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