Related papers: Explicit solutions to the Oppenheim conjecture for…

Let $(q_{\alpha, n})_{n \geq 0}$ be the sequence of convergent denominators to the simple continued fraction expansion of $\alpha$. For certain specific choices of $\alpha$, this sequence is a Lehmer sequence. In this paper, we show that…

We present a number of results concerning infinite-exponent partition relations on linear orders of the form $\langle {}^\alpha 2,<_{\text{lex}}\rangle$ for $\alpha$ an ordinal, generalising the setting of the real line, working throughout…

We set up a dual variational framework to detect real standing wave solutions of the nonlinear Helmholtz equation $$ -\Delta u-k^2 u =Q(x)|u|^{p-2}u,\qquad u \in W^{2,p}(\mathbb{R}^N) $$ with $N\geq 3$, $\frac{2(N+1)}{(N-1)}<…

We prove that a finite dimensional algebra $\Lambda$ is $\tau-$tilting finite if and only if all the bricks over $\Lambda$ are finitely generated. This is obtained as a consequence of the existence of proper locally maximal torsion classes…

The theory of $(\mathbb{R},<,+,\mathbb{Z},\mathbb{Z} a)$ is decidable if $a$ is quadratic. If $a$ is the golden ratio, $(\mathbb{R},<,+,\mathbb{Z},\mathbb{Z} a)$ defines multiplication by $a$. The results are established by using the…

We prove that for any countable set $A$ of real numbers, the set of binary indefinite quadratic forms $Q$ such that the closure of $Q(\mathbb{Z}^2)$ is disjoint from $A$ has full Hausdorff dimension.

We study the bifurcation of solutions of semilinear elliptic boundary value problems of the form \begin{align*} \begin{aligned} -\Delta u &= f_\lambda(|x|,u,|\nabla u|) &&\text{in }\Omega, u &= 0 &&\text{on }\partial\Omega, \end{aligned}…

We show that the Auslander-Reiten Formula for a finite dimensional hereditary algebra is invariant under the Auslander-Reiten translate.

The ternary Goldbach conjecture states that every odd number n>=7 is the sum of three primes. The estimation of sums of the form \sum_{p\leq x} e(\alpha p), \alpha = a/q + O(1/q^2), has been a central part of the main approach to the…

We show that each member of a doubly infinite sequence of highly nonlinear expressions of Bernoulli polynomials, which can be seen as linear combinations of certain higher-order convolutions, is a multiple of a specific product of linear…

In this paper, we study the discrete fractional Schr\"{o}dinger equation $$ (-\Delta)^\alpha u+h(x) u=f(x,u),\quad x\in \mathbb{Z}^d,$$ where $d\in\mathbb{N}^*,\,\alpha \in(0, 1)$ and the nonlocal operator $(-\Delta)^\alpha $ is defined by…

The direct or algorithmic approach for the Jacobian problem, consisting of the direct construction of the inverse polynomials is proposed. The so called principle and derived Jacobi conditions are proposed and discussed. The algorithmic…

For the ternary quadratic form Q(x) = x^2 + y^2 - z^2 and a non-zero Pythagorean triple x_0 in Z^3 lying on the cone Q(x) = 0, we consider an orbit O = x_0 Gamma of a finitely generated subgroup Gamma < SO_Q(Z) with critical exponent…

We consider a fractional elliptic equation in an unbounded set with both Dirichlet and fractional normal derivative datum prescribed. We prove that the domain and the solution are necessarily radially symmetric. The extension of the result…

In this paper we prove a conjecture regarding the form of the Born-Infeld Lagrangian with a U(1)^2n gauge group after the elimination of the auxiliary fields. We show that the Lagrangian can be written as a symmetrized trace of Lorentz…

A formal description of a functional analysis approach to the Riemann zeta-functional equation that provides in principle an infinity of different proofs based on work by the author on the existence of dilation-invariant unitary operators…

We present an elementary method for proving enumeration formulas which are polynomials in certain parameters if others are fixed and factorize into distinct linear factors over Z. Roughly speaking the idea is to prove such formulas by…

We consider the following conjecture (from Huang, et al): Let $\Delta^+$ denote the upper half disc in $\mathbb{C}$ and let $\gamma = ( - 1, 1)$ (viewed as an interval in the real axis in $\mathbb{C}$). Assume that $F$ is a holomorphic…

We investigate the quantitative unique continuation properties of real-valued solutions to planar Schr\"odinger equations with potential functions that exhibit pointwise decay at infinity. That is, for equations of the form $-\Delta u + V u…

We obtain an essentially optimal estimate for the moment of order 32/3 of the exponential sum having argument $\alpha x^3+\beta x^2$. Subject to modest local solubility hypotheses, we thereby establish that pairs of diagonal Diophantine…