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We establish several formulas relating periods of modular forms on quaternion algebras over number fields to special values of L-functions. Our main inputs are the cohomological techniques for working with periods introduced in [Mol21],…
Submodular functions are a fundamental object of study in combinatorial optimization, economics, machine learning, etc. and exhibit a rich combinatorial structure. Many subclasses of submodular functions have also been well studied and…
We give some results and conjectures about recurrence relations for certain sequences of binomial sums.
We show that indefinite theta series on cones converge and provide an explicit modular completion. Our completion rests on a convolution of the Gaussian with a piecewise constant function supported on the cone. Our main innovation is to…
In this article we present certain formulas involving arithmetical functions. In the first part we study properties of sums and product formulas for general type of arithmetic functions. In the second part we apply these formulas to the…
We introduce certain linear positive operators and study some approximation properties of these operators in the space of functions, continuous on a compact set, of two variables. We also find the order of this approximation by using…
In the theory of harmonic Maass forms and mock modular forms, mock theta functions are distinguished examples which arose from $q$-hypergeometric examples of Ramanujan. Recently, there has been a body of work on higher depth mock modular…
We extend the relation between quasi-modular forms and modular forms to a wider class of functions. We then relate both forms to vector-valued modular forms with symmetric power representations, and prove a general structure theorem for…
In this note we deduce well known modular identities for Jacobi theta functions using the spectral representations associated with the real valued Brownian motion taking values on $[-1,+1]$. We consider two cases: (i) reflection at $-1$ and…
The rational function approximation method, density functional theory, and NVT Monte Carlo simulation are used to obtain the density profiles of multicomponent hard-sphere mixtures near a planar hard wall. Binary mixtures with a size ratio…
Compounding submodular monotone (i.e. 2-alternating) set functions on a finite set preserves this property, as shown in 2010. A natural generalization to k-alternating functions was presented in 2018, however hardly readable because of page…
We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes.
In this paper we offer some new identities associated with mock theta functions and establish new Bailey pairs related to indefinite quadratic forms. We believe our proof is instructive use of changing base of Bailey pairs, and offers new…
We introduce meromorphic nearby cycle functors and study their functorial properties. Moreover we apply them to monodromies of meromorphic functions in various situations. Combinatorial descriptions of their reduced Hodge spectra and Jordan…
Ramanujan's last letter to Hardy explored the asymptotic properties of modular forms, as well as those of certain interesting $q$-series which he called \emph{mock theta functions}. For his mock theta function $f(q)$, he claimed that as $q$…
We consider a sequence of composite Bernstein operators and the quadrature formulae associated with them. Upper bounds for the approximation error of continuous functions and for the approximation of integrals of continuous functions are…
We recently formulated important Modular Bourgain-Tzafriri Restricted Invertibility Conjectures and Modular Johnson-Lindenstrauss Flattening Conjecture in the Appendix of \textit{[arXiv: 2207.12799.v1]}. For the sake of wide accessibility…
We verify a finiteness conjecture of Feit on sources of simple modules over group algebras for various classes of finite groups related to the symmetric groups.
In a recent paper, Merca posed three conjectures on congruences for specific convolutions of a sum of odd divisor functions with a generating function for generalized $m$-gonal numbers. Extending Merca's work, we complete the proof of these…
We offer some further applications of some Bailey pairs related to some mock theta functions which were established in a recent study. We discuss and offer some double-sum $q$-series, with new relationships among mock theta functions. We…