相关论文: Real roots of real cubics and optimization

Our main result is that any real cubic algebraic number has a continued fraction expansion with polynomial coefficients. Some generalizations are mentioned.

We describe a combinatorial approach for investigating properties of rational numbers. The overall approach rests on structural bijections between rational numbers and familiar combinatorial objects, namely rooted trees. We emphasize that…

The metohod of ortogonal rotations introduced in the previous papers of the author is used for construction of the explicit form the generators of the simple roots for quantum (and ussual) semisimple algebras. All calculations are presented…

The paper deals with the resolution of third and fourth degree equations by means of radicals. It is a survey of some historical details about this fundamental problem. Moreover, it explains practical methods for the resolution of third and…

The usual methods for root finding of polynomials are based on the iteration of a numerical formula for improvement of successive estimations. The unpredictable nature of the iterations prevents to search roots inside a pre-specified region…

Real root finding of polynomial equations is a basic problem in computer algebra. This task is usually divided into two parts: isolation and refinement. In this paper, we propose two algorithms LZ1 and LZ2 to refine real roots of univariate…

A problem concerning the shift of roots of a system of homogeneous algebraic equations is investigated. Its conservation and decomposition of a multiple root into simple roots are discussed.

The optimal calculation order of a computational graph can be represented by a set of algebraic expressions. Computational graph and algebraic expression both have close relations and significant differences, this paper looks into these…

Square root is a useful tool to study the properties of (ordered) algebraic structures. In this article, we are going to employ this tool to study hoop algebras. To do so, we define square root and make the first attempt to explore the…

Recent advancements in quantum computing and quantum-inspired algorithms have sparked renewed interest in binary optimization. These hardware and software innovations promise to revolutionize solution times for complex problems. In this…

In this Master of Science Thesis I introduce geometric algebra both from the traditional geometric setting of vector spaces, and also from a more combinatorial view which simplifies common relations and operations. This view enables us to…

This will is an expository description of quadratic rational maps. Sections 2 through 6 are concerned with the geometry and topology of such maps. Sections 7--10 survey of some topics from the dynamics of quadratic rational maps. There are…

We consider multidimensional optimization problems, which are formulated and solved in terms of tropical mathematics. The problems are to minimize (maximize) a linear or nonlinear function defined on vectors over an idempotent semifield,…

We investigate the method of conjugate gradients, exploiting inaccurate matrix-vector products, for the solution of convex quadratic optimization problems. Theoretical performance bounds are derived, and the necessary quantities occurring…

We study the existence of formal power series solutions to q-algebraic equations. When a solution exists, we give a sufficient condition on the equation for this solution to have a positive radius of convergence. We emphasize on the case…

In this paper we consider the spatial semi-discretization of conservative PDEs. Such finite dimensional approximations of infinite dimensional dynamical systems can be described as flows in suitable matrix spaces, which in turn leads to the…

Eventually positive matrices are real matrices whose powers become and remain strictly positive. As such, eventually positive matrices are a fortiori matrix roots of positive matrices, which motivates us to study the matrix roots of…

Cylindrical algebraic decomposition (CAD) is an important tool for the study of real algebraic geometry with many applications both within mathematics and elsewhere. It is known to have doubly exponential complexity in the number of…

The notion of 'bifurcating continued fractions' is introduced. Two coupled sequences of non-negative integers are obtained from an ordered pair of positive real numbers in a manner that generalizes the notion of continued fractions. These…

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…