The stability of the exponential equation

We generalize the well-known Baker's superstability result for exponential mappings with values in the field of complex numbers to the case of an arbitrary commutative complex semisimple Banach algebra. It was shown by Ger that the superstability phenomenon disappears if we formulate the stability question for exponential complex-valued functions in a more natural way. We improve his result by showing that the maximal possible distance of an -approximately exponential function to the set of all exponential functions tends to zero as tends to zero. In order to get this result we have to prove a stability theorem for real-valued functions additive modulo the set of all integers .

     

Theoretical study of addition reactions of heavy carbenes to carbon and boron nitride nanotubes

In an effort to gain insight into the activation energies and reaction enthalpies of the chemical functionalization of carbon and boron nitride nanotubes, calculations using density functional theory have been carried out for the cycloaddition of a heavy carbene to a single-walled carbon (SWCNT; C(130)H(20)) and a boron nitride (SWBNNT; B(65)N(65)H(20)) nanotube. The (CH(3))(2)X + SWCNT and (CH(3))(2)X + SWBNNT (X = C, Si, Ge, Sn, and Pb) reactions are the subject of the present study. All the stationary points were determined at the B3LYP/LANL2DZ level of theory. The major conclusions that can be drawn from this work are as follows: (i) Considering both the activation barrier and reaction enthalpy based on the model calculations presented here, it is found that the order of (CH(3))(2)X reactivity is X = C > Si > Ge > Sn > Pb, irrespective of whether cycloaddition is to a SWCNT or a SWBNNT sidewall. That is to say, (CH(3))(2)C and (CH(3))(2)Si can readily add to the sidewalls of SWCNT and SWBNNT, whereas (CH(3))(2)Ge, (CH(3))(2)Sn, and (CH(3))(2)Pb are unreactive. (ii) Since the chemical reactivities of SWCNT and SWBNNT sidewalls closely resemble those of the small C(16)H(10) and B(8)N(8)H(10) molecules, at least in a qualitative sense, the use of the above small molecules as models is sufficient to provide qualitatively correct results. (iii) Our theoretical observations indicate that all the (5,5) SWCNT and SWBNNT cycloadducts favor opened rather than closed three-membered ring structures. (iv) The theoretical investigations demonstrate that the singlet-triplet splitting of the carbene species (R(2)X) as well as that of the small model molecules can be used as a diagnostic tool to predict the addition reactivities of carbene analogues and sidewalls of various nanotubes, respectively. Moreover, the results obtained in this work allow a number of predictions to be made.     

Mechanism of abstraction reactions of dimetallenes (R2X=XR2; X = C, Si, Ge, Sn, Pb) with halocarbons: a theoretical study

Potential energy surfaces for the abstraction reactions of dimetallenes with halocarbons have been studied using density functional theory (B3LYP). Five dimetallene species, (SiH(3))(2)X=X(SiH(3))(2), where X = C, Si, Ge, Sn, and Pb, have been chosen in this work as model reactants. The present theoretical investigations suggest that the relative dimetallenic reactivity increases in the order C=C < Si=Si < Ge=Ge < Sn=Sn < Pb=Pb. That is to say, for halocarbon abstractions there is a very clear trend toward lower activation barriers and more exothermic reactions on going from C to Pb. Moreover, for a given dimetallene, the overall barrier heights are determined to be in the order CF(4) > CCl(4) > CBr(4) > CI(4). That is, the heavier the halogen atom (Y), the more facile its abstraction from CY(4). Halogen abstraction is always predicted to be much faster than the abstraction of a CY(3) group irrespective of the dimetallene or halocarbon involved. Our model conclusions are consistent with some available experimental findings. Furthermore, both a configuration mixing model based on the work of Pross and Shaik and bonding dissociation energies can be used to rationalize the computational results. These results allow a number of predictions to be made.     

Theoretical study of halophilic reactions of stable silylenes with chloro- and bromocarbons

A theoretical study of the mechanism of the reaction of stable silylenes with halocarbons has been carried out using the B3LYP density functional method. The main findings are as follows: (1) Lewis acid-base complexes formed between silylenes and halocarbons do not play a role in silylene insertion chemistry into halocarbons; therefore, the acid-base complex mechanism proposed by West et al. (J. Am. Chem. Soc. 2002, 124, 4186) is not appropriate to describe the disilane formation reaction. (2) The disilane formation reactions follow the energetically favorable general reaction pathway (X = halogen): (i) Y2Si: + HCX3 --> TS1 --> Y2XSi-CHX2. (ii) Y2Si: + Y2XSi-CHX2 --> TS2 --> Y2XSi-SiY2CHX2. (3) The observed preference of stable silylenes to undergo C-X bond insertion rather than C-H bond has been investigated. The theoretical findings suggest that this preference is a result of the thermodynamic factor. (4) Stable silylenes prefer to insert into a C-Br rather than a C-Cl bond because the energy barrier to insertion is lower, and the reaction is more exothermic.     

On alienation of two functional equations of quadratic type

Aequationes mathematicae - &nbsp; We deal with an alienation problem for an Euler–Lagrange type functional equation $$\begin{aligned} f(\alpha x + \beta y) + f(\alpha x - \beta y) =...     

Homomorphisms and boundedness in topological groups

We consider homomorphisms from a normed space into a topological group. Assuming their boundedness in a neighbourhood (in relative topology) of an extremal point of the unit sphere, we derive their linearity (whenever it makes sense), closedness of the graph or continuity.     

The spin-quadrupole interaction and the 0<Superscript>+</Superscript> excitations in <Superscript>158</Superscript>Gd

In this work, the effect of spin-quadrupole forces on the 0⁺ sates in ¹⁵⁸Gd has been investigated. For this purpose, the model Hamiltonian including monopole pairing, quadrupole-quadrupole and spin-quadrupole forces has been diagonalized in one phonon basis. In conclusion, for the distribution of energies of the states and their collective properties, fairly good results have been obtained.