A Canonical Ensemble Approach to the Fermion/Boson Random Point Processes and its Applications

We introduce the boson and the fermion point processes from the elementary quantum mechanical point of view. That is, we consider quantum statistical mechanics of canonical ensemble for a fixed number of particles which obey Bose-Einstein, Fermi-Dirac statistics, respectively, in a finite volume. Focusing on the distribution of positions of the particles, we have point processes of the fixed number of points in a bounded domain. By taking the thermodynamic limit such that the particle density converges to a finite value, the boson/fermion processes are obtained. This argument is a realization of the equivalence of ensembles, since resulting processes are considered to describe a grand canonical ensemble of points. Random point processes corresponding to para-particles of order two are discussed as an application of the formulation. A statistics of a system of composite particles at zero temperature are also considered as a model of determinantal random point processes.Comment: 26pages, Some typos are corrected, to be published in Commun. Math. Phy

Legendre-Tau approximations for functional differential equations

The numerical approximation of solutions to linear functional differential equations are considered using the so called Legendre tau method. The functional differential equation is first reformulated as a partial differential equation with a nonlocal boundary condition involving time differentiation. The approximate solution is then represented as a truncated Legendre series with time varying coefficients which satisfy a certain system of ordinary differential equations. The method is very easy to code and yields very accurate approximations. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline approximations is made

Legendre-tau approximation for functional differential equations. Part 2: The linear quadratic optimal control problem

The numerical scheme based on the Legendre-tau approximation is proposed to approximate the feedback solution to the linear quadratic optimal control problem for hereditary differential systems. The convergence property is established using Trotter ideas. The method yields very good approximations at low orders and provides an approximation technique for computing closed-loop eigenvalues of the feedback system. A comparison with existing methods (based on averaging and spline approximations) is made

Chandrasekhar equations for infinite dimensional systems

Chandrasekhar equations are derived for linear time invariant systems defined on Hilbert spaces using a functional analytic technique. An important consequence of this is that the solution to the evolutional Riccati equation is strongly differentiable in time and one can define a strong solution of the Riccati differential equation. A detailed discussion on the linear quadratic optimal control problem for hereditary differential systems is also included

Continuous vacua in bilinear soliton equations

We discuss the freedom in the background field (vacuum) on top of which the solitons are built. If the Hirota bilinear form of a soliton equation is given by A(D_{\vec x})\bd GF=0,\, B(D_{\vec x})(\bd FF - \bd GG)=0 where both $A$ and $B$ are even polynomials in their variables, then there can be a continuum of vacua, parametrized by a vacuum angle $\phi$. The ramifications of this freedom on the construction of one- and two-soliton solutions are discussed. We find, e.g., that once the angle $\phi$ is fixed and we choose $u=\arctan G/F$ as the physical quantity, then there are four different solitons (or kinks) connecting the vacuum angles $\pm\phi$, $\pm\phi\pm\Pi2$ (defined modulo $\pi$). The most interesting result is the existence of a ``ghost'' soliton; it goes over to the vacuum in isolation, but interacts with ``normal'' solitons by giving them a finite phase shift.Comment: 9 pages in Latex + 3 figures (not included