Revista Matemática Complutense最新文献

A finite-dimensional (textsf{RCD}) space can be foliated into sufficiently regular leaves, where a differential calculus can be performed. Two important examples are given by the measure-theoretic boundary of the superlevel set of a function of bounded variation and the needle decomposition associated to a Lipschitz function. The aim of this paper is to connect the vector calculus on the lower dimensional leaves with the one on the base space. In order to achieve this goal, we develop a general theory of integration of (L^0)-Banach (L^0)-modules of independent interest. Roughly speaking, we study how to ‘patch together’ vector fields defined on the leaves that are measurable with respect to the foliation parameter.

Let (textsf{Hom}^{0}(Gamma ,G)) be the connected component of the identity of the variety of representations of a finitely generated nilpotent group (Gamma ) into a connected reductive complex affine algebraic group G. We determine the mixed Hodge structure on the representation variety (textsf{Hom}^{0}(Gamma ,G)) and on the character variety (textsf{Hom}^{0}(Gamma ,G)/!!/G). We obtain explicit formulae (both closed and recursive) for the mixed Hodge polynomial of these representation and character varieties.

We present a new proof of the caloric smoothing related to the fractional Gauss–Weierstrass semi–group in Triebel-Lizorkin spaces. This property will be used to prove existence and uniqueness of mild and strong solutions of the Cauchy problem for a fractional nonlinear heat equation.

In this note we give a re-interpretation of the algebraic fundamental group for proper schemes that is rather close to the original definition of the fundamental group for topological spaces. The idea is to replace the standard interval from topology by what we call interval schemes. This leads to an algebraic version of continuous loops, and the homotopy relation is defined in terms of the monodromy action. Our main results hinge on Macaulayfication for proper schemes and Lefschetz type results.

In this paper we are concerned with Triebel–Lizorkin–Morrey spaces ({mathcal {E}}^{s}_{u,p,q}(Omega )) of positive smoothness s defined on (special or bounded) Lipschitz domains (Omega subset {{mathbb {R}}}^{d}) as well as on ({{mathbb {R}}}^{d}). For those spaces we prove new equivalent characterizations in terms of local oscillations which hold as long as some standard conditions on the parameters are fulfilled. As a byproduct, we also obtain novel characterizations of ({mathcal {E}}^{s}_{u,p,q}(Omega )) using differences of higher order. Special cases include standard Triebel–Lizorkin spaces (F^s_{p,q} (Omega )) and hence classical (L_p)-Sobolev spaces (H^s_p(Omega )).

We realize specific classical symmetric spaces, like the semi-Kähler symmetric spaces discovered by Berger, as cotangent bundles of symmetric flag manifolds. These realizations enable us to describe these cotangent bundles’ geodesics and Lagrangian submanifolds. As a final application, we present the first examples of vector bundles over simply connected manifolds with nonnegative curvature that cannot accommodate metrics with nonnegative sectional curvature, even though their associated unit sphere bundles can indeed accommodate such metrics. Our examples are derived from explicit bundle constructions over symmetric flag spaces.

In this note, we prove that the boundary of a ((W^{1, p}, BV))-extension domain is of volume zero under the assumption that the domain ({Omega }) is 1-fat at almost every (xin partial {Omega }). Especially, the boundary of any planar ((W^{1, p}, BV))-extension domain is of volume zero.

We prove that every smooth closed connected manifold admits a smooth real-valued function with only two critical values such that the set of minima (or maxima) can be arbitrarily prescribed, as soon as this set is a finite subcomplex of the manifold (we call a function of this type a Reeb function). In analogy with Reeb’s Sphere Theorem, we use such functions to study the topology of the underlying manifold. In dimension 3, we give a characterization of manifolds having a Heegaard splitting of genus g in terms of the existence of certain Reeb functions. Similar results are proved in dimension (nge 5).

We study actions of multiplicative subgroups of Clifford algebras on Riemann surfaces. We show that every Klein surface of algebraic genus greater than 1 is isomorphic to the orbit space of such an action. We obtain linear representations of fundamental groups of Klein surfaces by using the spinor representations of Clifford algebras.

In this manuscript, we will study the asymptotic behavior for a class of nonlocal diffusion equations associated with the weighted fractional (wp (cdot ))-Laplacian operator involving constant/variable exponent, with (wp ^{-}:=min _{(x,y) in {overline{Omega }}times {overline{Omega }}} wp (x,y)geqslant max left{ 2N/(N+2s),1right} ) and (sin (0,1).) In the case of constant exponents, under some appropriate conditions, we will study the existence of solutions and asymptotic behavior of solutions by employing the subdifferential approach and we will study the problem when (wp ) goes to (infty ). Already, for case the weighted fractional (wp (cdot ))-Laplacian operator, we will also study the asymptotic behavior of the problem solution when (wp (cdot )) goes to (infty ), in the whole or in a subset of the domain (the problem involving the fractional (wp (cdot ))-Laplacian presents a discontinuous exponent). To obtain the results of the asymptotic behavior in both problems it will be via Mosco convergence.