Jesús Muciño-Raymundo

Jesús Muciño-Raymundo
Universidad Nacional Autónoma de México | UNAM · Centro de Ciencias Matemáticas

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65
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Publications

Publications (65)
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Let $\Sigma(f)$ be critical points of a polynomial $f \in \mathbb{K}[x,y]$ in the plane $\mathbb{K}^2$, where $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C}$. Our goal is to study the critical point map $\mathfrak{S}_d$, by sending polynomials $f$ of degree $d$ to their critical points $\Sigma(f)$ . Very roughly speaking, a polynomial $f$ is essential...
Preprint
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Singular complex analytic vector fields on the Riemann surfaces enjoy several geometric properties (singular means that poles and essential singularities are admissible). We describe relations between singular complex analytic vector fields $\mathbb{X}$ and smooth vector fields $X$. Our approximation route studies three integrability notions for re...
Preprint
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We study transcendental meromorphic functions with essential singularities on Riemann surfaces. Every function $\Psi_X$ has associated a complex vector field $X$. In the converse direction, vector fields $X$ provide single valued or multivalued functions $\Psi_X$. Our goal is to understand the relationship between the analytical properties of $\Psi...
Article
We consider the family ℰ ( s , r , d ) of all singular complex analytic vector fields X ( z ) = Q ( z ) P ( z ) e E ( z ) ∂ ∂ z $X(z)=\frac{Q(z)}{P(z)}{{e}^{E(z)}}\frac{\partial }{\partial z}$ on the Riemann sphere where Q , P , ℰ are polynomials with deg Q = s , deg P = r and deg ℰ = d ≥ 1. Using the pullback action of the affine group Aut(ℂ) and...
Article
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We provided a detailed study of the Schur–Cohn stability algorithm for Schur stable polynomials of one complex variable. Firstly, a real analytic principal C×S1-bundle structure in the family of Schur stable polynomials of degree n is constructed. Secondly, we consider holomorphic C-actions A on the space of polynomials of degree n. For each orbit...
Preprint
Generically, the singular complex analytic vector fields $X$ on the Riemann sphere $\widehat{\mathbb{C}}_{z}$ belonging to the family $$ \mathscr{E}(r,d)=\Big\{ X(z)=\frac{1}{P(z)}\ \text{e}^{E(z)}\frac{\partial}{\partial z} \ \big\vert \ P, E\in\mathbb{C}[z], \ deg(P)=r, \ deg(E)=d \Big\}, $$ have an essential singularity of finite 1-order at infi...
Preprint
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Generically, the singular complex analytic vector fields X on the Riemann sphere Cz belonging to the family E (r, d) = X(z) = 1 P (z) e E(z) ∂ ∂z P, E ∈ C[z], deg P = r, deg E = d , have an essential singularity of finite 1-order at infinity and a finite number of poles on the complex plane. We describe X, particularly the singularity at ∞ ∈ Cz. In...
Preprint
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We consider the family $\mathcal{E}(s,r,d)=\Big\{ X(z)=\frac{Q(z)}{P(z)}\ e^{E(z)} \frac{\partial}{\partial z} \Big\},$ with $Q, P, E$ polynomials, deg${Q}=s$, deg$(P)=r$ and deg$(E)=d$, of singular complex analytic vector fields $X$ on the Riemann sphere $\widehat{\mathbb{C}}$. For $d\geq1$, $X\in\mathcal{E}(s,r,d)$ has $s$ zeros and $r$ poles on...
Chapter
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Motivated by the wild behavior of isolated essential singularities in complex analysis, we study singular complex analytic vector fields X on arbitrary Riemann surfaces M. By vector field singularities we understand zeros, poles, isolated essential singularities and accumulation points of the above kind. In this framework, a singular analytic vecto...
Preprint
Motivated by the wild behavior of isolated essential singularities in complex analysis, we study singular complex analytic vector fields $X$ on arbitrary Riemann surfaces $M$. By vector field singularities we understand zeros, poles, isolated essential singularities and accumulation points of the above kind. In this framework, a singular analytic v...
Preprint
Full-text available
A study of the geometry of dynamics of the shcur-Cohn stability algortim for one variable complex polynomials, using principal fiber bundles and meromorphic vector fields.
Article
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We tackle the problem of understanding the geometry and dynamics of singular complex analytic vector fields X with essential singularities on a Riemann surface M (compact or not). Two basic techniques are used. (a) In the complex analytic category on M, we exploit the correspondence between singular vector fields X, differential forms w_X (with w_X...
Article
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Let (F-1, . . . , F-n) : C-n -> C-n be a locally invertible polynomial map. We consider the canonical pull-back vector fields under this map, denoted by partial derivative/partial derivative F-1, . . . , partial derivative/partial derivative F-n. Our main result is the following: if n - 1 of the vector fields partial derivative/partial derivative F...
