Modified dispersion relations in Hořava–Lifshitz gravity and Finsler brane models

We study possible links between quantum gravity phenomenology encoding Lorentz violations as nonlinear dispersions, the Einstein–Finsler gravity models, EFG, and nonholonomic (non-integrable) deformations to Hořava–Lifshitz, HL, and/or Einstein’s general relativity, GR, theories. EFG and its scaling anisotropic versions formulated as Hořava–Finsler models, HF, are constructed as covariant metric compatible theories on (co) tangent bundle to Lorentz manifolds and respective anisotropic deformations. Such theories are integrable in general form and can be quantized following standard methods of deformation quantization, A-brane formalism and/or (perturbatively) as a nonholonomic gauge like model with bi-connection structure. There are natural warping/trapping mechanisms, defined by the maximal velocity of light and locally anisotropic gravitational interactions in a (pseudo) Finsler bulk spacetime, to four dimensional (pseudo) Riemannian spacetimes. In this approach, the HL theory and scenarios of recovering GR at large distances are generated by imposing nonholonomic constraints on the dynamics of HF, or EFG, fields.     

Einstein gravity in almost Kähler and Lagrange–Finsler variables and deformation quantization

A geometric procedure is elaborated for transforming (pseudo) Riemannian metrics and connections into canonical geometric objects (metric and nonlinear and linear connections) for effective Lagrange, or Finsler, geometries which, in turn, can be equivalently represented as almost Kähler spaces. This allows us to formulate an approach to quantum gravity following standard methods of deformation quantization. Such constructions are performed not on tangent bundles, as in usual Finsler geometry, but on spacetimes enabled with nonholonomic distributions defining 2+2$2+2$ splitting with associate nonlinear connection structure. We also show how the Einstein equations can be written in terms of Lagrange–Finsler variables and corresponding almost symplectic structures and encoded into the zero-degree cohomology coefficient for a quantum model of Einstein manifolds.     

Cusp geometry and the cobordism invariance of the index

The cobordism invariance of the index on closed manifolds is reproved using the calculus Ψ_c of cusp pseudodifferential operators on a manifold with boundary. More generally, on a compact manifold with corners, the existence of a symmetric cusp differential operator of order 1 and of Dirac type near the boundary implies that the sum of the indices of the induced operators on the hyperfaces is null.     

New conjugates of calix[4]arenes bearing dipyridine and indolizine heterocycles

A simple route to introduce various heterocycles, derivatives of dipyridyls and indolizines on the lower rim of the para-tert-butylcalix[4]arene via ester bond formation to afford 1,3-disubstituted conjugates is described. The conformation of the new compounds and some intermolecular interactions are discussed on the basis of X-ray and NMR analyses. Preliminary complexation properties of some of the new tert-butylcalix[4]arene heterocyclic conjugates with Cu (II), Co (II) and Ni (II) were studied by means of UV–Vis titration.     

Supramolecular synthesis based on piperidone derivatives and pharmaceutically acceptable co‐formers

The target complexes, bis{(E,E)‐3,5‐bis[4‐(diethylamino)benzylidene]‐4‐oxopiperidinium} butanedioate, 2C₂₇H₃₆N₃O⁺·C₄H₄O₄²⁻, (II), and bis{(E,E)‐3,5‐bis[4‐(diethylamino)benzylidene]‐4‐oxopiperidinium} decanedioate, 2C₂₇H₃₆N₃O⁺·C₁₀H₁₆O₄²⁻, (III), were obtained by solvent‐mediated crystallization of the active pharmaceutical ingredient (API) (E,E)‐3,5‐bis[4‐(diethylamino)benzylidene]‐4‐piperidone and pharmaceutically acceptable dicarboxylic (succinic and sebacic) acids from ethanol solution. They have been characterized by melting point, IR spectroscopy and single‐crystal X‐ray diffraction. For the sake of comparison, the structure of the starting API, (E,E)‐3,5‐bis[4‐(diethylamino)benzylidene]‐4‐piperidone methanol monosolvate, C₂₇H₃₅N₃O·CH₄O, (I), has also been studied. Compounds (II) and (III) represent salts containing H‐shaped centrosymmetric hydrogen‐bonded synthons, which are built from two parallel piperidinium cations and a bridging dicarboxylate dianion. In both (II) and (III), the dicarboxylate dianion resides on an inversion centre. The two cations and dianion within the H‐shaped synthon are linked by two strong intermolecular N⁺—H...⁻OOC hydrogen bonds. The crystal structure of (II) includes two crystallographically independent formula units, A and B. The cation geometries of units A and B are different. The main N—C₆H₄—C=C—C(=O)—C=C—C₆H₄—N backbone of cation A has a C‐shaped conformation, while that of cation B adopts an S‐shaped conformation. The same main backbone of the cation in (III) is practically planar. In the crystal structures of both (II) and (III), intermolecular N⁺—H...O=C hydrogen bonds between different H‐shaped synthons further consolidate the crystal packing, forming columns in the [100] and [10] directions, respectively. Salts (II) and (III) possess increased aqueous solubility compared with the original API and thus enhance the bioavailability of the API.     

