Dual Lie Bialgebras of Witt and Virasoro Types
Structures of dual Lie bialgebras on the one sided Witt algebra, the Witt algebra and the Virasoro algebra are investigated. As a result, we obtain some infinite dimensional Lie algebras.
math.QA↗
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Quantum Systems and Resolvent Algebras
This survey article is concerned with the modeling of the kinematical structure of quantum systems in an algebraic framework which eliminates certain conceptual and computational difficulties of the conventional approaches. Relying on the Heisenberg picture it is based on the resolvents of the basic canonically conjugate operators and covers finite and infinite quantum systems. The resulting C*-algebras, the resolvent algebras, have many desirable properties. On one hand they encode specific information about the dimension of the respective quantum system and have the mathematically comfortable feature of being nuclear, and for finite dimensional systems they are even postliminal. This comes along with a surprisingly simple structure of their representations. On the other hand, they are a convenient framework for the study of interacting as well as constrained quantum systems since they allow the direct application of C*-algebraic methods which often simplify the analysis. Some pertinent facts are illustrated by instructive examples.
math-ph↗
Saari's Conjecture and Variational Minimal Solutions for $N$-Body Problems
In this paper, we will prove Saari's conjecture in a particular case by using a arithmetic fact, and then, apply it to prove that for any given positive masses, the variational minimal solutions of the N-body problem in ${\mathbb{R}}^2$ are precisely a relative equilibrium solution whose configuration minimizes the function $IU^2$ in ${\mathbb{R}}^2$.
math-ph↗
Bunching coefficients in Echo-Enabled Harmonic Generation
Coulombian diffusion determines a dilution of bunching coefficients in Free Electron Laser seeded devices. From the mathematical point of view the effect can be modeled through a heat type equation, which can be merged with the ordinary Liouville equation, ruling the evolution of the longitudinal phase space beam distribution. We will show that the use of analytical tools like the Generalized Bessel Functions and algebraic techniques for the solution of evolution problems may provide a useful method of analysis and shine further light on the physical aspects of the underlying mechanisms.
physics.acc-ph↗
Renormalizable Models in Rank $d\geq 2$ Tensorial Group Field Theory
Classes of renormalizable models in the Tensorial Group Field Theory framework are investigated. The rank $d$ tensor fields are defined over $d$ copies of a group manifold $G_D=U(1)^D$ or $G_D= SU(2)^D$ with no symmetry and no gauge invariance assumed on the fields. In particular, we explore the space of renormalizable models endowed with a kinetic term corresponding to a sum of momenta of the form $p^{2a}$, $a\in ]0,1]$. This study is tailored for models equipped with Laplacian dynamics on $G_D$ (case $a=1$) but also for more exotic nonlocal models in quantum topology (case $0<a<1$). A generic model can be written $(_{\dim G_D}Φ^{k}_{d}, a)$, where $k$ is the maximal valence of its interactions. Using a multi-scale analysis for the generic situation, we identify several classes of renormalizable actions including matrix model actions. In this specific instance, we find a tower of renormalizable matrix models parametrized by $k\geq 4$. In a second part of this work, we focus on the UV behavior of the models up to maximal valence of interaction $k =6$. All rank $d\geq 3$ tensor models proved renormalizable are asymptotically free in the UV. All matrix models with $k=4$ have a vanishing $β$-function at one-loop and, very likely, reproduce the same feature of the Grosse-Wulkenhaar model [Commun. Math. Phys. {\bf 256}, 305 (2004)].
hep-th↗
A Curie-Weiss Model of Self-Organized Criticality : The Gaussian Case
We try to design a simple model exhibiting self-organized criticality, which is amenable to a rigorous mathematical analysis. To this end, we modify the generalized Ising Curie-Weiss model by implementing an automatic control of the inverse temperature. With the help of exact computations, we show that, in the case of a centered Gaussian measure with positive variance $σ^{2}$, the sum $S_n$ of the random variables has fluctuations of order $n^{3/4}$ and that $S_n/n^{3/4}$ converges to the distribution $C \exp(-x^{4}/(4σ^4))\,dx$ where $C$ is a suitable positive constant.
math.PR↗
Analytic solution of nonlinear fractional Burgers-type equation by invariant subspace method
In this paper we study the analytic solutions of Burgers-type nonlinear fractional equations by means of the Invariant Subspace Method. We first study a class of nonlinear equations directly related to the time-fractional Burgers equation. Some generalizations linked to the forced time-fractional Burgers equations and variable-coefficient diffusion are also considered. Finally we study a Burgers-type equation involving both space and time-fractional derivatives.
