The Mathieu Group M11 and the Modular Curve X(11)

In this paper, we prove that the modular curve X(11) over afield of characteristic 3 admits the Mathieu group M₁₁ as anautomorphism group. We also examine some aspects of the geometryof the curve X(11) in characteristic 3. In particular, we showthat every point of the curve is a point of inflection, thecurve has 110 hyperflexes and there are no inflectional trianglesand 11232 inflectional pentagons, of which 144 are self-conjugate.The hyperflexes correspond to the supersingular elliptic curves.We comment on the relationship of Ward's quadrilinear invariantfor M₁₂ to our work and announce for the first time the equationsfor Klein's A-curve of level 11. We also comment on the relationof our work to some unpublished work of Bott and Tate. 1991Mathematics Subject Classification: 11F32, 11G20, 14G10, 14H10,14N10, 20B25, 20C34.     

Classification of integral lattices with large class number

A detailed exposition of Kneser's neighbour method for quadratic lattices over totally real number fields, and of the sub-procedures needed for its implementation, is given. Using an actual computer program which automatically generates representatives for all isomorphism classes in one genus of rational lattices, various results about genera of -elementary lattices, for small prime level are obtained. For instance, the class number of -dimensional -elementary even lattices of determinant is ; no extremal lattice in the sense of Quebbemann exists. The implementation incorporates as essential parts previous programs of W. Plesken and B. Souvignier.

     

On Borchers' Theorem

We give a simplified proof of the following important theorem,which is due to Borchers. Let be a von Neumann algebra with cyclic and separating vector and let U(t), t R, be a group with positive generator, which leaves fixed and induces for positive arguments endomorphisms of . The modular group of (, ) acts then as dilatations on this group. Our proof can also be applied to half-sided modular inclusions.     

Crystal structures of 2-pyrazinecarboxylato silver(I) and ammine 2,3-pyrazinecarboxylato disilver(I)

X-ray analysis has shown that the complexes of silver(I) with the anionic forms of 2-pyrazinecarboxylic and 2,3-pyrazinedicarboxylic acids are three-dimensional network polymers. Ag(C₅H₃O₂N₂) (1) crystallizes in space groupPna2₁ witha=7.028(2),b=11.510(3),c=6.949(2) Å andZ=4. Silver tetrahedral coordination occurs through three ligands: the first one acts as a bidentate chelator (Ag–O=2.539(4), Ag–N=2.334(4)Å), the second one involves one N pyrazine atom (Ag–N=2.196(4)Å) and the third one bonds through one O-carboxylato atom (Ag–O=2.323(4)Å). The dinuclear complex Ag₂(C₆H₂O₄N₂)(NH₃) (2) crystallizes in space groupP2₁/c witha=7.775(2),b=16.455(2),c=7.174(1) Å, -115.21(1)° andZ =4. Each of the two silver atoms shows a trigonal environment. The first silver atom involves two O carboxylic atoms (Ag–O=2.312(4), 2.374(4)Å) and one N pyrazine atom (Ag–N=2.277(3)Å) belonging to three different ligands. The second one involves one O carboxylic atom (Ag–O=2.364(3)Å) and one N pyrazine atom (Ag–N=2.304(4)Å) from the same chelating ligand and one N ammine atom (Ag–N=2.158(5)Å).     

Moments of the Percus-Yevick hard-sphere correlation function

A simple recursive relation is derived for the momentsM _n ,n=1, 2,..., of the Percus-Yevick correlation functionh(r) for identical hard spheres. TheM _n are rational functions of the volume fractionw occupied by the spheres; the first ten are given explicitly, and a single-term asymptotic form is obtained to suffice for the rest. Applications of theM _n(w) include testing different approximations forh by numerical integration ofh(r) r ⁿ . We compare exact moments with shell approximationsM _n [h ^s ] corresponding to integration fromr=0 tos+1 fors=3–8, and with hybrid approximationsM _n [h ^s +h ^a ] which supplement the shell approximations with integrals of an asymptotic tail froms+1 to . For a givens, the hybrid approximation is better forw increasing than the shell approximation, andM _n [h ³+h ^a ] is even better thanM _n [h ⁸]     

