arXiv subjects

Tanay Wakhare

Publications and source records attributed to Tanay Wakhare.

At least 19 recordsLinked to original sources

arXiv2026

Graph Eigenvalues and Projection Constants

For an integer $k\ge2$, let $λ_k(G)$ denote the $k$th largest adjacency eigenvalue of a graph $G$. For every graph $G$ on $n$ vertices and every $2 \leq k \leq n$, we prove \[ λ_k(G) \le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1. \] Our bound is tight for $k\in\{2,3,4,8,24\}$. We obtain it by reducing the graph-eigenvalue problem to an extremal problem for orthogonal projections and then applying the general upper bound on the absolute projection constant $γ(r)$ due to Deręgowska and Lewandowska. We also give an alternative proof of their bound by repairing the Gegenbauer-polynomial argument of König and Tomczak-Jaegermann. The resulting slack identity yields a strict improvement in every even dimension $r\ge4$ for which $r+2$ is not a perfect square.

math.CO↗

arXiv2026

Tadpole Nahm sum as a Wronskian

We show that the tadpole Nahm sum is essentially a specialization of a series studied by Bartlett and Warnaar. This overlooked connection was first discovered by the internal model mathvision-harness-0.3 at MathVision AI. We then use Macdonald-type identities for the Bartlett-Warnaar series to express the principal tadpole Nahm sum and its twisted version in terms of a Wronskian of generalized theta series. The principal case confirms a conjecture of Milas and Wang.

math.NT↗

arXiv2026

On conjectural fermionic formulas for the Macdonald index in Argyres-Douglas theories

We prove a fermionic-bosonic duality relation for the Macdonald index in Argyres-Douglas theories of type $(A_1, D_{2k+1})$, thereby yielding a conjectural fermionic formula due to Andrews et al. Our duality is built upon a new conjugate Bailey pair to be established using techniques from orthogonal polynomials and basic hypergeometric series. In addition, this fermionic formula implies another sum-like expression independently conjectured by Andrews et al. and Kim et al. for the same Macdonald index.

math.CO↗

arXiv2026

Graph Eigenvalues and Projection Constants

Let $λ_1(G)\ge λ_2(G)\ge \cdots \ge λ_n(G)$ denote the adjacency eigenvalues of a graph $G$ of order $n$. We prove that for every $k\geq 2$ and every graph $G$ on $n\geq k$ vertices, $$ λ_k(G)\le \frac{λ_{\mathbb{R}}(k-1)}{2(k-1)}\,n-1, $$ where $$ λ_{\mathbb{R}}(r)=\sup_{N\ge r}\frac1N \max_{Q\in \mathcal P_r(N)}\sum_{i,j=1}^N |q_{ij}| $$ and $\mathcal P_r(N)$ denotes the set of rank-$r$ orthogonal projections in $\mathbb{R}^{N\times N}$. In Banach space theory, $λ_{\mathbb{R}}(r)$ is well known as the maximal absolute projection constant, which has been shown to equal the quasimaximal absolute projection constant $μ_{\mathbb{R}}(r)$. This yields a new conceptual connection: universal upper bounds on $λ_k(G)$ are controlled by the real maximal absolute projection constant $λ_{\mathbb{R}}(k-1)$. In dimensions where $λ_{\mathbb{R}}(k-1)$ is known explicitly, this gives explicit coefficients. In particular, for $k=3$ this recovers Tang's recent sharp bound $λ_3(G)\le n/3-1$. For $k=4$, using $λ_{\mathbb{R}}(3)=\frac{1+\sqrt5}{2}$ together with Linz's closed blowups of the icosahedral graph, we obtain the result $$ λ_4(G) \leq \frac{1+\sqrt5}{12}n-1. $$ The method allows us to transfer known upper bounds on $λ_{\mathbb{R}}(k-1)$ to match the best known upper bounds on $λ_k(G)$ for other values of $k$, such as $k=5$.

