On a Pre-Lie Algebra Defined by Insertion of Rooted Trees

Let K be a field of characteristic zero. For \({n \in \mathbb{N}^{*}}\) , let \({\mathcal{T}^{\,\prime}_{n}}\) be the vector space of non-planar rooted trees with n vertices (Foissy in Bull Sci Math 126, no. 3, 193–239; no. 4, 249–288, 2002). Let \({\vartriangleright}\) be the left pre-Lie product of insertion of a tree inside another defined on infinitesimal characters of the graded Hopf algebra \({\mathcal{H}}\) introduced by Calaque, Ebrahimi-Fard and Manchon. Let \({\mathcal{T}^{\,\prime}=\oplus_{n\geq 2}\mathcal{T}^{\,\prime}_{n}}\) . In this work, we first prove that \({(\mathcal{T}^{\,\prime}, \vartriangleright)}\) a pre-Lie algebra generated by the two ladders E ₁ and E ₂ where E ₁ is the ladder with one edge and E ₂ is the ladder with two edges. Second, we prove that \({(\mathcal{T}^{\,\prime}, \vartriangleright)}\) is not a free pre-Lie algebra, and we exhibit a family of relations.     

Synthesis of Tacrine Analogs Derived from N-Aryl-5-amino-4-cyanopyrazoles

Synthesis of 11 tacrine analogs derived from N-aryl-5-amino-4-cyanopyrazoles, by a Friedländer type reaction, is described. Their structures were confirmed by ¹H and ¹³C NMR spectroscopy, elemental analysis, and/or mass spectrometry.     

Théorèmes de trace de type Szegö dans le cas singulier

Let f be a function eiθ→f(eiθ)=|1−eiθ|^2αf₁(eiθ) with f₁ a regular strictly positive function and a real number α in ]−1/2,1/2[?{0}. In a previous paper for such a number α we have obtained the asymptotic behaviour of the entries of the inverse of the Toeplitz matrix TN,f when N goes to infinity. These results allow us to give trace formulas, which extend a classical expression of Szegö's limits theorems.     

Finding a Zero of The Sum of Two Maximal Monotone Operators

In this paper, the equivalence between variational inclusions and a generalized type of Weiner–Hopf equation is established. This equivalence is then used to suggest and analyze iterative methods in order to find a zero of the sum of two maximal monotone operators. Special attention is given to the case where one of the operators is Lipschitz continuous and either is strongly monotone or satisfies the Dunn property. Moreover, when the problem has a nonempty solution set, a fixed-point procedure is proposed and its convergence is established provided that the Brézis–Crandall–Pazy condition holds true. More precisely, it is shown that this allows reaching the element of minimal norm of the solution set.     

Coupling proximal methods and variational convergence

An approximation method which combines a data perturbation by variational convergence with the proximal point algorithm, is presented. Conditions which guarantee convergence, are provided and an application to the partial inverse method is given.     

Split Monotone Variational Inclusions

Based on the very recent work by Censor-Gibali-Reich (), we propose an extension of their new variational problem (Split Variational Inequality Problem) to monotone variational inclusions. Relying on the Krasnosel’skii-Mann Theorem for averaged operators, we analyze an algorithm for solving new split monotone inclusions under weaker conditions. Our weak convergence results improve and develop previously discussed Split Variational Inequality Problems, feasibility problems and related problems and algorithms.     

Absolute Valued Algebras with Involution

We study absolute valued algebras with involution, as defined in Urbanik (1961 Urbanik , K. ( 1961 ). Absolute valued algebras with an involution . Fundamenta Math. 49 : 247 – 258 . [Google Scholar]). We prove that these algebras are finite-dimensional whenever they satisfy the identity (x, x ², x) = 0, where (·, ·, ·) means associator. We show that, in dimension different from two, isomorphisms between absolute valued algebras with involution are in fact *-isomorphisms. Finally, we give a classification up to isomorphisms of all finite-dimensional absolute valued algebras with involution. As in the case of a parallel situation considered in Rochdi (2003 Rochdi , A. ( 2003 ). Eight-dimensional real absolute valued algebras with left unit whose automorphism group is trivial . Int. J. Math. Math. Sci. 70 : 4447 – 4454 .[Crossref] , [Google Scholar]), the triviality of the group of automorphisms of such an algebra can happen in dimension 8, and is equivalent to the nonexistence of 4-dimensional subalgebras.     

Investigation of the threshold intensity versus gas pressure in the breakdown of helium by 248 nm laser radiation

An investigation of the unexpectedly strong dependence of the threshold intensity on the gas pressure in the experimental study on the breakdown of He by short laser wavelength (Turcu et al., in Opt Commun, 134:66–68, 1997) is presented. A modified electron cascade model is applied (Evans and Gamal, in J Phys D Appl Phys, 13:1447–1458, 1980). Computations revealed reasonable agreement between the calculated thresholds and the measured ones. Moreover, the calculated electron energy distribution function and its parameters proved that multiphoton ionization of ground and excited atoms is the main source for the seed electrons, which contributes to the breakdown of helium. The effect of diffusion losses over pressures <1,000 Torr elucidated the origin of the strong dependence of the threshold intensity on the gas pressure. Collisional ionization dominates only at high pressures. No evidence for recombination losses is observed for pressures up to 3,000 Torr.