Inhomogeneous Diophantine approximation over the field of formal Laurent series

De Mathan [B. de Mathan, Approximations diophantiennes dans un corps local, Bull. Soc. Math. France, Suppl. Mém. 21 (1970)] proved that Khintchine's theorem on homogeneous Diophantine approximation has an analogue in the field of formal Laurent series. Kristensen [S. Kristensen, On the well-approximable matrices over a field of formal series, Math. Proc. Cambridge Philos. Soc. 135 (2003) 255–268] extended this metric theorem to systems of linear forms and gave the exact Hausdorff dimension of the corresponding exceptional sets. In this paper, we study the inhomogeneous Diophantine approximation over a field of formal Laurent series, the analogue Khintchine's theorem and Jarnik–Besicovitch theorem are proved.     

Cycle Orders

Let X, T and C be, respectively, a finite set with at least three points, a set of ordered triples of distinct points from X, and a cyclic ordering of the points in X. Define T C to mean that, for every a b c T, the elements a, b, c occur in that cyclic order in C, and let C(T) denote the set of cyclic orderings of X for which T C. We say that T is noncyclic if C(T) is empty, cyclic if C(T) is nonempty, uniquely cyclic if | C(T) | = 1, a partial cycle order if it is cyclic and T ={a b c :{a b c} C for all C C(T)}, and a total cycle order if it is a uniquely cyclic partial cycle order. Many years ago E. V. Huntington axiomatized total cycle orders by independent necessary and sufficient conditions on T. The present paper studies the more relaxed structures of cyclic T sets and partial cycle orders. We focus on conditions for cyclicity, a theory of cycle dimension of partial cycle orders, and extremal problems that address combinatorial structures of T sets.     

Structure of 1,4-dihydro-1-ethyl-4-iminecinnoline-3-carboxylic acids, classical isosters of quinolone antibacterials: Crystal structures of hydrochlorides of 7-chloro and 7-methyl derivatives

Crystal structures of hydrochlorides of 7-chloro- and 7-methyl-4-iminecinnoline analogs of antibacterial quinolones have been determined by X-ray diffraction methods. The cell parameters for the 7-chloro (1) analog in the space group P2₁/c are a = 9.061(1), b = 19.062(1), c = 7.310(1)Å, = 104.92(1)°, Z = 4, and D _calc = 1.569 g/cm³ and for the 7-methyl (2) analog in the space group P are a = 7.277(5), b = 9.080(5), c = 10.058(5) Å, = 106.10(1), = 102.38(1), = 90.18(1)°, Z = 2, and D _calc = 1.429 g/cm³. Despite geometrical equivalency of methyl and chlorine and some resemblance of their packings, the crystal structures are not isostructural. Each compound forms a strong intramolecular hydrogen bond between protonated 4-imine (a donor) and 3-carboxylic (an acceptor) groups. Compounds with a similar bond, but with reversed functionality and orientation of the 3-carboxylic group, form common quinolones, being mostly 4-oxoquinoline-3-carboxylic acids. The difference should exclude chemical affinity of 4-imine analogs to the guanine base of a bacterial DNA in DNA-gyrase complex, as proposed by Shen et al. ² for 4-oxoquinoline-3-carboxylic acids showing antibacterial activity. Also the free acidic function of a carboxyl group may significantly lower permeability of 4-imine-3-carboxylic analogs of quinolones. Surprisingly, they have demonstrated antibacterial activity comparable with that of nalidixic acid.     

1,2,3,4,5,6,7,8-Octahydroanthracene-9,10- dicarbonitrile at 172, 216, 256, and 297 K

Octahydroanthracenedicarbonitrile is monoclinic, space group P2₁/c, at 172 K, a = 4.723(7), b = 8.254(3), c = 16.190(7) Å, = 91.76(8)°, Z = 2, V = 630.8(12) ų. The molecule, located on a center of symmetry, is disordered between the two senses of puckering in the aliphatic ring. The structure was also determined at 216, 256, and 297 K. The disorder increases with increasing temperature and can be described by a van't Hoff relationship with H = 5.2(3) kJ mol^–1 and S = 13.8(12) J K^–1 mol^–1.     

Denjoy's theorem with exponents

If is the (unique) minimal set for a diffeomorphism of the circle without periodic orbits, , then the upper box dimension of is at least . The method of proof is to introduce the exponent into the proof of Denjoy's theorem.

     

Convexity properties of holomorphic mappings in

Not many convex mappings on the unit ball in for are known. We introduce two families of mappings, which we believe are actually identical, that both contain the convex mappings. These families which we have named the ``Quasi-Convex Mappings, Types A and B' seem to be natural generalizations of the convex mappings in the plane. It is much easier to check whether a function is in one of these classes than to check for convexity. We show that the upper and lower bounds on the growth rate of such mappings is the same as for the convex mappings.

     

Isoperimetric Estimates on Sierpinski Gasket Type Fractals

For a compact Hausdorff space that is pathwise connected, we can define the connectivity dimension to be the infimum of all such that all points in can be connected by a path of Hausdorff dimension at most . We show how to compute the connectivity dimension for a class of self-similar sets in that we call point connected, meaning roughly that is generated by an iterated function system acting on a polytope such that the images of intersect at single vertices. This class includes the polygaskets, which are obtained from a regular -gon in the plane by contracting equally to all vertices, provided is not divisible by 4. (The Sierpinski gasket corresponds to .) We also provide a separate computation for the octogasket (), which is not point connected. We also show, in these examples, that , where the infimum is taken over all paths connecting and , and denotes Hausdorff measure, is equivalent to the original metric on . Given a compact subset of the plane of Hausdorff dimension and connectivity dimension , we can define the isoperimetric profile function to be the supremum of , where is a region in the plane bounded by a Jordan curve (or union of Jordan curves) entirely contained in , with . The analog of the standard isperimetric estimate is . We are particularly interested in finding the best constant and identifying the extremal domains where we have equality. We solve this problem for polygaskets with . In addition, for we find an entirely different estimate for as , since the boundary of has infinite measure. We find that the isoperimetric profile function is discontinuous, and that the extremal domains have relatively simple polygonal boundaries. We discuss briefly the properties of minimal paths for the Sierpinski gasket, and the isodiametric problem in the intrinsic metric.

     

Conformal measures and Hausdorff dimension for infinitely renormalizable quadratic polynomials

We show in this paper that iff is a quadratic infinitely many times renormalizable polynomial of sufficient high combinatorial type, then: HD (J(f))= inf{: -conformal measure for f} We use Lyubich's construction of the principal nest ([Lyu97]) in order to prove this result.Partially supported by CNP q-Brazil grant # 300534/96-5     

对称双方势阱能级无简并

证明了一维对称有限双方势阱的束缚态能级是不存在简并的。     

Means of extreme points and F-spaces

Given a topological space T and a strictly convex real normed space X, let be the space of continuous and bounded functions from T into X, with its uniform norm. This paper is devoted to the study of the relation between the fact of T being an F-space and the property that every element in the unit ball of has a representation as a mean of two extreme points.