scholarly journals Affine Independence in Vector Spaces

2010 ◽  
Vol 18 (1) ◽  
pp. 87-93 ◽  
Author(s):  
Karol Pąk
Keyword(s):  
Linear Space ◽  
Vector Spaces ◽  
Barycentric Coordinates ◽  
Independent Subset ◽  
Linear Combinations ◽  
Affine Hull ◽  
Real Linear Space ◽  
Real Linear ◽  
The Given

Affine Independence in Vector Spaces In this article we describe the notion of affinely independent subset of a real linear space. First we prove selected theorems concerning operations on linear combinations. Then we introduce affine independence and prove the equivalence of various definitions of this notion. We also introduce the notion of the affine hull, i.e. a subset generated by a set of vectors which is an intersection of all affine sets including the given set. Finally, we introduce and prove selected properties of the barycentric coordinates.

scholarly journals Nearly directed subspaces of partially ordered linear spaces

1968 ◽  
Vol 16 (2) ◽  
pp. 135-144
Author(s):  
G. J. O. Jameson
Keyword(s):  
Linear Space ◽  
Transitive Relation ◽  
Linear Spaces ◽  
Real Linear Space ◽  
Partially Ordered ◽  
Real Linear ◽  
Image Position ◽  
Ordered Linear Spaces

Let X be a partially ordered linear space, i.e. a real linear space with a reflexive, transitive relation ≦ such that

About the Polarity in a Real Linear Space

1977 ◽  
Vol 77 (1) ◽  
pp. 181-185 ◽  
Author(s):  
Jacques Bair
Keyword(s):  
Linear Space ◽  
Real Linear Space ◽  
Real Linear

On the Hahn-Banach Extension Property

1970 ◽  
Vol 13 (1) ◽  
pp. 9-13
Author(s):  
Ting-On To
Keyword(s):  
Linear Function ◽  
Linear Space ◽  
Linear Extension ◽  
Normed Linear Space ◽  
Extension Property ◽  
Bounded Linear ◽  
Linear Spaces ◽  
Real Linear Space ◽  
Real Linear

In this paper, we consider real linear spaces. By (V:‖ ‖) we mean a normed (real) linear space V with norm ‖ ‖. By the statement "V has the (Y, X) norm preserving (Hahn-Banach) extension property" we mean the following: Y is a subspace of the normed linear space X, V is a normed linear space, and any bounded linear function f: Y → V has a linear extension F: X → V such that ‖F‖ = ‖f‖. By the statement "V has the unrestricted norm preserving (Hahn-Banach) extension property" we mean that V has the (Y, X) norm preserving extension property for all Y and X with Y ⊂ X.

Linear Maps on Selfadjoint Operators Preserving Invertibility, Positive Definiteness, Numerical Range

2003 ◽  
Vol 46 (2) ◽  
pp. 216-228 ◽  
Author(s):  
Chi-Kwong Li ◽  
Leiba Rodman ◽  
Peter Šemrl
Keyword(s):  
Linear Space ◽  
Numerical Range ◽  
Positive Definiteness ◽  
Linear Operators ◽  
Bounded Linear ◽  
Real Linear Space ◽  
The Real ◽  
Selfadjoint Operators ◽  
Linear Maps ◽  
Real Linear

AbstractLet H be a complex Hilbert space, and be the real linear space of bounded selfadjoint operators on H. We study linear maps ϕ: → leaving invariant various properties such as invertibility, positive definiteness, numerical range, etc. The maps ϕ are not assumed a priori continuous. It is shown that under an appropriate surjective or injective assumption ϕ has the form , for a suitable invertible or unitary T and ξ ∈ {1, −1}, where Xt stands for the transpose of X relative to some orthonormal basis. Examples are given to show that the surjective or injective assumption cannot be relaxed. The results are extended to complex linear maps on the algebra of bounded linear operators on H. Similar results are proved for the (real) linear space of (selfadjoint) operators of the form αI + K, where α is a scalar and K is compact.

scholarly journals ON EXTREME POINTS OF A CERTAIN LINEAR SPACE OF LOCALLY UNIVALENT FUNCTIONS

