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In this article we consider the extended version of a real counterpart of qubit density matrices, called the Möbius matrix, and we see that it is a normalized Lorentz boost. Using the isomorphic gyrogroup structures between the set $\PP$ of all Lorentz boosts and the Einstein gyrogroup on the open unit ball $\B$ of $\R^n$ we give an explicit formula of the fidelity for Möbius matrices in terms of Lorentz gamma factors.
\emph{Mechanical systems} called by use, \emph{mechanical}$\left(ρ,η\right) $\emph{-systems, Lagrange mechanical}$\left(ρ,η\right) $\emph{-systems} or \emph{Finsler mechanical}$\left(ρ,η\right) $\emph{-systems} are presented. The canonical $\left(ρ,η\right) $\emph{-}semi(spray) associated to a mechanical $\left(ρ,η\right) $-system is obtained. New and important results are obtained in the particular case of Lie algebroids. The Lagrange mechanical $(ρ,η)$% -systems are the spaces necessary to develop a new Lagrangian formalism. We obtain the $(ρ,η)$-semispray associated to a regular Lagrangian $L$ and external force $F_{e}$ and we derive the equations of Euler-Lagrange type. So, a new solution for the Weinstein's Problem in the general framework of generalized Lie algebroids is presented.
Rational solutions of the inhomogeneous Painleve-II equation and of a related coupled Painleve-II system have recently arisen in studies of fluid vortices and of the sine-Gordon equation. For the sine-Gordon application in particular it is of interest to understand the large-degree asymptotic behavior of the rational Painleve-II functions. We explicitly compute the leading-order large-degree asymptotics of these two families of rational functions valid in the whole complex plane with the exception of a neighborhood of a certain piecewise-smooth closed curve. We obtain rigorous error bounds by using the Deift-Zhou nonlinear steepest-descent method for Riemann-Hilbert problems.
Based on the well-posedness of the stationary Wigner equation with inflow boundary conditions given in (A. Arnold, H et al. J. Math. Phys., 41, 2000), we prove without any additional prerequisite conditions that the solution of the Wigner equation with symmetric potential and inflow boundary conditions will be symmetric. This improve the result in (D. Taj et al. Europhys. Lett., 74, 2006) which depends on the convergence of solution formulated in the Neumann series. By numerical studies, we present the convergence of the numerical solution to the symmetric profile for three different numerical schemes. This implies that the upwind schemes can also yield a symmetric numerical solution, on the contrary to the argument given in (D. Taj et al. Europhys. Lett., 74, 2006).
The Hankel determinant appears in the representation of solutions to several integrable systems. Asymptotic expansion of the Hankel determinant thus plays a key role for investigating asymptotic analysis of such integrable system. In this paper, an asymptotic expansion formula of a certain Casorati determinant is presented as an extension of the Hankel case. It is also shown that an application of it to an asymptotic analysis of the discrete hungry Lotka-Volterra system, which is one of basic models in mathematical biology.
Capacity drop at active bottlenecks is one of the most puzzling traffic phenomena, but a thorough understanding is practically important for designing variable speed limit and ramp metering strategies. In this study, we attempt to develop a simple model of capacity drop within the framework of kinematic wave theory based on the observation that capacity drop occurs when an upstream queue forms at an active bottleneck. In addition, we assume that the fundamental diagrams are continuous in steady states. This assumption is consistent with observations and can avoid unrealistic infinite characteristic wave speeds in discontinuous fundamental diagrams. A core component of the new model is an entropy condition defined by a discontinuous boundary flux function. For a lane-drop area, we demonstrate that the model is well-defined, and its Riemann problem can be uniquely solved. We theoretically discuss traffic stability with this model subject to perturbations in density, upstream demand, and downstream supply. We clarify that discontinuous flow-density relations, or so-called "discontinuous" fundamental diagrams, are caused by incomplete observations of traffic states. Theoretical results are consistent with observations in the literature and are verified by numerical simulations and empirical observations. We finally discuss potential applications and future studies.