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We study vector fields on the plane having only isochronous centres. The most familiar examples are isochronous vector fields, they are the real parts of complex polynomial vector fields on having all their zeroes of centre type. We describe the number of topologically inequivalent isochronous (singular) foliations that can appear for degree s, up...
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In this paper we study the flat geometry and real dynamics of meromorphic vector fields on compact Riemann surfaces. Necessary and sufficient conditions to assert the existence of meromorphic vector fields with prescribed singularities are given. A characterization of the real dynamics of meromorphic vector fields is also given. Several explicit ex...
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Using singular flat metrics associated to meromorphic differential forms on Riemann surfaces, a converse of the classical residue theorem is due. Also a dynamical interpretation of residues using real geodesic flows is given. 2000 Mathematics Subject Classification: 32G34.
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Pairs of real analytic Hamiltonian vector fields Xh, Xg in Poisson involution over (not necessarily compact) symplectic manifolds are considered. We address the following problem: describe how a two-dimensional orbit {L} of the induced (R2, +)-action falls to an isolated common zero of Xh and Xg. A generalization of the Poincaré-Hopf index is intro...
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Let M be a complex manifold provided with a holomorphic vector field X. We say that X is complete if its flow Φ: ℂ × M → M is well defined for all complex values of the time, otherwise X is incomplete.
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Let {Xθ} be a family of rotated singular real foliations in the Riemann sphere which is the result of the rotation of a meromorphic vector field X with zeros and poles of multiplicity one. We prove that the set of bifurcation values, in the circle {θ}, is for each family a set with at most a finite number of accumulation points. A condition which i...
Article
In this work the Yang–Mills functional with gauge group SU(2) in compactified Minkowski space is reduced to a mechanical Lagrangian which enables one to find new classes of exact dual and nonself-dual solutions. The technique used in the reduction allows a simple interpretation of meron-type solutions. The formalism is then used to produce a good s...
Article
Connection 1-forms on principal fiber bundles with arbitrary structure groups are considered, and a characterization of gauge-equivalent connections in terms of their associated holonomy groups is given. These results are then applied to invariant connections in the case where the symmetry group acts transitively on fibers, and both local and globa...
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Making use of the general theory of connections invariant under a symmetry group which acts transitively on fibers, explicit solutions are derived for SU(2)SU(2)-symmetric multi-instantons over S 2S 2, with SU(2) structure group. These multi-instantons correspond to a principal fiber bundle characterized by a second Chern number given by 2m 2, with...
Article
Connection 1-forms on principal fiber bundles with arbitrary characteristic groups are considered, and a characterization of gauge-equivalent connections in terms of their associated holonomy groups is given. The authors then apply these results to group-invariant connections, which give an algebraic procedure for obtaining solutions to the gauge f...
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A fiber-bundle treatment for Kaluza-Klein-type geometric unification of gravitation with the bosonic sector of the standard electroweak theory was presented by Rosenbaum et al. Here we show that it admits spontaneously compactified solutions where the dimensions of the internal space are of the order of the Planck length. Furthermore, the model is...
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A fiber bundle treatment for Kaluza–Klein-type geometric unification of gravitation with the bosonic sector of the standard electroweak theory is presented. The most general G-invariant quadratic Lagrangian is constructed explicitly, and it is shown that the Higgs field sector, including the symmetry-breaking potential, arise naturally from torsion...
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In this note we describe a procedure for obtaining explicit Yang-Mills connections in principal fiber bundles P, with structural group G, over a homogeneous space M. We use connections which are invariant under a Lie group action in P. Explicit solutions over M=S 2 ×S 2 , where G=SU(2) are given, and their second Chern number computed.

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