Stein’s method is used to study discrete representations of multidimensional distributions that arise as approximations of states of quantum harmonic oscillators. These representations model how quantum effects result from the interaction of finitely many classical ‘worlds’, with the role of sample size played by the number of worlds. Each approximation arises as the ground state of a Hamiltonian involving a particular interworld potential function. Our approach, framed in terms of spherical coordinates, provides the rate of convergence of the discrete approximation to the ground state in terms of Wasserstein distance. Applying a novel Stein’s method technique to the radial component of the ground state solution, the fastest rate of convergence to the ground state is found to occur in three dimensions.

We prove an extension of the homology version of the Hofer–Zehnder conjecture proved by Shelukhin to the weighted projective spaces which are symplectic orbifolds. In particular, we prove that if the number of fixed points counted with their isotropy order as multiplicity of a non-degenerate Hamiltonian diffeomorphism of such a space is larger than the minimum number possible, then there are infinitely many periodic points.

We present a general solution approach for analysis of transversely isotropic cylindrical tubes and circular plates. On the basis of Hamiltonian state space formalism in a systematic way, rigorous solutions of the twisting problems are determined by means of separation of variables and symplectic eigenfunction expansion.

We present an exact analysis of the displacement and stress fields in an elastic 2-D cantilever subjected to axial force, shear force and moment, in which the end conditions are exactly satisfied. The problem is formulated on the basis of the state space formalism for 2-D deformation of an orthotropic body. Upon delineating the Hamiltonian characteristics of the formulation and by using eigenfunction expansion, a rigorous solution which satisfies the end conditions is determined. The results show that the end condition alters the stress significantly only near the end, and elementary solutions in the form of polynomials can give sufficiently accurate results except near the ends. Such a system would give rise to localized stresses and displacements in the immediate neighborhood of the ends, and the effect may be expected to diminish with distance on account of geometrical divergence.

We demonstrate that a piecewise linear slow-fast Hamiltonian system with an equilibriumof the saddle-center type can have a sequence of small parameter values for which aone-round homoclinic orbit to this equilibrium exists. This contrasts with the well-knownfindings by Amick and McLeod and others that solutions of such type do not exist inanalytic Hamiltonian systems, and that the separatrices are split by the exponentiallysmall quantity. We also discuss existence of homoclinic trajectories to small periodicorbits of the Lyapunov family as well as symmetric periodic orbits near the homoclinicconnection. Our further result, illustrated by simulations, concerns the complicatedstructure of orbits related to passage through a non-smooth bifurcation of a periodicorbit.

We develop a Discrete Element Method (DEM) for elastodynamics using polyhedral elements. We show that for a given choice of forces and torques, we recover the equations of linear elastodynamics in small deformations. Furthermore, the torques and forces derive from a potential energy, and thus the global equation is an Hamiltonian dynamics. The use of an explicit symplectic time integration scheme allows us to recover conservation of energy, and thus stability over long time simulations. These theoretical results are illustrated by numerical simulations of test cases involving large displacements.

We apply in this study an area preserving level set method to simulate gas/water interface flow. For the sake of accuracy, the spatial derivative terms in the equations of motion for an incompressible fluid flow are approximated by the fifth-order accurate upwinding combined compact difference (UCCD) scheme. This scheme development employs two coupled equations to calculate the first- and second-order derivative terms in the momentum equations. For accurately predicting the level set value, the interface tracking scheme is also developed to minimize phase error of the first-order derivative term shown in the pure advection equation. For the purpose of retaining the long-term accurate Hamiltonian in the advection equation for the level set function, the time derivative term is discretized by the sixth-order accurate symplectic Runge-Kutta scheme. Also, to keep as a distance function for ensuring the front having a finite thickness for all time, the re-initialization equation is used. For the verification of the optimized UCCD scheme for the pure advection equation, two benchmark problems have been chosen to investigate in this study. The level set method with excellent area conservation property proposed for capturing the interface in incompressible fluid flows is also verified by solving the dam-break, Rayleigh-Taylor instability, two-bubble rising in water, and droplet falling problems.

We study optimal control problems for (time-)delayed stochastic differential equations with jumps. We establish sufficient and necessary stochastic maximum principles for an optimal control of such systems. The associated adjoint processes are shown to satisfy a (time-)advanced backward stochastic differential equation (ABSDE). Several results on existence and uniqueness of such ABSDEs are shown. The results are illustrated by an application to optimal consumption from a cash flow with delay.

A Halin graph is a graph $H\,=\,T\,\cup \,C$, where $T$ is a tree with no vertex of degree two, and $C$ is a cycle connecting the end-vertices of $T$ in the cyclic order determined by a plane embedding of $T$. In this paper, we define classes of generalized Halin graphs, called $k$-Halin graphs, and investigate their Hamiltonian properties.

The state of a patient is an important concept in biomedical sciences. While analytical methods for predicting and exploring treatment strategies of disease dynamics have proven to have useful applications in public health policy and planning, the state of a patient has attracted less attention, at least mathematically. As a result, models constructed in relation to treatment strategies may not be very informative. We derive a patient-dependent parameter from an age-physiology dependent population model, and show that a single treatment strategy is not always optimal. Also, we derive a function which increases with the patient dependence parameter and describes the effort expended to be in good health.

We prove that an inherently nonfinitely based algebra cannot generate an abelian variety. On the other hand, we show by example that it is possible for an inherently nonfinitely based algebra to generate a strongly solvable variety.