Counting isomorphism classes of pointed hyperelliptic curves of genus 4 over finite fields with even characteristic

This paper is devoted to counting the number of isomorphism classes of pointed hyperelliptic curves over finite fields. We deal with the genus 4 case and the finite fields are of even characteristics. The number of isomorphism classes is computed and the explicit formulae are given. This number can be represented as a polynomial in q of degree 7, where q is the order of the finite field. The result can be used in the classification problems and it is useful for further studies of hyperelliptic curve cryptosystems, e.g. it is of interest for research on implementing the arithmetics of curves of low genus for cryptographic purposes. It could also be of interest for point counting problems; both on moduli spaces of curves, and on finding the maximal number of points that a pointed hyperelliptic curve over a given finite field may have.     

Club isomorphisms on higher Aronszajn trees

We prove the consistency, assuming an ineffable cardinal, of the statement that CH holds and any two normal countably closed ${\omega }_2$-Aronszajn trees are club isomorphic. This work generalizes to higher cardinals the property of Abraham–Shelah [1] that any two normal ${\omega }_1$-Aronszajn trees are club isomorphic, which follows from PFA. The statement that any two normal countably closed ${\omega }_2$-Aronszajn trees are club isomorphic implies that there are no ${\omega }_2$-Suslin trees, so our proof also expands on the method of Laver–Shelah [5] for obtaining the ${\omega }_2$-Suslin hypothesis.     

Jordan isomorphisms of finitary incidence algebras

Let X be a partially ordered set, R a commutative 2-torsionfree unital ring and FI(X, R) the finitary incidence algebra of X over R. In this note we prove that each R-linear Jordan isomorphism of FI(X, R) onto an R-algebra A is the near-sum of a homomorphism and an anti-homomorphism.     

因子von Neumann代数中套子代数上的Jordan同构

研究了因子yon Neumann代数中套子代数上的Jordan同构,证明了套子代数algMβ和algMγ之间的每一个Jordan同构φ:要么是同构;要么是反同构.     

On the Determination Problem for P 4-Transformation of Graphs

We prove that the P ₄-transformation is one-to-one on the set of graphs with minimum degree at least 3, and if graphs G and G ^' have minimum degree at least 3 then any isomorphism from the P ₄-graph P ₄(G) to the P ₄-graph P ₄(G ^') is induced by a vertex-isomorphism from G to G ^' unless G and G ^' both belong to a special family of graphs. Supported by NSFC, PCSIRT and the “973” program.     

On x 6 + x + a in Characteristic Three

Let F be a field of characteristic 3 and 0 a F. We show that the 10 ways to factor x ⁶ + x + a into two cubics over the algebraic closure F are in natural Galois bijection with the 10 roots of x ¹⁰ + ^ax + 1. We use this to (1) prove the two polynomials have the same splitting field; (2) prove that a difference set constructed by Arasu and Player using the polynomial x ⁶ + x + a is isomorphic to a difference set constructed by Dillon using the polynomial x ¹⁰ + x + a; (3) obtain a natural realization for the accidental isomorphism between the alternating group A ₆ and the special linear group PSL₂(9); and (4) characterize how x ⁶ + x + a factors when F = GF (3^m) with m odd. For example, x ⁶ + x + a is irreducible if and only if a can be written as ^{– 36} + ⁴ with F ^× and Tr(⁵) 0.     

Jordan isomorphisms of nest algebras

Let and be two nest algebras. A Jordan isomorphism from onto is a bijective linear map such that for every . In this note, we prove that every Jordan isomorphism of nest algebras is of the form or and then is, in fact, an isomorphism or an anti-isomorphism.

     

Application of the electronegativity indices of organic molecules to tasks of chemical informatics

An efficient structure filtration method for the operation with chemical databases containing information on the structures and properties of organic molecules was proposed. The technique involves the use of electronegativity indices for generation of identification keys and for isomorphism tests of the molecular graphs corresponding to the structural formulas. The test set for the method proposed included a total of 95,000,000 molecules containing up to sixty carbon atoms. Tests revealed a high discriminating capability of the electronegativity indices and high efficiency of the method for solving both general problems (recognition of chemical structures, chemical database management systems) and specific tasks (generation of molecular graphs, etc.) in chemical informatics. Dedicated to Academician N. S. Zefirov on the occasion of his 70th birthday. Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 9, pp. 2166–2176, September, 2005.     

Antisymmetry and stability of water systems. I. Planar cyclic clusters

All topologically distinct configurations of planar cyclic water clusters consisting of three to six molecules are calculated. The symmetry of configurations is analyzed using an additional operation of antisymmetry that changes the directions of all hydrogen bonds. It is concluded that the concept of antisymmetry and the presence of similar in properties but inequivalent “configurations-antipodes” reflects a new fundamental feature of water systems, namely, the internal molecular asymmetry.     

Crystallochemistry of mercury chalcogenides. IV. Crystalline structure of two polymorphous modifications of the mercury sulfochalcogenide Hg3S2Br1.5Cl0.5

Two modifications of a new mercury sulfohalide of Hg₃S₂Br_2−x Cl_x (x = 0.5) composition have been grown from the gas phase and explored by X-ray structural analysis. The compounds were obtained at an attempt to synthesize an analogue of the rare mineral arzakite Hg₃S₂(Br, Cl)₂ (Br > Cl). The refinement of the crystalline structures of monoclinic (I) and cubic (II) phases (I: a = 17.824(4) Å, b = 9.238(2) Å, c = 10.269(2) Å, β = 115.69(1)°, V = 1523.8(5) ų, space group C2/m, Z = 8, R = 0.0513; II: a = 18.248(2) Å, V = 6076.4(12) ų, space group Pm3ˉn, Z = 32, R = 0.038) has shown that they are polymorphous modifications of the compound of Hg₃S₂Br_1.5Cl_0.5 formula. The monoclinic modification I is isostractural to the synthetic compound α-Hg₃S₂Br₂. Modification II is isostructural to synthetic β-Hg₃S₂Cl₂. In both structures, each atom S has in its surrounding three atoms of Hg forming umbrella-type groups SHg₃ with spaces Hg—S 2.366–2.430 Å and angles HgSHg 95.66–97.60°. SHg₃-fragments are bound by Hg-apices with the formation of isolated cubic groups [Hg₁₂S₈]. Like that in other structures of mercury chalcohalides, the main role in structure-forming of the investigated compounds is played by atoms of halogens creating a cubic sublattice in which radicals Hg—S are arranged. Original Russian Text Copyright ? 2006 by N. V. Pervukhina, S. A. Magarill, D. Yu. Naumov, S. V. Borisov, V. I. Vasil’yev, and B. G. Nenashev __________ Translated from Zhurnal Strukturnoi Khimii, Vol. 47, No. 2, pp. 318–323, March–April, 2006.