We address several issues regarding the derivation and implementation of the Cauchy-Born approximation of the stress at finite temperature. In particular, an asymptotic expansion is employed to derive a closed form expression for the first Piola-Kirchhoff stress. For systems under periodic boundary conditions, a derivation is presented, which takes into account the translational invariance and clarifies the removal of the zero phonon modes. Also revealed by the asymptotic approach is the role of the smoothness of the interatomic potential. Several numerical examples are provided to validate this approach.
We obtain complete classification of in-equivalent realizations of the Virasoro algebra by Lie vector fields over the three-dimensional field of real numbers. As an application we construct new classes of nonlinear second-order partial differential equations possessing infinite-dimensional Lie symmetries.
The analytic solution for the kinetic description of binary reactions can be seen as the continuum version of a basic discrete iterate mapping. This fact allows a clear definition of the reaction characteristic time which takes the backward effect into account.
A model is presented, consisting of a single structureless particle on the line subject to a potential with three minima, with an exactly soluble ground level. In this model the ground level probability density becomes more sensitive to the global shape of the potential as the distance between the minima increases, so that for big enough distances small variations in the potential bring a qualitative change in the probability density, taking it from a unimodal, localized, distribution, to a bimodal one. We conjecture that this effect, of which we have not found any precedent in the literature, may be relevant in the design and characterization of mesoscopic devices such as triple quantum well systems.
A difficult problem in the theory of random tensors is to calculate the expectation values of polynomials in the tensor entries, even in the large N limit and in a Gaussian distribution. Here we address this issue, focusing on a family of polynomials labeled by permutations, which naturally generalize the single-trace invariants of random matrix models. Through Wick's theorem, we show that the Feynman graph expansion of the expectation values of those polynomials enumerates meandric systems whose lower arch configuration is obtained from the upper arch configuration by a permutation on half of the arch feet. Our main theorem reduces the calculation of expectation values to those of polynomials labeled by stabilized-interval-free permutations (SIF) which are proved to enumerate irreducible meandric systems. This together with explicit calculations of expectation values associated to SIF permutations allows to exactly evaluate large N expectation values beyond the so-called melonic polynomials.
In 1974 Jürgen Moser has shown that the classical Moment Problem plays a fundamental role for the theory of completely integrable systems, by proving that the simplest case of the finite Toda lattice is described exhaustively in its terms. In particular, the Jacobi matrix defined by the Flaschka variables corresponds to the Schrödinger operator related to the Korteweg-de Vries operator. In the present paper we use a recent breakthrough in the area of the multidimensional Moment Problem in order to develop a multidimensional generalization of the finite Toda lattice. Our construction is based on the notion of pseudo-positive measure and the notion of multidimensional Markov-Stieltjes transform naturally defined on the Klein-Dirac quadric. An important consequence of the properties of the Markov-Stieltjes transform is a new method for summation of multidimensional divergent series on the Klein-Dirac quadric.
Soliton theory and the theory of Hankel (and Toeplitz) operators have stayed essentially hermetic to each other. This paper is concerned with linking together these two very active and extremely large theories. On the prototypical example of the Cauchy problem for the Korteweg-de Vries (KdV) equation we demonstrate the power of the language of Hankel operators in which symbols are conveniently represented in terms of the scattering data for the Schrodinger operator associated with the initial data for the KdV equation. This approach yields short-cuts to already known results as well as to a variety of new ones (e.g. wellposedness beyond standard assumptions on the initial data) which are achieved by employing some subtle results for Hankel operators.
A general approach to provide approximate parameterizations of the "small" scales by the "large" ones, is developed for stochastic partial differential equations driven by linear multiplicative noise. This is accomplished via the concept of parameterizing manifolds (PMs) that are stochastic manifolds which improve in mean square error the partial knowledge of the full SPDE solution $u$ when compared to the projection of $u$ onto the resolved modes, for a given realization of the noise. Backward-forward systems are designed to give access to such PMs in practice. The key idea consists of representing the modes with high wave numbers (as parameterized by the sought PM) as a pullback limit depending on the time-history of the modes with low wave numbers. The resulting manifolds obtained by such a procedure are not subject to a spectral gap condition such as encountered in the classical theory. Instead, certain PMs can be determined under weaker non-resonance conditions. Non-Markovian stochastic reduced systems are then derived based on such a PM approach. Such reduced systems take the form of SDEs involving random coefficients that convey memory effects via the history of the Wiener process, and arise from the nonlinear interactions between the low modes, embedded in the "noise bath." These random coefficients follow typically non-Gaussian statistics and exhibit an exponential decay of correlations whose rate depends explicitly on gaps arising in the non-resonances conditions. It is shown on a stochastic Burgers-type equation, that such PM-based reduced systems can achieve very good performance in reproducing statistical features of the SPDE dynamics projected onto the resolved modes, such as the autocorrelations and probability functions of the corresponding modes amplitude.
