Solution of a problem in the optimal estimation of the states of linear dynamical systems

A problem in the optimal estimation of a linear dynamical system is solved under the condition that the effects on the system of noise and observation error belong to a prescribed parallelepiped region.Translated fromVychislitel'naya i Prikladnaya Matematika, No. 69, pp. 110–114, 1989.     

Intermolecular interactions of p-nitrophenol with butyl acetate,according to IR spectroscopic data

Infrared difference spectroscopy has been used to study the association of p-nitrophenol (NP) with butyl acetate (BA) in CCl₄ and in undiluted BA. It has been shown that in CCl₄, an associate is formed, [NP·BA] ·BA (Ia), in which one molecule of BA is linked to the NP by a hydrogen bond, and the second BA molecule is linked to the H-complex NP·BA by outer-sphere (dipole-dipole) interaction. In undiluted BA with low concentrations of NP, (Ia) is formed; with high concentrations of NP, an associate [(NP)₂BA] ·mBA is formed, in which two molecules of NP are bonded successively to each other and with a BA molecule; and the number of outer-sphere BA molecules may depend on the concentration of free BA molecules in the solutions.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 3, pp. 593–599, March, 1991.     

Calibration of a basket option model applied to company valuation

Applying real options thinking to company valuation seems theoretically and intuitively appealing. However, the real option analogy of a single European option as well as the compound option proxy perform poorly when applied to company valuation. We therefore suggest to rework the building blocks of real option applications to corporate valuation.  We introduce a framework to delineate the distribution of the underlying asset in the risk neutral world, which is important in order to value any derivative. This is achieved by an algorithm to calibrate a basket option model using real world data of observed share prices. The fitting takes account of the class of stable distributions. The index of stability of asymmetric α stable distribution serves as an over-all parameter to characterise the specific distribution.     

Asymptotic expansions of integrals of two Bessel functions via the generalized hypergeometric and Meijer functions

Asymptotic expansions of certain finite and infinite integrals involving products of two Bessel functions of the first kind are obtained by using the generalized hypergeometric and Meijer functions. The Bessel functions involved are of arbitrary (generally different) orders, but of the same argument containing a parameter which tends to infinity. These types of integrals arise in various contexts, including wave scattering and crystallography, and are of general mathematical interest being related to the Riemann—Liouville and Hankel integrals. The results complete the asymptotic expansions derived previously by two different methods — a straightforward approach and the Mellin-transform technique. These asymptotic expansions supply practical algorithms for computing the integrals. The leading terms explicitly provide valuable analytical insight into the high-frequency behavior of the solutions to the wave-scattering problems.     

Rigidity of the scattering length spectrum

In this paper we consider properties of obstacles satisfying some non-degeneracy conditions that can be recovered from the scattering length spectrum (SLS). Clearly the latter tells us whether the obstacle K is trapping or non-trapping. If the set of trapped points is relatively small, then the SLS also determines the volume of the obstacle, the number of its connected components, and whether its boundary is convex everywhere or it has non-trivial concavities. Under the additional assumption that the curvature of the obstacle does not vanish of infinite order, it is proved that from the SLS one can recover certain information about the number of reflection points of any simply reflecting ray in the exterior of the obstacle. Finally, for some special classes of obstacles (e.g. star-shaped ones), it is shown that the SLS completely determines the obstacle. Received: 2 March 1999 / Revised version: 16 January 2001 / Published online: 5 September 2002     

On the interpretation of the RHEED intensity oscillations during the growth of vicinal faces

A quantitative treatment of the crystallization kinetics in MBE growth of vicinal surfaces results the relation l ²=2D _sK between the surface diffusion coefficient D _s, the time for a monolayer deposition and the interstep distance l at which the RHEED intensity oscillations disappear. The correction factor K depends on the size and the energy of the two-dimensional critical nucleus and it is estimated to be smaller than 10^–2. The currently used interpretation of the RHEED intensity oscillations ignores the correction factor K and, therefore, the calculated values of D _s are several orders of magnitude smaller than its real values. The surface transport during the time of growth interruption is discussed in connection with the tendency to three dimensional growth at every second interface (where a deposition of the material with strong intermolecular bonds starts) of a small period superlattice.     

Cationic complexes of Eu(III) and Sr(II) with diphosphine dioxides in organic extracts

IR spectroscopy was used to study the structure and composition of Eu(III) and Sr(II) complexes formed by cation-exchange extraction of these metals from their aqueous nitrate solutions with dichlorethane solutions of mixtures of chlorinated cobalt(III) dicarbollide (CCD) as a superacid with diphenyldiphosphine dioxides containing a methyl (Me-DPDO), ethyl (Et-DPDO), or polyoxyethylene bridge between two phosphorus atoms of phosphine oxide groups. At molar ratios DPDO/CCD ≤ 1, [Eu(H₂O)_nL₄]³⁺ complexes are formed in organic phases, whereas with an excess of DPDO relative to CCD, Eu(NO₃)L ₄ ²⁺ complexes are formed, where L = Me-or Et-DPDO. Polyoxyethylenediphosphine dioxide forms anhydrous complexes of composition Eu:L = 1:1 and 1:2 with Eu(III) and outer-spheric complexes of composition Sr:L = 1:1 and 1:2 with Sr(II), where the organic ligand molecules envelop the hydrated Sr(H₂O) _n ²⁺ cation. The peculiarities of extraction of the complexes are explained based on data about their composition and structure.     

State of Molecules and Ions in the Structural Channels of Synthetic Beryl with an Ammonium Impurity

The contents of the structural channels of beryl, grown hydrothermally from an ammonium-containing solution, were investigated by IR and EPR spectroscopy. Using IR spectroscopy we found that water molecules, ammonium ions, and a small number of HCl molecules enter the structural channels of beryl in the course of mineral growth. In these beryls, the ammonium ions play the role of alkali cations. The ammonium ions are as rigidly fixed in the lattice as are water molecules; they are eliminated by calcination at high temperatures close to the decomposition temperature. On exposure to radiation at 77 K, the paramagnetic NH ₃ ⁺ and H⁰ radicals are stabilized in the structural channels of beryl. In addition to the known H⁰ radical, other states of atomic hydrogen, interacting with medium protons, are observed as well. For one of the additional radicals, H_b, we suggest the model of atomic hydrogen stabilized at the center of a silicon-oxygen ring with two water molecules in adjacent cavities.