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Research
Research
Research
228+ open-access research outputs.
Let $\mathcal{OP}_n$ be the monoid of all orientation-preserving full transformations on $X_n=\{1,\dots, n\}$ with the natural order. For $\alpha \in \mathcal{OP}_n$, let $F(\alpha)=\{y\in X_n: y\alph…
For a simple graph $G$ with $n$ vertices, let $A_G$ denote the adjacency matrix of $G$, and let $\lambda_1(G) \geq \lambda_2(G) \geq \dots \geq \lambda_n(G)$ be its eigenvalues. For an integer $p \geq…
We prove an asymptotic formula for the second moment of the first derivative of quadratic twists of modular $L$-functions with three leading order main terms. It improves the previous result of Kumar …
Let $\mathcal R_{n}$ be the set of all rational functions of the type $r(z) = f(z)/w(z)$, where $f(z)$ is a polynomial of degree at most $n$ and $w(z) = \prod_{j=1}^{n}(z-\beta_j)$, $|\beta_j|>1$ for …
In this paper, we study inequalities involving polynomials and quasimodular forms. More precisely, we focus on the monotonicity of the functions of the form $t \mapsto t^m F(it)$ where $F$ is a quasim…
For any modulus of continuity $\omega$ that fails the Osgood condition, we construct a divergence-free velocity field $v \in C_t C^\omega_x$ for which the associated ODE admits at least two distinct f…
As a consequence of the Littlewood-Richardson (LR) commuters coincidence and the Kumar-Torres branching model via Kushwaha-Raghavan-Viswanath flagged hives, we have solved the Lecouvey- -Lenart conjec…
This paper develops our previous works concerning the classical Peterson hit problem for the polynomial algebra on five variables over the mod--2 Steenrod algebra $\mathscr A$ in a generic family of d…
Glaisher's theorem states that the number of partitions of $n$ into parts which repeat at most $m-1$ times is equal to the number of partitions of $n$ into parts which are not divisible by $m$. The $m…
Recently, Choie and Kumar extensively studied the Herglotz-Zagier-Novikov function $\mathfrak{F}(z;u,v)$, defined as \begin{align*} \mathfrak{F}(z;u,v) = \int_{0}^{1} \frac{\log(1-ut^z)}{v^{-1}-t} dt,…
Let $f: X\to Y$ be a proper surjective morphism of varieties defined over an algebraically closed field of positive characteristic. We prove that if $f$ has geometrically connected fibers then the ind…
The main goal of this paper is to extend two fundamental combinatorial results in Schubert calculus on flag manifolds from equivariant cohomology and $K$-theory to equivariant elliptic cohomology. The…
We develop and analyze a general class of Euler-type numerical schemes for Levy-driven McKean-Vlasov stochastic differential equations (SDEs), where the drift, diffusion and jump coefficients grow sup…
The notion of local weak convergence, or Benjamini--Schramm convergence, was introduced by Benjamini and Schramm. The local weak limit of sparse Erd\H os--R\'enyi graphs is the Galton--Watson measure …
These notes address two problems. First, we investigate the question of ``how many'' are (in Baire sense) vector fields in $L^1_t L^q_x$, $q \in [1, \infty)$, for which existence and/or uniqueness of …
In this expository note, by using the Kostant-Kumar method, we prove the Poincar\'e duality of the elliptic classes associated to Schubert varieties.…
The Pappas-Rapoport coherence conjecture, proved by Zhu, states that the dimensions of spaces of sections of certain line bundles coincide. The two sides of the equality correspond to the line bundles…
In this article, the candidate optimal paths for a Dubins vehicle on a sphere are analytically derived. In particular, the arc angles for segments in $CGC$, $CCC$, $CCCC$, and $CCCCC$ paths, which hav…
Cohn and Kumar showed that the permutation on the set of the classes of left associated Hurwitz primes above an odd prime $p$ induced through metacommutation by a Hurwitz prime $\xi$ of norm $q$ has e…
Let $\overline{bt}(n)$ denote the number of overcubic partition triples of $n$. Nayaka, Dharmendra and Kumar proved some congruences modulo 8, 16 and 32 for $\overline{bt}(n)$. Recently, Saikia and Sa…
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