The formula for the number of order-preserving selfmappings of a fence

LetY be a fence of sizem andr=?m?1/2?. The numberb(m) of order-preserving selfmappings ofY is equal toA _r-B_r-C_r-D_r, where, ifm is odd, $$\begin{gathered} A_r = 2(r + 1)\sum\limits_{s = 0}^r {\left( {\begin{array}{*{20}c} {r + s} \\ {2s} \\ \end{array} } \right)} 4^s , B_r = 2r\sum\limits_{s = 1}^r {\left( {\begin{array}{*{20}c} {r + s} \\ s \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {r - 1} \\ {s - 1} \\ \end{array} } \right),} \hfill \\ C_r = 4r\sum\limits_{s = 0}^{r - 1} {\left( {\begin{array}{*{20}c} {r + s} \\ s \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {r - 1} \\ s \\ \end{array} } \right), D_r = \sum\limits_{s = 0}^{r - 1} {(2s + 1)} \left( {\begin{array}{*{20}c} {r + s - 1} \\ s \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {r - 1} \\ s \\ \end{array} } \right)} \hfill \\ \end{gathered} $$ . Ifm is even, a similar formula forb(m) is true. The key trick in the proof is a one-to-one correspondence between order-preserving selfmappings ofY and pairs consisted of a partition ofY and a strictly increasing mapping of a subfence ofY toY.     

Multifunctions and the fixed point property for products of ordered sets

Sufficient conditions for the fixed point property for products of two partially ordered sets are proved. These conditions are formulated in terms of multifunctions (functions with non-empty sets as values).     

Hydration of gas-phase ytterbium ion complexes studied by experiment and theory

Hydration of ytterbium (III) halide/hydroxide ions produced by electrospray ionization was studied in a quadrupole ion trap mass spectrometer and by density functional theory (DFT). Gas-phase YbX₂ ⁺ and YbX(OH)⁺ (X?=?OH, Cl, Br, or I) were found to coordinate from one to four water molecules, depending on the ion residence time in the trap. From the time dependence of the hydration steps, relative reaction rates were obtained. It was determined that the second hydration was faster than both the first and third hydrations, and the fourth hydration was the slowest; this ordering reflects a combination of insufficient degrees of freedom for cooling the hot monohydrate ion and decreasing binding energies with increasing hydration number. Hydration energetics and hydrate structures were computed using two approaches of DFT. The relativistic scalar ZORA approach was used with the PBE functional and all-electron TZ2P basis sets; the B3LYP functional was used with the Stuttgart relativistic small-core ANO/ECP basis sets. The parallel experimental and computational results illuminate fundamental aspects of hydration of f-element ion complexes. The experimental observations??kinetics and extent of hydration??are discussed in relationship to the computed structures and energetics of the hydrates. The absence of pentahydrates is in accord with the DFT results, which indicate that the lowest energy structures have the fifth water molecule in the second shell.     

Ab initio studies of electron acceptor-donor interactions with blue- and red-shifted hydrogen bonds.

The features of blue- and red-shifted electron acceptor-donor (ACH/B) hydrogen bonds have been compared by using quantum chemical calculations. The geometry, the interaction energy and the vibrational frequencies of both blue- (ACH=F3CH, Cl3CH with B=FCD3) and red-shifted (ACH=F3CH, Cl3CH with B=NH3 and ACH=CH3CCH with B=FCD3, NH3) complexes were obtained by using ab initio MP2(Full)/6-31+G(d,p) calculations with the a priori basis-set superposition error (BSSE) correction method. One-dimensional potential energy and dipole moment functions of the dimensionless normal coordinate Q1, corresponding to the CH stretching mode of ACH, have been compared for both types of complexes. Contributions of separate components of the interaction energy to the frequency shift and the effect of electron charge transfer were examined for a set of intermolecular distances by using the symmetry-adapted perturbation theory (SAPT) approach and natural bond orbitals (NBO) population analysis.     

The mass spectra of some 2- and 4-pyridylethylamines

The mass spectra of some N,N-disubstituted 2-aminoethylpyridines are reported and their principle fragmentation routes analysed.     

Nonlinear effects in photonic crystal fibers filled with nematic liquid crystals

We have investigated the nonlinear propagation of light in photonic crystal fibers filled with nematic liquid crystals. We analyzed a configuration with a periodic modulation of the refractive index corresponding to a matrix of waveguides. Matrices of coupled waveguides allow observing a variety of new phenomena both for low power light beam propagation and with an existence of nonlinear effects. The opportunity for the creation of solitary waves caused by the interplay between diffraction and nonlinear effects in these kinds of fibers is investigated. At low power the propagating light beam spreads as it couples to more and more waveguides. When the intensity is increased the light modifies the refractive index distribution, inducing a defect in the periodic structure. The creation of such a defect can lead to a situation in which the light becomes self-localized and its diffractive broadening is eliminated. Eventually, in the case of positive Kerr-type nonlinearity, a discrete soliton can be created. In the case of negative nonlinearity the refractive index decreases with the optical power and can lead to bandgap shifting. The incident beam, with a frequency initially within the bandgap, is then turned outside the bandgap resulting in the changing of the propagation effect for the discrete diffraction effect. As a consequence the delocalization of the light can be observed. Presented at 9-th International Workshop on Nonlinear Optics Application, NOA 2007, May 17–20, 2007, Świnoujscie, Poland     

The fixed point property for small sets

There exist exactly eleven (up to isomorphism and duality) ordered sets of size 10 with the fixed point property and containing no irreducible elements.The great part of the work presented here has been done when the author was visiting Ivan Rival at the University of Ottawa, Department of Computer Science.     

Car-Parrinello molecular dynamics study of the blue-shifted F3CH...FCD3 system in liquid N2.

Fluoroform, as confirmed by both experimental and theoretical studies, can participate in improper H-bond formation, which is characterized by a noticeable increase in the fundamental stretching frequency nu(C-H) (so-called blue frequency shift), an irregular change of its integral intensity, and a C-H bond contraction. A Car-Parrinello molecular dynamics simulation was performed for a complex formed by fluoroform (F3CH) and deuterated methyl fluoride (FCD3) in liquid nitrogen. Vibrational analysis based on the Fourier transform of the dipole moment autocorrelation function reproduces the blue shift of the fundamental stretching frequency nu(C-H) and the decrease in the integral intensity. The dynamic contraction of the C-H bond is also predicted. The stoichiometry of the solvated, blue-shifted complexes and their residence times are examined.