Cyclic torsion of elliptic curves

Let be an elliptic curve over a number field such that
and let denote the number of roots of unity in . Ross proposed a question: Is isogenous over to an elliptic curve such that is cyclic of order dividing ? A counter-example of this question is given. We show that is isogenous to such that . In case has complex multiplication and , we obtain certain criteria whether or not is isogenous to such that .

     

Ab initio calculations on peptide-derived oxazoles and thiazoles: Improved molecular mechanics parameters for the AMBER* force field

Ab initio calculations at the RHF/6-31G* and MP2/6- 31G*//RHF/6-31G* levels of theory are performed for 2-methyl-4-carboxamido-oxazoles and -thiazoles, including rotational profiles for the ring-carboxamide bond, which showed the expected conjugation and hydrogen bonding effects. On the basis of these data, newly optimised stretch, bend and torsional parameters for the AMBER* force field are derived, along with CHELPG-fitted partial atomic charges.     

Determination of Multiple φ-Torsion Angles in Proteins by Selective and Extensive C Labeling and Two-Dimensional Solid-State NMR

We describe an approach to efficiently determine the backbone conformation of solid proteins that utilizes selective and extensive ¹³C labeling in conjunction with two-dimensional magic-angle-spinning NMR. The selective ¹³C labeling approach aims to reduce line broadening and other multispin complications encountered in solid-state NMR of uniformly labeled proteins while still enhancing the sensitivity of NMR spectra. It is achieved by using specifically labeled glucose or glycerol as the sole carbon source in the protein expression medium. For amino acids synthesized in the linear part of the biosynthetic pathways, [1-¹³C]glucose preferentially labels the ends of the side chains, while [2-¹³C]glycerol labels the C^α of these residues. Amino acids produced from the citric-acid cycle are labeled in a more complex manner. Information on the secondary structure of such a labeled protein was obtained by measuring multiple backbone torsion angles φ simultaneously, using an isotropic–anisotropic 2D correlation technique, the HNCH experiment. Initial experiments for resonance assignment of a selectively ¹³C labeled protein were performed using ¹⁵N–¹³C 2D correlation spectroscopy. From the time dependence of the ¹⁵N–¹³C dipolar coherence transfer, both intraresidue and interresidue connectivities can be observed, thus yielding partial sequential assignment. We demonstrate the selective ¹³C labeling and these 2D NMR experiments on a 8.5-kDa model protein, ubiquitin. This isotope-edited NMR approach is expected to facilitate the structure determination of proteins in the solid state.     

Thermodynamic calculations for molecules with asymmetric internal rotors. II. Application to the 1,2-dihaloethanes

The thermodynamic properties of three halocarbon molecules relevant in atmospheric and public health applications are presented from ab initio calculations. Our technique makes use of a reaction path-like Hamiltonian to couple all the vibrational modes to a large-amplitude torsion for 1,2-difluoroethane, 1,2-dichloroethane, and 1,2-dibromoethane, each of which possesses a heavy asymmetric rotor. Optimized ab initio energies and Hessians were calculated at the CCSD(T) and MP2 levels of theory, respectively. In addition, to investigate the contribution of electronically excited states to thermodynamic properties, several excited singlet and triplet states for each of the halocarbons were computed at the CASSCF/MRCI level. Using the resulting potentials and projected frequencies, the couplings of all the vibrational modes to the large-amplitude torsion are calculated using the new STAR-P 2.4.0 software platform that automatically parallelizes our codes with distributed memory via a familiar MATLAB interface. Utilizing the efficient parallelization scheme of STAR-P, we obtain thermodynamic properties for each of the halocarbons, with temperatures ranging from 298.15 to 1000 K. We propose that the free energies, entropies, and heat capacities obtained from our methods be used to supplement theoretical and experimental values found in current thermodynamic tables.     

Dependence of the rate constants on the treatment of internal rotation modes: the reaction OH + CH3SH --> CH3S + H2O as an example

The title reaction has been used as an example to test the importance of using a hindered rotor treatment instead of a harmonic oscillator model for calculating vibrational partition functions corresponding to low-frequency internal rotation modes. First, a normal-mode analysis according to the Ayala and Schlegel's algorithm has been used to identify the internal rotation modes of methanethiol and the transition state structure. Then, after calculation of the energy barrier for each internal rotation, the corresponding hindered rotor partition functions have been calculated following the CW scheme of Chuang and Truhlar. The results show that the anharmonic treatment produces a rather modest improvement of the rate constants at room temperature or below.     

