Metric Diophantine approximation over a local field of positive characteristic

The conjectures of Sprind?uk in the metric theory of Diophantine approximation are established over a local field of positive characteristic. In the real case, these were settled by D. Kleinbock and G.A. Margulis using a new technique which involved nondivergence estimates for quasi-polynomial flows on the space of lattices. We extend their technique to the positive characteristic setting.     

Stereoisomeric flavour compounds XXXVIII: Direct chirospecific analysis of γ-lactones using on-line coupled LC-GC with a chiral separation column

Summary Chiral -lactones from the raw flavour extract of strawberries and commercially available fruit-containing products are stereoanalyzed directly by on-line coupling of liquid chromatography (LC) to gas chromatography (GC) and enantiomer separation with 2,6-di-O-pentyl-3-O-acetyl--cyclodextrin as the chiral stationary phase.     

The critical temperature and gap solution in the Bardeen-Cooper-Schrieffer theory of superconductivity

The Letter studies the problem of numerical approximations of the critical transition temperature and the energy gap function in the Bardeen-Cooper-Schrieffer equation arising in superconductivity theory. The positive kernel function leads to a phonon-dominant state at zero temperature. Much attention is paid to the equation defined on a bounded region. Two discretized versions of the equation are introduced. The first version approximates the desired solution from below, while the second, from above. Numerical examples are presented to illustrate the efficiency of the method. Besides, the approximations of a full space solution and the associated critical temperature by solution sequences constructed on bounded domains are also investigated.Part of this work was done while this author was visiting the Department of Mathematics, Carnegie Mellon University, Pittsburgh, PA, 15213, USA     

On completions of symmetric and antisymmetric block diagonal partial matrices

A partial matrix is a matrix where only some of the entries are given. We determine the maximum rank of the symmetric completions of a symmetric partial matrix where only the diagonal blocks are given and the minimum rank and the maximum rank of the antisymmetric completions of an antisymmetric partial matrix where only the diagonal blocks are given.     

Local Tree-Width,Excluded Minors,and Approximation Algorithms

The local tree-width of a graph G=(V,E) is the function ltw^G : that associates with every r the maximal tree-width of an r-neighborhood in G. Our main grapht heoretic result is a decomposition theorem for graphs with excluded minors, which says that such graphs can be decomposed into trees of graphs of almost bounded local tree-width.As an application of this theorem, we show that a number of combinatorial optimization problems, suchas Minimum Vertex Cover, Minimum Dominating Set, and Maximum Independent Set have a polynomial time approximation scheme when restricted to a class of graphs with an excluded minor.     

On the two-dimensional orthogonal drawing of series-parallel graphs

It has been known that every planar 4-graph has a 2-bend 2-D orthogonal drawing, with the only exception being the octahedron, every planar 3-graph has a 1-bend 2-D orthogonal drawing with the only exception being K₄, and every outerplanar 3-graph with no triangles has a 0-bend 2-D orthogonal drawing. We show in this paper that every series-parallel 4-graph has a 1-bend 2-D orthogonal drawing.     

Fekete-like polynomials

In 2001, Borwein, Choi, and Yazdani looked at an extremal property of a class of polynomial with ±1 coefficients. Their key result was:

Theorem. (See Borwein, Choi, Yazdani, 2001.) Letf(z)=±z±z²±?±zN−1, and ζ a primitive Nth root of unity. If N is an odd positive integer then

     

Minors in ‐ Chromatic Graphs,II

Let denote the graph obtained from the complete graph by deleting the edges of some ‐subgraph. The author proved earlier that for each fixed s and , every graph with chromatic number has a minor. This confirmed a partial case of the corresponding conjecture by Woodall and Seymour. In this paper, we show that the statement holds already for much smaller t, namely, for .     

Diophantine approximation and badly approximable sets

Let (X,d) be a metric space and (Ω,d) a compact subspace of X which supports a non-atomic finite measure m. We consider ‘natural’ classes of badly approximable subsets of Ω. Loosely speaking, these consist of points in Ω which ‘stay clear’ of some given set of points in X. The classical set Bad of ‘badly approximable’ numbers in the theory of Diophantine approximation falls within our framework as do the sets Bad(i,j) of simultaneously badly approximable numbers. Under various natural conditions we prove that the badly approximable subsets of Ω have full Hausdorff dimension. Applications of our general framework include those from number theory (classical, complex, p-adic and formal power series) and dynamical systems (iterated function schemes, rational maps and Kleinian groups).