A geometric formula for Haefliger knots

Haefliger has shown that a smooth embedding of the (4k−1)-sphere in the 6k-sphere can be knotted in the smooth sense. In this paper, we give a formula with which we can detect the isotopy class of such a Haefliger knot. The formula is expressed in terms of the geometric characteristics of an extension, analogous to a Seifert surface, of the given embedding. In particular, the Hopf invariant associated to the extension plays a crucial role. This leads us to a new characterisation of Haefliger knots.     

Achirality of knots

A theoretic and diagrammatic relationship between knots and planar graphs has enabled us to establish necessary condition for achirality. It is shown that the regions, crossings and consequently the number of vertices, edges, and faces in the corresponding LR-graph are same and invariant. Establishment of new but pivotal moves such as R*-move, 2π-twist and π-twist enabled us to prove that the black regions can be changed into white regions via Reidemeister moves. Consequently, the equivalence of the companion graphs, necessary and sufficient conditions for achirality. This revised version was published online in June 2006 with corrections to the Cover Date.     

Small latin squares,quasigroups, and loops

We present the numbers of isotopy classes and main classes of Latin squares, and the numbers of isomorphism classes of quasigroups and loops, up to order 10. The best previous results were for Latin squares of order 8 (Kolesova, Lam, and Thiel, 28 ), quasigroups of order 6 (Bower, 7 ), and loops of order 7 (Brant and Mullen, 8 ). The loops of order 8 have been independently found by “QSCGZ” and Guérin (unpublished, 25 ). We also report on the most extensive search so far for a triple of mutually orthogonal Latin squares (MOLS) of order 10. Our computations show that any such triple must have only squares with trivial symmetry groups. © 2006 Wiley Periodicals, Inc. J Combin Designs     

Real quadratic flexible division algebras

In this paper we give a new process called vectorial isotopy, in order to classify the eight-dimensional real quadratic flexible division algebras, and we solve the isomorphism problem.     

Isotopy and invariants of Albert algebras

Let k be a field with characteristic different from 2 and 3. Let B be a central simple algebra of degree 3 over a quadratic extension K/k, which admits involutions of second kind. In this paper, we prove that if the Albert algebras and have same and invariants, then they are isotopic. We prove that for a given Albert algebra J, there exists an Albert algebra J' with , and . We conclude with a construction of Albert division algebras, which are pure second Tits' constructions. Received: December 9, 1997.     

On the Haefliger-Hirsch-Wu invariants for embeddings and immersions

We prove beyond the metastable dimension the PL cases of the classical theorems due to Haefliger, Harris, Hirsch and Weber on the deleted product criteria for embeddings and immersions. The isotopy and regular homotopy versions of the above theorems are also improved. We show by examples that they cannot be improved further. These results have many interesting corollaries, e.g.? 1) Any closed homologically 2-connected smooth 7-manifold smoothly embeds in .? 2) If and then the set of PL embeddings up to PL isotopy is in 1-1 correspondence with?. Received: July 6, 2000     

Properties and structure of aqueous urea up to the singular temperature of overcooled water: Isotopy effects

In equations for the partial volume, dilatability, heat capacity, and adiabatic compressibility of aqueous urea, the properties are “fitted” to the singular temperature T _s of overcooled water T _s = 227.15 K (R. J. Speedy, J. Phys. Chem., 91, 3354 (1987)). The equations for the characteristics of D₂O-(ND₂)₂CO and T₂O-(NT₂)₂CO systems were obtained by scaling the temperature to 6 and 9.4 K, respectively. The properties of infinitely dilute solutions did not reveal the sign reversal of the isotope effect. The isoconcentrates of the partial volumes and dilatability form wagging lines in the region of low temperatures. The temperature dependences of the properties have extrema and inflection points, while the isotope effects in solutions of finite concentrations experience sign reversal. The maximum density temperature (MDT) of the solution decreases as the urea concentration increases, and also on passing from D₂O-(ND₂)₂CO to H₂O-(NH₂)₂CO. Thus, for the 8m solution, the MDT is 259.6 K (protium system) and 266.7 K (deuterium system). The difference is almost equal to the shift of the MDT after the H₂O → D₂O transition (7.2 K).     

关于辛同痕问题的一个注记

结合Gromov的思想和McDuff的工作，本文证明了在具有正第一陈类的有理直纹面的给定同调类中的任意两个辛嵌入球面都辛同痕，这样扩充了Gromov的一个经典结果．     

Guaranteed consistency of surface intersections and trimmed surfaces using a coupled topology resolution and domain decomposition scheme

We describe a method that serves to simultaneously determine the topological configuration of the intersection curve of two parametric surfaces and generate compatible decompositions of their parameter domains, that are amenable to the application of existing perturbation schemes ensuring exact topological consistency of the trimmed surface representations. To illustrate this method, we begin with the simpler problem of topology resolution for a planar algebraic curve F(x,y)=0 in a given domain, and then extend concepts developed in this context to address the intersection of two tensor-product parametric surfaces p(s,t) and q(u,v) defined on (s,t)∈[0,1]² and (u,v)∈[0,1]². The algorithms assume the ability to compute, to any specified precision, the real solutions of systems of polynomial equations in at most four variables within rectangular domains, and proofs for the correctness of the algorithms under this assumption are given. Mathematics subject classification (2000)  65D17