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Functional integrals are central to modern theories ranging from quantum mechanics and statistical thermodynamics to biology, chemistry, and finance. In this work we present a new method for calculating functional integrals based on a…
Field-theoretic construction of functional representations of solutions of stochastic differential equations and master equations is reviewed. A generic expression for the generating function of Green functions of stochastic systems is put…
In this short survey article, we showcase a number of non-trivial geometric problems that have recently been resolved by marrying methods from functional calculus and real-variable harmonic analysis. We give a brief description of these…
Estimating the coefficient functionals on various classes of holomorphic functions traditionally forms an important field of geometric complex analysis and its mathematical and physical applications. These coefficients reflect fundamental…
In this paper, we determine the almost sure multifractal spectrum of a class of random functions constructed as sums of pulses with random dilations and translations. In addition, the continuity modulii of these functions is investigated.
This article handles in a short manner a few Laplace transform pairs and some extensions to the basic equations are developed. They can be applied to a wide variety of functions in order to find the Laplace transform or its inverse when…
We review some of the basic mathematical results about density functional theory.
In this article we extend the framework of execution of concurrent functions on different abstract levels from previous work with communication between the concurrent functions. We classify the communications and identify problems that can…
The meromorphic functional calculus developed in Part I overcomes the nondiagonalizability of linear operators that arises often in the temporal evolution of complex systems and is generic to the metadynamics of predicting their behavior.…
The goal of this paper is to show that there exists a simple, yet universal statistical logic of spectral graph analysis by recasting it into a nonparametric function estimation problem. The prescribed viewpoint appears to be good enough to…
In this course of lectures we give an account of the growth theory of subharmonic functions, which is directed towards its applications to entire functions of one and several complex variables.
An analytical approach to convolution of functions, which appear in perturbative calculations, is discussed. An extended list of integrals is presented.
In stochastic analysis, the flow of information through time is typically modelled using a filtration. We introduce some of the basic ideas involving enlargements of filtration. Here, we focus mainly on initial enlargements, where a given…
Analytical solutions are constructed for an assembly of any finite number of bubbles in steady motion in a Hele-Shaw channel. The solutions are given in the form of a conformal mapping from a bounded multiply connected circular domain to…
Fractal percolation exhibits a dramatic topological phase transition, changing abruptly from a dust-like set to a system spanning cluster. The transition points are unknown and difficult to estimate. In many classical percolation models the…
The Bayesian approach to data analysis provides a powerful way to handle uncertainty in all observations, model parameters, and model structure using probability theory. Probabilistic programming languages make it easier to specify and fit…
We consider bifurcation of critical points from a trivial branch for families of functionals that are invariant under the orthogonal action of a compact Lie group. Based on a recent construction of an equivariant spectral flow by the…
Functionals are an important research subject in Mathematics and Computer Science as well as a challenge in Information Technologies where the current programming paradigm states that only symbolic computations are possible on higher order…
With the goal of providing the foundations for a rigorous study of modules of bicomplex holomorphic functions, we develop a general theory of functional analysis with bicomplex scalars. Even though the basic properties of bicomplex number…
Spectral functions of symmetric matrices -- those depending on matrices only through their eigenvalues -- appear often in optimization. A cornerstone variational analytic tool for studying such functions is a formula relating their…