The relativistic theory of chemical binding

Starting from Breit’s relativistic equation for a system of two electrons, it is shown that for a hydrogen molecule (or for a system of two electrons moving in a field of cylindrical symmetry) the component of the total angular momentum (J _x ) along the axis of the molecule (axis of symmetry) is a constant of motion. Thus every eigenstate of the system is simultaneously an eigenstate of J _x also, and a state of the system will specify, besides its energy, only the eigenvalue of the component of the angular momentum parallel to the axis of symmetry. The form of the four large components of the wave function relating to their dependence on the azimuthal co-ordinates has been given. The case of Russel-Saunders approximation has been considered in detail and the nature of the components of the wave function for the singlet and triplet states has been discussed. It is shown that the wave function for the ground state of the hydrogen molecule could be expressed as a sum of a set of symmetric functions of which the first term is the Heitler-London function, and that the wave function for a triplet state should be a superposition of anti-symmetric molecular orbitals. It is shown that relativistic theory brings about in a natural manner the facts relating to the ground state of the molecules C₂ and O₂. Finally, some remarks are made concerning the case of molecules for which the spinorbit and the spin-spin couplings are strong.     

Focussing and defocussing of ballistic phonons in diamond and Nb3Sn

The ballistic propagation of phonons in diamond and Nb₃Sn at 40 and at 4.2 K is examined. The nature of variation of the phonon magnification factor has been analysed both in the wave vector as well as group velocity spaces. Using the Polar Schmidt-net with Pole at (π/2,π/2), the mappings of phonon focussing and defocussing are made. It is shown that the mapping for theFTA mode for Nb₃Sn exhibits islands acting as impenetrable barriers for phonon propagation.     

SU(5) gut phase transitions via 't Hooft's duality relation

't Hooft's duality relation is used to investigate the possible dynamical symmetry breaking pattern SU(5)→ SU(4) ? U(1) where the gauge fields of SU(4) may be in one of four possible phases: (i) confinement phase, (ii) Higgs phase, (iii) “self-dual phase”, (iv) Coulomb phase. It is found that the duality relation involving the electric and magnetic free energies is satisfied in all these cases.     

Tin-free intermolecular addition of primary alkyls to imines via the dimethylzinc-air radical process

[reaction: see text] A dimethylzinc-air initiator was applied to the generation of primary alkyl radicals from alkyl iodides. The addition of the generated primary alkyl radicals to N-tosylimines was accelerated by the action of boron trifluoride-diethyl etherate and copper(II) triflate to give the corresponding adducts in good yields after 2-3 h. Air oxygen was essential for the reaction to proceed, showing involvement of a radical process in the reaction.     

Scenario reduction for stochastic programs with Conditional Value-at-Risk

In this paper we discuss scenario reduction methods for risk-averse stochastic optimization problems. Scenario reduction techniques have received some attention in the literature and are used by practitioners, as such methods allow for an approximation of the random variables in the problem with a moderate number of scenarios, which in turn make the optimization problem easier to solve. The majority of works for scenario reduction are designed for classical risk-neutral stochastic optimization problems; however, it is intuitive that in the risk-averse case one is more concerned with scenarios that correspond to high cost. By building upon the notion of effective scenarios recently introduced in the literature, we formalize that intuitive idea and propose a scenario reduction technique for stochastic optimization problems where the objective function is a Conditional Value-at-Risk. Numerical results presented with problems from the literature illustrate the performance of the method and indicate the cases where we expect it to perform well.     

Mathematical programming models for revenue management under customer choice

We study a network airline revenue management problem with discrete customer choice behavior. We discuss a choice model based on the concept of preference orders, in which customers can be grouped according to a list of options in decreasing order of preference. If a customer’s preferred option is not available, the customer moves to the next choice on the list with some probability. If that option is not available, the customer moves to the third choice on the list with some probability, and so forth until either the customer has no other choice but to leave or his/her request is accepted. Using this choice model as an input, we propose some mathematical programs to determine seat allocations. We also propose a post-optimization heuristic to refine the allocation suggested by the optimization model. Simulation results are presented to illustrate the effectiveness of our method, including comparisons with other models.     

