Arithmetical properties of wendt's determinant

Wendt's determinant of order n is the circulant determinant W_n whose (i,j)-th entry is the binomial coefficient , for 1?i,j?n, where n is a positive integer. We establish some congruence relations satisfied by these rational integers. Thus, if p is a prime number and k a positive integer, then and . If q is another prime, distinct from p, and h any positive integer, then . Furthermore, if p is odd, then . In particular, if p?5, then . Also, if m and n are relatively prime positive integers, then W_mW_n divides W_mn.     

On the range of a covering function

Let be a finite system of residue classes with the moduli n₁,…,n_k distinct. By means of algebraic integers we show that the range of the covering function is not contained in any residue class with modulus greater one. In particular, the values of w(x) cannot have the same parity.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates for the uniform norm of solutions of quasilinear SPDE's

In this paper we prove L^p estimates (p≥2) for the uniform norm of the paths of solutions of quasilinear stochastic partial differential equations (SPDE) of parabolic type. Our method is based on a version of Moser's iteration scheme developed by Aronson and Serrin in the context of non-linear parabolic PDE.     

Volume collapsed three-manifolds with a lower curvature bound

In this paper we determine the topology of three-dimensional complete orientable Riemannian manifolds with a uniform lower bound of sectional curvature whose volume is sufficiently small.     

Compactifications of contractible affine 3-folds into smooth Fano 3-folds with <Emphasis Type="Italic">B</Emphasis><Subscript>2</Subscript>=2

After the contributions of Furushima, Nakayama, Peternell and Schneider, in 1993, Furushima [Fur93] finally succeeded in the classification of the compactifications of the affine 3-space into smooth Fano 3-folds with B₂=1. In this paper, we consider the compactifications of the contractible affine 3-folds X (not necessarily X=) into smooth Fano 3-folds V with B₂=2. Consequently, we classify all such compactifications X↪(V,D₁∪D₂) in the case where K_V+D₁+D₂ is not nef. Furthermore, we see that infinitely many mutually non-isomorphic exotic 's can be compactified into Fano 3-folds with B₂=2. This phenomenon never occurs when B₂=1. During this research the author was supported as a Twenty-First Century COE Kyoto Mathematics Fellow.     

Compactification of Strictly Convex Real Projective Structures

In this paper we compactify the space of convex real projective structures, known as Hitchin’s component in a representation variety. We give geometric meanings to the boundary points.Mathematics Subject Classifications (2000). 51M10, 57S25.     

Spectral studies on mixed ligand complexes of sodium dinitro-bis(acetylacetonato) cobaltate(iii) with imida-zole ligands

A number of mixed ligand complexes, nitro(imidazole) bis(acetylacetonato) cobaltate(III), [Co(acac)₂ (NO₂ )₂ (L)] (L = imidazole or substituted imidazole) have been synthesised and characterised on the basis of chemical analyses, IR, electronic, mass and nuclear (¹H and ¹³C) magnetic resonance spectra as well as TGA and DTA data. All complexes are non-electrolytic and possess a trans octahedral structure. The NO₂^? group is coordinated through the N atom. Though the ¹H NMR data of fresh CDCl₃ solutions of complexes confirm the trans structure, slow isomerisation to the cis configuration is observed at room temperature.     

A disjoint disks property for 3-manifolds

We show that the map separation property (MSP), a concept due to H.W. Lambert and R.B. Sher, is an appropriate analogue of J.W. Cannon’s disjoint disks property (DDP) for the class $\text{C}$ of compact generalized 3-manifolds with zero-dimensional singular set, modulo the Poincaré conjecture. Our main result is that the Poincaré conjecture (in dimension three) is equivalent to the conjecture that every X?$\text{C}$ with the MSP is a topological 3-manifold.     

Characteristic polynomials of symmetric matrices III: some counterexamples

When is a monic polynomial the characteristic polynomial of a symmetric matrix over an integral domain D? Known necessary conditions are shown to be insufficient when D is the field of 2-adic numbers and when D is the rational integers. The latter counterexamples lead to totally real cubic extensions of the rationals whose difierents are not narrowly equivalent to squares. Furthermorex³-4x+1 is the characteristic polynomial of a rational symmetric matrix and is the characteristic polynomial of an integral symmetric p-adic matrix for every prime p, but is not the characteristic polynomial of a rational integral symmetric matrix.