math.CO↗

arXiv2025

A divisor generating $q$-series and cumulants arising from random graphs

Uchimura, in 1987, introduced a probability generating function for a random variable $X$ and using properties of this function he discovered an interesting $q$-series identity. He further showed that the $m$-th cumulant with respect to the random variable $X$ is nothing but the generating function for the generalized divisor function $σ_{m-1}(n)$. Simon, Crippa, and Collenberg, in 1993, explored the $G_{n,p}$-model of a random acyclic digraph and defined a random variable $γ_n^{*}(1)$. Quite interestingly, they found links between limit of its mean and the generating function for the divisor function $d(n)$. Later in 1997, Andrews, Crippa and Simon extended these results using $q$-series techniques. They calculated limit of the mean and variance of the random variable $γ_n^{*}(1)$ which correspond to the first and second cumulants. In this paper, we generalize the result of Andrews, Crippa and Simon by calculating limit of the $t$-th cumulant in terms of the generalized divisor function. Furthermore, we also discover limit forms for identities of Uchimura and Dilcher. This provides a fourth side to the Uchimura-Ramanujan-divisor type three way partition identities expounded by the first four authors recently.

math.NT↗

arXiv2025

Romik's Conjecture for the Jacobi Theta Function

Dan Romik recently considered the Taylor coefficients of the Jacobi theta function around the complex multiplication point $i$. He then conjectured that the Taylor coefficients $d(n)$ either vanish or are periodic modulo any prime ${p}$; this was proved by the combined efforts of Scherer and Guerzhoy-Mertens-Rolen, who considered arbitrary half integral weight modular forms. We refine previous work for $p \equiv 1 \pmod{4}$ by displaying a concise algebraic relation between $d\left( n+ \frac{p-1}{2} \right)$ and $d(n)$ related to the $p$-adic factorial, from which we can deduce periodicity with an effective period.

math.NT↗

arXiv2025

Iterated Entropy Derivatives and Binary Entropy Inequalities

We embark on a systematic study of the $(k+1)$-th derivative of $x^{k-r}H(x^r)$, where $H(x):=-x\log x-(1-x)\log(1-x)$ is the binary entropy and $k>r\geq 1$ are integers. Our motivation is the conjectural entropy inequality $α_k H(x^k)\geq x^{k-1}H(x)$, where $0<α_k<1$ is given by a functional equation. The $k=2$ case was the key technical tool driving recent breakthroughs on the union-closed sets conjecture. We express $ \frac{d^{k+1}}{dx^{k+1}}x^{k-r}H(x^r)$ as a rational function, an infinite series, and a sum over generalized Stirling numbers. This allows us to reduce the proof of the entropy inequality for real $k$ to showing that an associated polynomial has only two real roots in the interval $(0,1)$, which also allows us to prove the inequality for fractional exponents such as $k=3/2$. The proof suggests a new framework for proving tight inequalities for the sum of polynomials times the logarithms of polynomials, which converts the inequality into a statement about the real roots of a simpler associated polynomial.

cs.IT↗

arXiv2024

Prompts have evil twins

We discover that many natural-language prompts can be replaced by corresponding prompts that are unintelligible to humans but that provably elicit similar behavior in language models. We call these prompts "evil twins" because they are obfuscated and uninterpretable (evil), but at the same time mimic the functionality of the original natural-language prompts (twins). Remarkably, evil twins transfer between models. We find these prompts by solving a maximum-likelihood problem which has applications of independent interest.

cs.CL↗

arXiv2022

The Inverse Eigenvalue Problem for Linear Trees

We prove the sufficiency of the Linear Superposition Principle for linear trees, which characterizes the spectra achievable by a real symmetric matrix whose underlying graph is a linear tree. The necessity was previously proven in 2014. This is the most general class of trees for which the inverse eigenvalue problem has been solved. We explore many consequences, including the Degree Conjecture for possible spectra, upper bounds for the minimum number of eigenvalues of multiplicity $1$, and the equality of the diameter of a linear tree and its minimum number of distinct eigenvalues, etc.

math.SP↗

arXiv2022

Special classes of $q$-bracket operators

We study the $q$-bracket operator of Bloch and Okounkov when applied to $f(λ)=\sum_{λ_i \in λ}g(λ_i)$ and $f(λ)=\sum_{\substack{λ_i \in λλ_i \text{distinct} }}g(λ_i)$. We use these expansions to derive convolution identities for the functions $f$ and link both classes of $q$-brackets through divisor sums. As a result, we generalize Euler's classic convolution identity for the partition function and obtain an analogous identity for the totient function. As corollaries, we generalize Stanley's theorem as well as provide several new combinatorial results.

math.CO↗

arXiv2022

Dirichlet Series Under Standard Convolutions: Variations on Ramanujan's Identity for Odd Zeta Values

Inspired by a famous identity of Ramanujan, we propose a general formula linearizing the convolution of Dirichlet series as the sum of Dirichlet series with modified weights; its specialization produces new identities and recovers several identities derived earlier in the literature, such as the convolution of squares of Bernoulli numbers by A. Dixit and collaborators, or the convolution of Bernoulli numbers by Y. Komori and collaborators.

