Related papers: Harmonic Analysis on the Positive Rationals II: Mu…
These lecture notes discuss several related features of the exactly solvable two-dimensional corner growth model with exponentially distributed weights. A key property of this model is the availability of a fairly explicit stationary…
We provide a simple and new induction based treatment of the problem of distinguishing cusp forms from the growth of the Fourier coefficients of modular forms. Our approach gives the best possible ranges of the weights for this problem, and…
The authors review results implicit in their recent paper [2] on the product/quotient representation of rationals by rationals of the type $( an + b )/ ( An+ B )$ and give a detailed account of a particular related non-intuitive…
Associated to each random variable $Y$ having a finite moment generating function, we introduce a different generalization of the Stirling numbers of the second kind. Some characterizations and specific examples of such generalized numbers…
Given noisy data, function estimation is considered when the unknown function is known apriori to consist of a small number of regions where the function is either convex or concave. When the regions are known apriori, the estimate is…
Auxiliary variable is extensively used in survey sampling to improve the precision of estimates. Whenever there is availability of auxiliary information, we want to utilize it in the method of estimation to obtain the most efficient…
Functional equations satisfied by additive functions have a special interest not only in the theory of functional equations, but also in the theory of (commutative) algebra because the fundamental notions such as derivations and…
We study the simultaneous sign change of Fourier coefficients of a pair of distinct normalized newforms of integral weight supported on primes power indices, we also prove some equidistribution results. Finally, we consider an analogous…
We study the regularity and the growth rates of solutions to two-dimensional Monge-Amp\`ere equations with the right-hand side exhibiting polynomial growth. Utilizing this analysis, we demonstrate that the translators for the flow by…
Let $f : \mathbf{N} \rightarrow \mathbf{C}$ be a bounded multiplicative function. Let $a$ be a fixed integer (say $a = 1$). Then $f$ is well-distributed on the progression $n \equiv a \pmod{q} \subset \{1,\dots, X\}$, for almost all primes…
In this paper we prove the mean values of some multiplicative functions connected with the divisor function on the short interval of summation.
This thesis studies modular forms from a classical and adelic viewpoint. We use this interplay to obtain results about the arithmetic of the Fourier coefficients of modular forms and their generalisations. In Chapter 2, we compute lower…
In this paper we consider two properties of variadic functions, namely associativity and preassociativity, that are pertaining to several data and language processing tasks. We propose parameterized relaxations of these properties and…
The article shows how two choices are possible whenever computing the geometric mean, and the repetition of this process can in general yield 2-to-the power N different values when the choices are compounded in the first N steps of…
We describe recent nonlinear analytic approximation tools in the classical setting of Hardy spaces in the upper half plane and show how to transfer them to the higher dimensional real setting of harmonic functions in upper half spaces. It…
Results involving various mean value properties are reviewed for harmonic, biharmonic and metaharmonic functions. It is also considered how the standard mean value property can be weakened to imply harmonicity and belonging to other classes…
Inspired by the recent works of Srivastava et al. (HMS-AKM-PG), Frasin and Aouf (BAF-MKA) and others (Ali-Ravi-Ma-Mina-class,Caglar-Orhan,Goyal-Goswami,Xu-HMS-AML,Xu-HMS-AMC), we propose to investigate the coefficient estimates for a…
Let $L$ be a degree-$2$ $L$-function associated to a Maass cusp form. We explore an algorithm that evaluates $t$ values of $L$ on the critical line in time $O(t^{1+\varepsilon})$. We use this algorithm to rigorously compute an abundance of…
In this paper, we describe the automorphic properties of the Fourier coefficients of meromorphic Jacobi forms. Extending results of Dabholkar, Murthy, and Zagier, and Bringmann and Folsom, we prove that the canonical Fourier coefficients of…
We evaluate binomial series with harmonic number coefficients, providing recursion relations, integral representations, and several examples. The results are of interest to analytic number theory, the analysis of algorithms, and…