scholarly journals Components in vector lattices and extreme extensions of quasi-measures and measures

1993 ◽  
Vol 35 (2) ◽  
pp. 153-162 ◽  
Author(s):  
Z. Lipecki
Keyword(s):  
Convex Set ◽  
Vector Lattice ◽  
Convex Combination ◽  
Extreme Points ◽  
Main Body ◽  
Projection Property ◽  
Vector Lattices ◽  
Definition Of ◽  
Image Position ◽  
Theoretical Results

We develop some ideas contained in the author's paper [8] which was, in turn, inspired by Bierlein and Stich [5]. The main body of the present paper is divided into three sections. Section 2 is concerned with some vector-lattice-theoretical results. They are then applied to extensions of quasi-measures and measures in Sections 3 and 4, respectively.Let X be a vector lattice, let x ε X+ and let S be a non-empty set. Theorems 1 and 2 describe some properties of the convex set(see Section 2 for the definition of the sum above). The extreme points of Dx,s are characterized in terms of the components of x. It is also shown that if X has the principal projection property and S is countable, then extr Dx,s is, in some sense, large in Dx,s. Furthermore, for finite S, each point in Dx,s is then a sσ-convex combination of extreme ones.

On the Extreme Rays of the Metric Cone

1980 ◽  
Vol 32 (1) ◽  
pp. 126-144 ◽  
Author(s):  
David Avis
Keyword(s):  
Convex Set ◽  
Convex Combination ◽  
Extreme Points ◽  
Convex Polyhedra ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Permutation Matrices ◽  
Image Position ◽  
Polyhedral Convex Set ◽  
Extreme Rays

A classical result in the theory of convex polyhedra is that every bounded polyhedral convex set can be expressed either as the intersection of half-spaces or as a convex combination of extreme points. It is becoming increasingly apparent that a full understanding of a class of convex polyhedra requires the knowledge of both of these characterizations. Perhaps the earliest and neatest example of this is the class of doubly stochastic matrices. This polyhedron can be defined by the system of equationsBirkhoff [2] and Von Neuman have shown that the extreme points of this bounded polyhedron are just the n × n permutation matrices. The importance of this result for mathematical programming is that it tells us that the maximum of any linear form over P will occur for a permutation matrix X.

Extreme Points of the Convex set of Stochastic Maps on A C*-Algebra

1998 ◽  
Vol 01 (04) ◽  
pp. 599-609 ◽  
Author(s):  
K. R. Parthasarathy
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Convex Set ◽  
Convex Combination ◽  
Extreme Points ◽  
Bounded Operators ◽  
Completely Positive ◽  
C Algebra ◽  
Permutation Matrices ◽  
Normal Maps

Let [Formula: see text] be a unital C*-subalgebra of the C*-algebra ℬ(ℋ) of all bounded operators on a complex separable Hilbert space ℋ. Let [Formula: see text] denote the convex set of all unital, linear, completely positive and normal maps of [Formula: see text] into itself. Using Stinespring's theorem, we present a criterion for an element [Formula: see text] to be extremal. When [Formula: see text], this criterion leads to an explicit description of the set of all extreme points of [Formula: see text]. We also obtain a quantum probabilistic analogue of the classical Birkhoff's theorem2 that every bistochastic matrix can be expressed as a convex combination of permutation matrices.

scholarly journals A note on vector lattices

1967 ◽  
Vol 7 (1) ◽  
pp. 32-38 ◽  
Author(s):  
Sadayuki Yamamuro
Keyword(s):  
Vector Lattice ◽  
Vector Lattices ◽  
Image Position

Let E be a vector lattice in the sense of Birkhoff [1]. We use the following notations:

An unpleasant set in a non-locally-convex vector lattice

1973 ◽  
Vol 18 (3) ◽  
pp. 229-233
Author(s):  
J. D. Pryce
Keyword(s):  
Convex Hull ◽  
Vector Lattice ◽  
Linear Topological Space ◽  
Real Vector ◽  
Vector Lattices ◽  
Köthe Spaces ◽  
The Common ◽  
Solid Hull ◽  
Image Position ◽  
Locally Convex

In a linear topological space E one often carries out various “ smoothing ” operations on a subset A, such as taking the convex hull co A and the closure A-. If E is also a (real) vector lattice, the solid hullis also a natural “ smoothing out ” of A. If sol A = A then A is called solid, and if E has a base of solid neighbourhoods of 0 as do all the common topological vector lattices such as C(X), Lp, Köthe spaces and so on—then E is called a locally solid space.

scholarly journals On Bilinear Narrow Operators

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2892
Author(s):  
Marat Pliev ◽  
Nonna Dzhusoeva ◽  
Ruslan Kulaev
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Vector Lattice ◽  
Cartesian Product ◽  
Bilinear Operator ◽  
Projection Property ◽  
New Class ◽  
Bilinear Operators ◽  
Vector Lattices ◽  
Order Continuous

In this article, we introduce a new class of operators on the Cartesian product of vector lattices. We say that a bilinear operator T:E×F→W defined on the Cartesian product of vector lattices E and F and taking values in a vector lattice W is narrow if the partial operators Tx and Ty are narrow for all x∈E,y∈F. We prove that, for order-continuous Köthe–Banach spaces E and F and a Banach space X, the classes of narrow and weakly function narrow bilinear operators from E×F to X are coincident. Then, we prove that every order-to-norm continuous C-compact bilinear regular operator T is narrow. Finally, we show that a regular bilinear operator T from the Cartesian product E×F of vector lattices E and F with the principal projection property to an order continuous Banach lattice G is narrow if and only if |T| is.

