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We consider the question of which nonconvex sets can be represented exactly as the feasible sets of mixed-integer convex optimization problems. We state the first complete characterization for the case when the number of possible integer…

(1) There is a finitely presented group with a word problem which is a uniformly effectively inseparable equivalence relation. (2) There is a finitely generated group of computable permutations with a word problem which is a universal…

Models of computation operating over the real numbers and computing a larger class of functions compared to the class of general recursive functions invariably introduce a non-finite element of infinite information encoded in an arbitrary…

For a natural number $k>1$, let $f_k(n)$ denote the number of distinct representations of a natural number $n$ of the form $p^k+q^k$ for primes $p,q$. We prove that, for all $k>1$, $$\limsup_{n\to\infty}f_k(n)=\infty.$$ This positively…

We give two trees allowing to represent all positive rational numbers. These trees can be seen as ternary and quinary analogues of the Calkin-Wilf tree. For each of these two trees, we give recurrence formulas allowing to compute the…

We present an explicit bijection between finite-decimal real numbers and natural numbers ($\mathbb{N} = \{1, 2, 3, ...\}$) using a systematic 4-tuple parametrization with closed-form mathematical formulas for enumeration. Our enumeration…

Let $\mathcal{P}$ and $\mathbb{N}$ be the sets of all primes and natural numbers, respectively. In this article, it is proved that there has a positive lower density of the natural numbers which can be represented by the form…

We study the logic obtained by endowing the language of first-order arithmetic with second-order measure quantifiers. This new kind of quantification allows us to express that the argument formula is true in a certain portion of all…

We present a method, illustrated by several examples, to find explicit counts of permutations containing a given multiset of three letter patterns. The method is recursive, depending on bijections to reduce to the case of a smaller…

We conjecture that for a strongly minimal theory T in a finite signature satisfying the Zilber Trichotomy, there are only three possibilities for the recursive spectrum of T: all countable models of T are recursively presentable; none of…

It is shown that a nonclassicality witness, whose expectation value can be measured for all quantum states, can be constructed from every nonclassicality filter. This finding leads to a set of universal witnesses, parameterized by only…

We first consider various methods for the indirect implementation of unitary gates. We apply these methods to rederive the universality of 4-qubit measurements based on a scheme much simpler than Nielsen's original construction…

We describe those unipotent representations of a finite group of Lie type which are defined over the rational numbers.

Quantified constraints over the reals appear in numerous contexts. Usually existential quantification occurs when some parameter can be chosen by the user of a system, and univeral quantification when the exact value of a parameter is…

Given a totally real number field $F$, we show that there are only finitely many totally real extensions of $K$ of a fixed degree that admit a universal quadratic form defined over $F$. We further obtain several explicit classification…

We study systematically groups whose marked finite quotients form a recursive set. We give several definitions, and prove basic properties of this class of groups, and in particular emphasize the link between the growth of the depth…

The $3x+1$ Problem asks if whether for every natural number $n$, there exists a finite number of iterations of the piecewise function $$f(2n)=n, \quad f(2n-1)=6n-2, $$ with an iterate equal to the number $1$, or in other words, every…

We consider the representation of primes as a sum of a prime and twice a triangular number. We prove that a subset of the primes having density 1 is expressible in this form. We conjecture that every odd prime number is expressible as a sum…

Let $m\geq 3$ be a positive integer. We prove that there are uncountably many non-commensurable metabelian uniform pro-$p$ groups of dimension $m$. Consequently, there are uncountably many non-commensurable finitely presented pro-$p$ groups…

We present a conjecture on the irreducibility of the tensor products of fundamental representations of quantized affine algebras. This conjecture implies in particular that the irreducibility of the tensor products of fundamental…