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Topological algebras with an orthogonal total sequence

100%

Render H.

Colloquium Mathematicum

1997

tom 72

nr 2

215-222

The aim of this paper is an investigation of topological algebras with an orthogonal sequence which is total. Closed prime ideals or closed maximal ideals are kernels of multiplicative functionals and the continuous multiplicative functionals are given by the "coefficient functionals". Our main result states that an orthogonal total sequence in a unital Fréchet algebra is already a Schauder basis. Further we consider algebras with a total sequence $(x_n)_{n∈ℕ}$ satisfying $x^2_n=x_n$ and $x_n x_{n+1} = x_{n+1}$ for all n ∈ ℕ.

On Orthogonal Projections on the Space of Consistent Pairwise Comparisons Matrices

88%

Koczkodaj W. W. , Smarzewski R. , Szybowski J.

Fundamenta Informaticae

2020

tom Vol. 172, nr 4

379--397

In this study, the orthogonalization process for different inner products is applied to pairwise comparisons. Properties of consistent approximations of a given inconsistent pairwise comparisons matrix are examined. A method of a derivation of a priority vector induced by a pairwise comparison matrix for a given inner product has been introduced. The mathematical elegance of orthogonalization and its universal use in most applied sciences has been the motivating factor for this study. However, the finding of this study that approximations depend on the inner product assumed, is of considerable importance.

Multipliers of topological algebras

78%

Husain T.

Instytut Matematyczny Polskiej Akademii Nauk

CONTENTS 1. Introduction.............................................................................................5 2. Multipliers of topological algebras...........................................................6 3. Multipliers of topological algebras with orthogonal bases......................11 4. π-Algebras............................................................................................18 5. Orthogonal decomposition of topological algebras................................21 6. Π-Algebras associated with orthogonal decompositions........................24 7. Perturbation of orthogonal bases..........................................................27 8. Banach algebras with unconditional bases and multipliers....................31 References................................................................................................36

Orthogonal bases for spaces of complex spherical harmonics

75%

Menegatto V. A. , Oliveira C. P.

Journal of Applied Analysis

2005

tom Vol. 11, nr 1

113-132

This paper proposes an inductive method to construct bases for spaces of spherical harmonics over the unit sphere Ω 2q of Cq. The bases are shown to have many interesting properties, among them orthogonality with respect to the inner product of L²(Ω 2q). As a bypass, we study the inner product [f,g] = f(D)(g(z))(0) over the space P(Cq) of polynomials in the variables [wzór], in which f(D) is the differential operator with symbol f(z). On the spaces of spherical harmonics, it is shown that the inner product [. , .] reduces to a multiple of the L²(Ω 2q) inner product. Bi-orthogonality in (F(Cq), [. , .] ) is fully investigated.

The use of logical product and the orthogonal base of planes f[x,y]-a1x plus b1 and f[x,y]-a2y plus b2 to recognize objects in sonar picture

51%

Borawski M.

Polish Journal of Environmental Studies

2005

tom 14

nr Suppl.1

This paper descibes the application of the method of logical product to detect changes in sonar picture. It also presents the possibility of joining the orthogonal basis of planes ƒ(x,y) = a1x+b1 and ƒ(x,y) = a2y+b2, with scalar product to improve recognition results. It provides a comparative analysis of the results obtained by the use of scalar product itself and of the combination of both the methods on the ground of real data obtained from the tests carried out by the Akademia Morska /Maritime Academy/ in Szczecin on the fairway Szczecin-Świnoujście.