Operateurs essentiellement reguliers dans les espaces de Banach

Given a closed operatorA acting in a Banach spaceX, we define the regular (respectively the essentialy regular) spectrum σ _r (A) (respectively σ _e,r (A)) ofA. We prove that σ _r (A) and σ _e,r (A) are a closed subsets of the classical spectrum σ(A) ofA. Morever ifA is bounded we prove that σ _r (A) and σ _e,r (A)) satisfies the spectral mapping theorem.     

Thermodynamic properties of the NdBr3–MBr binary systems (M = Na,K)

Thixoforming involves heating different types of alloys to the semisolid state at high heating rates and forming in die-casting machines or conventional presses. At temperatures higher than the solidus and lower than the liquidus, the mush metal behaves like a high-viscosity thixotropic material. Therefore, determining the thermodynamic behavior of the solid-to-liquid transition is paramount to control thixoforming processes. This article describes a simple, novel experimental setup based on differential thermal analysis (DTA) for analyzing the phase transitions in an alloy heated using high heating rates typical of industrial applications. A365 alloy was chosen to demonstrate the effectiveness of the method as the phase transformations for this alloy in semisolid materials (SSM) processing are well understood. Samples were heated to 750 °C using constant linear heating rates of 5, 10, 15, 20, 25, 50, 75, 100 and 125 °C min in a Norax 25 kW 8 kHz induction furnace with an Omron E5CK temperature controller. AISI 316 austenitic stainless steel was used as the inert reference. Comparison of the results of DTA using the proposed method and the results of simulation with Thermo-Calc^® indicates that the proposed in situ DTA device and its method is suitable for analyzing phase transitions when high heating rates are used.     

Index of B-Fredholm operators and generalization of a Weyl theorem

The aim of this paper is to show that if and are commuting B-Fredholm operators acting on a Banach space , then is a B-Fredholm operator and , where means the index. Moreover if is a B-Fredholm operator and is a finite rank operator, then is a B-Fredholm operator and We also show that if is isolated in the spectrum of , then is a B-Fredholm operator of index if and only if is Drazin invertible. In the case of a normal bounded linear operator acting on a Hilbert space , we obtain a generalization of a classical Weyl theorem.     

B-Fredholm Properties of Closed Invertible Operators

In this paper, we study B-Fredholm spectral properties of an invertible closed linear operator in relation with the B-Fredholm spectral properties of its bounded inverse. Precisely, for such operator, we characterize its B-Fredholm spectrum and other related spectra in terms of the corresponding spectra of its bounded inverse. As an application, we show that every normal operator with nonempty resolvent set, in particular self-adjoint Schrödinger operators, satisfies generalized Weyl’s theorem.

     

Tracing locations of new coating material during spark anodizing of titanium

The growth of anodic coatings on titanium, under sparking conditions, is investigated in tracer experiments, using alkaline silicate and phosphate electrolytes. Coatings are formed sequentially in each electrolyte, with phosphorus and silicon located by energy-dispersive X-ray analysis and glow discharge optical emission spectroscopy. The coatings, containing anatase, rutile and amorphous oxide, with incorporated phosphorus and silicon species, are shown to grow by discrete thickening at sites of dielectric breakdown. New material is found near the metal, within the coating bulk and at the coating surface. Approximately 10–30% of the new material is located near to the coating surface and about 40–60% near to the metal. The findings are attributed to the formation of breakdown channels allowing access of electrolyte species to the inner parts of the coating and to subsequent rapid formation of coating material, under high temperatures, associated with increased local current density, and high pressures, associated with volume constraints on oxide growth and gas generation.     

On the Property (gw)

In this note we introduce and study the property (gw), which extends property (w) introduced by Rakoc̆evic in [23]. We investigate the property (gw) in connection with Weyl type theorems. We show that if T is a bounded linear operator T acting on a Banach space X, then property (gw) holds for T if and only if property (w) holds for T and Π _a (T) = E(T), where Π _a (T) is the set of left poles of T and E(T) is the set of isolated eigenvalues of T. We also study the property (gw) for operators satisfying the single valued extension property (SVEP). Classes of operators are considered as illustrating examples. The second author was supported by Protars D11/16 and PGR- UMP.     

