice of the parameterizations [3]. In this paper, we discuss further the completeness of another parameterization of the Bethe ansatz equations and its reduction to the parallel boundary field case. The numerical results show that when the relative angle between the boundary fields turns to zero, both the T-Q relations and the Bethe ansatz equations are reduced to the ones obtained by the conventional Bethe ansatz methods. This allows us to establish a one-to-one correspondence between the Bethe roots of the unparallel boundary field case and those of the parallel boundary field case. In the thermodynamic limit, those two sets of Bethe roots tend to the same and the contribution of the relative angle to the energy is in the order of 1/N.

cond-mat.stat-mech↗
arXiv2013

A new transfer-matrix algorithm for exact enumerations: self-avoiding walks on the square lattice

We recently published [J. Phys A: Math. Theor. {\bf 45} 115202 (2012)] a new and more efficient implementation of a transfer-matrix algorithm for exact enumerations of self-avoiding polygons. Here we extend this work to the enumeration of self-avoiding walks on the square lattice. A detailed comparison with our previous best algorithm shows very significant improvement in the running time of the new algorithm. The new algorithm is used to extend the enumeration of self-avoiding walks to length 79 from the previous record of 71 and for metric properties, such as the average end-to-end distance, from 59 to 71.

math-ph↗
arXiv2013

Infinite Dimensional Choi-Jamiolkowski States and Time Reversed Quantum Markov Semigroups

We propose a definition of infinite dimensional Choi-Jamiolkowski state associated with a completely positive trace preserving map. We introduce the notion of Theta-KMS adjoint of a quantum Markov semigroup, which is identified with the time reversed semigroup. The break down of Theta-KMS symmetry (or Theta-standard quantum detailed balance in the sense of Fagnola-Umanita) is measured by means of the von Neumann relative entropy of the Choi-Jamiolkowski states associated with the semigroup and its Theta-KMS adjoint.

math-ph↗
arXiv2013

Topological analysis and Boolean functions. I. Methods and application to classical systems

We aim to completely formalize the rough topological analysis of integrable Hamiltonian systems admitting analytical solutions such that the initial phase variables along with the time derivatives of the auxiliary variables are expressed as rational functions (in fact, as polynomials) in some set of radicals depending on one variable each. We suggest a method to define the admissible regions in the integral constants space, the segments of oscillation of the separated variables and the number of connected components of integral manifolds and critical integral surfaces. This method is based on some algorithms of processing the tables of some Boolean vector-functions and of reducing the matrices of linear Boolean vector-functions to some canonical form. From this point of view we consider here the topologically richest classical problems of the rigid body dynamics. The article will be continued with the investigation of some new integrable problems.

nlin.SI↗
arXiv2013

Averaging and spectral properties for the 2D advection-diffusion equation in the semi-classical limit for vanishing diffusivity

We consider the two-dimensional advection-diffusion equation on a bounded domain subject to either Dirichlet or von Neumann boundary conditions and study both time-independent and time-periodic cases involving Liouville integrable Hamiltonians that satisfy conditions conducive to applying the averaging principle. Transformation to action-angle coordinates permits averaging in time and angle, leading to an underlying eigenvalue equation that allows for separation of the angle and action coordinates. The result is a one-dimensional second-order equation involving an anti-symmetric imaginary potential. For radial flows on a disk or an annulus, we rigorously apply existing complex-plane WKBJ methods to study the spectral properties in the semi-classical limit for vanishing diffusivity. In this limit, the spectrum is found to be a complicated set consisting of lines related to Stokes graphs. Eigenvalues in the neighborhood of these graphs exhibit nonlinear scaling with respect to diffusivity leading to convection-enhanced rates of dissipation (relaxation, mixing) for initial data which are mean-free in the angle coordinate. These branches coexist with a diffusive branch of eigenvalues that scale linearly with diffusivity and contain the principal eigenvalue (no dissipation enhancement).

physics.flu-dyn↗
arXiv2013

Infinite energy solutions to inelastic homogeneous Boltzmann equation

This paper is concerned with the existence, shape and dynamical stability of infinite-energy equilibria for a general class of spatially homogeneous kinetic equations in space dimensions $d \geq 3$. Our results cover in particular Bobylëv's model for inelastic Maxwell molecules. First, we show under certain conditions on the collision kernel, that there exists an index $α\in(0,2)$ such that the equation possesses a nontrivial stationary solution, which is a scale mixture of radially symmetric $α$-stable laws. We also characterize the mixing distribution as the fixed point of a smoothing transformation. Second, we prove that any transient solution that emerges from the NDA of some (not necessarily radial symmetric) $α$-stable distribution converges to an equilibrium. The key element of the convergence proof is an application of the central limit theorem to a representation of the transient solution as a weighted sum of i.i.d. random vectors.