Gravity and CP violation

The non-conservation of CP symmetry in the decay of neutralK mesons was discovered almost three decades ago. The origin of this unique phenomenon is still not well understood. There have been attempts to link it with energy splitting of theK and in the earth's gravitational field. In this essay we examine these attempts critically. A more natural way in which a gravitational field can couple asymmetrically to the system is through torsion, which coupling to isospin can, in addition to parity violation, also give rise to a C violation. The magnitude of this effect is estimated in the vicinity of a kaon and found to give a CP-violating parameter of the right value. Moreover, it is predicted that the CP violating parameter for the new system is much smaller 10^–6. Implications for baryogenesis in the early universe is also studied. In this paper we work not with the Newtonian gravitational constantG but with a torsion coupling constant, which is energy-dependent and can be related to the Fermi weak interaction constant.     

Geometric Hermite interpolation by spatial Pythagorean-hodograph cubics

It is shown that, depending upon the orientation of the end tangents t₀,t₁ relative to the end point displacement vector p=p₁–p₀, the problem of G¹ Hermite interpolation by PH cubic segments may admit zero, one, or two distinct solutions. For cases where two interpolants exist, the bending energy may be used to select among them. In cases where no solution exists, we determine the minimal adjustment of one end tangent that permits a spatial PH cubic Hermite interpolant. The problem of assigning tangents to a sequence of points p₀,...,p_n in R³, compatible with a G¹ piecewise-PH-cubic spline interpolating those points, is also briefly addressed. The performance of these methods, in terms of overall smoothness and shape-preservation properties of the resulting curves, is illustrated by a selection of computed examples.     

On the -torsion of elliptic curves and elliptic surfaces in characteristic

We study the extension generated by the -coordinates of the -torsion points of an elliptic curve over a function field of characteristic . If is a non-isotrivial elliptic surface in characteristic with a -torsion section, then for 11$"> our results imply restrictions on the genus, the gonality, and the -rank of the base curve , whereas for such a surface can be constructed over any base curve . We also describe explicitly all occurring in the cases where the surface is rational or or the base curve is rational, elliptic or hyperelliptic.

     

Pseudoinstantons in Metric-Affine Field Theory

In abstract Yang–Mills theory the standard instanton construction relies on the Hodge star having real eigenvalues which makes it inapplicable in the Lorentzian case. We show that for the affine connection an instanton-type construction can be carried out in the Lorentzian setting. The Lorentzian analogue of an instanton is a spacetime whose connection is metric compatible and Riemann curvature irreducible (pseudoinstanton). We suggest a metric-affine action which is a natural generalization of the Yang–Mills action and for which pseudoinstantons are stationary points. We show that a spacetime with a Ricci flat Levi-Civita connection is a pseudoinstanton, so the vacuum Einstein equation is a special case of our theory. We also find another pseudoinstanton which is a wave of torsion in Minkowski space. Analysis of the latter solution indicates the possibility of using it as a model for the neutrino.     

The Ziegler Spectrum of a Locally Coherent Grothendieck Category

The general theory of locally coherent Grothendieck categoriesis presented. To each locally coherent Grothendieck categoryC a topological space, the Ziegler spectrum of C, is associated.It is proved that the open subsets of the Ziegler spectrum ofC are in bijective correspondence with the Serre subcategoriesof coh C the subcategory of coherent objects of C. This is aNullstellensatz for locally coherent Grothendieck categories.If R is a ring, there is a canonical locally coherent Grothendieckcategory RC (respectively, CR) used for the study of left (respectively,right) R-modules. This category contains the category of R-modulesand its Ziegler spectrum is quasi-compact, a property used toconstruct large (not finitely generated) indecomposable modulesover an artin algebra. Two kinds of examples of locally coherentGrothendieck categories are given: the abstract category theoreticexamples arising from torsion and localization and the examplesthat arise from particular modules over the ring R. The dualitybetween coh-(RC) and coh-CR is shown to give an isomorphismbetween the topologies of the left and right Ziegler spectraof a ring R. The Nullstellensatz is used to give a proof ofthe result of Crawley-Boevey that every character : K₀(coh-C) Z is uniquely expressible as a Z-linear combination of irreduciblecharacters. 1991 Mathematics Subject Classification: 16D90,18E15.