Related papers: The propagator for the step potential using the pa…
Transformer has achieved great success in computer vision, while how to split patches in an image remains a problem. Existing methods usually use a fixed-size patch embedding which might destroy the semantics of objects. To address this…
We investigate random walks on a lattice with imperfect traps. In one dimension, we perturbatively compute the survival probability by reducing the problem to a particle diffusing on a closed ring containing just one single trap. Numerical…
Motivated by extending the functional stochastic calculus, to important functionals to which it does not apply, a notion of functional derivative along a curve is introduced. This new setting is developed by incorporating path-dependent…
We obtain a remainder estimate for the truncated Taylor expansion for differential equations driven by weakly geometric $\Pi $-rough paths for $\Pi =\left( p_{1},\cdots ,p_{k}\right) $, $p_{i}\geq 1$. When there exists $ p\geq 1$ such that…
We show how to construct path integrals for quantum mechanical systems where the space of configurations is a general non-compact symmetric space. Associated with this path integral is a perturbation theory which respects the global…
We construct a one-dimensional model for the stepped leader in the filamental discharge by simplifying an electric-circuit model of discharge. We find that the leader of the discharge moves stepwise by direct numerical simulations, and then…
I discuss the connection between the Hamiltonian and path integral approaches for fermionic fields. I show how the temporal Wilson projection operators appear naturally in a lattice action. I also carefully treat the insertion of a chemical…
We investigate the analytic continuation of wave equations into the complex position plane. For the particular case of electromagnetic waves we provide a physical meaning for such an analytic continuation in terms of a family of closely…
Making use of the expression for the propagator in terms of path integrals, we study the effect of certain variational symmetries of a Lagrangian on the corresponding propagator. We also show that by considering a point transformation that…
The backpropagation algorithm for neural networks is widely felt hard to understand, despite the existence of some well-written explanations and/or derivations. This paper provides a new derivation of this algorithm based on the concept of…
The scattering properties of quantum particles on fractal potentials at different stages of fractal growth are obtained by means of the transfer matrix method. This approach can be easily adopted for project assignments in introductory…
We begin our analysis with the study of two collections of lattice paths in the plane, denoted $\mathcal{D}_{[n,i,j]}$ and $\mathcal{P}_{[n,i,j]}$. These paths consist of sequences of $n$ steps, where each step allows movement in three…
Starting from a Langevin description of active particles that move with constant speed in infinite two-dimensional space and its corresponding Fokker-Planck equation, we develop a systematic method that allows us to obtain the…
We construct an effective potential for the complex Langevin equation on a lattice. We show that the minimum of this effective potential gives the space-time and Langevin time average of the complex Langevin field. The loop expansion of the…
These short notes present to the reader (students, in particular) a concise approach to the derivation of the propagator of Hamiltonians with position-dependent kinetic energy. The formalism is applied to the von Roos Hamiltonian with…
We study finite-dimensional integrals in a way that elucidates the mathematical meaning behind the formal manipulations of path integrals occurring in quantum field theory. This involves a proper understanding of how Wick's theorem allows…
In this paper we develop a quantization method for flat compact manifolds based on path integrals. In this method the Hilbert space of holomorphic functions in the complexification of the manifold is used. This space is a reproducing kernel…
Extensions to the trapezoidal rule using derivative information are studied for periodic integrands and integrals along the entire real line. Integrands which are analytic within a half plane or within a strip containing the path of…
We consider flows of ordinary differential equations (ODEs) driven by path differentiable vector fields. Path differentiable functions constitute a proper subclass of Lipschitz functions which admit conservative gradients, a notion of…
We introduce a model for stochastic transport on a one-dimensional substrate with particles assuming different conformations during their stepping cycles. These conformations correspond to different footprints on the substrate: in order to…