Related papers: On the two conjectures

We conjecture two combinatorial interpretations for the symmetric function $\Delta_{e_k} e_n$, where $\Delta_f$ is an eigenoperator for the modified Macdonald polynomials defined by Bergeron, Garsia, Haiman, and Tesler. Both interpretations…

In 1952, Littlewood stated a conjecture about the average growth of spherical derivatives of polynomials, and showed that it would imply that for entire function of finite order, "most" preimages of almost all points are concentrated in a…

Let $f:\mathbb{N}\to \mathbb{D}$ be a multiplicative function. Under the merely necessary assumption that $f$ is non-pretentious (in the sense of Granville and Soundararajan), we show that for any pair of distinct integer shifts $h_1,h_2$…

We introduce matching functions as a means of summing heavy-quark logarithms to any order. Our analysis is based on Witten's approach, where heavy quarks are decoupled one at a time in a mass-independent renormalization scheme. The outcome…

We show that the Generalized Vanishing Conjecture $$\forall_{m \ge 1} [\Lam^m f^m = 0] \Longrightarrow \forall_{m \gg 0} [\Lam^m (g f^m) = 0]$$ for a fixed differential operator $\Lam \in k[\partial]$ follows from a special case of it,…

We investigate possible quantifications of strictly singular operators, $l_{p}$-strictly singular operators, $c_{0}$-strictly singular operators, strictly cosingular operators, $l_{p}$-strictly cosingular operators. We prove quantitative,…

We obtain the off-diagonal Muckenhoupt-Wheeden conjecture for Calder\'on-Zygmund operators. Namely, given $1<p<q<\infty$ and a pair of weights $(u,v)$, if the Hardy-Littlewood maximal function satisfies the following two weight…

Let $F$ be a totally real number field, $p$ a rational prime, and $\chi$ a finite order totally odd abelian character of Gal$(\bar{F}/F)$ such that $\chi(\mathfrak{p})=1$ for some $\mathfrak{p}|p$. Motivated by a conjecture of Stark, Gross…

This article proves a conjecture by Zuber about the enumeration of fully packed loops (FPLs). The conjecture states that the number of FPLs whose link pattern consists of two noncrossing matchings which are separated by $m$ nested arches is…

We develop techniques to deal with monotonicity of sequences z_{n+1}/z_n and \sqrt[n]{z_n}. A series of conjectures of Zhi-Wei Sun and of Amdeberhan et al. are verified in certain unified approaches.

We prove some results related to a conjecture of Hivert and Thi\'ery about the dimension of the space of q-harmonics. In the process we compute the actions of the involved operators on symmetric and alternating functions, which have some…

In this note, it is proved the existence of an infinitely generated multiplicative group consisting of entire functions that are, except for the constant function 1, hypercyclic with respect to the convolution operator associated to a given…

In this article, we conduct a study of integral operators defined in terms of non-convolution type kernels with singularities of various degrees. The operators that fall within our scope of research include fractional integrals, fractional…

We prove the existence of common hypercyclic, entire functions for certain families of translation operators.

In 2007, Dmytrenko, Lazebnik and Williford posed two related conjectures about polynomials over finite fields. Conjecture~1 is a claim about the uniqueness of certain monomial graphs. Conjecture~2, which implies Conjecture~1, deals with…

At the end of 1960's, Lawrence Zalcman posed a conjecture that the coefficients of univalent functions $f(z) = z + \sum\limits_2^\infty a_n z^n$ on the unit disk satisfy the sharp inequality $|a_n^2 - a_{2n-1}| \le (n-1)^2$, with equality…

The Uniform convergence of double Fourier-Legendre series of function of bounded Harmonic variation and bounded partial $\Lambda $-variation are investigated.

This article studies the rearrangement problem for Fourier series introduced by P.L. Ulyanov, who conjectured that every continuous function on the torus admits a rearrangement of its Fourier coefficients such that the rearranged partial…

We note that the Fubini theorem may be used to prove that an $L^1$ function is determined by its Fourier coefficients.

We resolve a question of Bank and Laine on the zeros of solutions of $w'' + Aw = 0$ where $A$ is an entire function of finite order.