Related papers: Proving it is impossible; on Erd\H{o}s problem $\#…
We investigate the class of regular-ordered word equations. In such equations, each variable occurs at most once in each side and the order of the variables occurring in both sides is the preserved (the variables can be, however, separated…
The aim of this paper is to study negative classical solutions to a $k$-Hessian equation involving a nonlinearity with a general weight \begin{equation} \label{Eq:Ma:0} \tag{$P$} \begin{cases} S_k(D^2u)= \lambda \rho(|x|) (1-u)^q &\mbox{in…
We present an explicit subset $A\subseteq \mathbb{N} = \{0,1,\ldots\}$ such that $A + A = \mathbb{N}$ and for all $\varepsilon > 0$, \[\lim_{N\to \infty}\frac{\big|\big\{(n_1,n_2): n_1 + n_2 = N, (n_1,n_2)\in…
In this paper, we formulate and prove several variants of the Erd\H{o}s-Tur\'{a}n additive bases conjecture.
The Erd\"{o}s-Moser equation $ \sum_{i=1}^{m - 1} i^k = m^k $ is a longstanding challenge in number theory, with the only known integer solution being $ (k,m) = (1,3) $. Here, we investigate whether other solutions might exist by using the…
In this paper we study the existence of continuous solutions and their constructions for a second order iterative functional equation, which involves iterate of the unknown function and a nonlinear term. Imposing Lipschitz conditions to…
We prove that the power word problem for certain metabelian subgroups of $\mathsf{GL}(2,\mathbb{C})$ (including the solvable Baumslag-Solitar groups $\mathsf{BS}(1,q) = \langle a,t \mid t a t^{-1} = a^q \rangle$) belongs to the circuit…
In this paper, we study the well-posedness of Poisson-Nernst-Planck system with no-flux boundary condition and singular permanent charges in two dimension. The main difficulty comes from the lack of integrability of singular permanent…
We use recent results about linking the number of zeros on algebraic varieties over $\mathbb{C}$, defined by polynomials with integer coefficients, and on their reductions modulo sufficiently large primes to study congruences with products…
The Finiteness Problem is shown to be unsolvable for any sufficiently large class of modular lattices.
Wormhole solutions in gravitational theories typically require exotic matter. Here we present a wormhole solution to the field equations of Einsteinian Cubic Gravity -- a phenomenological competitor to general relativity that includes terms…
Erd\H{o}s first showed that the number of positive integers up to $x$ which can be written as a product of two number less than $\sqrt{x}$ has zero density. Ford then found the correct order of growth of the set of all these integers. We…
We show that for any sequence $f: {\bf N} \to \{-1,+1\}$ taking values in $\{-1,+1\}$, the discrepancy $$ \sup_{n,d \in {\bf N}} \left|\sum_{j=1}^n f(jd)\right| $$ of $f$ is infinite. This answers a question of Erd\H{o}s. In fact the…
This article proposes a unified analytical approach leading to a partial resolution of the Erdos-Straus, Sierpinski conjectures, and their generalization. We introduce an equivalent reformulation of these conjectures while constructing two…
Goedel Incompleteness Theorem leaves open a way around it, vaguely perceived for a long time but not clearly identified. (Thus, Goedel believed informal arguments can answer any math question.) Closing this loophole does not seem obvious…
We establish a variety of extensions to the Erdos-Rado Theorem, particularly involving ordinal numbers, and always involving ordinary partition relations. Most of the results can be regarded as consequences of the Ramification Principle,…
In this paper, we consider the Cauchy problem for the 3D Euler equations with the Coriolis force in the whole space. We first establish the local-in-time existence and uniqueness of solution to this system in $B^s_{p,r}(\R^3)$. Then we…
We consider non-linear electrodynamics (NED) minimally coupled to general relativity. We derive novel electrically charged, spherically symmetric, black-hole solutions having, for some set of parameters, all their NED fields (the electric…
This paper addresses some fundamental issues in nonconvex analysis. By using pure complementary energy principle proposed by the author, a class of fully nonlinear partial diforerential equations in nonlinear elasticity is able to converted…
How can a stack of identical blocks be arranged to extend beyond the edge of a table as far as possible? We consider a generalization of this classic puzzle to blocks that differ in width and mass. Despite the seemingly simple premise, we…