相关论文: SICs and Algebraic Number Theory
Numerical homotopy continuation of solutions to polynomial equations is the foundation for numerical algebraic geometry, whose development has been driven by applications of mathematics. We use numerical homotopy continuation to investigate…
This paper concerns the notion of a symmetric algebra and its generalization to a quasi-symmetric algebra. We study the structure of these algebras in respect to their hull-kernel regularity and existence of some ideals, especially the…
We show that the variety of symmetric implication algebras is generated from cubic implication algebras and Boolean algebras. We do this by developing the notion of a locally symmetric implication algebra that has properties similar to…
A representation of the quadratic Dirac equation and the Maxwell equations in terms of the three-dimensional universal complex Clifford algebra is given. The investigation considers a subset of the full algebra, which is isomorphic to the…
A complete set of mutually unbiased bases for a Hilbert space of dimension N is analogous in some respects to a certain finite geometric structure, namely, an affine plane. Another kind of quantum measurement, known as a symmetric…
Apart from a few remarks on lattice systems with global or gauge symmetries, most of this talk is devoted to some interesting ancient examples of symmetries and their breakdowns in elasticity theory and hydrodynamics. Since Galois Theory is…
All 4D gauge and gravitational theories in asymptotically flat spacetimes contain an infinite number of non-trivial symmetries. They can be succinctly characterized by generalized 2D currents acting on the celestial sphere. A complete…
The aim of these notes is to present an accessible overview of some topics in classical algebraic geometry which have applications to aspects of discrete integrable systems. Precisely, we focus on surface theory on the algebraic geometry…
It is widely known that numerically integrated orbits are more precise than analytical theories for celestial bodies. However, calculation of the positions of celestial bodies via numerical integration at time $t$ requires the amount of…
We describe those group algebras over fields of characteristic different from 2 whose units symmetric with respect to the classical involution, satisfy some group identity.
We propose a duality between quiver gauge theories and the combinatorics of dimer models. The connection is via toric diagrams together with multiplicities associated to points in the diagram (which count multiplicities of fields in the…
This article is firstly a historic review of the theory of Riemann-Hilbert problems with particular emphasis placed on their original appearance in the context of Hilbert's 21st problem and Plemelj's work associated with it. The secondary…
By the example of a Fourier transform, the possibilities of Hilbert space geometry applications for statistical model construction are analyzed. In accordance with Bohr's complementarity principle, mutually-complementary coordinate and…
We consider pairs of quantum observables (POVMs) and analyze the relation between the notions of non-disturbance, joint measurability and commutativity. We specify conditions under which these properties coincide or differ---depending for…
One of the basic questions in number theory is to determine semi-simple l-adic representations of the absolute Galois group of a number field. In this paper, we discuss the question for two dimensional representations over a totally real…
We study symplectic linear algebra over the ring $\Rt$ of Colombeau generalized numbers. Due to the algebraic properties of $\Rt$ it is possible to preserve a number of central results of classical symplectic linear algebra. In particular,…
The space of unitary $C_{0}$-semigroups on separable infinite dimensional Hilbert space, when viewed under the topology of uniform weak convergence on compact subsets of $\mathbb{R}_{+}$, is known to admit various interesting residual…
Algebraic statistics is concerned with the study of probabilistic models and techniques for statistical inference using methods from algebra and geometry. This article presents a list of open mathematical problems in this emerging field,…
In this paper, we study groups of automorphisms of algebraic systems over a set of $p$-adic integers with different sets of arithmetic and coordinate-wise logical operations and congruence relations modulo $p^k,$ $k\ge 1.$ The main result…
The theory of symmetric-hyperbolic systems is useful for constructing smooth solutions of nonlinear wave equations, and for studying their singularities, including shock waves. We present the main techniques which are required to apply the…