Research in the Mathematical Sciences最新文献

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.

We relate the Brasselet number of a complex analytic function-germ defined on a complex analytic set to the critical points of its real part on the regular locus of the link. Similarly we give a new characterization of the Euler obstruction in terms of the critical points on the regular part of the link of the projection on a generic real line. As a corollary, we obtain a new proof of the relation between the Euler obstruction and the Gauss–Bonnet measure, conjectured by Fu.

Abstract

An algebraic variety X is called a homogeneous variety if the automorphism group ({{,textrm{Aut},}}(X)) acts on X transitively, and a homogeneous space if there exists a transitive action of an algebraic group on X. We prove a criterion of smoothness of a suspension to construct a wide class of homogeneous varieties. As an application, we give criteria for a Danielewski surface to be a homogeneous variety and a homogeneous space. Also, we construct affine suspensions of arbitrary dimension that are homogeneous varieties but not homogeneous spaces.

We propose a way to study the caustics of surfaces in non-flat Riemannian 4-space form from the viewpoint of singularity theory in this paper. As an application of the theory of Lagrangian singularity, we study the contact of surfaces with the families of hyperspheres, which is measured by the singularities of functions defined on the surfaces.

The Reeb space of a continuous function on a topological space is the space of connected components of the level sets. In this paper we characterize those smooth functions on closed manifolds whose Reeb spaces have the structure of a finite graph. We also give several explicit examples of smooth functions on closed manifolds such that they themselves or their Reeb spaces have some interesting properties.

Abstract

A network of coupled dynamical systems is represented by a graph whose vertices represent individual cells and whose edges represent couplings between cells. Motivated by the impact of synchronization results of the Kuramoto networks, we introduce the generalized class of Laplacian networks, governed by mappings whose Jacobian at any point is a symmetric matrix with row entries summing to zero. By recognizing this matrix with a weighted Laplacian of the associated graph, we derive the optimal estimates of its positive, null and negative eigenvalues directly from the graph topology. Furthermore, we provide a characterization of the mappings that define Laplacian networks. Lastly, we discuss stability of equilibria inside synchrony subspaces for two types of Laplacian network on a ring with some extra couplings.