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We obtain an explicit simple formula for the coefficients of the asymptotic expansion for the factorial of a natural number,in terms of derivatives of powers of an elementary function. The unique explicit expression for the coefficients…
We have established a coherent framework for applying variational methods to partial differential equations on hypergraphs, which includes the propositions of calculus and function spaces on hypergraphs. Several results related to the…
Regularization plays a key role in a variety of optimization formulations of inverse problems. A recurring theme in regularization approaches is the selection of regularization parameters, and their effect on the solution and on the optimal…
In this paper we propose a general method to derive an upper bound for the contraction rate of the posterior distribution for nonparametric inverse problems. We present a general theorem that allows us to derive con- traction rates for the…
We develop closed-form expansions for the implied volatility of VIX options within the class of forward variance models. Our approach builds on weak-approximation techniques for VIX option prices and yields explicit implied volatility…
Explicit representations of the eigenvalues of the peridynamic operator have been recently derived in [5]. These representations are given in terms of generalized hypergeometric functions. Asymptotic analysis of the hypergeometric functions…
Vector autoregressions (VARs) are a widely used tool for modelling multivariate time-series. It is common to assume a VAR is stationary; this can be enforced by imposing the stationarity condition which restricts the parameter space of the…
In recent work, we derived a direct expression for one-loop tensor reduction using generating functions and Feynman parametrization in projective space, avoiding recursive relations. However, for practical applications, this expression…
Given a directed acyclic graph $G,$ and a set of values $y$ on the vertices, the Isotonic Regression of $y$ is a vector $x$ that respects the partial order described by $G,$ and minimizes $||x-y||,$ for a specified norm. This paper gives…
We extend first-order logic to include variadic function symbols, and prove a substitution lemma. Two applications are given: one to bounded quantifier elimination and one to the definability of certain Borel sets.
The chapter presents mathematical models intended for creating a topological drawing of a non-separable non-planar graph based on the methods of G. Ringel's vertex rotation theory. The induced system of cycles generates a topological…
Explicit formulae for Weber-Schafheitlin's type integrals with exponent 1 are derived. The results of these integrals are distributions on R_+.
This paper introduces a notion of decompositions of integral varifolds into countably many integral varifolds, and the existence of such decomposition of integral varifolds whose first variation is representable by integration is…
The spectral decomposition for an explicit second-order differential operator $T$ is determined. The spectrum consists of a continuous part with multiplicity two, a continuous part with multiplicity one, and a finite discrete part with…
We consider estimation procedures which are recursive in the sense that each successive estimator is obtained from the previous one by a simple adjustment. We propose a wide class of recursive estimation procedures for the general…
First-order automatic differentiation is a ubiquitous tool across statistics, machine learning, and computer science. Higher-order implementations of automatic differentiation, however, have yet to realize the same utility. In this paper I…
The relationship between Feynman diagrams and hypergeometric functions is discussed. Special attention is devoted to existing techniques for the construction of the $\epsilon$-expansion. As an example, we present a detailed discussion of…
Indicator variograms and madograms are structural tools used in many disciplines of the natural sciences and engineering to describe random sets and random fields. To date, several necessary conditions are known for a function to be a valid…
This pedagogical document explains three variational representations that are useful when comparing the efficiencies of reversible Markov chains: (i) the Dirichlet form and the associated variational representations of the spectral gaps;…
We consider the class $\mathcal{E}_t(Y)$ of Appell polynomials whose generating function is given by means of a real power $t$ of the moment generating function of a certain random variable $Y$. For such polynomials, we obtain explicit…