math.NT↗

arXiv2021

The Toughness of Kneser Graphs

The \textit{toughness} $t(G)$ of a graph $G$ is a measure of its connectivity that is closely related to Hamiltonicity. Brouwer proved the lower bound $t(G) > \ell / λ- 2$ on the toughness of any connected $\ell$-regular graph, where $ λ$ is the largest nontrivial eigenvalue of the adjacency matrix. He conjectured that this lower bound can be improved to $\ell / λ-1$ and this conjecture is still open. Brouwer also observed that many families of graphs (in particular, those achieving equality in the Hoffman ratio bound for the independence number) have toughness exactly $\ell / λ$. Cioabă and Wong confirmed Brouwer's observation for several families of graphs, including Kneser graphs $K(n,2)$ and their complements, with the exception of the Petersen graph $K(5,2)$. In this paper, we extend these results and determine the toughness of Kneser graphs $K(n,k)$ when $k\in \{3,4\}$ and $n\geq 2k+1$ as well as for $k\geq 5$ and sufficiently large $n$ (in terms of $k$). In all these cases, the toughness is attained by the complement of a maximum independent set and we conjecture that this is the case for any $k\geq 5$ and $n\geq 2k+1$.

math.CO↗

arXiv2021

Extremal Graphs for a Spectral Inequality on Edge-Disjoint Spanning Trees

Liu, Hong, Gu, and Lai proved if the second largest eigenvalue of the adjacency matrix of graph $G$ with minimum degree $δ\ge 2m+2 \ge 4$ satisfies $λ_2(G) < δ- \frac{2m+1}{δ+1}$, then $G$ contains at least $m+1$ edge-disjoint spanning trees, which verified a generalization of a conjecture by Cioabă and Wong. We show this bound is essentially the best possible by constructing $d$-regular graphs $\mathcal{G}_{m,d}$ for all $d \ge 2m+2 \ge 4$ with at most $m$ edge-disjoint spanning trees and $λ_2(\mathcal{G}_{m,d}) < d-\frac{2m+1}{d+3}$. As a corollary, we show that a spectral inequality on graph rigidity by Cioabă, Dewar, and Gu is essentially tight.

math.CO↗

arXiv2021

Structural properties of multiple zeta values

We study some classical identities for multiple zeta values and show that they still hold for zeta functions built on the zeros of an arbitrary function. We introduce the complementary zeta function of a system, which naturally occurs when lifting identities for multiple zeta values to identities for quasisymmetric functions.

math.NT↗

arXiv2020

The Proportion of Trees that are Linear

We study several enumeration problems connected to linear trees, a broad class which includes stars, paths, generalized stars, and caterpillars. We provide generating functions for counting the number of linear trees on $n$ vertices, characterize the asymptotic growth rate of the number of nonisomorphic linear trees, and show that the distribution of $k$-linear trees on $n$ vertices follows a central limit theorem.

math.CO↗

arXiv2020

Taylor coefficients of the Jacobi $θ_{3}\left( q \right)$ function

We extend some results recently obtained by Dan Romik about the Taylor coefficients of the theta function $θ_{3}\left(1\right)$ to the case $θ_{3}\left(q\right)$ of an arbitrary value of the elliptic modulus $k.$ These results are obtained by carefully studying the properties of the cumulants associated to a $θ_{3}$ (or discrete normal) distributed random variable. This article also states some congruence conjectures about integers sequences that generalize the one studied by D. Romik.

math.NT↗

arXiv2019

Analytic Continuation for Multiple Zeta Values using Symbolic Representations

We introduce a symbolic representation of $r$-fold harmonic sums at negative indices. This representation allows us to recover and extend some recent results by Duchamp et al., such as recurrence relations and generating functions for these sums. This approach is also applied to the study of the family of extended Bernoulli polynomials, which appear in the computation of harmonic sums at negative indices. It also allows us to reinterpret the Raabe analytic continuation of the multiple zeta function as both a constant term extension of Faulhaber's formula, and as the result of a natural renormalization procedure for Faulhaber's formula.

math.NT↗

arXiv2019

A Generalized Newton-Girard Identity

We present a generalization of the Newton-Girard identities, along with some applications. As an addendum, we collect many evaluations of symmetric polynomials to which these identities apply.

math.NT↗