Related papers: On a Conjecture of Bahri-Xu

We prove that the known sufficient conditions on the real parameters $(p,q)$ for which the matrix power mean inequality $((A^p+B^p)/2)^{1/p}\le((A^q+B^q)/2)^{1/q}$ holds for every pair of matrices $A,B>0$ are indeed best possible. The proof…

In this paper, we show how to reduce the Penrose conjecture to the known Riemannian Penrose inequality case whenever certain geometrically motivated systems of equations can be solved. Whether or not these special systems of equations have…

In 2020, Kang and Park conjectured a "level $2$" Alder-type partition inequality which encompasses the second Rogers-Ramanujan Identity. Duncan, Khunger, the fourth author, and Tamura proved Kang and Park's conjecture for all but finitely…

The function $Q(x):=\sum_{n\ge 1} (1/n) \sin(x/n)$ was introduced by Hardy and Littlewood [10] in their study of Lambert summability, and since then it has attracted attention of many researchers. In particular, this function has made a…

Let $\Gamma\subset\mathbb{Q}^*$ be a finitely generated subgroup and let $p$ be a prime such that the reduction group $\Gamma_p$ is a well defined subgroup of the multiplicative group $\mathbb{F}_p^*$. We prove an asymptotic formula for the…

A conjecture of Khang Tran [6] claims that for an arbitrary pair of polynomials $A(z)$ and $B(z)$, every zero of every polynomial in the sequence $\{P_n(z)\}_{n=1}^\infty$ satisfying the three-term recurrence relation of length $k$…

We provide conjectural necessary and (separately) sufficient conditions for the Hilbert scheme of points of a given length to have the maximum dimension tangent space at a point. The sufficient condition is claimed for 3D and reduces the…

We give a systematic method of providing numerical evidence for higher order Stark-type conjectures such as (in chronological order) Stark's conjecture over $\mathbb{Q}$, Rubin's conjecture, Popescu's conjecture, and a conjecture due to…

Let f be a non-constant meromorphic function and a = a(z) be a small function of f. Under certain essential conditions, we obtained similar type conclusion of Bruck Conjecture, when f and its differential polynomial P[f] shares a with…

In this paper we use algebraic curves and other algebraic number theory methods to show the validity of a permutation polynomial conjecture regarding $f(X)=X^{q(p-1)+1} +\alpha X^{pq}+X^{q+p-1}$, on finite fields $\mathbb{F}_{q^2}, q=p^k$,…

We prove the following conjecture, due to Tomaszewski (1986): Let $X= \sum_{i=1}^{n} a_{i} x_{i}$, where $\sum_i a_i^2=1$ and each $x_i$ is a uniformly random sign. Then $\Pr[|X|\leq 1] \geq 1/2$. Our main novel tools are local…

To verify the universal validity of the "two-sided" monotonicity condition introduced in [8], we will apply it to include more classical examples. The present paper selects the $L^{p}$ convergence case for this purpose. Furthermore, Theorem…

We study the Lane-Emden conjecture, which asserts the non-existence of non-trivial, non-negative solutions to the Lane-Emden system \[ -\Delta u = v^p, \quad -\Delta v = u^q, \quad x \in \mathbb{R}^n\] in the subcritical regime. By…

Eisenbud and Harris conjectured in 1982 that an algebraic curve of high genus lies on a surface of low degree (which they proved for curves of very large degree). They observed constraints on the Hilbert function of a general hyperplane…

A sharp estimation of the $L^p$-norms of some matrix coefficients of the square integrable representations is conjectured. The conjecture can be proved for integer values of $p$ using a result of J. Burbea.

We study the existence of non--trivial solutions to the Yamabe equation: $$-\Delta u+ a(x)= \mu u|u|^\frac4{n-2} \hbox{} \mu >0, x\in \Omega \subset {\mathbf R}^n \hbox{with} n\geq 4,$$ $$ u(x)=0 \hbox{on} \partial \Omega$$ under weak…

A famous conjecture bearing the name of Vandiver states that $p \nmid h_p^+$ in the $p$ - cyclotomic extension of $\Q$. Heuristics arguments of Washington, which have been briefly exposed in Lang (1978), p. 261 and Washington (1996), p. 158…

A famous conjecture of Graham asserts that every set $A \subseteq \mathbb{Z}_p \setminus \{0\}$ can be ordered so that all partial sums are distinct. Although this conjecture was recently proved for sufficiently large primes by Pham and…

In this paper, we generalize the Cosmetic Surgery Conjecture to an $n$-cusped hyperbolic $3$-manifold and prove it under the assumption of another well-known conjecture in number theory, so called the Zilber-Pink Conjecture. For $n=1$ and…

We say that a cyclotomic polynomial \Phi_{n}(x) has order three if n is the product of three distinct primes, p<q<r. Let A(n) be the largest absolute value of a coefficient of \Phi_{n}(x) and M(p) be the maximum of A(pqr). In 1968, Sister…