1992 ◽  
Vol 23 (4) ◽  
pp. 321-325
Author(s):  
KUALIDA INAYAT NOOR
Keyword(s):  
Linear Space ◽  
Unit Disc ◽  
Univalent Functions ◽  
Extreme Points ◽  
Space Structure ◽  
Real Linear Space ◽  
The Real ◽  
Bounded Boundary Rotation ◽  
Boundary Rotation ◽  
Real Linear

Let $H = (H, \oplus, \odot)$ denote the real linear space of locally univalent normalized functions in the unit disc as defined by Hornich. For $-1\le B <A\le 1$, $k>2$, the classes $V_k[A,B]$ of functions with bounded boundary rotation are introduced and this linear space structure is used to determine the extreme points of the classes $V_k[A,B]$.

Intersections of Loci on an Algebraic V4

1937 ◽  
Vol 33 (4) ◽  
pp. 425-437 ◽  
Author(s):  
J. A. Todd
Keyword(s):  
Algebraic Geometry ◽  
Linear Space ◽  
Considerable Importance ◽  
Classical Paper ◽  
Residual Intersection ◽  
Pass Through ◽  
The Given

The problem of residual intersections is one of considerable importance in algebraic geometry. In its most general form the problem may be stated as follows. On a given variety V of d dimensions two varieties A and B, of respective dimensions k and k′, pass through a variety C whose dimension r is not less than r′ = k + k′ – d. It is required to determine the variety D of dimension r′ which forms the residual intersection of A and B. The classical paper on this subject is that of Severi*. He considers the case in which V is a linear space, and obtains a large variety of enumerative results connecting the characters of the residual intersection with those of the given loci.

scholarly journals On matrices whose real linear combinations are non-singular

1965 ◽  
Vol 16 (2) ◽  
pp. 318-318
Author(s):  
J. F. Adams ◽  
Peter D. Lax ◽  
Ralph S. Phillips
Keyword(s):  
Linear Combinations ◽  
Real Linear

scholarly journals Old and New Identities for Bernoulli Polynomials via Fourier Series

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Luis M. Navas ◽  
Francisco J. Ruiz ◽  
Juan L. Varona
Keyword(s):  
Fourier Series ◽  
Linear Combination ◽  
Legendre Polynomials ◽  
Fourier Coefficients ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Gegenbauer Polynomials ◽  
Linear Combinations ◽  
The Given

The Bernoulli polynomialsBkrestricted to[0,1)and extended by periodicity haventh sine and cosine Fourier coefficients of the formCk/nk. In general, the Fourier coefficients of any polynomial restricted to[0,1)are linear combinations of terms of the form1/nk. If we can make this linear combination explicit for a specific family of polynomials, then by uniqueness of Fourier series, we get a relation between the given family and the Bernoulli polynomials. Using this idea, we give new and simpler proofs of some known identities involving Bernoulli, Euler, and Legendre polynomials. The method can also be applied to certain families of Gegenbauer polynomials. As a result, we obtain new identities for Bernoulli polynomials and Bernoulli numbers.

scholarly journals A Family of Derivative-Free Methods with High Order of Convergence and Its Application to Nonsmooth Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Alicia Cordero ◽  
José L. Hueso ◽  
Eulalia Martínez ◽  
Juan R. Torregrosa
Keyword(s):  
Fourth Order ◽  
Order Of Convergence ◽  
Nonsmooth Equations ◽  
Order Convergence ◽  
Linear Combinations ◽  
Numerical Examples ◽  
Derivative Free ◽  
Derivative Free Methods ◽  
Seventh Order ◽  
The Given

A family of derivative-free methods of seventh-order convergence for solving nonlinear equations is suggested. In the proposed methods, several linear combinations of divided differences are used in order to get a good estimation of the derivative of the given function at the different steps of the iteration. The efficiency indices of the members of this family are equal to 1.6266. Also, numerical examples are used to show the performance of the presented methods, on smooth and nonsmooth equations, and to compare with other derivative-free methods, including some optimal fourth-order ones, in the sense of Kung-Traub’s conjecture.

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