If each critical point is degenerate, then the conclusion that M is homeomorphic to a sphere is true. This was proved by Milnor [8] (1959) and by Rosen [10] (1960). With the aid of some techniques of Dyer and Hamstrom, recent results of Kirby, Edwards, and Cernavskii on spaces of homeomorphism on manifolds, and a selection theorem of Michael it is not difficult to establish the following topolo...

In this paper, we propose a new counting scheme for m × n contingency tables. Our scheme is a modification of Dyer and Greenhill’s scheme for two rowed contingency tables [5]. We can estimate not only the sizes of error, but also the sizes of the bias of the number of tables obtained by our scheme, on the assumption that we have an approximate sampler. key words: counting, approximation, #P-com...

In the early 90’s, Perrin-Riou (Ann Inst Fourier 43(4):945–995, 1993) introduced an important refinement of Mazur–Swinnerton-Dyer p-adic L-function elliptic curve E over $$\mathbb {Q}$$ , taking values in its de Rham cohomology. She then formulated a analogue Birch and Swinnerton-Dyer conjecture for this L-function, which formal group logarithms global points on make intriguing appearance. The ...

Given any integer N>1 prime to 3, we denote by C N the elliptic curve x 3 +y =N. We first study 3-adic valuation of algebraic part value Hasse–Weil L-function L(C ,s) over ℚ at s=1, and exhibit a relation between 3-part its Tate–Shafarevich group number distinct divisors which are inert in imaginary quadratic field K=ℚ(-3). In case where ,1)≠0 is product split primes K, show that order as predi...