nlin.SI↗
Constructing 3-Lie algebras
3-Lie algebras are constructed by Lie algebras, derivations and linear functions, associative commutative algebras, whose involutions and derivations. Then the 3-Lie algebras are obtained from group algebras $F[G]$. An infinite dimensional simple 3-Lie algebra $(A, [,,]_{ω, δ_0})$ and a non-simple 3-Lie algebra $(A, [,,]_{ω_1, δ})$ are constructed by Laurent polynomials $A=F[t, t^{-1}]$ and its involutions $ω$ and $ω_1$ and derivations $δ$ and $δ_0$. At last of the paper, we summarize the methods of constructing $n$-Lie algebras for $n\geq 3$ and provide a problem.
math-ph↗
A Variational Analysis of a Gauged Nonlinear Schrödinger Equation
This paper is motivated by a gauged Schrödinger equation in dimension 2 including the so-called Chern-Simons term. The study of radial stationary states leads to the nonlocal problem: $$ - Δu(x) + \left(ω+ \frac{h^2(|x|)}{|x|^2} + \int_{|x|}^{+\infty} \frac{h(s)}{s} u^2(s)\, ds \right) u(x) = |u(x)|^{p-1}u(x), $$ where $$ h(r)= \frac{1}{2}\int_0^{r} s u^2(s) \, ds. $$ This problem is the Euler-Lagrange equation of a certain energy functional. In this paper the study of the global behavior of such functional is completed. We show that for $p\in(1,3)$, the functional may be bounded from below or not, depending on $ω$. Quite surprisingly, the threshold value for $ω$ is explicit. From this study we prove existence and non-existence of positive solutions.
math.AP↗
A symmetry property for q-weighted Robinson-Schensted algorithms and other branching insertion algorithms
In O'Connell-Pei(2013) a q-weighted version of the Robinson-Schensted algorithm was introduced. In this paper we show that this algorithm has a symmetry property analogous to the well known symmetry property of the normal Robinson-Schensted algorithm. The proof uses a generalisation of the growth diagram approach introduced by Fomin(1979,1986,1994,1995). This approach, which uses "growth graphs", can also be applied to a wider class of insertion algorithms which have a branching structure, including some of the other q-weighted versions of the Robinson-Schensted algorithm which have recently been introduced by Borodin-Petrov(2013).
math.CO↗
On SUSY curves
In this note we give a summary of some elementary results in the theory of super Riemann surfaces (SUSY curves).
math.AG↗
Solvable limits of a 4D noncommutative QFT
In previous work we have shown that the (θ->\infty)-limit of ϕ^4_4-quantum field theory on noncommutative Moyal space is an exactly solvable matrix model. In this paper we translate these results to position space. We show that the Schwinger functions are symmetric and invariant under the full Euclidean group. The Schwinger functions only depend on matrix correlation functions at coinciding indices per topological sector, and clustering is violated. We prove that Osterwalder-Schrader reflection positivity of the Schwinger two-point function is equivalent to the question whether the diagonal matrix two-point function is a Stieltjes function. Numerical investigations suggest that this can at best be expected for the wrong sign of the coupling constant. The corresponding Wightman functions would describe particles which interact without momentum transfer. The theory differs from a free theory by the presence of non-trivial topological sectors.
math-ph↗
Hardy space on the polydisk and scattering in layered media
Hardy space on the polydisk provides the setting for a global description of scattering in piecewise-constant layered media, giving a simple qualitative interpretation for the nonlinear dependence of the Green's function on reflection coefficients and layer depths. Using explicit formulas for amplitudes, we prove that the power spectrum of the Green's function is approximately constant. In addition we exploit a connection to Jacobi polynomials to derive formulas for computing reflection coefficients from partial amplitude data. Unlike most approaches to layered media, which variously involve scaling limits, approximations or iterative methods, the formulas and methods in the present paper are exact and direct.
