Ray Theorem Research Articles

Sub-Feller semigroups generated by pseudodifferential operators on symmetric spaces of noncompact type

We consider global pseudodifferential operators on symmetric spaces of noncompact type, defined using spherical functions. The associated symbols have a natural probabilistic form that extend the notion of the characteristic exponent appearing in Gangolli's Lévy–Khinchine formula to a function of two variables. The Hille–Yosida–Ray theorem is used to obtain conditions on such a symbol so that the corresponding pseudodifferential operator has an extension that generates a sub-Feller semigroup, generalising existing results for Euclidean space.

Feller semigroups and degenerate elliptic operators III

AbstractThis paper is devoted to the functional analytic approach to the problem of construction of Feller semigroups in the characteristic case via the Fichera function. Probabilistically, our result may be stated as follows: We construct a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves continuously in the interior of the state space, without reaching the boundary. We make use of the Hille–Yosida–Ray theorem that is a Feller semigroup version of the classical Hille–Yosida theorem in terms of the positive maximum principle. Our proof is based on a method of elliptic regularizations essentially due to Oleĭnik and Radkevič. The weak convergence of approximate solutions follows from the local sequential weak compactness of Hilbert spaces and Mazur's theorem in normed linear spaces.

Novel Design of Primary Optical Elements Based on a Linear Fresnel Lens for Concentrator Photovoltaic Technology

In this paper, we present a design and optical simulation of a novel linear Fresnel lens. The lens can be applied to a concentrator photovoltaic (CPV) system as a primary optical element (POE) to increase the concentration ratio and improve the uniformity of irradiance distribution over the receiver. In addition, the CPV system can use the proposed lens as a concentrator without involving a secondary optical element (SOE). The designed lens, which is a combination of two linear Fresnel lenses placed perpendicular to each other, can collect and distribute the direct sunlight on two dimensions. The lens is first designed in the MATLAB program, based on the edge ray theorem, Snell’s law, and the conservation of the optical path length, and then drawn in three dimensions (3D) by using LightToolsTM. Furthermore, in order to optimize the structure and investigate the performance of the lens, the ray tracing and the simulation are also performed in LightToolsTM. The results show that the newly designed lens can achieve a high concentration ratio of 576 times, a high optical efficiency of 82.4%, an acceptable tolerance of 0.84°, and high uniform irradiance of around 77% for both horizontal and vertical investigation lines over the receiver.

Design of Curved Fresnel Lens with High Performance Creating Competitive Price Concentrator Photovoltaic

In this paper, the design of a curved Fresnel lens applying to the concentrator photovoltaic system is proposed by using the edge ray theorem, the Snell’s law, and the conservation of optical path length. The new structure of the curved Fresnel lens can improve significantly the uniformity of sunlight distribution over the solar cell while the concentration ratio can reach a high value of 900 times. The good uniform distribution can be obtained by using the novel idea. The novel idea is based on the uniform sunlight distribution of every groove of the lens so that the whole lens also distributes uniformly the sunlight beam over the receiver. The structure of the lens is built by two surfaces: input surface (or upper surface) as a part of spherical surface and output surface (or lower surface) that consists all grooves of the lens. Matlab program is used to design the input and the output surfaces of the lens. The input surface and the output surface are independent to each other in construction in Matlab. LighttoolsTM software is used to optimize the structure of the novel lens. Furthermore, LighttoolsTM is also used to perform a simulation to examine the efficiency of the lens in concentrator photovoltaic (CPV) system by using the light source with a wide spectrum of 380 – 1600 nm. The results show that the lens has an acceptance angle of 0.80 and good optical efficiency (>85%).

Daylighting System Based on Novel Design of Linear Fresnel lens

In this paper, we present a design and optical simulation of a daylighting system using a novel design of linear Fresnel lens, which is constructed based on the conservation of optical path length and edge ray theorem. The linear Fresnel lens can achieve a high uniformity by using a new idea of design in which each groove of the lens distributes sunlight uniformly over the receiver so that the whole lens also uniformly distributes sunlight over the receiver. In this daylighting system, the novel design of linear Fresnel lens significantly improves the uniformity of collector and distributor. Therefore, it can help to improve the performance of the daylighting system. The structure of the linear Fresnel lenses is designed by using Matlab. Then, the structure of lenses is appreciated by ray tracing in LightToolsTM to find out the optimum lens shape. In addition, the simulation is performed by using LightToolsTM to estimate the efficiency of the daylighting system. The results show that the designed collector can achieve the efficiency of ~80% with the tolerance of ~0.60 and the concentration ratio of 340 times, while the designed distributor can reach a high uniformity of >90%.

