Related papers: On the minima of positive definite binary hamilton…

Following de Loera and Santos, the P\'olya exponent of a $n$-ary real form (i.e. a homogeneous polynomial in $n$ variables with real coefficients) $f$ is the infimum of the upward closed set of nonnegative integers $m$ such that $(x_1 +…

For standard algorithms verifying positive definiteness of a matrix $A\in\mathbb{M}_n(\mathbb{R})$ based on Sylvester's criterion, the computationally pessimistic case is this when $A$ is positive definite. We present two algorithms…

Let $-D$ be a fundamental discriminant. We express the number of representations of an integer by a positive definite binary quadratic form of discriminant $-D$ with an odd class number $h(-D)$ as a rational linear expression involving the…

For a totally positive definite quadratic form over the ring of integers of a totally real number field $K$, we show that there are only finitely many totally real field extensions of $K$ of a fixed degree over which the form is universal…

We study the problem of computing minimal distinguishing formulas for non-bisimilar states in finite LTSs. We show that this is NP-hard if the size of the formula must be minimal. Similarly, the existence of a short distinguishing trace is…

In this paper, we study the unary Hermitian lattices over imaginary quadratic fields. Let $E=\mathbb{Q}\big(\sqrt{-d}\big)$ be an imaginary quadratic field for a square-free positive integer $d$, and let $\mathcal{O}$ be its ring of…

Motivated by a question of Defant and Propp (2020) regarding the connection between the degrees of noninvertibility of functions and those of their iterates, we address the combinatorial optimization problem of minimizing the sum of squares…

The positive semidefinite rank of a convex body $C$ is the size of its smallest positive semidefinite formulation. We show that the positive semidefinite rank of any convex body $C$ is at least $\sqrt{\log d}$ where $d$ is the smallest…

We discuss different cases of dissipative Hamiltonian differential-algebraic equations and the linear algebraic systems that arise in their linearization or discretization. For each case we give examples from practical applications. An…

Recently, the authors showed that for every irrational number $\alpha$, there exist infinitely many positive integers $n$ represented by any given positive definite binary quadratic form $Q$, satisfying $||\alpha n||<n^{-(1/2-\varepsilon)}$…

Let $D$ be a totally definite quaternion algebra over a totally real number field $F$, and $\mathcal{O}$ be an $O_F$-order (of full rank) in $D$. The type number $t(\mathcal{O})$ is an important arithmetic invariant of $\mathcal{O}$ that…

We compute the dimension of the Hilbert scheme of subvarieties of positive dimension in projective space which are cut by maximal minors of a matrix with polynomial entries.

We consider the problem of finding a local minimum of a binary quadratic function, and show by an elementary construction that every descending local search algorithm takes exponential time in the worst case.

The contributions in this note begin with a new characterization of (positive) scalar multiples of minimal tripotents in a general JB$^*$-triple $E$, proving that a non-zero element $a\in E$ is a positive scalar multiple of a minimal…

For the minimal O(N) sigma model, which is defined to be generated by the O(N) scalar auxiliary field alone, all n-point functions, till order 1/N included, can be expressed by elementary functions without logarithms. Consequently, the…

For any given positive definite binary quadratic form $Q$ with integer coefficients, we establish two results on Diophantine approximation with integers represented by $Q$. Firstly, we show that for every irrational number $\alpha$, there…

We investigate quantum algorithms for classification, a fundamental problem in machine learning, with provable guarantees. Given $n$ $d$-dimensional data points, the state-of-the-art (and optimal) classical algorithm for training…

We study relatively minimal subgroups in topological groups. We find, in particular, some natural relatively minimal subgroups in unipotent groups which are defined over "good" rings. By "good" rings we mean archimedean absolute valued (not…

In this paper we give an algorithm for enumerating all primitive (positive) definite maximal Z-valued quadratic forms Q in n >= 3 variables with bounded class number h(Q) <= B. We do this by analyzing the exact mass formula [GHY], and…

We generate all the Orthogonal Arrays (OAs) of a given size n and strength t as the union of a collection of OAs which belong to an inclusion-minimal set of OAs. We derive a formula for computing the (Generalized) Word Length Pattern of a…