Research in the Mathematical Sciences最新文献

In this paper we construct new classes of mixed singularities that provide realizations of real algebraic links in the 3-sphere. Especially, we describe this construction in the case of semiholomorphic polynomials, which are mixed polynomials that are holomorphic in one variable. Classifications and characterizations of real algebraic links are still open. These new classes of mixed singularities may help to shed light on the Benedetti–Shiota conjecture, which states that any fibered link on the 3-sphere is a real algebraic link.

Abstract

In this work, we show that Hölder equivalence of analytic functions germs (({mathbb {R}}^2,0)rightarrow ({mathbb {R}},0)) admits continuous moduli.

This is a complete study of the dynamics of polynomial planar vector fields whose linear part is a multiple of the identity and whose nonlinear part is a contracting homogeneous polynomial. The contracting nonlinearity provides the existence of an invariant circle and allows us to obtain a classification through a complete invariant for the dynamics, extending previous work by other authors that was mainly concerned with the existence and number of limit cycles. The general results are also applied to some classes of examples: definite nonlinearities, ({textbf {Z}}_2oplus {textbf {Z}}_2) symmetric systems and nonlinearities of degree 3, for which we provide complete sets of phase-portraits.

In Cisneros-Molina et al. (São Paulo J Math Sci, 2023. https://doi.org/10.1007/s40863-023-00370-y) it was proved the existence of fibrations à la Milnor (in the tube and in the sphere) for real analytic maps (f:({mathbb {R}}^n,0) rightarrow ({mathbb {R}}^k,0)), where (nge kge 2), with non-isolated critical values. In the present article we extend the existence of the fibrations given in Cisneros-Molina et al. (São Paulo J Math Sci, 2023. https://doi.org/10.1007/s40863-023-00370-y) to differentiable maps of class (C^{ell }), (ell ge 2), with possibly non-isolated critical value. This is done using a version of Ehresmann fibration theorem for differentiable maps of class (C^{ell }) between smooth manifolds, which is a generalization of the proof given by Wolf (Michigan Math J 11:65–70, 1964) of Ehresmann fibration theorem. We also present a detailed example of a non-analytic map which has the aforementioned fibrations.

Abstract

In this paper we propose a de Rham-Witt version of the derived KZ equations and their hypergeometric realizations.

We investigate geometric properties of a special kind of evolutes, so-called horocyclic evolutes, of smooth curves in hyperbolic plane from the viewpoint of duality. To do that, we first review the basic notions of (spacelike) frontals in hyperbolic plane, which developed by the first and the third authors by using basic Legendrian duality theorem developed by the second author. Moreover, two kinds of horocyclic evolutes are defined and the relationship between these two different evolutes are studied. As results, they are Legendrian dual to each other.

Let (F: {mathbb {R}}^2rightarrow {mathbb {R}}^2) be a polynomial mapping. We consider the image of the compositions (F^k) of F. We prove that under some condition then the image of the iterated map (F^k) is stable when k is large.

We introduced an (tilde{mathcal {A}})-invariant for quasi-ordinary parameterizations, and we consider it to describe quasi-ordinary surfaces with one generalized characteristic exponent admitting a countable moduli.

This is a survey article, in which we explore how the presence of singularities affects the geometry and topology of complex projective hypersurfaces.

The purpose of this paper is to present an algebraic theoretical basis for the study of (omega )-Hamiltonian vector fields defined on a symplectic vector space ((V,omega )) with respect to coordinates that are not necessarily symplectic. We introduce the concepts of (omega )-symplectic and (omega )-semisymplectic groups, and describe some of their properties that may not coincide with the classical context. We show that the Lie algebra of such groups is a useful tool in the recognition and construction of (omega )-Hamiltonian vector fields.