J. Muñoz Masqué
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View article: Lifting diffeomorphisms to vector bundles
Lifting diffeomorphisms to vector bundles Open
Criteria for a diffeomorphism of a smooth manifold $M$ to be lifted to a linear automorphism of a given real vector bundle $p\colon V\rightarrow M$, are stated. Examples are included and the metric and complex vector-bundle cases are also …
View article: Quadratic maps in two variables on arbitrary fields
Quadratic maps in two variables on arbitrary fields Open
Let F be a field of characteristic different from 2 and 3, and let V be a vector space of dimension 2 over F. The generic classification of homogeneous quadratic maps f : V → V under the action of the linear group of V , is given and effic…
View article: The Square-Zero Basis of Matrix Lie Algebras
The Square-Zero Basis of Matrix Lie Algebras Open
A method is presented that allows one to compute the maximum number of functionally-independent invariant functions under the action of a linear algebraic group as long as its Lie algebra admits a basis of square-zero matrices even on a fi…
View article: Differential p-forms and q-vector fields with constant coefficients
Differential p-forms and q-vector fields with constant coefficients Open
Differential $p$-forms and $q$-vector fields with constant coefficients are\nstudied. Differential $p$-forms of degrees $p=1,2,n-1,n$ with constant\ncoefficients on a smooth $n$-dimensional manifold $M$ are characterized. In the\ncontravar…
View article: Structure of Gauge-Invariant Lagrangians
Structure of Gauge-Invariant Lagrangians Open
The theory of gauge fields in Theoretical Physics poses several mathematical problems of interest in Differential Geometry and in Field Theory. Below we tackle one of these problems: The existence of a finite system of generators of gauge-…
View article: Quadratic Maps in Two Variables on Arbitrary Fields
Quadratic Maps in Two Variables on Arbitrary Fields Open
Let $\mathbb{F}$ be a field of characteristic different from $2$ and $3$, and let $V$ be a vector space of dimension $2$ over $\mathbb{F}$. The generic classification of homogeneous quadratic maps $f\colon V\to V$ under the action of the l…
View article: First-order invariants of differential 2-forms
First-order invariants of differential 2-forms Open
Let $M$ be a smooth manifold of dimension $2n$, and let $O_{M}$ be the dense open subbundle in $\wedge^{2}T^{\ast}M$ of $2$-covectors of maximal rank. The algebra of $\operatorname*{Diff}M$-invariant smooth functions of first order on $O_{…
View article: A new look at the classification of the tri-covectors of a 6-dimensional symplectic space
A new look at the classification of the tri-covectors of a 6-dimensional symplectic space Open
Let $\\mathbb{F}$ be a field of characteristic $\\neq 2$ and $3$, let $V$ be a\n$\\mathbb{F}$-vector space of dimension $6$, and let $\\Omega \\in \\wedge ^2V^\\ast\n$ be a non-degenerate form. A system of generators for polynomial invaria…
View article: The Classification Problem for 2-Forms in Four Variables
The Classification Problem for 2-Forms in Four Variables Open
The notion of type of a differential 2-form in four variables is introduced and for 2-forms of type < 4, local normal models are given. If the type of a 2-form $Ω$ is 4, then the equivalence under diffeomorphisms of $Ω$ is reduced to the e…
View article: A group law for PKC purposes.
A group law for PKC purposes. Open
Let $\mathbb{F}$ be a field, let $V=\mathbb{F}^3$, and let $A\colon V\to V$ a linear map. The polynomial $P(x)=\det (x_1I+x_2A+x_3A^2)$ does not depend on $A$ but only on its characteristic polynomial $\chi(X)$. A law of composition $\oplu…
View article: A characterization of nonprime powers
A characterization of nonprime powers Open
A criterion is presented in order to decide whether agiven integer is a prime power or not. The criterion associatesto each positive integer $m$ a finite set of integers$\mathcal{S}(m)$, each of them $\le m $ and the propertiesof this set …