math-ph↗
arXiv2013

On the dynamics of WKB wave functions whose phase are weak KAM solutions of H-J equation

In the framework of toroidal Pseudodifferential operators on the flat torus $\Bbb T^n := (\Bbb R / 2π\Bbb Z)^n$ we begin by proving the closure under composition for the class of Weyl operators $\mathrm{Op}^w_\hbar(b)$ with simbols $b \in S^m (\mathbb{T}^n \times \mathbb{R}^n)$. Subsequently, we consider $\mathrm{Op}^w_\hbar(H)$ when $H=\frac{1}{2} |η|^2 + V(x)$ where $V \in C^\infty (\Bbb T^n;\Bbb R)$ and we exhibit the toroidal version of the equation for the Wigner transform of the solution of the Schrödinger equation. Moreover, we prove the convergence (in a weak sense) of the Wigner transform of the solution of the Schrödinger equation to the solution of the Liouville equation on $\Bbb T^n \times \Bbb R^n$ written in the measure sense. These results are applied to the study of some WKB type wave functions in the Sobolev space $H^{1} (\mathbb{T}^n; \Bbb C)$ with phase functions in the class of Lipschitz continuous weak KAM solutions (of positive and negative type) of the Hamilton-Jacobi equation $\frac{1}{2} |P+ \nabla_x v_\pm (P,x)|^2 + V(x) = \bar{H}(P)$ for $P \in \ell \Bbb Z^n$ with $\ell >0$, and to the study of the backward and forward time propagation of the related Wigner measures supported on the graph of $P+ \nabla_x v_\pm$.

math.AP↗
arXiv2013

Infinity Links L, infinity-4-Manifolds M_L and Kirby Categories

We construct what we call a Kirby category, a monoidal category whose morphisms are smooth 4-manifolds, projecting down to another monoidal category whose morphisms are orientable 3-manifolds, the projection being induced by the boundary map on manifolds. We construct a higher categorical generalization of such concepts and introduce the notion of ribbon $\infty$-categories, a generalization of braided monoidal $\infty$-categories (\cite{Lu1}), which gives rise to the concepts of $\infty$-links, $\infty$-4-manifolds as well as the more general notion of walled $\infty$-4-manifolds if one focuses attention on $\infty$-4-manifolds built from gluing thickened sheets on ribbons. These fall into a larger class of constrained $\infty$-4-manifolds whose classical 4-dimensional counterparts are constrained 4-manifolds on which we consider physical theories. We regard pairs of constrained 4-manifolds and Lagrangians densities depicting physical theories defined on such spaces as morphism objects in an enhanced Kirby category, whose objects are regarded as events. We define a universal category $Λ$ of all events that we relate to the $\infty$-category of ribbon $\infty$-categories and conclude in part that Lagrangian field theories can be superseded by using $\infty$-categories.

math.GT↗
arXiv2013

On the Evolution of Structure in Turbulence Phenomena

We study the evolution of coherent structures in arbitrary turbulence phenomena, developing some tools, from non-archimedean analysis and algebraic geometry, in order to model its display. We match the scale-dependent, topological structure of the physical fields subsumed by these coherent structures with the phase portraits of some convenient, non-archimedean dynamical systems whose set we endow with a law of action. Finally, we obtain the proposition of an extremum principle for this action to define a geodesic path, as the modelized coherent structures evolve, and we delineate the conception of two experiments, derived from the fields of fluid and plasma turbulence, to test its efectiveness.

math-ph↗
arXiv2013

Translation invariant pure state on $\otimes_{\IZ}/!M_d(\IC)$ and Haag duality

We prove Haag duality property of any translation invariant pure state on $\clb = \otimes_{\IZ}/!M_d(\IC), \;d \ge 2$, where $/!M_d(\IC)$ is the set of $d \times d$ dimensional matrices over the field of complex numbers. We also prove a necessary and sufficient condition for a translation invariant factor state to be pure on $\clb$.

math.OA↗
arXiv2013

Free convection over a non-isothermal axisymmetric body immersed in a porous medium saturated with an electrically conducting non-Newtonian fluid

The study investigates the problem of magnetohydrodynamic (MHD) free convection over a non-isothermal axisymmetric body under the action of transverse magnetic field. The body is embedded in a porous medium saturated with electrically conducting non-Newtonian power law fluid. In order to obtain similarity solution, it is assumed that the viscosity of the fluid decays exponentially with temperature. The qualitative results are illustrated for a vertical flat plate, horizontal cylinder and sphere.

math-ph↗