In a recent paper, Selberg-Tesfahun proved that the abelian Chern-Simons-Higgs system (CSH) is globally well-posed for finite energy initial data under the Lorenz gauge condition. It has been suspected by Huh, however, that such a result should hold in the Coulomb gauge as well. In this note, we give an affirmative answer to this question by first establishing low regularity local well-posededness of (CSH) in the Coulomb gauge for initial data set $(f, g) \in H^γ_{x} \times H^{γ-1}_{x}$ for any $γ> 3/4$. Then by conservation of energy, global well-posedness for (CSH) in the energy space $(f, g) \in H^{1}_{x} \times L^{2}_{x}$ follows rather immediately.
This Ph.D. thesis is devoted to the constructions of Lagrangian formulation on Finsler and Kawaguchi manifolds. While Finsler geometry is a natural extension of Riemannian geometry, Kawaguchi geometry is the extension of Finsler geometry to higher order derivatives and to k-dimensional parameter space. The latter extension is also called areal geometry in some references. On Finsler (Kawaguchi) manifold, we can define a reparameterisation invariant 1 (k)-dimensional area by the Hilbert (Kawaguchi) form, which we take as an action. The equation of motion obtained from such action also has the property of reparameterisation invariance. In this framework, the solution manifold of the Euler-Lagrange equation is realised as a submanifold of Finsler/Kawaguchi manifold, and no fibered structure over the parameter space is needed. We also show that for the case of first order k-dimensional parameter space and second order 1-dimensional parameter space, a global Lagrangian could be constructed. For second order k-dimensional parameter space, Lagrangian is not global but it still has the reparameterisation invariant property. Such theory is expected to provide the ideal stage for formulating fundamental theories of physics, especially for cases such as when one needs to consider the mixing of spacetime and field variables. It is shown that locally, any conventional Lagrangian could be reformulated by the parameter independent Lagrangian. Furthermore, the parameter independent property will gives us the freedom of choosing a parameter, which in some cases turns out to be useful in finding solutions and symmetries.
We consider a model for heterogeneous 'gene regulatory networks' that is a generalization of the model proposed by Chatterjee and Durrett (2011) as an "annealed approximation" of Kauffmann's (1969) random Boolean networks. In this model, genes are represented by the nodes of a random directed graph on n vertices with specified in-degree distribution (resp. out-degree distribution or joint distribution of in-degree and out-degree), and the expression bias (the expected fraction of 1's in the Boolean functions) p is same for all nodes. Following a standard practice in the physics literature, we use a discrete-time threshold contact process with parameter q=2p(1-p) (in which a vertex with at least one 'occupied' input at time t will be occupied at time t+1 with probability q, and 'vacant' otherwise) on the above random graph to approximate the dynamics of the Boolean network. We show that there is a parameter r (which can be written explicitly in terms of first few moments of the degree distribution) such that, with probability tending to 1 as n goes to infinity, if 2p(1-p)r>1, then starting from all occupied sites the threshold contact process maintains a positive ({\it quasi-stationary}) density of occupied sites for time which is exponential in n, whereas if 2p(1-p)r<1, then the persistence time of the threshold contact process is at most logarithmic in n. These two phases correspond to the 'chaotic' and 'ordered' behavior of the gene networks.
This paper provides theoretical consistency results for compressed modes. We prove that as L1 regularization term in certain non-convex variational optimization problems vanishes, the solutions of the optimization problem and the corresponding eigenvalues converge to Wannier-like functions and the eigenvalues of the Hamiltonian, respectively.