Correction to: Preface

Foundations of Computational Mathematics -     

Scanning moiré at high magnification using optical methods

Methods of employing scanning moiré at high magnification are developed and demonstrated. Modern lithographic techniques for producing custom moiré gratings with a frequency up to 250l/mm are described. On a probing station equipped with a video system, pseudo-color moiré fringes are produced using the scannning lines of the color charge-coupled-device (CCD) camera. Fringe multiplication from 1 to 5 is possible with correct combinations of magnification and grating pitch. An analysis is given to show that strain sensitivity depends only on the number of scanning lines used to record the image. The grating pitch and the magnification are important because they reduce the gage length of the strain measurement. The high-magnification scanning moiré was used to study plastic- strain fields in an aluminum tensile specimen. Local disturbances in the strain field were observed at 2 to 2.5 percent applied strain. These discontinuities became more significant at higher levels of applied strain.     

Interpretation of scoring functions using 3D molecular fields. Mapping the diacyl-hydrazine-binding pocket of an insect ecdysone receptor

A method is presented for the interpretation of receptor docking score values (rough measures of binding affinities) of ligands in terms of 3D molecular field interaction contributions. The FlexX and FlexX-Pharm methods were used to dock the structures of designed sets of ligands into the ligand-binding pocket of a selected receptor. In the next step the relationship was investigated between the FlexX and CScore scores and 3D molecular fields obtained for the docked conformations of the ligands, using the CoMFA (Comparative Molecular Field Analysis) and CoMSIA (Comparative Molecular Similarity Indices Analysis) methods. The approach yielded highly significant CoMFA and CoMSIA models demonstrating that a high portion of the variance in the docking score values of the ligands can be explained by steric, electrostatic, hydrophobic, and hydrogen bond donor and acceptor molecular field interaction contributions. The approach was exemplified by using the crystal structure of the ligand-binding domain of the ecdysone receptor (EcR) of the moth Heliotis virescens as well as virtual molecule libraries of analogues of known diacyl-hydrazine (DAH) type ecdysteroid agonists. By docking appropriately designed virtual compound libraries into the DAH binding pocket of EcR followed by CoMFA and CoMSIA of the docked conformations, hitherto unexplored regions of the receptor cavity could be mapped. By mapping the significant molecular field interaction contributions onto the model of the receptor-ligand complex, important receptor-ligand interactions could be highlighted that may help the design of novel highly scored receptor ligands. An advantage of the method is that no experimental biological activity data are required to exhaustively map the receptor-binding site.     

Fusion-fission of light nuclear systems

Considerable interest has been devoted to fusion reactions between light heavy ions specially between weakly bound ones, due to the anomalous decrease of the fusion cross sections when compared to the total reaction cross section in the energy region around the barrier [1–4]. While the exact nature of the process responsible for the fusion cross section limitation at barrier energies is still unclear, this study shows an inhibition of the yield as the system mass decreases, resulting from the progressive increase of the barrier height and decrease of the effective barrier radius [3]. Furthermore, extensive efforts have been made recently in the study of energy-damped binary yields from light heavy-ion collisions [2,4]. Based on the substantial amount of data accumulated so far, it is now generally accepted and supported by the transition state model [4], that the observed yields arise mostly from a fusion-fission process. Data on complete fusion, fusion-fission and ‘elastic fission’ for the ⁹Be, ^10,11B+^10,11B; ^16,17,18O + ^10,11B; ¹⁹F+¹²C; ^6,7Li+⁹Be, ¹²C reactions among others, are presented. For the loosely bound nuclei it was found that the severe fusion cross section limitation is due to a low survival probability of the weakly bound nuclei until the instant of the collision [1].     

A bound for orders in differential Nullstellensatz

We give the first known bound for orders of differentiations in differential Nullstellensatz for both partial and ordinary algebraic differential equations. This problem was previously addressed in [A. Seidenberg, An elimination theory for differential algebra, Univ. of California Publ. in Math. III (2) (1956) 31–66] but no complete solution was given. Our result is a complement to the corresponding result in algebraic geometry, which gives a bound on degrees of polynomial coefficients in effective Nullstellensatz [G. Hermann, Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, Math. Ann. 95 (1) (1926) 736–788; E.W. Mayr, A.W. Meyer, The complexity of the word problems for commutative semigroups and polynomial ideals, Adv. Math. 46 (3) (1982) 305–329; W.D. Brownawell, Bounds for the degrees in the Nullstellensatz, Ann. of Math. 126 (3) (1987) 577–591; J. Kollár, Sharp effective Nullstellensatz, J. Amer. Math. Soc. 1 (4) (1988) 963–975; L. Caniglia, A. Galligo, J. Heintz, Some new effectivity bounds in computational geometry, in: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Rome, 1988, in: Lecture Notes in Comput. Sci., vol. 357, Springer, Berlin, 1989, pp. 131–151; N. Fitchas, A. Galligo, Nullstellensatz effectif et conjecture de Serre (théorème de Quillen–Suslin) pour le calcul formel, Math. Nachr. 149 (1990) 231–253; T. Krick, L.M. Pardo, M. Sombra, Sharp estimates for the arithmetic Nullstellensatz, Duke Math. J. 109 (3) (2001) 521–598; Z. Jelonek, On the effective Nullstellensatz, Invent. Math. 162 (1) (2005) 1–17; T. Dubé, A combinatorial proof of the effective Nullstellensatz, J. Symbolic Comput. 15 (3) (1993) 277–296].This paper is dedicated to the memory of Eugeny Pankratiev, who was the advisor of the first three authors at Moscow State University.     

Preface

Foundations of Computational Mathematics -     

A new possibility of identifying high-spin states at high excitation energies

A new method of identifying high-spin states at high excitation energies is presented.     

Study of Be + C elastic scattering at energies near the Coulomb barrier

In this work, angular distribution measurements for the elastic channel were performed for the ⁹Be + ¹²C reaction at the energies ELab=13.0$E_{\mathit{Lab}}=13.0$, 14.5, 17.3, 19.0 and 21.0 MeV, near the Coulomb barrier. The data have been analyzed in the framework of the double folding São Paulo potential. The experimental elastic scattering angular distributions were well described by the optical potential at forward angles for all measured energies. However, for the three highest energies, an enhancement was observed for intermediate and backward angles. This can be explained by the elastic transfer mechanism.