Related papers: Universality criterion sets for quadratic forms ov…

The Doob convergence theorem implies that the set of divergence of any martingale has measure zero. We prove that, conversely, any $G\_{\delta\sigma}$ subset of the Cantor space with Lebesgue-measure zero can be represented as the set of…

We investigate here the representability of integers as sums of triangular numbers, where the $n$-th triangular number is given by $T_n = n(n + 1)/2$. In particular, we show that $f(x_1,x_2,..., x_k) = b_1 T_{x_1} +...+ b_k T_{x_k}$, for…

We give an alternative proof of the theorem by Kuznetsov and Lunts, stating that any separated scheme of finite type over a field of characteristic zero admits a categorical resolution of singularities. Their construction makes use of the…

As an approach to a Theory of Everything a framework for developing a coherent theory of mathematics and physics together is described. The main characteristic of such a theory is discussed: the theory must be valid and and sufficiently…

The Schur-Horn theorem is a classical result in matrix analysis which characterizes the existence of positive semidefinite matrices with a given diagonal and spectrum. In recent years, this theorem has been used to characterize the…

This short introductory category theory textbook is for readers with relatively little mathematical background (e.g. the first half of an undergraduate mathematics degree). At its heart is the concept of a universal property, important…

Given a compact subset $\Sigma$ of the real numbers obeying some technical conditions, we consider the set of algebraic integers whose conjugates all lie in $\Sigma$. The distribution of conjugates of such an integer defines a probability…

In this paper, we study partitions of totally positive integral elements $\alpha$ in a real quadratic field $K$. We prove that for a fixed integer $m \geq 1$, an element with $m$ partition exists in almost all $K$. We also obtain an upper…

The Fundamental Theorem of Algebra can be thought of as a statement about the real numbers as a space, considered as an algebraic set over the real numbers as a field. This paper introduces what it means for an algebraic set or affine…

The concept of the {\em half density matrix} is proposed. It unifies the quantum states which are described by density matrices and physical processes which are described by completely positive maps. With the help of the half-density-matrix…

The purpose of this paper is to explain how the identities of various fundamental lemmas fall within the scope of the transfer principle, a general result that allows to transfer theorems about identities of p-adic integrals from one…

In this paper we develop in detail the geometric constructions that lead to many uniqueness results for the determination of polyhedral sets, typically scatterers, by a finite minimal number of measurements. We highlight how unique…

In a previous article, a universal linear algebraic model was proposed for describing homogeneous conformal geometries, such as the spherical, Euclidean, hyperbolic, Minkowski, anti-de Sitter and Galilei planes. This formalism was…

A description of physical reality in which wholeness is the foundation is discussed along with the motivation for such an attempt. As a possible mathematical framework within which a physical theory based on wholeness may be expressed,…

We prove that in each degree divisible by 2 or 3, there are infinitely many totally real number fields that require universal quadratic forms to have arbitrarily large rank.

We give an overview of the fundamental definitions and results concerning hypersurface singularities, defined by convergent power series over an arbitrary real valued field. This approach combines, on the one hand, the classical case of…

Root systems are sets with remarkable symmetries and therefore they appear in many situations in mathematics. Among others, denominator formulae of root systems are very beautiful and mysterious equations which have several meanings from a…

The complex numbers are an important part of quantum theory, but are difficult to motivate from a theoretical perspective. We describe a simple formal framework for theories of physics, and show that if a theory of physics presented in this…

We consider the problem of identifying the quantum spin states that are the optimal sensors of a given transformation averaged over all possible orientations of the spin system. Our geometric approach to the problem is based on a fidelity…

Universality in unitary invariant random matrix ensembles with complex matrix elements is considered. We treat two general ensembles which have a determinant factor in the weight. These ensembles are relevant, e.g., for spectra of the Dirac…