The correlations of finite Desarguesian planes, Part II: The classification (I)

This paper continues the classification of the correlations of planes of odd nonsquare order. Part I (Generalities) – see reference [1]-included introductory definitions and results (Section 1), algebraic preliminaries (Section 2), as well as a discussion of equivalent correlations (Section 3) and of their general properties (Section 4). The classification proper revolves around a special polynomial which can have one, two, or q + 1 zeros, or no zeros at all, and each of these four possibilities leads to different families of correlations. The present article contains Section 5, devoted to the cases in which the correlation is defined by a diagonal matrix (Subsection 5.1) or the polynomial in the preceding paragraph possesses q + 1 zeros (Subsection 5.2), one zero (Subsection 5.3) and two zeros (Subsection 5.4). Subsection 5.5 presents certain results to be used in the subsequent sections.     

Commutator Criteria for Magnetic Pseudodifferential Operators

The gauge covariant magnetic Weyl calculus has been introduced and studied in previous works. We prove criteria in terms of commutators for operators to be magnetic pseudodifferential operators of suitable symbol classes; neither the statements nor the proofs depend on a choice of a vector potential. We apply this criteria to inversion problems, functional calculus, affiliation results and to the study of the evolution group generated by a magnetic pseudodifferential operator.     

A note on the stabilizability of stochastic heat equations with multiplicative noiseSur la stabilisabilité des équations de la chaleur stochastiques avec bruit multiplicatif

We show that a stochastic heat equation with multiplicative noise on a bounded domain $\text{D}$ can be stabilized by a control acting only on a subdomain $\text{O}\text{?}\text{D}$ if $\text{D}\text{?}\text{O}$ is sufficiently ‘thin’. We consider both linear and semilinear stochastic heat equations. To cite this article: V. Barbu et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 311–316.     

Algebra Structures Arising from Yang–Baxter Systems

Yang–Baxter operators from algebra structures appeared for the first time in [11 D?sc?lescu , S. , Nichita , F. ( 1999 ). Yang–Baxter operators arising from (co)algebra structures . Comm. Algebra 27 : 5833 – 5845 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], 22 Nichita , F. ( 1999 ). Self-inverse Yang–Baxter operators from (co)algebra structures . J. Algebra 218 : 738 – 759 .[Crossref], [Web of Science ®] , [Google Scholar], 23 Nuss , P. ( 1997 ). Noncommutative descent and non-abelian cohomology . K-Theory 12 ( 1 ): 23 – 74 .[Crossref] , [Google Scholar]]. Later, Yang–Baxter systems from entwining structures were constructed in [8 Brzeziński , T. , Nichita , F. F. ( 2005 ). Yang–Baxter systems and Entwining Structures . Comm. Algebra 33 : 1083 – 1093 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. In fact, Yang–Baxter systems are equivalent with braid systems. In this paper we show that braidings and entwinings of various algebraic structures—in particular, algebra factorisations—can be constructed from a braid system, whence from a Yang–Baxter system as well.     

The Heat Equation for the Generalized Hermite and the Generalized Landau Operators

We give a formula for the one-parameter strongly continuous semigroups ${e^{-tL^{\lambda}}}We give a formula for the one-parameter strongly continuous semigroups e^-tLl{e^{-tL^{\lambda}}} and e^{-t [(A)\tilde]}{e^{-t \tilde{A}}}, t > 0 generated by the generalized Hermite operator L^l, l ? R\{0}{L^{\lambda}, \lambda \in {\bf R}\backslash \{0\}} respectively by the generalized Landau operator ?. These formula are derived by means of pseudo-differential operators of the Weyl type, i.e. Weyl transforms, Fourier-Wigner transforms and Wigner transforms of some orthonormal basis for L ²(R ²ⁿ ) which consist of the eigenfunctions of the generalized Hermite operator and of the generalized Landau operator. Applications to an L ² estimate for the solutions of initial value problems for the heat equations governed by L ^λ respectively ?, in terms of L ^p norm, 1 ≤ p ≤ ∞ of the initial data are given.     

An approximation theorem for sections of reflexive sheaves

 We show that, if X is a Stein manifold and D ? X an open set (not necessarily Stein) such that the restriction map has dense image, then, for any reflexive coherent analytic sheaf ℱ on X, the map has dense image, too. We also characterize the reflexivity of a torsion-free coherent sheaf on complex manifolds in terms of absolute gap sheaves or Kontinuit?tssatz. Received: 14 September 2001 / Revised version: 29 January 2002     

Necessary conditions for nonconvex distributed control problems governed by elliptic variational inequalities

In 1956, R. Penrose studied best-approximate solutions of the matrix equation AX = B. He proved that A⁺B (where A⁺ is the Moore-Penrose inverse) is the unique matrix of minimal Frobenius norm among all matrices which minimize the Frobenius norm of AX ? B. In particular, A⁺ is the unique best-approximate solution of AX = I. The vector version of Penrose's result (that is, the fact that the vector A⁺b is the best-approximate solution in the Euclidean norm of the vector equation Ax = b) has long been generalized to infinite dimensional Hilbert spaces.In this paper, an infinite dimensional version of Penrose's full result is given. We show that a straightforward generalization is not possible and provide new extremal characterizations (in terms of the Hermitian order) of A⁺ and of the classes of generalized inverses associated with minimal norm solutions of consistent operator equations or with least-squares solutions. For a certain class of operators, we can phrase our characterizations in terms of a whole class of norms (including the Hilbert-Schmidt and the trace norms), thus providing new extremal characterizations even in the matrix case. We treat both operators with closed range and with not necessarily closed range. Finally, we characterize A⁺ as the unique inner inverse of minimal Hilbert-Schmidt norm if ∥A⁺∥₂ < ∞. We give an application of the new extremal characterization to the compensation problem in systems analysis in infinite-dimensional Hilbert spaces.     

Investigation of thermal decomposition of yttrium–aluminum-based precursors for YAG phosphors

Three types of precursors were prepared using the wet-chemical synthesis route, starting from yttrium?Ceuropium?Caluminum nitrate solution and different precipitating agents (urea, oxalic acid, and ammonium carbonate). The precursors were fired at 1200?°C in nitrogen atmosphere in order to obtain europium-doped yttrium aluminate Y₃Al₅O₁₂:Eu³⁺ phosphor with garnet structure (YAG:Eu). The processes involved in the thermal decomposition of precursors and their composition were put in evidence using thermal analysis (TG?CDTA) and FT-IR spectroscopy. The GA?CDTA curves possess typical features for basic-oxalate, -nitrate, and -carbonates as formed with oxalic acid, urea, and ammonium carbonate, respectively. Correlation between the thermal decomposition steps, mass loss, and composition of gases evolved during the thermal treatment was established using TG?CDTA?CFT-IR coupling. It was found that the different composition of precursors reflects on the luminescent characteristics of the corresponding phosphors. Urea and ammonium carbonate lead to the formation of YAG type phosphors, with garnet structure and specific red emission. As for the oxalic acid, this precipitating agent generates a non-homogeneous powder that contains yttrium oxide as impurity phase. This phosphor is a mixture of Y₂O₃:Eu³⁺, Y₄Al₂O₉:Eu³⁺, and Y₃Al₅O₁₂:Eu³⁺ that explain the relative higher emission intensity.