A new entry to enantiopure polysubstituted cyclopropanes: stereoselective denitrogenation of sulfinylpyrazolines under Yb(OTf)3 catalysis

[Structure: see text] Chemoselective and completely stereoselective denitrogenation of optically pure pyrazolines, derived from 3-sulfinylfuran-2(5H)-ones, into cyclopropanes can be achieved under substoichiometric Yb(OTf)3 catalysis. Reactions evolve in almost quantitative yields with complete retention of the configuration at both carbons flanking the nitrogen atoms. The resulting enantiomerically pure polysubstituted cyclopropanes, containing up to five substituens, can be desulfinylated with Ra-Ni providing polysubstituted cyclopropanecarboxylic acid derivatives.     

A constructive approach to cubic Hermite Fractal Interpolation Function and its constrained aspects

The theory of splines is a well studied topic, but the kinship of splines with fractals is novel. We introduce a simple explicit construction for a -cubic Hermite Fractal Interpolation Function (FIF). Under some suitable hypotheses on the original function, we establish a priori estimates (with respect to the L _p -norm, 1≤p≤∞) for the interpolation error of the -cubic Hermite FIF and its first derivative. Treating the first derivatives at the knots as free parameters, we derive suitable values for these parameters so that the resulting cubic FIF enjoys global smoothness. Consequently, our method offers an alternative to the standard moment construction of -cubic spline FIFs. Furthermore, we identify appropriate values for the scaling factors in each subinterval and the derivatives at the knots so that the graph of the resulting -cubic FIF lies within a prescribed rectangle. These parameters include, in particular, conditions for the positivity of the cubic FIF. Thus, in the current article, we initiate the study of the shape preserving aspects of fractal interpolation polynomials. We also provide numerical examples to corroborate our results.     

Bimetallic Co–Ni/TiO<Subscript>2</Subscript> catalysts for selective hydrogenation of cinnamaldehyde

Bimetallic Co–Ni catalysts in the composition range Co_(1?x)Ni_x with x?=?0.0, 0.2, 0.3, 0.4, 0.5, 0.6, 0.8 and 1.0, with total metal loading of 15% w/w and supported on TiO₂-P25, have been prepared by chemical reduction of the metal acetates by glucose in aqueous alkaline medium and characterized by XRD, TEM, TPR, XPS and H₂-TPD techniques. Selective hydrogenation of cinnamaldhyde (CAL) to hydrocinnamaldehyde (HCAL), cinnamyl alcohol (COL) and hydrocinnamyl alcohol (HCOL) has been investigated at 20 bar pressure, in the temperature range 120–140 °C. Co/Ni crystallite sizes in the range 6.0?±?1 nm are observed by TEM. TPR and XPS results indicate the formation of nanoscale Co–Ni alloys, which tend to weaken M–H bond strength, as revealed by H₂-TPD measurements. Ni/TiO₂ displays very high conversion of CAL (86.9%) with high selectivity (78.7%) towards HCAL formation at 140 °C. Co/TiO₂, on the other hand, exhibits relatively lower CAL conversion (55%) and higher selectivity (61.3%) for COL formation at the same temperature. However, bi-metallic Co–Ni catalysts in the composition range x?=?0.3–0.6 display very high conversion (>?98%) due to alloy formation and weakening of M–H bonds. Bimetallic Co_0.7Ni_0.3 catalyst displays high conversion of CAL (98.1%) and high selectivity (82.9%) towards HCOL. Overall CAL hydrogenation activity at 140 °C, when expressed as TOF, displays a maximum value at the composition Co_0.5Ni_0.5. Activity and selectivity patterns have been rationalized based on the reaction pathways observed on the catalysts and the influence of Co–Ni alloy formation and M–H bond strength. Thus, a synergetic effect, originating from an appropriate composition of base metal catalysts and reaction conditions, could result in hydrogenation activity comparable with noble metal based catalysts.