On Ray's theorem for weak firmly nonexpansive mappings in Hilbert Spaces

In this work, we introduce notions of generalized firmly nonexpansive (G-firmly non expansive) and fundamentally firmly nonexpansive (F-firmly nonexpansive) mappings and utilize to the same to prove Ray's theorem for G-firmly and F-firmly nonexpansive mappings in Hilbert Spaces. Our results extend the result due to F. Kohsaka [ Ray's theorem revisited: a fixed point free firmly nonexpansive mapping in Hilbert spaces, Journal of Inequalities and Applications (2015) 2015:86 ].

THE FIXED POINT PROPERTY AND UNBOUNDED SETS IN BANACH SPACES

Let $E$ be a smooth, strictly convex and reflexive Banach space, let $J$ be the duality mapping of $E$ and let $C$ be a nonempty closed convex subset of $E$. Then, a mapping $S:C\to C$ is said to be nonspreading \cite{koh-tak3} if \[ \phi(Sx, Sy)+\phi(Sy, Sx)\le\phi(Sx, y)+\phi(Sy, x) \] for all $x, y\in C$, where $\phi(x,y)= \|x\|^2-2\langle x, Jy\rangle +\|y\|^2$ for all $x, y\in E$. In this paper, we prove that every nonspreading mapping of $C$ into itself has a fixed point in $C$ if and only if $C$ is bounded. This theorem extends Ray's theorem \cite{ray} in a Hilbert space to that in a Banach space.

Ray's Theorem for Firmly Nonexpansive-Like Mappings and Equilibrium Problems in Banach Spaces

AbstractWe prove that every firmly nonexpansive-like mapping from a closed convex subset "Equation missing" of a smooth, strictly convex and reflexive Banach pace into itself has a fixed point if and only if "Equation missing" is bounded. We obtain a necessary and sufficient condition for the existence of solutions of an equilibrium problem and of a variational inequality problem defined in a Banach space.

New perspectives on Ray's theorem for the local times of diffusions

A new global isomorphism theorem is obtained that expresses the local times of transient regular diffusions under $P^{x,y}$, in terms of related Gaussian processes. This theorem immediately gives an explicit description of the local times of diffusions in terms of $0$th order squared Bessel processes similar to that of Eisenbaum and Ray's classical description in terms of certain randomized fourth order squared Bessel processes. The proofs given are very simple. They depend on a new version of Kac's lemma for $h$-transformed Markov processes and employ little more than standard linear algebra. The global isomorphism theorem leads to an elementary proof of the Markov property of the local times of diffusions and to other recent results about the local times of general strongly symmetric Markov processes. The new version of Kac's lemma gives simple, short proofs of Dynkin's isomorphism theorem and an unconditioned isomorphism theorem due to Eisenbaum.

Application of the Weyl–Hörmander calculus to generators of Feller semigroups

AbstractIn this article we apply the S(M, g)–calculus of L. Hörmander and, in particular, results concerning the spectral invariance of the algebra of operators of order zero in ℒ︁(L2(ℝn)) to study generators of Feller semigroups. The core of the article is the proof of the invertibility of λ Id + P for a strongly elliptic operator P in Ψ(M, g) and suitable weight functions M and metrics g. The proof depends highly on precise estimates of the remainder term in asymptotic expansions of the product symbol in Weyl and Kohn–Nirenberg quantization. Due to the Hille–Yosida–Ray theorem and a theorem of Courrège, the result concerning the invertibility of λ Id + P is applicable to obtain sufficient conditions for an operator to extend to a generator of a Feller semigroup. Moreover, we discuss Sobolev spaces adapted to our weight functions and apply our results to study the generator of a subordinate Feller semigroup. As for the Feller semigroups and the subordination in the sense of Bochner, we build on results of N. Jacob and R. L. Schilling.