Orthogonality between scales and wavelets in a representation for correlation functions. The lattice dipole gas and (∇φ)4 models

Exact formulas for the correlation functions of lattice scalar field models in Z^d,d3, such as the dipole gas and anharmonic crystal are derived in terms of the effective action generated aftern applications of the block renormalization group transformation. Utilizing the orthogonality between different momentum scales (relations due to the wavelets implicit in the structure of the block renormalization group transformation), the formulas are quite simple, isolate the dominant term, and, in the thermodynamic andn limits, reduce the analysis to local estimates of the effective action. Based on a large-small field analysis, the two-point function is determined and it is shown how to extend the results to general correlations. The results proved here show the usefulness of the orthogonality-of-scales property for the study of correlation functions.     

若干三核羧酸配合物的振动光谱及其与结构的某些相关规律

报导了通式为［Ｆｅ_２Ｍ（μ_３－Ｏ）（μ－Ｏ_２ＣＲ）_６·Ｌ_３］·ＸＨ_２Ｏ［其中Ｍ＝Ｍｎ（Ⅱ），Ｃｏ（Ⅱ），Ｎｉ（Ⅱ），Ｆｅ（Ⅲ）（这时骨架为阳离子）；Ｒ＝ＣＨ_３，Ｃ_２Ｈ_５；Ｌ＝Ｈ_２Ｏ，ｐｙ］的３类１１个配合物的振动光谱。在谱带经验归属及简正坐标分析计算的基础上，分别详细讨论了配体及骨架的振动光谱与结构的某些相关规律。这些规律具有一定的普遍性，因而有其理论和实际应用价值。     

Quantum statistical mechanics for superstable interactions: Bose-Einstein statistics

We study quantum systems of interacting Bose particles confined to a bounded region ofR ^v . For any superstable and (strong) lower regular interaction, we obtain uniform bounds on the expectations of exponentials of local number operators for any activity and for any temperature. The method we use here is an improvement over our previous method on the same subject. As a consequence of the bounds, any infinite volume limit states are entire analytic and locally normal. Furthermore under an integrability condition on the interaction, the limit states are modular states. In this case, we use the Green's function method to construct an infinite volume limit Hilbert spece, a strongly continuous time evolution group of unitary operators and an invariant vector. Moreover we prove the existence of the pressure and its independence of boundary conditions.     

Kinetic Spectrofluorometric Determination of Thioctic Acid in Bulk and Pharmaceutical Preparations via its Oxidation with Cerium(IV)

A highly simple and sensitive kinetic spectrofluorometric method was developed for the determination of thioctic acid. The method is based on the oxidation of the studied drug with cerium(IV) ammonium sulfate in acidic medium. The fluorescence of the produced Ce(III) was measured at 365 nm after excitation at 255 nm. The different experimental parameters affecting the development and stability of the reaction product were carefully studied and optimized. The method is applicable over the concentration range of 0.02 to 0.12 μg/mL with a detection limit of 6.06 × 10^?3 μg/mL and a quantification limit of 0.02 μg/mL. The method was successfully applied for the assay of the studied drug in pharmaceutical formulations. The results obtained were in good agreement with those obtained with the reference method.     

Thermal Stability of Sol-Gel Derived 6 Mole%CaO-ZrO2 Ceramics

The thermal behavior of the hyperfine interaction at Zr sites on two sol-gel derived 6 mole% CaO-ZrO₂ powders, obtained from the particulate and non-particulate regimes, has been studied by Perturbed Angular Correlations and complementary techniques. The aim was to get experimental support on the thermal stability of the obtained material in order to see the advantages of the different preparation regimes. The results could be interpreted in terms of the different microstructures and nanoscopic configurations exhibited by the resulting powders. After crystallization both powders showed the hyperfine nanoconfigurations of metastable tetragonal zirconia. In the sample obtained following the hydrolysis and the condensation processes via a particulate regime, the undesirable phase transformation towards the monoclinic form of zirconia is inhibited up to higher temperatures.