The epsilon algorithm in a non-commutative algebra

INTERVAL ACCELERATION OF CONVERGENCE

The notion of an essentially generalized λ-slant Hankel operator of kth-order on the space L2 is introduced and its algebraic properties are discussed. The study is further carried to compression and essentially compression of generalized λ-slant Hankel operator of kth-order on the space H2.

The main purpose of this paper is to introduce the reader to the theory of normed division algebras. It is written with a graduate student of mathematics in mind, who only had exposure to Advanced Calculus and to an introduction to Functional Analysis. We establish the equivalence of the Mazur and Gelfand-Mazur theorems using only elementary properties of normed algebras. Mazur's theorem [19] states that every normed division algebra over the field of real numbers is isomorphic to either the field R of real numbers, the field C of complex numbers, or the non-commutative algebra Q of quaternions. Gelfand [15] proved that every normed division algebra over the field C is isomorphic to C. He named this theorem, which is fundamental for the development of the theory of Banach Algebras, the Gelfand-Mazur theorem. An analysis of Mazur's proof (see [27], p. 18–23) shows that he proved that theorem firstly for algebras over C (using the theory of Analytic Functions) and then generalized the proof to the case of algebras over R (using the theory of Harmonic Functions.) We prove the theorems that are essential for the development in detail. These include the construction of the algebra of quaternions and the theorems of Frobenius [14], Arens [1], and Stone [26].

A table algebra was defined in [1] in order to consider in a uniform way the common properties of conjugacy classes and irreducible characters. Non-commutative table algebras were introduced in [5]. They generalize properties of such well-known objects as coherent and Hecke algebras. Here we extend the main definition of a non-commutative table algebra by letting the ground field be an integral domain. We call these algebrasgeneralized table algebras (GT-algebras, in brief). It is worth mentioning that this class of algebras includes generic Hecke-Iwahori algebras of finite Coxeter groups. We develop the general theory for this type of algebras which includes their representation theory and theory of closed subsets. We also study the properties of primitive integral table algebras.

The rapid proliferation of coordinate measuring machines (CMM) triggered the need for precise and rigorous formulations of each tolerance concept. In this paper, we employ the concept of configuration space of symmetric features to define each type of datum features and orientation tolerances. The establishment of datum features and orientation tolerances are formulated in an linear programming approach. Using properties of Lie groups and Lie algebra, a simple, unified and coordinate free algorithm for datum establishment and orientation tolerance is developed. The results of the algorithm comply to the definitions stipulated in the standard ANSI Y14.5M.

Hankel Matrices and Their Applications to the Numerical Factorization of Polynomials

A general iterative scheme including relaxation and a corresponding problem class are presented. Some global convergence results are given. The acceleration of convergence is discussed, The scheme comprises a lot of known iterative methods such as subgradient methods and methods of successive orthogonal projections with relaxation. Applications to convex optimization, convex feasibility problems, systems of convex inequalities, variational inequalities, operator equations and systems of linear equations are given.

Algebraic properties of Toeplitz + Hankel operators

The $\epsilon$-algorithm has been proposed by Wynn on a number of occasions as a convergence acceleration device for vector sequences; however, little is known concerning its effect upon systems of equations. In this paper, we prove that the algorithm applied to the Picard sequence ${{\text {x}}_{i + 1}} = F({{\text {x}}_i})$ of an analytic function $F:{{\text {R}}^n} \supset D \to {{\text {R}}^n}$ provides a quadratically convergent iterative method; furthermore, no differentiation of $F$ is needed. Some examples illustrate the numerical performance of this method and show that convergence can be obtained even when $F$ is not contractive near the fixed point. A modification of the method is discussed and illustrated.

We study algebraic properties of Poisson brackets on non-associative non-commutative algebras, compatible with their multiplicative structure. Special attention is paid to the Poisson brackets of the Lie-Poisson type, related with the special Lie-structures on the differential-topological torus and brane algebras, generalizing those studied before by Novikov-Balinsky and Gelfand-Dorfman. Illustrative examples of Lie and Balinsky-Novikov algebras are discussed in detail. The non-associative structures (induced by derivation and endomorphism) of commutative algebras related to Lie and Balinsky-Novikov algebras are described in depth.

Algebraic properties of small Hankel operators on the Fock space