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Random constraint satisfaction problems (CSPs) such as random $3$-SAT are conjectured to be computationally intractable. The average case hardness of random $3$-SAT and other CSPs has broad and far-reaching implications on problems in…
We introduce a finite difference and $q$-difference analogues of the Asymptotic Iteration Method of Ciftci, Hall, and Saad. We give necessary, and sufficient condition for the existence of a polynomial solution to a general linear…
The distribution of $\alpha p$ modulo one, where $p$ runs over the rational primes and $\alpha$ is a fixed irrational real, has received a lot of attention. It is natural to ask for which exponents $\nu>0$ one can establish the infinitude…
We propose a new analytical method to solve for nonexactly soluble Schrodinger equation via expansions through some existing quantum numbers. Successfully, it is applied to the rational non-polynomial oscillator potential. Moreover, a…
Let $p/q$ ($p, q \in \mathbb{N}^*$) be a positive rational number such that $p > q^2$. We show that for any $\epsilon > 0$, there exists a set $A(\epsilon) \subset [0, 1[$, with finite border and with Lebesgue measure $< \epsilon$, for…
Conditionally on the Tate--Shafarevich and Bloch--Kato Conjectures, we give an explicit upper bound on the size of the $p$-adic Chabauty--Kim locus, and hence on the number of rational points, of a smooth projective curve $X/\mathbb{Q}$ of…
For $q$ a prime power and $\phi$ a rational function with coefficients in $\mathbb{F}_q$, let $p(q,\phi)$ be the proportion of $\mathbb{P}^1(\mathbb{F}_q)$ that is periodic with respect to $\phi$. And if $d$ is a positive integer, let $Q_d$…
In this article, we show new effective approximation measures for the shifted logarithm of algebraic numbers which belong to a number field of arbitrary degree. Our measures refine previous ones including those for usual logarithms. We…
By making use of the multiplicate form of the extended Carlitz inverse series relations, we establish two general `dual' theorems of Jackson's summation formula for well--poised $_8\phi_7$-series. Their duplicate forms under the partition…
We give a complete conjectural formula for the number $e_r(d,m)$ of maximum possible ${\mathbb{F}}q$-rational points on a projective algebraic variety defined by $r$ linearly independent homogeneous polynomial equations of degree $d$ in…
In this paper, we establish new irrationality criteria for certain sparse power series. As applications of these criteria, we generalize a result of Erd\H{o}s and obtain several irrationality results for various infinite series involving…
In a previous paper with the same title, we gave an upper bound for the exponent of uniform rational approximation to a quadruple of $\mathbb{Q}$-linearly independent real numbers in geometric progression. Here, we explain why this upper…
Let $\mathcal{G}$ be the greedy algorithm that, for each $\theta\in (0,1]$, produces an infinite sequence of positive integers $(a_n)_{n=1}^\infty$ satisfying $\sum_{n=1}^\infty 1/a_n = \theta$. For natural numbers $p < q$, let…
Quantum signal processing (QSP) and its extensions are increasingly popular frameworks for developing quantum algorithms. Yet QSP implementations still struggle to complete a classical pre-processing step ('QSP-processing') that determines…
The logic LAE discussed in this paper is based on an approximate entailment relation. LAE generalises classical propositional logic to the effect that conclusions can be drawn with a quantified imprecision. To this end, properties are…
This paper provides a local convergence analysis of the proximal augmented Lagrangian method (PALM) applied to a class of non-convex conic programming problems. Previous convergence results for PALM typically imposed assumptions such as…
This work derives 5 methods to evaluate families of odd zeta values by combining a power of $\pi$ with Lambert series whose ratios of successive terms tend to $e^{-\pi\sqrt{a}}$ with integers $a\ge7$, outperforming Ramanujan's results with…
V.I. Arnold has experimentally established that the limit of the statistics of incomplete quotients of partial continued fractions of quadratic irrationalities coincides with the Gauss--Kuz'min statistics. Below we briefly prove this fact…
Recently, authors have proposed under-approximate logics for reasoning about programs. So far, all such logics have been confined to reasoning about individual program behaviours. Yet there exist many over-approximate relational logics for…
We apply the specialization technique based on the decomposition of the diagonal to find an explicit example over $\mathbb{Q}$ of a quadric and cubic hypersurface in $\mathbb{P}^6$ such that their intersection is a smooth stably irrational…