Computing Final Polynomials and Final Syzygies Using Buchberger’s Gröbner Bases Method

Final polynomials and final syzygies provide an explicit representation of polynomial identities promised by Hilbert’s Nullstellensatz. Such representations have been studied independently by Bokowski [2,3,4] and Whiteley [23,24] to derive invariant algebraic proofs for statements in geometry. In the present paper we relate these methods to some recent developments in computational algebraic geometry. As the main new result we give an algorithm based on B. Buchberger’s Gröbner bases method for computing final polynomials and final syzygies over the complex numbers. Degree upper bound for final polynomials are derived from theorems of Lazard and Brownawell, and a topological criterion is proved for the existence of final syzygies. The second part of this paper is expository and discusses applications of our algorithm to real projective geometry, invariant theory and matrix theory.     

Gelfand numbers of operators with values in a Hilbert space

In the paper we prove two inequalities involving Gelfand numbers of operators with values in a Hilbert space. The first inequality is a Rademacher version of the main result in [Pa-To-1] which relates the Gelfand numbers of an operator from a Banach spaceX intol ₂ ⁿ with a certain Rademacher average for the dual operator. The second inequality states that the Gelfand numbers of an operatoru froml ₁ ^N into a Hilbert space satisfy the inequality

Investigation of the character of the exchange interaction in the system Hg1−xMnxSe

A new method is proposed for determining the magnetic characteristics (magnitude and sign of the exchange interaction energy and the average size of clusters of magnetic ions) of dilute solid solutions of semimagnetic semiconductors at low temperatures based on oscillation measurements. The method makes it possible to find the magnetic characteristics of the indicated systems at temperatures between the point of the transition into the spin glass state and the temperature corresponding to the characteristic binding energy of magnetic atoms in clusters, for which standard methods based on the measurement of the magnetic susceptibility are not effective. The method is used to study the character of the exchange interaction in the system of solid solutions Hg_1–xMn_xSe as a function of their composition as well as under conditions of hydrostatic compression. To this end the oscillations of the magnetoresistance (Shubnikov-de Haas (SH) effect) in single-crystalline samples of Hg_1–xMn_xSe in the region of compositions 0.001 x 0.23 in magnetic fields H up to 65 kOe at temperatures T = (0.4–20) K and pressures up to 16 kbar were studied.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 29–39, November, 1989.     

Reaction Engineering Aspects of α-l,4-D-Glucan Phosphorylase Catalysis

Some important process properties of α-l,4-D-ghican phosphorylases isolated from the bacteriumCorynebacterium callunae and potato tubers (Solatium tuberosum) were compared. Apart from minor differences in their stability and specificity (represented by the maximum degree of maltodextrin conversion) and a 10-fold higher affinity of the plant phosphorylase for maltodextrin (K _M of 1.3 g/L at 300 mM of orthophosphate), the performances of both enzymes in a continuous ultrafiltration membrane reactor were almost identical. Product synthesis was carried out over a time course of 300–400 h in the presence or absence of auxiliary pullulanase (increasing the accessibility of the glucan substrate for phosphorolytic attack up to 15–20%). The effect of varied dilution rate and reaction temperature on the resulting productivities was quantitated, and a maximum operational temperature of 40°C was identified.     

Insights into the structure of cyclohexane from femtosecond degenerate four-wave mixing spectroscopy and ab initio calculations

In this paper, we report the use of femtosecond time-resolved degenerate four-wave mixing rotationally resolved spectroscopy to obtain very accurate structural information on the symmetric top cyclohexane. Apart from highlighting the versatility of this method in determining accurate structures of large and complex molecules without dipole moment, the present study also details the comparison of the experimentally determined rotational constant B(0) with that obtained from high-level ab initio calculations. The theoretical calculations, which were carried out at both the second-order M?ller-Plesset (MP2) and coupled-cluster with single, double, and perturbative triple substitutions [CCSD(T)] levels of theory, also take into account vibrational averaging effects. A detailed investigation of the vibrational averaging effects reveals that the corrections emerge from only the six highly symmetric A(1g) modes, a justification of which is provided by an analysis of these modes.     

X-Ray Study of the Deformation Density in Tetrafluoroterephthalodinitrile: Weak Bonding Density in the C,F-Bond

A careful deformation density study of tetrafluoroterephthalodinitrile at 98K has been made from X-ray diffraction measurements. Prominent ‘bonding density’ peaks are found at or near the mid-points of the C, C- and C, N-bonds but not for the C, F-bonds, which show only weak density. Similarly weak bonding densities for C, F-bonds are also found for 1, 1, 4, 4-tetrafluorocyclohexane. The possible significance of these results in terms of bonding theory is briefly discussed.     

Improved separation of palladium species in biological matrices by using a combination of gel permeation chromatography and isotachophoresis

The binding of palladium to high-molecular-mass compounds in palladium-treated lettuce is investigated as an example for a biological matrix. The total palladium concentration in lettuce leaves is 10.3 ng/g wet weight. After homogenization, high-molecular-mass compounds (> 10 kDa) are isolated by ultrafiltration. For separation of these palladium species a combination of preparative gel permeation chromatography (GPC) and preparative isotachophoresis (ITP) is used. Palladium is determined in separated fractions by using a highly sensitive total reflection X-ray fluorescence (TXRF) method after preconcentration. After GPC separation, four main fractions of palladium species are collected, each containing palladium in ng quantities (3-10 ng). Two of these fractions are further separated by ITP, yielding at least three main peaks per GPC fraction, each containing palladium in the range of 0.3-3 ng. These palladium containing peaks are characterized by high-performance size exclusion chromatography (HPSEC) and capillary isotachophoresis (cITP) in parallel. HPSEC enables the estimation of the molecular mass of six main palladium peaks, covering a molecular mass range of 69-200 kDa. It is also shown that the estimation of molecular mass after separation is more reliable than the respective estimation directly in the first GPC run. However, cITP reveals that each of the separated peaks is still a mixture of at least five different compounds.