Infinite Dimensional Parametric Optimal Control Problems

In this paper we study parametric optimal control problems monitored by nonlinear evolution equations. The parameter appears in all the data, including the nonlinear operator. First we show that for every value of the parameter, the optimal control problem has a solution. Then we study how these solutions as well as the value of the problem respond to changes in the parameter. Finally, we work out in detail two examples of nonlinear parabolic optimal control systems.     

Fractional almost Kähler–Lagrange geometry

The goal of this paper is to encode equivalently the fractional Lagrange dynamics as a nonholonomic almost Kähler geometry. We use the fractional Caputo derivative generalized for nontrivial nonlinear connections (N-connections) originally introduced in Finsler geometry, with further developments in Lagrange and Hamilton geometry. For fundamental geometric objects induced canonically by regular Lagrange functions, we construct compatible almost symplectic forms and linear connections completely determined by a “prime” Lagrange (in particular, Finsler) generating function. We emphasize the importance of such constructions for deformation quantization of fractional Lagrange geometries and applications in modern physics.     

Syntheses, electronic structures, and EPR/UV-vis-NIR spectroelectrochemistry of nickel(II), copper(II), and zinc(II) complexes with a tetradentate ligand based on S-methylisothiosemicarbazide

Template condensation of 3,5-di-tert-butyl-2-hydroxybenzaldehyde S-methylisothiosemicarbazone with pentane-2,4-dione and triethyl orthoformate at elevated temperatures resulted in metal complexes of the type M(II)L, where M = Ni and Cu and H(2)L = a novel tetradentate ligand. These complexes are relevant to the active site of the copper enzymes galactose oxidase and glyoxal oxidase. Demetalation of Ni(II)L with gaseous hydrogen chloride in chloroform afforded the metal-free ligand H(2)L. Then by the reaction of H(2)L with Zn(CH(3)COO)(2)·2H(2)O in a 1:1 molar ratio in 1:2 chloroform/methanol, the complex Zn(II)L(CH(3)OH) was prepared. The three metal complexes and the prepared ligand were characterized by spectroscopic methods (IR, UV-vis, and NMR spectroscopy), X-ray crystallography, and DFT calculations. Electrochemically generated one-electron oxidized metal complexes [NiL](+), [CuL](+), and [ZnL(CH(3)OH)](+) and the metal-free ligand cation radical [H(2)L](+?) were studied by EPR/UV-vis-NIR and DFT calculations. These studies demonstrated the interaction between the metal ion and the phenoxyl radical.     

Structural investigation of xAg2O·(100−x)·[2B2O3·As2O3] glasses doped with manganese ions

Structural analysis of x[(100−y)Ag₂O·yMnO]·(100−x)[2B₂O₃·As₂O₃] glasses, with x=10 mol% and 0≤y≤10 mol%, was performed by means of FT-IR and FT-Raman spectroscopies. The purpose of this work is to investigate the structural changes that appear in the xAg₂O·(100−x)·[2B₂O₃·As₂O₃] glasses with the addition and increase in manganese ions content. FT-IR measurements revealed the presence of pyro-, ortho-, di-, tri-, tetra- and penta-borate groups and structural units characteristic to As₂O₃ in the structure of the studied glasses. FT-IR spectroscopy measurements also show that BO₃ units are the main structural units of the glass system. The presence of structural units characteristic to Ag₂O were not directly evidenced by FT-IR spectroscopy. In addtition, the FT-Raman analysis evidenced the presence of boroxol rings in the structure of the studied glasses.