Research in the Mathematical Sciences最新文献

This paper describes how the elliptic and hyperbolic regions of a surface are related to stable Gauss maps on closed orientable surfaces immersed in three-dimensional space. We will show that for certain connected, closed, orientable surfaces containing a finite number of embedded circles that delineate two distinct types of regions, if all regions of one type are homeomorphic to a cylinder, then there exists an immersion (f: M rightarrow mathbb {R}^3) for which the Gauss map is a fold Gauss map.

Let (g_t) be a loop in the space of monic complex polynomials in one variable of fixed degree n. If the roots of (g_t) are distinct for all t, they form a braid (B_1) on n strands. Likewise, if the critical points of (g_t) are distinct for all t, they form a braid (B_2) on (n-1) strands. In this paper we study the relationship between (B_1) and (B_2). Composing the polynomials (g_t) with the argument map defines a pseudo-fibration map on the complement of the closure of (B_1) in ({mathbb {C}}times S^1), whose critical points lie on (B_2). We prove that for (B_1) a T-homogeneous braid and (B_2) the trivial braid this map can be taken to be a fibration map. In the case of homogeneous braids we present a visualization of this fact. Our work implies that for every pair of links (L_1) and (L_2) there is a mixed polynomial (f:{mathbb {C}}^2rightarrow {mathbb {C}}) in complex variables u, v and the complex conjugate (overline{v}) such that both f and the derivative (f_u) have a weakly isolated singularity at the origin with (L_1) as the link of the singularity of f and (L_2) as a sublink of the link of the singularity of (f_u).

We determine the crossing numbers of (prime) amphicheiral knots. This problem dates back to the origin of knot tables by Tait and Little at the end of the nineteenth century. The proof is the most substantial application of the semiadequacy formulas for the edge coefficients of the Jones polynomial.

In this work, we extend the concept of the Lipschitz saturation of an ideal to the context of modules in some different ways, and we prove they are generically equivalent.

There is a natural duality between line congruences in (mathbb {R}^3) and surfaces in (mathbb {R}^4) that sends principal lines into asymptotic lines. The same correspondence takes the discriminant curve of a line congruence into the parabolic curve of the dual surface. Moreover, it takes the ridge curves to the flat ridge curves, while the subparabolic curves of a line congruence are taken to certain curves on the surface that we call flat subparabolic curves. In this paper, we discuss these relations and describe the generic behavior of the subparabolic curves at the discriminant curve of the line congruence, or equivalently, the parabolic curve of the dual surface. We also discuss Loewner’s conjectures under the duality viewpoint.

The Casas–Alvero conjecture predicts that every univariate polynomial over a field of characteristic zero having a common factor with each of its derivatives (H_i(f)) is a power of a linear polynomial. One approach to proving the conjecture is to first prove it for polynomials of some small degree d, compile a list of bad primes for that degree (namely, those primes p for which the conjecture fails in degree d and characteristic p) and then deduce the conjecture for all degrees of the form (dp^ell ), (ell in mathbb {N}), where p is a good prime for d. In this paper, we calculate certain distinguished monomials appearing in the resultant (R(f,H_i(f))) and obtain a (non-exhaustive) list of bad primes for every degree (din mathbb {N}setminus {0}).

The main objective of this paper is to compute RO(G)-graded cohomology of G-orbits for the group (G=C_n), where n is a product of distinct primes. We compute these groups for the constant Mackey functor (underline{mathbb {Z}}) and the Burnside ring Mackey functor (underline{A}). Among other results, we show that the groups (underline{H}^alpha _G(S^0)) are mostly determined by the fixed point dimensions of the virtual representations (alpha ), except in the case of (underline{A}) coefficients when the fixed point dimensions of (alpha ) have many zeros. In the case of (underline{mathbb {Z}}) coefficients, the ring structure on the cohomology is also described. The calculations are then used to prove freeness results for certain G-complexes.

We prove that if two germs of plane curves (C, 0) and ((C',0)) with at least one singular branch are equivalent by a (real) smooth diffeomorphism, then C is complex isomorphic to (C') or to (overline{C'}). A similar result was shown by Ephraim for irreducible hypersurfaces before, but his proof is not constructive. Indeed, we show that the complex isomorphism is given by the Taylor series of the diffeomorphism. We also prove an analogous result for the case of non-irreducible hypersurfaces containing an irreducible component that is non-factorable. Moreover, we provide a general overview of the different classifications of plane curve singularities.