scholarly journals THE METRIC PROJECTIONS ONTO CLOSED CONVEX CONES IN A HILBERT SPACE

2021 ◽  
pp. 1-34
Author(s):  
Yanqi Qiu ◽  
Zipeng Wang
Keyword(s):  
Hilbert Space ◽  
Convex Cone ◽  
Real Hilbert Space ◽  
Metric Projection ◽  
Convex Cones ◽  
Closed Convex Cone ◽  
Sufficient Condition ◽  
Best Approximations ◽  
General Convex ◽  
Nonnegative Coefficients

Abstract We study the metric projection onto the closed convex cone in a real Hilbert space $\mathscr {H}$ generated by a sequence $\mathcal {V} = \{v_n\}_{n=0}^\infty $ . The first main result of this article provides a sufficient condition under which the closed convex cone generated by $\mathcal {V}$ coincides with the following set: $$ \begin{align*} \mathcal{C}[[\mathcal{V}]]: = \bigg\{\sum_{n=0}^\infty a_n v_n\Big|a_n\geq 0,\text{ the series }\sum_{n=0}^\infty a_n v_n\text{ converges in } \mathscr{H}\bigg\}. \end{align*} $$ Then, by adapting classical results on general convex cones, we give a useful description of the metric projection onto $\mathcal {C}[[\mathcal {V}]]$ . As an application, we obtain the best approximations of many concrete functions in $L^2([-1,1])$ by polynomials with nonnegative coefficients.

Positive and Z-operators on Closed Convex Cones

2018 ◽  
Vol 34 ◽  
pp. 444-458
Author(s):  
Michael Orlitzky
Keyword(s):  
Game Theory ◽  
Hilbert Space ◽  
Dynamical Systems ◽  
Linear Operator ◽  
Positive Operator ◽  
Convex Cone ◽  
Real Hilbert Space ◽  
Closed Convex Cone ◽  
Nonnegative Matrices ◽  
Finite Dimensional

Let $K$ be a closed convex cone with dual $\dual{K}$ in a finite-dimensional real Hilbert space. A \emph{positive operator} on $K$ is a linear operator $L$ such that $L\of{K} \subseteq K$. Positive operators generalize the nonnegative matrices and are essential to the Perron-Frobenius theory. It is said that $L$ is a \emph{\textbf{Z}-operator} on $K$ if % \begin{equation*} \ip{L\of{x}}{s} \le 0 \;\text{ for all } \pair{x}{s} \in \cartprod{K}{\dual{K}} \text{ such that } \ip{x}{s} = 0. \end{equation*} % The \textbf{Z}-operators are generalizations of \textbf{Z}-matrices (whose off-diagonal elements are nonpositive) and they arise in dynamical systems, economics, game theory, and elsewhere. In this paper, the positive and \textbf{Z}-operators are connected. This extends the work of Schneider, Vidyasagar, and Tam on proper cones, and reveals some interesting similarities between the two families.

Remarks on CCR and CAR flows over closed convex cones

2022 ◽  
Author(s):  
Anbu Arjunan
Keyword(s):  
Convex Cone ◽  
Convex Cones ◽  
Closed Convex Cone ◽  
Cocycle Conjugacy

For a closed convex cone [Formula: see text] in [Formula: see text] which is spanning and pointed, i.e. [Formula: see text] and [Formula: see text] we consider a family of [Formula: see text]-semigroups over [Formula: see text] consisting of a certain family of CCR flows and CAR flows over [Formula: see text] and classify them up to the cocycle conjugacy.

scholarly journals Minimising quadratic functionals over closed convex cones

1989 ◽  
Vol 39 (1) ◽  
pp. 15-20 ◽  
Author(s):  
M. Seetharama Gowda
Keyword(s):  
Complementarity Problems ◽  
Real Hilbert Space ◽  
Linear Complementarity Problems ◽  
Linear Complementarity ◽  
Convex Cones ◽  
Closed Convex Cone ◽  
Quadratic Functional ◽  
Quadratic Functionals ◽  
Finite Dimensional ◽  
Bounded Below

In this article we show that, under suitable conditions a quadratic functional attains its minimum on a closed convex cone (in a finite dimensional real Hilbert space) whenever it is bounded below on the cone. As an application, we solve Generalised Linear Complementarity Problems over closed convex cones.

scholarly journals On the existence of E0-semigroups — the multiparameter case

2018 ◽  
Vol 21 (02) ◽  
pp. 1850007 ◽  
Author(s):  
S. P. Murugan ◽  
S. Sundar
Keyword(s):  
Hilbert Space ◽  
Convex Cone ◽  
Separable Hilbert Space ◽  
Closed Convex Cone ◽  
Product System ◽  
Infinite Dimensional ◽  
Intersection Formula

Let [Formula: see text] be a closed convex cone. Assume that [Formula: see text] is pointed, i.e. the intersection [Formula: see text] and [Formula: see text] is spanning, i.e. [Formula: see text]. Denote the interior of [Formula: see text] by [Formula: see text]. Let [Formula: see text] be a product system over [Formula: see text]. We show that there exists an infinite-dimensional separable Hilbert space [Formula: see text] and a semigroup [Formula: see text] of unital normal ∗-endomorphisms of [Formula: see text] such that [Formula: see text] is isomorphic to the product system associated to [Formula: see text].

scholarly journals New Inertial Forward-Backward Mid-Point Methods for Sum of Infinitely Many Accretive Mappings, Variational Inequalities, and Applications

