Related papers: Sums of Cubes in Quaternion Rings
Several topics on the implementation of spin qubits in quantum dots are reviewed. We first provide an introduction to the standard model of quantum computing and the basic criteria for its realization. Other alternative formulations such as…
We provide a spectrum of results for the Universal Guard Problem, in which one is to obtain a small set of points ("guards") that are "universal" in their ability to guard any of a set of possible polygonal domains in the plane. We give…
We show that almost every positive integer can be expressed as a sum of four squares of integers represented as the sums of three positive cubes.
We give an estimate of exponential sums over singular binary quintic forms in a characteristic-free form, based on the Waring decomposition of binary forms. This extends the method on our preceding result on the space of binary quartics to…
In this paper, we will interest in finding the number of zeros of the quadratic forms over finite fields. We will apply the tool for finding the number of rational points of supersingular curves in [6]. We will give some more tools for…
The review of modern study of algebraic, geometric and differential properties of quaternionic (Q) numbers with their applications. Traditional and "tensor" formulation of Q-units with their possible representations are discussed and groups…
We find the number of compositions over finite abelian groups under two types of restrictions: (i) each part belongs to a given subset and (ii) small runs of consecutive parts must have given properties. Waring's problem over finite fields…
We prove upper bounds for the number of rational points on non-singular cubic curves defined over the rationals. The bounds are uniform in the curve and involve the rank of the corresponding Jacobian. The method used in the proof is a…
In this note we give a theoretical support by means of quotient polynomial rings for the computation formulas of the dimension of abelian codes.
The paper explores the basic geometrical properties of the observables characterizing two-qubit systems by employing a novel projective ring geometric approach. After introducing the basic facts about quantum complementarity and maximal…
We examine a few problems of enumerative geometry and present their solutions in the framework of deformed (quantum) cohomology rings.
We investigate the average order of the divisor function at values of totally reducible binary cubic forms and discuss some applications.
This article considers the problem of designing adaption and optimisation techniques for training quantum learning machines. To this end, the division algebra of quaternions is used to derive an effective model for representing computation…
In this paper, we investigate exceptional sets in the Waring-Goldbach problem for unlike powers. For example, estimates are obtained for sufficiently large integers below a parameter subject to the necessary local conditions that do not…
Basic facts and definitions of conformal moduli of rings and quadrilaterals are recalled. Some computational methods are reviewed. For the case of quadrilaterals with polygonal sides, some recent results are given. Some numerical…
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…
In this paper we obtain explicit estimates and existence results on the number of $\mathbb{F}_q$-rational solutions of certain systems defined by families of diagonal equations over finite fields. Our approach relies on the study of the…
We consider a puzzle such that a set of colored cubes is given as an instance. Each cube has unit length on each edge and its surface is colored so that what we call the Surface Color Condition is satisfied. Given a palette of six colors,…
In this paper, we establish the explicit lower bound estimates for the rank of universal quadratic forms in some certain families of real cubic fields under the condition of density one. The more general results that represent all multiples…
We describe a large-scale computational experiment to study structure in the numbers of real solutions to osculating instances of Schubert problems. This investigation uncovered Schubert problems whose computed numbers of real solutions…