Feynman Path Integral Using Operator Integration in Banach Space

Feynman-Path Integral in Banach Space: In 1940, R.P. Feynman attempted to find a mathematical representation to express quantum dynamics of the general form for a double-slit experiment. His intuition on several slits with several walls in terms of Lagrangian instead of Hamiltonian resulted in a magnificent work. It was known as Feynman Path Integrals in quantum physics, and a large part of the scientific community still considers them a heuristic tool that lacks a sound mathematical definition. This paper aims to refute this prejudice, by providing an extensive and self-contained description of the mathematical theory of Feynman Path Integration, from the earlier attempts to the latest developments, as well as its applications to quantum mechanics. About a hundred years after the beginning of modern physics, it was realized that light could in fact show behavioral characteristics of both waves and particles. In 1927, Davisson and Germer demonstrated that electrons show the same dual behavior, which was later extended to atoms and molecules. We shall follow the method of integration with some modifications to construct a generalized Lebesgue-Bochner-Stieltjes (LBS) integral of the form , where u is a bilinear operator acting in the product of Banach spaces, f is a Bochner summable function, and μ is a vector-valued measure. We will demonstrate that the Feynman Path Integral is consistent and can be justified mathematically with LBS integration approach.

Variation of Parameters for Causal Operator Differential Equations

The operator T from a domain D into the space of measurable functions is called a nonanticipating (causal) operator if the past information is independent from the future outputs. We will study the solution x(t) of a nonlinear operator differential equation where its changes depends on the causal operator T, and semigroup of operator A(t), and all initial parameters (t0, x0) . The initial information is described x(t)=φ(t) for almost all t ≤t0 and φ(t0) =φ0. We will study the nonlinear variation of parameters (NVP) for this type of nonanticipating operator differential equations and develop Alekseev type of NVP.

Computation, Modeling, and Simulation of HIV-AIDS Epidemics with Vaccination

The principles of the HIV-AIDS epidemics are established based on the subpopulation 1) Susceptible;2) HIV-infected;3) AIDS-infected;4) Immunized. The immunized subset of the population in this paper is the total individuals who were infected and cured or immunized by vaccination. The immunized group can be identified by removing individuals from the susceptible group. A general mathematical model is developed for HIV-AIDS epidemics with Vaccination to understand the spread of the virus throughout the population. Particularly we use numerical simulation with some values of parameters to predict the number of infected individuals during a certain period in a population and the effect of vaccine to reduce infected group and increase the number of immunized individuals. Further, we expand the research to special cases with no vaccinations. A special case is when the removal subset of the population is empty, or there is no recovery in this epidemic. We also can consider the total infected number is equal to the sum of the HIV infected and the number of AIDS infected. As a result, one can reduce four-stage HIV-AIDS investigation to a three-stage of SIR. With this introduction and modification, the numerical simulation can be developed the Monte Carlo simulation method in SIR case to verify the Validity of the HIV-AIDS model.