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 466
Author(s):  
Li Wei ◽  
Yingzi Shang ◽  
Ravi P. Agarwal
Keyword(s):  
Hilbert Space ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Real Hilbert Space ◽  
Iterative Algorithms ◽  
Metric Projection ◽  
Special Cases ◽  
Accretive Mappings ◽  
Number Sequences ◽  
Zero Points

Some new inertial forward-backward projection iterative algorithms are designed in a real Hilbert space. Under mild assumptions, some strong convergence theorems for common zero points of the sum of two kinds of infinitely many accretive mappings are proved. New projection sets are constructed which provide multiple choices of the iterative sequences. Some already existing iterative algorithms are demonstrated to be special cases of ours. Some inequalities of metric projection and real number sequences are widely used in the proof of the main results. The iterative algorithms have also been modified and extended from pure discussion on the sum of accretive mappings or pure study on variational inequalities to that for both, which complements the previous work. Moreover, the applications of the abstract results on nonlinear capillarity systems are exemplified.

scholarly journals ON GEOMETRY OF FINSLER CAUSALITY: FOR CONVEX CONES, THERE IS NO AFFINE-INVARIANT LINEAR ORDER (SIMILAR TO COMPARING VOLUMES)

2020 ◽  
pp. 49-55
Author(s):  
Olga Kosheleva ◽  
Vladik Kreinovich
Keyword(s):  
Convex Body ◽  
Linear Order ◽  
Convex Cone ◽  
Causal Structure ◽  
Convex Cones ◽  
The Body ◽  
Quadratic Cone ◽  
Affine Invariant ◽  
General Convex ◽  
The One

Some physicists suggest that to more adequately describe the causal structure of space-time, it is necessary to go beyond the usual pseudoRiemannian causality, to a more general Finsler causality. In this general case, the set of all the events which can be influenced by a given event is, locally, a generic convex cone, and not necessarily a pseudo-Reimannian-style quadratic cone. Since all current observations support pseudo-Riemannian causality, Finsler causality cones should be close to quadratic ones. It is therefore desirable to approximate a general convex cone by a quadratic one. This can be done if we select a hyperplane, and approximate intersections of cones and this hyperplane. In the hyperplane, we need to approximate a convex body by an ellipsoid. This can be done in an affine-invariant way, e.g., by selecting, among all ellipsoids containing the body, the one with the smallest volume; since volume is affine-covariant, this selection is affine-invariant. However, this selection may depend on the choice of the hyperplane. It is therefore desirable to directly approximate the convex cone describing Finsler causality with the quadratic cone, ideally in an affine-invariant way. We prove, however, that on the set of convex cones, there is no affine-covariant characteristic like volume. So, any approximation is necessarily not affine-invariant.

scholarly journals A note on Gordan's theorem over cone domains

1982 ◽  
Vol 5 (4) ◽  
pp. 809-812 ◽  
Author(s):  
Brad Skarpness ◽  
V. A. Sposito
Keyword(s):  
Convex Cone ◽  
Convex Cones ◽  
Closed Convex Cone ◽  
Polar Cones ◽  
Natural Way

This note presents a proof of Gordan's Theorem over general closed, convex cone domains which follows in a natural way appealing to the standard definitions of closed convex cones and their respective polar cones.

A mass-conserved tumor invasion system with quasi-variational degenerate diffusion

2021 ◽  
pp. 1-66
Author(s):  
Akio Ito
Keyword(s):  
Hilbert Space ◽  
Tumor Cells ◽  
Tumor Invasion ◽  
Real Hilbert Space ◽  
Initial Boundary ◽  
Inner Product ◽  
Boundary Problems ◽  
Theory Of Evolution ◽  
Cross Diffusion ◽  
Variational Structure

This paper deals with a nonlinear system (S) composed of three PDEs and one ODE below: [Formula: see text] The system (S) was proposed as one of the mathematical models which describe tumor invasion phenomena with chemotaxis effects. The most important and interesting point is that the diffusion coefficient of tumor cells, denoted by [Formula: see text], is influenced by both nonlocal effect of a chemical attractive substance, denoted by [Formula: see text], and the local one of extracellular matrix, denoted by [Formula: see text]. From this point, the first PDE in (S) contains a nonlinear cross diffusion. Actually, this mathematical setting gives an inner product of a suitable real Hilbert space, which governs the dynamics of the density of tumor cells [Formula: see text], a quasi-variational structure. Hence, the first purpose in this paper is to make it clear what this real Hilbert space is. After this, we show the existence of strong time local solutions to the initial-boundary problems associated with (S) when the space dimension is [Formula: see text] by applying the general theory of evolution inclusions on real Hilbert spaces with quasi-variational structures. Moreover, for the case [Formula: see text] we succeed in constructing a strong time global solution.

scholarly journals The projection operator in a Hilbert space and its directional derivative. Consequences for the theory of projected dynamical systems

2004 ◽  
Vol 2 (1) ◽  
pp. 71-95 ◽  
Author(s):  
George Isac ◽  
Monica G. Cojocaru
Keyword(s):  
Hilbert Space ◽  
Dynamical Systems ◽  
Representation Theorem ◽  
Projection Operator ◽  
Directional Derivative ◽  
Metric Projection ◽  
Pseudomonotone Operators ◽  
Projected Dynamical Systems ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space

In the first part of this paper we present a representation theorem for the directional derivative of the metric projection operator in an arbitrary Hilbert space. As a consequence of the representation theorem, we present in the second part the development of the theory of projected dynamical systems in infinite dimensional Hilbert space. We show that this development is possible if we use the viable solutions of differential inclusions. We use also pseudomonotone operators.

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