Problems of the Theory of Rated Stability

Consider a new concept of stability of the so-called rated motion of a system of an ordinary differential equations. A rated motion need not be a solution of this system at all. In terms of Lyapunovs direct method, formulate and prove certain statements about rated stability, asymptotically rated stability, and rated unstability of zero motion of a system of an ordinary differential equations.AMS Subject Classification (1991): 34D05 34D35 93D30 93D20     

Towards an explanation of the mesoscopic double-slit experiment: A new model for charging of a quantum Dot

For a quantum dot (QD) in the intermediate regime between integrable and fully chaotic, the widths of single-particle levels naturally differ by orders of magnitude. In particular, the width of one strongly coupled level may be larger than the spacing between other, very narrow, levels. In this case many consecutive Coulomb blockade peaks are due to occupation of the same broad level. Between the peaks the electron jumps from this level to one of the narrow levels, and the transmission through the dot at the next resonance essentially repeats that at the previous one. This offers a natural explanation to the recently observed behavior of the transmission phase in an interferometer with a QD.     

Non-linear applications of microstructured optical fibres

The enhancement of different non-linear processes in microstructured optical fibres can be achieved through manipulation of the dispersion characteristics of the fibre. This is demonstrated by extending the region of short wavelength operation of high power supercontinuum generation through four wave mixing in a cascaded fibre geometry where the dispersion of each fibre decreased on propagation. The technique is further refined in a demonstration utilizing long lengths of dispersion decreasing tapered microstructured fibres, where the supercontinuum extends to around 300 nm with spectral power densities in excess of 2 mW/nm in the uv. These long length tapers can also be utilized for adiabatic soliton pulse compression in new spectral regions, allowing the compression of 655 fs pulses to 45 fs.     

Theoretical ROVibrational Energies (TROVE): A robust numerical approach to the calculation of rovibrational energies for polyatomic molecules

We present a new computational method with associated computer program TROVE (Theoretical ROVibrational Energies) to perform variational calculations of rovibrational energies for general polyatomic molecules of arbitrary structure in isolated electronic states. The (approximate) nuclear kinetic energy operator is represented as an expansion in terms of internal coordinates. The main feature of the computational scheme is a numerical construction of the kinetic energy operator, which is an integral part of the computation process. Thus the scheme is self-contained, i.e., it requires no analytical pre-derivation of the kinetic energy operator. It is also general, since it can be used in connection with any internal coordinates. The method represents an extension of our model for pyramidal XY₃ molecules reported previously [S.N. Yurchenko, M. Carvajal, P. Jensen, H. Lin, J.J. Zheng, W. Thiel, Mol. Phys. 103 (2005) 359]. Non-rigid molecules are treated in the Hougen-Bunker-Johns approach [J.T. Hougen, P.R. Bunker, J.W.C. Johns, J. Mol. Spectrosc. 34 (1970) 136]. In this case, the variational calculations employ a numerical finite basis representation for the large-amplitude motion using basis functions that are generated by Numerov-Cooley integration of the appropriate one-dimensional Schrödinger equation.     

On congruences with products of variables from short intervals and applications

We obtain upper bounds on the number of solutions to congruences of the type (x ₁ + s)... (x _ν + s) ≡ (y ₁ + s)... (x _ν + s) ? 0 (mod p) modulo a prime p with variables from some short intervals. We give some applications of our results and in particular improve several recent estimates of J. Cilleruelo and M.Z. Garaev on exponential congruences and on cardinalities of products of short intervals, some double character sum estimates of J. Friedlander and H. Iwaniec and some results of M.-C. Chang and A.A. Karatsuba on character sums twisted with the divisor function.     

Cationation of dimethylallyl alcohols by B(C6F5)3 as models of the (Re)initiation reaction in the bio‐inspired cationic polymerization of isoprene

The polymerization of isoprene initiated by dimethylallyl alcohols (DMAOH) in the presence of Lewis acids (LAs) as coinitiators has been selected as a potential model of the proposed cationic mechanisms involved in natural rubber (NR) biosynthesis. In view to investigate the activation/ionization mechanism of the allyl alcohol chain terminus (PIAllOH), which was shown to exist in NR, different isomers of DMAOH were used as simple models of polyisoprene chain‐end structures in the presence of tris(pentafluorophenyl)borane (BLA) as a LA. It is shown that cationation of 3,3‐DMAOH by BLA proceeds by direct OH abstraction. However, this process is strongly retarded in the presence of 2,6‐di‐tert‐butylpyridine (d^tBP), due to the formation of different BLA complexes, one with 3,3‐DMAOH, active for cationation, and a dormant one involving its ionized form 3,3‐DMAO^?. The monomer generated in situ by ionization/proton elimination steps subsequently adds on the primary allylic carbocation form, then resulting in the formation of oligoisoprenes, whereas the tertiary carbocation form of the allylic carbocation yields exclusively to proton elimination and isoprene formation, a possible mechanism of chain‐end termination. © 2011 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2011.     

The vertex isoperimetric problem for the powers of the diamond graph

We introduce a new graph for all whose cartesian powers the vertex isoperimetric problem has nested solutions. This is the fourth kind of graphs with this property besides the well-studied graphs like hypercubes, grids, and tori. In contrast to the mentioned graphs, our graph is not bipartite. We present an exact solution to the vertex isoperimetric problem on our graph by introducing a new class of orders that unifies all known isoperimetric orders defined on the cartesian powers of graphs.     

trans‐1,2,3‐Tris(1‐hydroxycyclo­propyl)cyclopropane

The central three‐membered ring in the title compound, trans‐1,1′,1′′‐cyclo­propane‐1,2,3‐triyl­tris­(cyclo­propanol), C₁₂H₁₈O₃, shows pronounced asymmetry of the bond lengths, which is induced by the different orientations of the substituents. A network of hydrogen bonds links the mol­ecules into sheets.