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Motivated by representation theory and geometry, we introduce and develop an equivariant generalization of Ehrhart theory, the study of lattice points in dilations of lattice polytopes. We prove representation-theoretic analogues of…
Different representations of dissipative Hamiltonian and port-Hamiltonian differential-algebraic equations (DAE) systems are presented and compared. Using global geometric and algebraic points of view, translations between the different…
Determinantal point processes are characterized by a special structural property of the correlation functions: they are given by minors of a correlation kernel. However, unlike the correlation functions themselves, this kernel is not…
The equation of motion of a massless Dirac particle in the C-metric leads to the general Heun equation (GHE) for the radial and the polar variables. The GHE, under certain parametric conditions, has been cast in terms of a new set of…
In this paper, we consider the Dirichlet problem for a new class of augmented Hessian equations. Under sharp assumptions that the matrix function in the augmented Hessian is regular and there exists a smooth subsolution, we establish global…
The quantum cosmology of a higher-derivative derivative gravity theory arising from the heterotic string effective action is reviewed. A new type of Wheeler-DeWitt equation is obtained when the dilaton is coupled to the quadratic curvature…
A global solution of the Schr\"odinger equation for explicitly time-dependent Hamiltonians is derived by integrating the non-linear differential equation associated with the time-dependent wave operator. A fast iterative solution method is…
Many basis sets for electronic structure calculations evolve with varying external parameters, such as moving atoms in dynamic simulations, giving rise to extra derivative terms in the dynamical equations. Here we revisit these derivatives…
A new discrete model for energy relaxation of a quantum particle is described via a projection operator, causing the wave function collapse. Power laws for the evolution of the particle coordinate and momentum dispersions are derived. A new…
We first prove a new separating hyperplane theorem characterizing when a pair of compact convex subsets $K, K'$ of the Euclidean space intersect, and when they are disjoint. The theorem is distinct from classical separation theorems. It…
We find kernel functions of the $q$-Heun equation and its variants. We apply them to obtain $q$-integral transformations of solutions to the $q$-Heun equation and its variants. We discuss special solutions of the $q$-Heun equation from the…
A detailed study of solutions to the first order partial differential equation H(x,y,z_x,z_y)=0, with special emphasis on the eikonal equation z_x^2+z_y^2=h(x,y), is made near points where the equation becomes singular in the sense that…
Tracy and Widom showed that fundamentally important kernels in random matrix theory arise from differential equations with rational coefficients. More generally, this paper considers symmetric Hamiltonian systems abd determines the…
In a digraph, a kernel is a subset of vertices that is both independent and absorbing. Kernels have important applications in combinatorics and outside. Kernels do not always exist and finding sufficient conditions ensuring their existence…
We introduce a new class of two(multi)-matrix models of positive Hermitean matrices coupled in a chain; the coupling is related to the Cauchy kernel and differs from the exponential coupling more commonly used in similar models. The…
The effective-interaction theory has been one of the useful and practical methods for solving nuclear many-body problems based on the shell model. Various approaches have been proposed which are constructed in terms of the so-called…
A recently proposed discrete version of the Schrodinger spectral problem is considered. The whole hierarchy of differential-difference nonlinear evolution equations associated to this spectral problem is derived. It is shown that a discrete…
In this paper, we present numerical procedures to compute solutions of partial differential equations posed on fractals. In particular, we consider the strong form of the equation using standard graph Laplacian matrices and also weak forms…
Using a theorem of partial differential equations, we present a general way of deriving the conserved quantities associated with a given classical point mechanical system, denoted by its Hamiltonian. Some simple examples are given to…
This paper introduces a kernel discrepancy-based framework for rerandomization to enhance the precision of causal inference in controlled experiments. We demonstrate that the kernel discrepancy is the key part of the variance upper bound…