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In this paper we show several applications of the general theory developed in \cite{MV_I}, where we studied smoothing and ergodicity for infinite dimensional Markovian systems with hypocoercive type generator.
We develop a version of dipolar conformal field theory in a simply connected domain with the Dirichlet-Neumann boundary condition and central charge one. We prove that all correlation functions of the fields in the OPE family of Gaussian free field with a certain boundary value are martingale-observables for dipolar SLE(4).
Using the results on the $1/n$-expansion of the Verblunsky coefficients for a class of polynomials orthogonal on the unit circle with $n$ varying weight, we prove that the local eigenvalue statistic for unitary matrix models is independent of the form of the potential, determining the matrix model. Our proof is applicable to the case of four times differentiable potentials and of supports, consisting of one interval.
Consider the problem "Give the equation of the conceptional rotations in ${\mathbb R}^3$ without using the parameter expressing individual rotations", just as the conceptional motion of constant velocity along straight lines (Galiley motions) is expressed by an elementary differential equation. In this we try to give an answer to the topic related with the above issue.
For a connected Lie group G, we show that a complex structure on the total space TG of the tangent bundle of G that is left invariant and has the property that each left translation G-orbit is a totally real submanifold is induced from a smooth immersion of TG into the complexification of G. For G compact and connected, we then characterize left invariant and biinvariant complex structures on the total space T*G of the cotangent bundle of G which combine with the tautological symplectic structure to a Kaehler structure.
In this note, I describe a formalism for treating knots as geometric spaces, and make an application to a simple statistical mechanics computation. The motivation for this study is the natural visual symmetry of the knot, and I describe how this might be carried out. The direct approach, however, fails due to limits of the visual symmetry, but by recasting the problem in terms of the geometry of contours of the knot, the resulting permutation operators provide a better analytical tool.
We study underlying geometric structures for integral variational functionals, depending on submanifolds of a given manifold. Applications include (first order) variational functionals of Finsler and areal geometries with integrand the Hilbert 1-form, and admit immediate extensions to higher-order functionals.
We determine the trace formula for the fourth order operator on the circle. This formula is similar to the famous trace formula for the Hill operator obtained by Dubrovin, Its-Matveev and McKean-van Moerbeke.
It is well known that the symmetric group has an important role (via Young tableaux formalism) both in labelling of the representations of the unitary group and in construction of the corresponding basis vectors (in the tensor product of the defining representations). We show that orthogonal group has a very similar role in the context of positive energy representations of osp(1|2n,R). In the language of parabose algebra, we essentially solve the long standing problem of reducibility of Green's ansatz representations.
On a bounded three-dimensional smooth domain, we consider the generalized oscillon equation with Dirichlet boundary conditions, with time-dependent damping and time-dependent squared speed of propagation. Under structural assumptions on the damping and the speed of propagation, which include the relevant physical case of reheating phase of inflation, we establish the existence of a pullback global attractor of optimal regularity, and finite-dimensionality of the kernel sections.
A parabolic integro differential operator operator L suitable to describe many phenomena in various physical fields,is considered. By means of equivalence between L and the third order equation which describe the evolution inside an exponentially shaped Josephson junction (ESJJ), an asymptotic analysis for (ESJJ) is achieved, evaluating explicitly boundary contributions related to the Dirichlet problem.
The aim of this paper is to provide a mathematical model for spatial distribution of membrane electrical potential changes by fluorescence diffuse optical tomography. We derive the resolving power of the imaging method in the presence of measurement noise. The proposed mathematical model can be used for cell membrane tracking with the resolution of the optical microscope.
We consider semi-excited resonances created by a periodic orbit of hyperbolic type for Schrödinger type operators with a small "Planck constant". They are defined within an analytic framework based on the semi-classical quantization of Poincaré map in action-angle variables.
Integral representation for harmonic function in the ring is constructed in this work. We prove the existence and uniqueness of solutions of the Cauchy problem for the Laplace equation in the ring. Integral representation for the solution of the Cauchy problem was found.
We suggest a new representation of Maslov's canonical operator in a neighborhood of the caustics using a special class of coordinate systems ("eikonal coordinates") on Lagrangian manifolds.
We propose a new Eulerian turbulence theory to obtain a closed set of equations for homogeneous, isotropic turbulent velocity field correlations and propagator functions by incorporating constraints of random Galilean invariance. This incorporation generates a few different equations for propagator and the present theory suggests a way to utilize them into the closure solutions of two-time and single-time velocity correlations' equations so as to properly account for random sweeping phenomena. The present theory yields exact solutions when applied to simple model problem of random oscillator.
Determinantal process is a dynamical extension of a determinantal point process such that any spatio-temporal correlation function is given by a determinant specified by a single continuous function called the correlation kernel. Noncolliding diffusion processes are important examples of determinantal processes. In the present lecture, we introduce determinantal martingales and show that if the interacting particle system (IPS) has determinantal-martingale representation, then it becomes a determinantal process. From this point of view, the reason why noncolliding diffusion processes and noncolliding random walk are determinantal is simply explained by the fact that the harmonic transform with the Vandermonde determinant provides a proper determinantal martingale. Recently O'Connell introduced an interesting IPS, which can be regarded as a stochastic version of a quantum Toda lattice. It is a geometric lifting of the noncolliding Brownian motion and is not determinantal, but Borodin and Corwin discovered a determinant formula for a special observable for it. We also discuss this new topic from the present view-point of determinantal martingale.
This paper devoted to proof the existence of stable quasi-periodic motions of the magnetic dipole that is under the action of the external magnetic field and homogeneous field of gravity. For proof this we used the group-theoretic methods of Hamiltonian mechanics, viz energy-momentum method. Numerical simulation shows the possibility of realization of stable motions with physically reasonable parameters of the system.