Related papers: The 8-universality Criterion is Unique

Let F be a perfect field. Then the diagonal quadratic form $a_iX_i^2$ over $F$ is universal over $M_2(F)$ if and only if atleast two of the $a_i$ are non-zero.

New sets (typically found by computer search) with Sidon constant equal to the square root of their cardinalities are given. For each integer $N$ there are only a finite number of groups of prime order containing $N$-element extreme sets.…

Motivated by quantum thermodynamics we first investigate the notion of strict positivity, that is, linear maps which map positive definite states to something positive definite again. We show that strict positivity is decided by the action…

In this paper we give a necessary and sufficient combinatorial condition for a monomial ideal to have a linear resolution over fields of characteristic 2. We also give a new proof of Fr\"oberg's theorem over fields of characteristic 2.

There have been several algorithms designed to optimise matrix multiplication. From schoolbook method with complexity $O(n^3)$ to advanced tensor-based tools with time complexity $O(n^{2.3728639})$ (lowest possible bound achieved), a lot of…

Together with David Schlang we computed the discriminants of the invariant Hermitian forms for all indicator $o$ even degree absolutely irreducible characters of the ATLAS groups supplementing the tables of orthogonal determinants computed…

In this paper we give necessary and sufficient trace conditions for an n by n matrix over any commutative and associative ring with unity to be a sum of k-th powers of matrices over that ring, where n,k are integers greater equal 2. We…

The u-invariant of a field is the supremum of the dimensions of anisotropic quadratic forms over the field. We define corresponding u-invariants for hermitian and generalised quadratic forms over a division algebra with involution in…

This work is motivated by the papers [EG85] and [Ngu15] in which the following two problems are solved. Let $\mathcal{O}$ is a finitely generated $\mathbb{Z}$-algebra that is an integrally closed domain of characteristic zero, consider the…

We show global uniqueness of the solution to a class of constrained variational problems, using scaling properties. This is used to establish the essential uniqueness of solutions of a large deviations problem in multiple dimensions. The…

We examine the existence of universal elements in classes of infinite abelian groups. The main method is using group invariants which are defined relative to club guessing sequences. We prove, for example: Theorem: For $n\ge 2$, there is a…

We prove Kitaoka's conjecture for all totally real number fields of degree 4 -- namely, there is no positive definite classical quadratic form in three variables which is universal. To achieve this, we study the fields (often without…

In this paper, a class of combinatorial identities is proved. A method is used which is based on the following rule: counting elements of a given set in two ways and making equal the obtained results. This rule is known as "counting in two…

We discuss various universality aspects of numerical computations using standard algorithms. These aspects include empirical observations and rigorous results. We also make various speculations about computation in a broader sense.

This is a matricial description of all the fine group gradings on the exceptional Lie algebra $o(8,\mathbb C)$. There are fourteen.

We analyze the question of deciding whether a quadratic or a hyperbolic 0-1 programming instance has a unique optimal solution. Both uniqueness questions are known to be NP-hard, but are unlikely to be contained in the class NP. We…

We develop the necessary machinery in order to prove that hexagonal tilings are uniquely determined by their Tutte polynomial, showing as an example how to apply this technique to the toroidal hexagonal tiling.

Some new results concerning the equation $\sigma(N)=aM, \sigma(M)=bN$ are proved. As a corollary, there are only finitely many odd superperfect numbers with a fixed number of distinct prime factors.

A (positive definite and non-classic integral) quadratic form is called strongly $s$-regular if it satisfies a strong regularity property on the number of representations of squares of integers. In this article, we prove that for any…

We introduce O-systems (Definition \ref{DO}) of orthogonal transformations of ${\Bbb R}^{m}$, and establish $1-1$ correspondences both between equivalence classes of Clifford systems and that of O-systems, and between O-systems and…