Related papers: On a Conjecture of Bahri-Xu

We prove Manin's conjecture for a singular cubic surface S with a singularity of type E6. If U is the open subset of S obtained by deleting the unique line from S, then the number of rational points in U with anticanonical height bounded by…

In this paper, we propose the following conjecture which generalizes a theorem proved by Huang [Hua19] in his recent breakthrough proof of the sensitivity conjecture. We conjecture that for any Cayley graph $X = \Gamma(G,S)$ on a group $G$…

We obtain similar types of conclusions as that of Br\"{u}ck [1] for two differential polynomials which in turn radically improve and generalize several existing results. Moreover, a number of examples have been exhibited to justify the…

In 2005, B\"ottcher and Wenzel raised the conjecture that if $X,Y$ are real square matrices, then $||XY-YX||^2\leq 2||X||^2||Y||^2$, where $||\cdot||$ is the Frobenius norm. Various proofs of this conjecture were found in the last few years…

We propose a conjecture for exponential sums which generalizes both a conjecture by Igusa and a local variant by Denef and Sperber, in particular, it is without the homogeneity condition on the polynomial in the phase, and with new…

We prove a generalized Faulhaber inequality to bound the sums of the $j$-th powers of the first $n$ (possibly shifted) natural numbers. With the help of this inequality we are able to improve the known bounds for bracketing numbers of…

We consider probability mass functions $V$ supported on the positive integers using arguments introduced by Caputo, Dai Pra and Posta, based on a Bakry--\'{E}mery condition for a Markov birth and death operator with invariant measure $V$.…

Let $G$ be an arbitrary finite group and fix a prime number $p$. The McKay conjecture asserts that $G$ and the normalizer in $G$ of a Sylow $p$-subgroup have equal numbers of irreducible characters with degrees not divisible by $p$. The…

The Leopoldt conjecture is concerned with the image of the global units in the local units at the primes dividing p. In the definition of the global units the infinite place is distinguished. Exchanging p and infinity in the formulation one…

In this paper we show examples for applications of the Bombieri-Lang conjecture in additive combinatorics, giving bounds on the cardinality of sumsets of squares and higher powers of integers. Using similar methods we give bounds on the…

Let $G=(V,E)$ be a finite connected weighted graph, and assume $1\leq\alpha\leq p\leq q$. In this paper, we consider the following $p$-th Yamabe type equation $$-\Delta_pu+hu^{q-1}=\lambda fu^{\alpha-1}.$$ on $G$, where $\Delta_p$ is the…

We establish a general transference principle for the irrationality measure of points with $\mathbb{Q}$-linearly independent coordinates in $\mathbb{R}^{n+1}$, for any given integer $n\geq 1$. On this basis, we recover an important…

We resolve the 1978 Brian\c{c}on-Iarrobino Conjecture regarding the maximum singularity of $\mathcal{H}=\mathrm{Hilb}^{l}(\mathbb{A}^3)$, where $l$ is a tetrahedral number, by refining the work of Ramkumar-Sammartano in…

Remarkable progress has been made in recent years to establish log-Sobolev type inequalities under the assumption of discrete Ricci curvature bounds. More specfically, Salez and Youssef have proven that the log-Sobolev constant can be lower…

We argue that the conclusion about the invalidity of the Dirac conjecture, made in the paper by Wang, Li, and Wang (Int. J. Theor. Phys. 48 1894, 2009), was based upon a flawed analysis of the proposed counterexamples. In the case of the…

We investigate the analogue of the Andr\'e--Pink--Zannier conjecture in characteristic $p$. Precisely, we prove it for ordinary function field-valued points with big monodromy, in Shimura varieties of Hodge type. We also prove an algebraic…

This note establishes that the opposite Gaussian product inequality (GPI) of the type proved by Russell & Sun (2022a) in two dimensions, and partially extended to higher dimensions by Zhou et al. (2024), continues to hold for an arbitrary…

We give a condition for certain divisors on the blow up of P^3 at points in general position to be ample. The result extends a theorem of G. Xu on the blow up of the projective plane.

An assignment problem is the optimization problem of finding, in an m by n matrix of nonnegative real numbers, k entries, no two in the same row or column, such that their sum is minimal. Such an optimization problem is called a random…

Generalized $m$-gonal numbers are those $p_m(x)= [ (m - 2)x^2 - (m - 4)x ]/2 $ where $x$ and $m$ are integers with $m \geq 3$. If any nonnegative integer can be written in the form $ap_r(h)+bp_s(l)+cp_t(m)+dp_u(n)$, where $a,b,c,d$ are…