Related papers: Optimal strong approximation for quadratic forms

Given a prime $p\ge5$ and an integer $s\ge1$, we show that there exists an integer $M$ such that for any quadratic polynomial $f$ with coefficients in the ring of integers modulo $p^s$, such that $f$ is not a square, if a sequence…

For a positive integer $n$, let $\mathcal T(n)$ be the set of all integers greater than or equal to $n$. An integral quadratic form $f$ is called tight $\mathcal T(n)$-universal if the set of nonzero integers that are represented by $f$ is…

We prove an estimate on the smallest singular value of a multiplicatively and additively deformed random rectangular matrix. Suppose $n\le N \le M \le \Lambda N$ for some constant $\Lambda \ge 1$. Let $X$ be an $M\times n$ random matrix…

Building on the blueprint from Goemans and Williamson (1995) for the Max-Cut problem, we construct a polynomial-time approximation algorithm for orthogonally constrained quadratic optimization problems. First, we derive a semidefinite…

Given a numerical semigroup $S = < a_1, a_2,..., a_t>$ and $s\in S$, we consider the factorization $s = c_1 a_1 + c_2 a_2 +... + c_t a_t$ where $c_i\ge0$. Such a factorization is {\em maximal} if $c_1+c_2+...+c_t$ is a maximum over all such…

In the maximum independent set of convex polygons problem, we are given a set of $n$ convex polygons in the plane with the objective of selecting a maximum cardinality subset of non-overlapping polygons. Here we study a special case of the…

In this paper, we present the quadratic associative symmetry algebra of the 3D nondegenerate maximally quantum superintegrable system. This is the complete symmetry algebra of the system. It is demonstrated that the symmetry algebra…

We relate proper isometry classes of maximal lattices in a totally definite quaternary quadratic space (V,q) with trivial discriminant to certain equivalence classes of ideals in the quaternion algebra representing the Clifford invariant of…

We prove a new quantitative result on the degeneracy of the dimension of the subspace spanned by the best Diophantine approximations for a linear form.

In the present paper, optimal quadrature formulas in the sense of Sard are constructed for numerical integration of the integral $\int_a^be^{2\pi i\omega x}\varphi(x)d x$ with $\omega\in \mathbb{R}$ in the Sobolev space $L_2^{(m)}[a,b]$ of…

Let $p$ be an odd prime and let $f(x)=\sum_{i=1}^ka_ix^{p^{\alpha_i}+1}\in\Bbb F_{p^n}[x]$, where $0\le \alpha_1<...<\alpha_k$. We consider the exponential sum $S(f,n)=\sum_{x\in\Bbb F_{p^n}}e_n(f(x))$, where $e_n(y)=e^{2\pi…

We give an upper bound for the norm of the determinant of additively indecomposable, totally positive definite quadratic forms defined over the ring of integers of totally real number fields. We apply these results to find lower and upper…

We improve a result of H. L. Montgomery and J. P. Weinberger by establishing the existence of infinitely many fundamental discriminants $d>0$ for which the class number of the real quadratic field $\mathbb{Q}(\sqrt{d})$ exeeds…

In this paper we consider certain quaternary quadratic forms and octonary quadratic forms and by using the theory of modular forms, we find formulae for the number of representations of a positive integer by these quadratic forms.

Given vectors $v_1,\dots,v_n\in\mathbb{R}^d$ and a matroid $M=([n],I)$, we study the problem of finding a basis $S$ of $M$ such that $\det(\sum_{i \in S}v_i v_i^\top)$ is maximized. This problem appears in a diverse set of areas such as…

We utilize recently introduced linear programming bounds for the energy of periodic configurations in $\mathbb{R}^d$ to construct configurations which are universally optimal among those of the form $\omega_4+L_\beta$, where $\omega_4$ is a…

Let $X$ be a random vector valued in $\mathbb{R}^{m}$ such that $\|X\|_{2} \le 1$ almost surely. For every $k\ge 3$, we show that there exists a sigma algebra $\mathcal{F}$ generated by a partition of $\mathbb{R}^{m}$ into $k$ sets such…

Let $Q(x,y)$ be a quadratic form with discriminant $D\neq 0$. We obtain non trivial upper bound estimates for the number of solutions of the congruence $Q(x,y)\equiv\lambda \pmod{p}$, where $p$ is a prime and $x,y$ lie in certain intervals…

Let $S \subseteq \mathbb{N}$ be finite. Is there a positive definite quadratic form that fails to represent only those elements in $S$? For $S = \emptyset$, this was solved (for classically integral forms) by the $15$-Theorem of…

We establish an ideal-theoretic rigidity principle for quadratic distance images over integer residue rings. Specifically, we prove that near-extremal collapse of the distance set in $\mathbb{Z}_n^d$ forces strong algebraic structure…