Théorème de l'application spectrale pour le spectre essentiel quasi-Fredholm

In 1958, T. Kato proved that a closed semi-Fredholm operator in a Banach space can be written where is a nilpotent operator and is a regular one.

J. P. Labrousse studied and characterised this class of operators in the case of Hilbert spaces. He also defined a new spectrum named ``essential quasi-Fredholm spectrum' and denoted .

In this paper we prove that the essential quasi-Fredholm spectrum defined by J. P. Labrousse satisfies the mapping spectral theorem, i.e.: If is a bounded operator in a Hilbert space and an analytic function in a neighbourhood of the spectrum of , then .

RÉSUMÉ. En 1958, T. Kato a montré que si est un opérateur fermé dans un espace de Banach et semi-Fredholm, alors il existe tels que où est nilpotent et est régulier.

J. P. Labrousse a étudié et caractérisé cette classe d'opérateurs dans le cadre des espaces de Hilbert et a défini un nouveau spectre qu'on appelle ``spectre essentiel quasi-Fredholm' et noté par .

Dans ce travail nous allons démontrer que le spectre essentiel quasi-Fredholm défini par J. P. Labrousse vérifie le théorème de l'application spectrale, c'est à dire: Si est un opérateur bourné d'un espace de Hilbert dans lui même et une fonction analytique au voisinage du spectre de , alors .

     

Effects of heat treatment and TiO2 content on the optical properties of Eu3+ doped TiO2–SiO2 thin films

The photoluminescence properties of Eu³⁺-doped TiO₂–SiO₂ thin films were investigated. The films were deposited on silicon substrates by the sol–gel process using the dip-coating method. The molar ratio of TiO₂ content was varied from 25% to 100%, while Europium concentration was fixed to 1%. The obtained films were calcinated at various temperatures ranging from 400 °C to 1300 °C, which allowed determining the optimal conditions for the Eu³⁺ luminescence. Meanwhile, the structure of TiO₂–SiO₂ powders, prepared in the same conditions as the films, was also studied by Raman spectroscopy. It revealed the role of Europium and SiO₂ on the stabilization of the anatase phase and the importance of the silica matrix in the control of titania particle size.     

Determination of the optimum conditions for the synthesis of praseodymium(III) chloride

In the present work, we have synthesized praseodymium(III) chloride, PrCl₃, from the praseodymium oxide, Pr₆O₁₁, by dry method in the presence of ammonium chloride, NH₄Cl. This study includes the establishment of an assembly synthesis under inert gas. The thermal decomposing process of pure NH₄Cl was investigated by TG–DTG. The results showed that NH₄Cl begins to lose weight at 188 °C, large loss of weight ending at 302 °C when NH₄Cl is heated at the rate of 10 °C/min under N₂ atmosphere. For chlorination, NH₄Cl participates directly in the reaction, and HCl decomposed from NH₄Cl also contributes to the chlorination reaction. The influence of various synthesis parameters (temperature, contact time and chemical composition) on the reaction yield was studied, and the optimum conditions for synthesis were, thus, determined and discussed.     

Phase diagram and electrical conductivity of the DyBr<Subscript>3</Subscript>–RbBr binary system

Phase equilibrium in the DyBr₃–RbBr binary system was established from differential scanning calorimetry measurements. This system exhibits three compounds, namely Rb₃DyBr₆, Rb₂DyBr₅, and RbDy₂Br₇, and two eutectics located at DyBr₃ molar fraction x = 0.116 (T = 886 K) and x = 0.458 (T = 702 K), respectively. Rb₃DyBr₆ undergoes a solid–solid phase transition at 726 K and melts congruently at 1,059 K. Rb₂DyBr₅ and RbDy₂Br₇ melt incongruently at 737 and 748 K, respectively. The electrical conductivity of DyBr₃–RbBr liquid mixtures was measured over the whole composition range. Results obtained are discussed in term of possible complex formation.