Search results for "45"
showing 10 items of 1896 documents
Bing meets Sobolev
2019
We show that, for each $1\le p < 2$, there exists a wild involution $\mathbb S^3\to \mathbb S^3$ in the Sobolev class $W^{1,p}(\mathbb S^3,\mathbb S^3)$.
Weakly controlled Moran constructions and iterated functions systems in metric spaces
2011
We study the Hausdorff measures of limit sets of weakly controlled Moran constructions in metric spaces. The separation of the construction pieces is closely related to the Hausdorff measure of the corresponding limit set. In particular, we investigate different separation conditions for semiconformal iterated function systems. Our work generalizes well known results on self-similar sets in metric spaces as well as results on controlled Moran constructions in Euclidean spaces.
Local Gromov-Witten invariants are log invariants
2019
We prove a simple equivalence between the virtual count of rational curves in the total space of an anti-nef line bundle and the virtual count of rational curves maximally tangent to a smooth section of the dual line bundle. We conjecture a generalization to direct sums of line bundles.
Least gradient functions in metric random walk spaces
2019
In this paper we study least gradient functions in metric random walk spaces, which include as particular cases the least gradient functions on locally finite weighted connected graphs and nonlocal least gradient functions on $\mathbb{R}^N$. Assuming that a Poincar\'e inequality is satisfied, we study the Euler-Lagrange equation associated with the least gradient problem. We also prove the Poincar\'e inequality in a few settings.
Fourier analysis of periodic Radon transforms
2019
We study reconstruction of an unknown function from its $d$-plane Radon transform on the flat $n$-torus when $1 \leq d \leq n-1$. We prove new reconstruction formulas and stability results with respect to weighted Bessel potential norms. We solve the associated Tikhonov minimization problem on $H^s$ Sobolev spaces using the properties of the adjoint and normal operators. One of the inversion formulas implies that a compactly supported distribution on the plane with zero average is a weighted sum of its X-ray data.
Deformations of Calabi-Yau manifolds in Fano toric varieties
2020
In this article, we investigate deformations of a Calabi-Yau manifold $Z$ in a toric variety $F$, possibly not smooth. In particular, we prove that the forgetful morphism from the Hilbert functor $H^F_Z$ of infinitesimal deformations of $Z$ in $F$ to the functor of infinitesimal deformations of $Z$ is smooth. This implies the smoothness of $H^F_Z $ at the corresponding point in the Hilbert scheme. Moreover, we give some examples and include some computations on the Hodge numbers of Calabi-Yau manifolds in Fano toric varieties.
Some considerations on the nonabelian tensor square of crystallographic groups
2011
The nonabelian tensor square $G\otimes G$ of a polycyclic group $G$ is a polycyclic group and its structure arouses interest in many contexts. The same assertion is still true for wider classes of solvable groups. This motivated us to work on two levels in the present paper: on a hand, we investigate the growth of the Hirsch length of $G\otimes G$ by looking at that of $G$, on another hand, we study the nonabelian tensor product of pro--$p$--groups of finite coclass, which are a remarkable class of solvable groups without center, and then we do considerations on their Hirsch length. Among other results, restrictions on the Schur multiplier will be discussed.
Harnack and Shmul'yan pre-order relations for Hilbert space contractions
2015
We study the behavior of some classes of Hilbert space contractions with respect to Harnack and Shmul'yan pre-orders and the corresponding equivalence relations. We give some conditions under which the Harnack equivalence of two given contractions is equivalent to their Shmul'yan equivalence and to the existence of an arc joining the two contractions in the class of operator-valued contractive analytic functions on the unit disc. We apply some of these results to quasi-isometries and quasi-normal contractions, as well as to partial isometries for which we show that their Harnack and Shmul'yan parts coincide. We also discuss an extension, recently considered by S.~ter~Horst [\emph{J. Operato…
Local structure of self-affine sets
2011
The structure of a self-similar set with open set condition does not change under magnification. For self-affine sets the situation is completely different. We consider planar self-affine Cantor sets E of the type studied by Bedford, McMullen, Gatzouras and Lalley, for which the projection onto the horizontal axis is an interval. We show that within small square neighborhoods of almost each point x in E, with respect to many product measures on address space, E is well approximated by product sets of an interval and a Cantor set. Even though E is totally disconnected, the limit sets have the product structure with interval fibres, reminiscent to the view of attractors of chaotic differentia…
Equivalence of viscosity and weak solutions for the $p(x)$-Laplacian
2010
We consider different notions of solutions to the $p(x)$-Laplace equation $-\div(\abs{Du(x)}^{p(x)-2}Du(x))=0$ with $ 1<p(x)<\infty$. We show by proving a comparison principle that viscosity supersolutions and $p(x)$-superharmonic functions of nonlinear potential theory coincide. This implies that weak and viscosity solutions are the same class of functions, and that viscosity solutions to Dirichlet problems are unique. As an application, we prove a Rad\'o type removability theorem.