math-ph↗
Toda lattice G-Strands
Hamilton's principle is used to extend for the Toda lattice ODEs to systems of PDEs called the Toda lattice strand equations (T-Strands). The T-Strands in the $n$-particle Toda case comprise $4n-2$ quadratically nonlinear PDEs in one space and one time variable. T-Strands form a symmetric hyperbolic Lie-Poisson Hamiltonian system of quadratically nonlinear PDEs with constant characteristic velocities. The travelling wave solutions for the two-particle T-Strand equations are solved geometrically, and their Lax pair is given to show how nonlinearity affects the solution. The three-particle T-Strands equations are also derived from Hamilton's principle. For both the two-particle and three-particle T-Strand PDEs the determining conditions for the existence of a quadratic zero-curvature relation (ZCR) exactly cancel the nonlinear terms in the PDEs. Thus, the two-particle and three-particle T-Strand PDEs do not pass the ZCR test for integrability.
nlin.SI↗
Some Remarks on the Logic of Quantum Gravity
We discuss some conceptual issues that any approach to quantum gravity has to confront. In particular, it is argued that one has to find a theory that can be interpreted in a realist manner, because theories with an instrumentalist interpretation are problematic for several well-known reasons. Since the Hilbert space formalism almost inevitably forces an instrumentalist interpretation on us, we suggest that a theory of quantum gravity should not be based on the Hilbert space formalism. We briefly sketch the topos approach, which makes use of the internal logic of a topos associated with a quantum system and comes with a natural (neo-)realist interpretation. Finally, we make some remarks on the relation between system logic and metalogic.
gr-qc↗
Reconstruction of the refractive index from experimental backscattering data using a globally convergent inverse method
The problem to be studied in this work is within the context of coefficient identification problems for the wave equation. More precisely, we consider the problem of reconstruction of the refractive index (or equivalently, the dielectric constant) of an inhomogeneous medium using one backscattering boundary measurement. The goal of this paper is to analyze the performance of a globally convergent algorithm of Beilina and Klibanov on experimental data acquired in the Microwave Laboratory at University of North Carolina at Charlotte. The main challenge working with experimental data is the the huge misfit between these data and computationally simulated data. We present data pre-processing steps to make the former somehow look similar to the latter. Results of both non-blind and blind targets are shown indicating good reconstructions even for high contrasts between the targets and the background medium.
math-ph↗
Hydrodynamics of the Kuramoto-Vicsek model of rotating self-propelled particles
We consider an Individual-Based Model for self-rotating particles interacting through local alignment and investigate its macroscopic limit. This model describes self-propelled particles moving in the plane and trying to synchronize their rotation motion with their neighbors. It combines the Kuramoto model of synchronization and the Vicsek model of swarm formation. We study the mean-field kinetic and hydrodynamic limits of this system within two different scalings. In the small angular velocity regime, the resulting model is a slight modification of the 'Self-Organized Hydrodynamic' model which has been previously introduced by the first author. In the large angular velocity case, a new type of hydrodynamic model is obtained. A preliminary study of the linearized stability is proposed.
math-ph↗
Investigation of the transfer and dissipation of energy in isotropic turbulence
A parallel pseudospectral code for the direct numerical simulation (DNS) of isotropic turbulence has been developed. The code has been extensively benchmarked using established results from literature. The code has been used to conduct a series of runs for freely-decaying turbulence. We explore the use of power-law decay of the total energy to determine an evolved time and compare with the use of dynamic quantities such as the peak dissipation rate, maximum transport power and velocity derivative skewness. Stationary turbulence has also been investigated, where we ensure that the energy input rate remains constant for all runs. We present results for Reynolds numbers up to Rλ = 335 on a 1024^3 lattice. An exploitation of the pseudospectral technique is used to calculate second and third-order structure functions from the energy and transfer spectra, with a comparison presented to the real-space calculation. An alternative to ESS is discussed, with the second-order exponent found to approach 2/3. The dissipation anomaly is considered for forced and free-decay. The Kármán-Howarth equation (KHE) is studied and a derivation of a new work term presented. The balance of energy represented by the KHE is then investigated. Based on the KHE, we develop a model for the behaviour of the dimensionless dissipation coefficient that predicts Cε = Cε(\infty) + C_L/R_L, with Cε(\infty) = 0.47 and C_L = 19.1 obtained from DNS data. Theoretical methods based on RG and statistical closures are still being developed to study turbulence. The dynamic RG procedure used by Forster, Nelson and Stephen (FNS) is considered in some detail and a disagreement in the literature is resolved here. The application of statistical closure and renormalized perturbation theory is discussed and a new two-time model probability density functional presented.
physics.flu-dyn↗