Fractional powers of the noncommutative Fourier's law by the S‐spectrum approach

Let e_?, for ? = 1,2,3, be orthogonal unit vectors in and let be a bounded open set with smooth boundary ?Ω. Denoting by a point in Ω, the heat equation, for nonhomogeneous materials, is obtained replacing the Fourier law, given by the following: into the conservation of energy law, here a, b, are given functions. With the S‐spectrum approach to fractional diffusion processes we determine, in a suitable way, the fractional powers of T. Then, roughly speaking, we replace the fractional powers of T into the conservation of energy law to obtain the fractional evolution equation. This method is important for nonhomogeneous materials where the Fourier law is not simply the negative gradient. In this paper, we determine under which conditions on the coefficients a, b, the fractional powers of T exist in the sense of the S‐spectrum approach. More in general, this theory allows to compute the fractional powers of vector operators that arise in different fields of science and technology. This paper is devoted to researchers working in fractional diffusion and fractional evolution problems, partial differential equations, and noncommutative operator theory.     

Synthesis and Characterization of Homoleptic and Heteroleptic Complexes Involving Dithiocarbamates,Triphenylphosphine, and Nickel(II)

Abstract

Six new nickel complexes of two dithiocarbamate ligands (cyfdtc = N-cyclohexyl-N- furfuryldithiocarbamate and bztpedtc = N-benzyl-N-[2-thiophenylethyl]dithiocarbamate) namely, (Ni[cyfdtc]₂) (1), (Ni[bztpedtc]₂) (2), (Ni[cyfdtc][NCS][PPh₃]) (3), (Ni[bztpedtc] [NCS][PPh₃]) (4), (Ni[cyfdtc][PPh₃]₂)ClO₄ (5), and (Ni[bztpedtc][PPh₃]₂)ClO₄ (6) have been prepared and characterized using IR, electronic, and NMR (¹H and ¹³C) spectra. A single crystal X-ray structural analysis was carried out for complex 3 and showed that nickel is in a distorted square planar arrangement with the NiS₂PN chromophore. The shift in ν_C?N of the heteroleptic complexes to higher frequencies compared with the parent complex is assigned to mesomeric delocalization of electron density from the

dithiocarbamate ligand toward the metal atom, which increases the contribution of polar thioureide form in mixed ligand complexes. Electronic spectral studies suggest square planar geometry for the complexes. In the ¹³C NMR spectra, the upfield shift of NCS₂ carbon signal for 3 and 4 from the chemical shift value of 1 and 2 is due to effect of PPh₃ on the mesomeric drift of electron density toward nickel throughout thioureide C?N bond.     

Asymmetric Hydrogenation of Unfunctionalized Tetrasubstituted Acyclic Olefins

Asymmetric hydrogenation has evolved as one of the most powerful tools to construct stereocenters. However, the asymmetric hydrogenation of unfunctionalized tetrasubstituted acyclic olefins remains the pinnacle of asymmetric synthesis and an unsolved challenge. We report herein the discovery of an iridium catalyst for the first, generally applicable, highly enantio‐ and diastereoselective hydrogenation of such olefins and the mechanistic insights of the reaction. The power of this chemistry is demonstrated by the successful hydrogenation of a wide variety of electronically and sterically diverse olefins in excellent yield and high enantio‐ and diastereoselectivity.     

Generating examples: focus on processes

Constructing an example can be a rich and complex activity, interesting to investigate mathematical thinking and with many potentialities in mathematics education. In this article, I analyse processes involved in example generation, with particular emphasis on production and transformation of signs representing mathematical objects and on generation of inferences. The richness and complexity of these processes will also be shown through the notions of prototypes, concept image and concept definition. This investigation reveals aspects that are significant both in education and for the reflection on cognitive and cultural aspects of mathematical thinking.     

Strongly correlated superconductivity and pseudogap phase near a multiband Mott insulator

Near a Mott transition, strong electron correlations may enhance Cooper pairing. This is demonstrated in the dynamical mean field theory solution of a twofold-orbital degenerate Hubbard model with an inverted on-site Hund rule exchange, favoring local spin-singlet configurations. Close to the Mott insulator (which here is a local version of a valence bond insulator) a pseudogap non-Fermi-liquid metal, a superconductor, and a normal metal appear, in striking similarity with the physics of cuprates. The strongly correlated s-wave superconducting state has a larger Drude weight than the corresponding normal state. The role of the impurity Kondo problem is underscored.