The maximum modulus of a trigonometric trinomial

Let Lambda be a set of three integers and let C_Lambda be the space of 2pi-periodic functions with spectrum in Lambda endowed with the maximum modulus norm. We isolate the maximum modulus points x of trigonometric trinomials T in C_Lambda and prove that x is unique unless |T| has an axis of symmetry. This permits to compute the exposed and the extreme points of the unit ball of C_Lambda, to describe how the maximum modulus of T varies with respect to the arguments of its Fourier coefficients and to compute the norm of unimodular relative Fourier multipliers on C_Lambda. We obtain in particular the Sidon constant of Lambda

The Necessary and Sufficient Conditions for Representing Lipschitz Bivariate Functions as a Difference of Two Convex Functions

In the article the necessary and sufficient conditions for a representation of Lipschitz function of two variables as a difference of two convex functions are formulated. An algorithm of this representation is given. The outcome of this algorithm is a sequence of pairs of convex functions that converge uniformly to a pair of convex functions if the conditions of the formulated theorems are satisfied. A geometric interpretation is also given

Hydrogen-bonding interactions in monomeric dimethylcuprates. A theoretical study

The hydrogen-bonding interactions between dimethylcuprate anions of monomeric lithium dimethylcuprate solvates and C–H bonds of molecules which solvate lithium cation were studied. The NBO analysis and analysis of internuclear distances are used to reveal these interactions in B3LYP and MP2 calculated monomeric lithium dimethylcuprates (LDMCs), and in the published X-ray structures of monomeric dimethylcuprates. The relevance of QTAIM analysis for the description of hydrogen-bonding interactions in LDMCs is evaluated. Different types of hydrogen bonds (HBs), e.g. C–H···Cu and C–H···C HBs as well as C–H···H dihydrogen bonds are revealed by the calculations of monomeric lithium dimethylcuprates with the cation surrounded by solvent molecules. All of these HB types were found to exist in the solid state between dimethylcuprate anion and C–H bonds of the solvent or ligand molecules. Multiple HBs of different types and strengths are simultaneously formed with dimethylcuprate anions in the monomeric cuprates studied. It is demonstrated that multiple hydrogen-bondings, accompanied by an unsymmetrical interaction of the dimethylcuprate anion with C–H bonds, can lead to impressive geometrical changes, including e.g. different C–Cu bond lengths in some “solvent-separated ion pairs”. In some monomeric cuprates the C–Cu–C unit of dimethylcuprate anion was found to bend even in the absence of Cu···Cu interactions

Structure of (Me2Cu)Li • 3DME and its oligomers [(Me2Cu)Li • 3DME]n (n = 2-5): a theoretical study

The structures of monomeric lithium dimethylcuprate solvated by three 1,2-dimethoxyethane (DME) molecules (Me2Cu)Li · 3DME and its oligomers [(Me2Cu)Li · 3DME] n (n = 2–5) in the gas phase were studied using the density functional theory. The structures of these systems are compared with that of (Me2Cu)Li · 3DME in the crystal. It is shown that numerous C-H…Cu and C-H…C hydrogen bonds, as well as C-H…H dihydrogen bonds between DME molecules, which solvate lithium cations, and corresponding dimethylcuprate anions significantly affect the formation and structure of these compounds in the gas and solid phases

The neutron field and radiation shielding efficiency measurement and the luminosity monitoring using the HF RADMON system of the CMS setup

The HF RADMON system monitors radiation environment in the region of the forward calorimeter of the CMS setup. Measurements of neutron fluxes at energies of 7, 8, and 13 GeV are carried out, estimates of the radiation shielding efficiency are obtained. Modernization of the system for operating under high luminosity conditions is completed and its capabilities to be used as an additional luminometer are confirmed

DFT Investigation of the η(6) ⇌ η(6)-Inter-ring Haptotropic Rearrangement of the Group 8 Metals Complexes [(graphene)MCp](+) (M = Fe, Ru, Os)

International audienceMetalcyclopentadienyl complexes (MCp) (M = Fe, Ru, Os) bound to the large polyaromatic hydrogenated hydrocarbon (PAH) CH used as a model for pristine graphene have been studied using a density functional theory (DFT) generalized gradient approximation (PBE functional) to reveal their structural features and dynamic behavior. The inter-ring haptotropic rearrangements (IRHRs) for these complexes were shown to occur via two transition states and one intermediate. The energy barriers of the η ⇌ η IRHRs of the (MCp) unit were found to be 30, 27, and 29 kcal/mol for M = Fe, Ru, and Os, respectively. These values are significantly lower than the values found previously for smaller PAHs. Both polar and nonpolar solvents were found not to affect significantly the energy barrier heights. Investigated transition metal complexes could be used in general as catalysts in the design of novel derivatives or materials with promising properties. Metalcyclopentadienyl complexes (MCp) of PAHs show catalytic properties mainly due to their structural details as well as their important characteristic of inter-ring haptotropic rearrangement. IRHRs take place usually by intramolecular mechanisms. During IRHRs, the ML organometallic groups (OMGs) undergo shifting along the PAH plane and could coordinate additional reagents, which is important for catalysis. Large PAHs such as graphene, fullerenes, and nanotubes possess intrinsic anticancer activity, and numerous arene complexes of Ru and Os have been proven to have anticancer properties as well. We suppose that coordinating Ru or Os to very large PAHs could synergistically increase the anticancer activity of resulting complexes