scholarly journals Atomic Operators in Vector Lattices

2020 ◽  
Vol 17 (5) ◽  
Author(s):  
Ralph Chill ◽  
Marat Pliev
Keyword(s):  
Nonlinear Operator ◽  
Vector Lattice ◽  
Projection Property ◽  
New Class ◽  
Vector Lattices ◽  
Complete Vector Lattice ◽  
Whole Space ◽  
Complete Vector ◽  
Orthogonally Additive ◽  
Orthogonally Additive Operators

Abstract In this paper, we introduce a new class of operators on vector lattices. We say that a linear or nonlinear operator T from a vector lattice E to a vector lattice F is atomic if there exists a Boolean homomorphism $$\Phi $$ Φ from the Boolean algebra $${\mathfrak {B}}(E)$$ B ( E ) of all order projections on E to $${\mathfrak {B}}(F)$$ B ( F ) such that $$T\pi =\Phi (\pi )T$$ T π = Φ ( π ) T for every order projection $$\pi \in {\mathfrak {B}}(E)$$ π ∈ B ( E ) . We show that the set of all atomic operators defined on a vector lattice E with the principal projection property and taking values in a Dedekind complete vector lattice F is a band in the vector lattice of all regular orthogonally additive operators from E to F. We give the formula for the order projection onto this band, and we obtain an analytic representation for atomic operators between spaces of measurable functions. Finally, we consider the procedure of the extension of an atomic map from a lateral ideal to the whole space.

scholarly journals Global Geometric Difference Between Separable and Positive Partial Transpose States

2014 ◽  
Vol 21 (04) ◽  
pp. 1450009
Author(s):  
Kil-Chan Ha ◽  
Seung-Hyeok Kye
Keyword(s):  
Interior Point ◽  
Convex Set ◽  
Convex Combination ◽  
Extreme Points ◽  
Positive Partial Transpose ◽  
Separable States ◽  
Positive Maps ◽  
Geometric Difference ◽  
Partial Transpose ◽  
Ppt States

In the convex set of all 3 ⊗ 3 states with positive partial transposes, we show that one can take two extreme points whose convex combinations belong to the interior of the convex set. Their convex combinations may be even in the interior of the convex set of all separable states. In general, we need at least mn extreme points to get an interior point by their convex combination, for the case of the convex set of all m ⊗ n separable states. This shows a sharp distinction between PPT states and separable states. We also consider the same questions for positive maps and decomposable maps.

A Class of Harmonic Starlike Functions defined by Multiplier Transformation

2020 ◽  
pp. 455-469
Author(s):  
Deepali Khurana ◽  
Raj Kumar ◽  
Sibel Yalcin
Keyword(s):  
Convex Combination ◽  
Univalent Functions ◽  
Extreme Points ◽  
Starlike Functions ◽  
Harmonic Mappings ◽  
Distortion Bounds ◽  
Radius Of Convexity ◽  
Univalent Harmonic Mappings ◽  
Harmonic Starlike ◽  
Multiplier Transformation

We define two new subclasses, $HS(k, \lambda, b, \alpha)$ and \linebreak $\overline{HS}(k, \lambda, b, \alpha)$, of univalent harmonic mappings using multiplier transformation. We obtain a sufficient condition for harmonic univalent functions to be in $HS(k,\lambda,b,\alpha)$ and we prove that this condition is also necessary for the functions in the class $\overline{HS} (k,\lambda,b,\alpha)$. We also obtain extreme points, distortion bounds, convex combination, radius of convexity and Bernandi-Libera-Livingston integral for the functions in the class $\overline{HS}(k,\lambda,b,\alpha)$.

Pascal's Prism

2012 ◽  
Vol 96 (536) ◽  
pp. 213-220
Author(s):  
Harlan J. Brothers
Keyword(s):  
Probability Theory ◽  
Fractal Geometry ◽  
Euclidean Geometry ◽  
Fibonacci Series ◽  
Pascal’S Triangle ◽  
Number Sequences ◽  
Definition Of ◽  
Image Position ◽  
Pascal's Triangle

Pascal's triangle is well known for its numerous connections to probability theory [1], combinatorics, Euclidean geometry, fractal geometry, and many number sequences including the Fibonacci series [2,3,4]. It also has a deep connection to the base of natural logarithms, e [5]. This link to e can be used as a springboard for generating a family of related triangles that together create a rich combinatoric object.2. From Pascal to LeibnizIn Brothers [5], the author shows that the growth of Pascal's triangle is related to the limit definition of e.Specifically, we define the sequence sn; as follows [6]:

scholarly journals Pseudoinversion of degenerate metrics

2003 ◽  
Vol 2003 (55) ◽  
pp. 3479-3501 ◽  
Author(s):  
C. Atindogbe ◽  
J.-P. Ezin ◽  
Joël Tossa
Keyword(s):  
Differential Geometry ◽  
Large Class ◽  
Smooth Manifold ◽  
Differential Operators ◽  
Type Operator ◽  
Lorentzian Space ◽  
Lightlike Hypersurface ◽  
Definition Of ◽  
Lightlike Hypersurfaces ◽  
Theoretical Results

Let(M,g)be a smooth manifoldMendowed with a metricg. A large class of differential operators in differential geometry is intrinsically defined by means of the dual metricg∗on the dual bundleTM∗of 1-forms onM. If the metricgis (semi)-Riemannian, the metricg∗is just the inverse ofg. This paper studies the definition of the above-mentioned geometric differential operators in the case of manifolds endowed with degenerate metrics for whichg∗is not defined. We apply the theoretical results to Laplacian-type operator on a lightlike hypersurface to deduce a Takahashi-like theorem (Takahashi (1966)) for lightlike hypersurfaces in Lorentzian spaceℝ1n+2.

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