scholarly journals The curl of a weighted network

2008 ◽  
Vol 2 (2) ◽  
pp. 241-254 ◽  
Author(s):  
E. Bendito ◽  
A. Carmona ◽  
A.M. Encinas ◽  
J.M. Gesto

In this work we introduce an accurate definition of the curl operator on weighted networks that completes the discrete vector calculus developed by the authors. This allows us to define the circulation of a vector field along a curve and to characterize the conservative fields. In addition, we obtain an adequate discrete version of the De Rham cohomology of a compact manifold, giving in particular discrete analogues of the Poincar? and Hodge's decomposition theorems.

Author(s):  
Franc¸ois Charles ◽  
Christian Schnell

This chapter surveys the theory of absolute Hodge classes. First, the chapter recalls the construction of cycle maps in de Rham cohomology, which is then used in the definition of absolute Hodge classes. The chapter then deals with variational properties of absolute Hodge classes. After stating the variational Hodge conjecture, the chapter proves Deligne's principle B and discusses consequences of the algebraicity of Hodge bundles and of the Galois action on relative de Rham cohomology. Finally, the chapter provides some important examples of absolute Hodge classes: a discussion of the Kuga–Satake correspondence as well as a full proof of Deligne's theorem which states that Hodge classes on abelian varieties are absolute.


2004 ◽  
Vol 01 (01n02) ◽  
pp. 33-48 ◽  
Author(s):  
E. J. BEGGS ◽  
TOMASZ BRZEZIŃSKI

Various aspects of the de Rham cohomology of Hopf algebras are discussed. In particular, it is shown that the de Rham cohomology of an algebra with the differentiable coaction of a cosemisimple Hopf algebra with trivial 0-th cohomology group, reduces to the de Rham cohomology of (co)invariant forms. Spectral sequences are discussed and the van Est spectral sequence for Hopf algebras is introduced. A definition of Hopf–Lie algebra cohomology is also given.


2013 ◽  
Vol 56 (2) ◽  
pp. 501-508
Author(s):  
Ana Cristina Ferreira

AbstractWe prove a vanishing theorem for the twisted de Rham cohomology of a compact manifold.


2019 ◽  
Vol 2019 (756) ◽  
pp. 101-149 ◽  
Author(s):  
Marius Crainic ◽  
Rui Loja Fernandes ◽  
David Martínez Torres

AbstractThis is the first in a series of papers dedicated to the study of Poisson manifolds of compact types (PMCTs). This notion encompasses several classes of Poisson manifolds defined via properties of their symplectic integrations. In this first paper we establish some fundamental properties and constructions of PMCTs. For instance, we show that their Poisson cohomology behaves very much like the de Rham cohomology of a compact manifold (Hodge decomposition, non-degenerate Poincaré duality pairing, etc.) and that the Moser trick can be adapted to PMCTs. More important, we find unexpected connections between PMCTs and symplectic topology: PMCTs are related with the theory of Lagrangian fibrations and we exhibit a construction of a non-trivial PMCT related to a classical question on the topology of the orbits of a free symplectic circle action. In subsequent papers, we will establish deep connections between PMCTs and integral affine geometry, Hamiltonian G-spaces, foliation theory, orbifolds, Lie theory and symplectic gerbes.


Author(s):  
Silvano Delladio

We establish that the intrinsic distance dE associated with an indecomposable plane set E of finite perimeter is infinitesimally Euclidean; namely, in E. By this result, we prove through a standard argument that a conservative vector field in a plane set of finite perimeter has a potential. We also provide some applications to complex analysis. Moreover, we present a collection of results that would seem to suggest the possibility of developing a De Rham cohomology theory for integral currents.


Author(s):  
Michael T. Postek

The term ultimate resolution or resolving power is the very best performance that can be obtained from a scanning electron microscope (SEM) given the optimum instrumental conditions and sample. However, as it relates to SEM users, the conventional definitions of this figure are ambiguous. The numbers quoted for the resolution of an instrument are not only theoretically derived, but are also verified through the direct measurement of images on micrographs. However, the samples commonly used for this purpose are specifically optimized for the measurement of instrument resolution and are most often not typical of the sample used in practical applications.SEM RESOLUTION. Some instruments resolve better than others either due to engineering design or other reasons. There is no definitively accurate definition of how to quantify instrument resolution and its measurement in the SEM.


Author(s):  
Carlos Ortiz de Landázuri

Heidegger, Zubiri, Apel y Polo habrían propuesto una definición más correcta de las respectivas nociones de sujeto relacional humano, a saber: “Dasein” o “ser-ahí”; “personeidad” o “esencia abierta”; “intersubjetividad” o “la llamada por parte de los entes a diversos interlocutores”; y, finalmente, “persona-núcleo” o “agente mediador entre los entes y el ser”. Se pretendía así evitar una vuelta a las paradojas del “sujeto transcendental” en Kant, del “yo absoluto” en Hegel o del “sujeto fenomenológico” en Husserl. Sin embargo en cada caso se siguieron estrategias heurísticas específicamente distintas a la hora de conceptualizar dicho sujeto relacional: Heidegger propuso una superación de la noción de “sujeto fenomenológico” en Husserl; Zubiri, en cambio, defendería una recuperación de la noción de “sujeto fenomenológico” en Husserl; por su parte, Apel propondría una reformulación semióticamente transformada del “Dasein” heideggeriano; finalmente, Polo propondría una reformulación gnoseológica de la noción de “Dasein” heideggeriano.Heidegger, Zubiri, Apel, and Polo have proposed a more accurate definition of the respective notions of human relational subject: “Dasein” or “being-there”; “Personhood” or “open essence”; “inter-subjectivity” or “entities’ appeal to diverse interlocutors”; and, finally, “nucleus-person” or “mediator between entities and being”. The aim is to avoid a return to Kant’s transcendental subject paradoxes and Hegel’s “absolute I” or Husserl´s “fenomenological subject”. But in each case specifically different heuristic strategies were followed when conceptualizing said relational subject: Heidegger proposed overcoming the notion of “phenomenological subject” in Husserl; Zubiri, however, defend the recovery of the notion of “phenomenological subject” in Husserl; meanwhile, Apel propose a transformed semiotically reformulation of Heidegger’s “Dasein”; finally, Polo propose a reformulation of the epistemological notion of Heidegger’s “Dasein”.


1989 ◽  
Vol 22 (1) ◽  
pp. 249-272 ◽  
Author(s):  
Wiesław Sasin

Author(s):  
Federico Scavia

Abstract Building upon work of Epstein, May and Drury, we define and investigate the mod p Steenrod operations on the de Rham cohomology of smooth algebraic stacks over a field of characteristic $p>0$ . We then compute the action of the operations on the de Rham cohomology of classifying stacks for finite groups, connected reductive groups for which p is not a torsion prime and (special) orthogonal groups when $p=2$ .


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