Related papers: Ordinary and logarithmical convexity of moment gen…
We present an overview of the role of generating functions in quantum mechanical contexts, mainly in the modern theory of polarization and in the study of quantum phase transitions. Generating functions enable the derivation of moments and…
In the paper, the authors review origins, motivations, and generalizations of a series of inequalities involving several exponential functions and sums, establish three new inequalities involving finite exponential functions and sums by…
We consider distributions on a closed compact manifold $M$ as maps on smoothing operators. Thus spaces of certain maps between $\Psi^{-\infty}(M)\to \mathcal{C}^{\infty}(M)$ are considered as generalized functions. For any collection of…
Lagrangian averaging theories, most notably the Generalised Lagrangian Mean (GLM) theory of Andrews & McIntyre (1978), have been primarily developed in Euclidean space and Cartesian coordinates. We re-interpret these theories using a…
Random tensors are the natural generalization of random matrices to higher order objects. They provide generating functions for random geometries and, assuming some familiarity with random matrix theory and quantum field theory, we discuss…
A general explicit form for generating functions for approximating fractional derivatives is derived. To achieve this, an equivalent characterisation for consistency and order of approximations established on a general generating function…
Order statistics find applications in various areas of communications and signal processing. In this paper, we introduce an unified analytical framework to determine the joint statistics of partial sums of ordered random variables (RVs).…
We investigate quantitative implications of the notion of log-concavity through a probabilistic interpretation. In particular, we derive concentration inequalities, moment and entropy bounds for random variables satisfying a precise degree…
We investigate several families of multiple orthogonal polynomials associated with weights for which the moment generating functions are hypergeometric series with slightly varying parameters. The weights are supported on the unit interval,…
We study semi-martingale obliquely reflected Brownian motion with drift in the first quadrant of the plane in the transient case. Our main result determines a general explicit integral expression for the moment generating function of…
The purpose of this note is to provide an expository introduction to some more curious integral formulas and transformations involving generating functions. We seek to generalize these results and integral representations which effectively…
We provide a generating functional for the gravitational field, associated to the relaxation of the primary constraints as extended to the quantum sector. This requirement of the theory, relies on the assumption that a suitable time…
We consider a sub-critical Gaussian multiplicative chaos (GMC) measure defined on the unit interval [0,1] and prove an exact formula for the fractional moments of the total mass of this measure. Our formula includes the case where…
We propose a new class of convex penalty functions, called \emph{variational Gram functions} (VGFs), that can promote pairwise relations, such as orthogonality, among a set of vectors in a vector space. These functions can serve as…
It is well-known that a random variable, i.e., a function defined on a probability space, with values in a Borel space, can be represented on the special probability space consisting of the unit interval with Lebesgue measure. We show an…
Lagrange multipliers are present in any gauge theory. They possess peculiar gauge transformation which is not generated by the constraints in the model as it is the case with the other variables. For rank one gauge theories we show how to…
We demonstrate the versatility of mean-field games (MFGs) as a mathematical framework for explaining, enhancing, and designing generative models. In generative flows, a Lagrangian formulation is used where each particle (generated sample)…
A Lagrangian depending on geometric variables (metric, affine connection, gauge group generators) is given which maintains compatibility with General Relativity. It generates the dynamics for Electromagnetism and other Gauge Fields along…
A unique set of correction functions for Flux Reconstruction is presented, with there derivation stemming from proving the existence of energy stability in the Lebesgue norm. The set is shown to be incredibly arbitrary with the only union…
We define two-parameter generalizations of Andrews' $(k+1)$-marked odd Durfee symbols and $2k$th symmetrized odd rank moments, and study the automorphic properties of some of their generating functions. When $k=0$ we obtain families of…