Weak topology | EPFL Graph Search

In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a topological vector space (such as a normed vector space) with respect to its continuous dual. The remainder of this article will deal with this case, which is one of the concepts of functional analysis. One may call subsets of a topological vector space weakly closed (respectively, weakly compact, etc.) if they are closed (respectively, compact, etc.) with respect to the weak topology. Likewise, functions are sometimes called weakly continuous (respectively, weakly differentiable, weakly analytic, etc.) if they are continuous (respectively, differentiable, analytic, etc.) with respect to the weak topology. History Starting in the early 1900s, David Hilbert and Marcel Riesz made extensive use of weak convergence. The early pioneer

Regularity of solutions for a fourth-order elliptic system via Conservation law

Changyu Guo

In this paper, we obtain interior Holder continuity for solutions of the fourth-order elliptic system Delta(2)u = Delta(V center dot del u) + div(w del u) + W center dot del u formulated by Lamm and Riviere [Comm. Partial Differential Equations 33 (2008) 245-262]. Boundary continuity is also obtained under a standard Dirichlet or Navier boundary condition. We also use conservation law to establish a weak compactness result which generalizes a result of Riviere for the second-order problem.

2019

Topological Structures on DMC spaces

Rajai Nasser

Two channels are said to be equivalent if they are degraded from each other. The space of equivalent channels with input alphabet X and output alphabet Y can be naturally endowed with the quotient of the Euclidean topology by the equivalence relation. We show that this topology is compact, path-connected and metrizable. A topology on the space of equivalent channels with fixed input alphabet X and arbitrary but finite output alphabet is said to be natural if and only if it induces the quotient topology on the subspaces of equivalent channels sharing the same output alphabet. We show that every natural topology is sigma-compact, separable and path-connected. On the other hand, if |X| \geq 2, a Hausdorff natural topology is not Baire and it is not locally compact anywhere. This implies that no natural topology can be completely metrized if |X| \geq 2. The finest natural topology, which we call the strong topology, is shown to be compactly generated, sequential and T_4. On the other hand, the strong topology is not first-countable anywhere, hence it is not metrizable. We show that in the strong topology, a subspace is compact if and only if it is rank-bounded and strongly-closed. We provide a necessary and sufficient condition for a sequence of channels to converge in the strong topology. We introduce a metric distance on the space of equivalent channels which compares the noise levels between channels. The induced metric topology, which we call the noisiness topology, is shown to be natural. We also study topologies that are inherited from the space of meta-probability measures by identifying channels with their posterior meta-probability distributions. We show that the weak-* topology is exactly the same as the noisiness topology and hence it is natural. We prove that if |X| \geq 2, the total variation topology is not natural nor Baire, hence it is not completely metrizable. Moreover, it is not locally compact anywhere. Finally, we show that the Borel sigma-algebra is the same for all Hausdorff natural topologies.

2017

Filming the formation and fluctuation of skyrmion domains by cryo-Lorentz transmission electron microscopy

Edoardo Baldini, Marco Cantoni, Fabrizio Carbone, Thierry Giamarchi, Ping Huang, Tatiana Latychevskaia, Arnaud Magrez, Giulia Fulvia Mancini, Yoshie Murooka, Henrik Moodysson Rønnow, Jonathan White

Magnetic skyrmions are promising candidates as information carriers in logic or storage devices thanks to their robustness, guaranteed by the topological protection, and their nanometric size. Currently, little is known about the influence of parameters such as disorder, defects, or external stimuli on the long-range spatial distribution and temporal evolution of the skyrmion lattice. Here, using a large (7.3×7.3 μm2) single-crystal nanoslice (150 nm thick) of Cu2OSeO3, we image up to 70,000 skyrmions by means of cryo-Lorentz transmission electron microscopy as a function of the applied magnetic field. The emergence of the skyrmion lattice from the helimagnetic phase is monitored, revealing the existence of a glassy skyrmion phase at the phase transition field, where patches of an octagonally distorted skyrmion lattice are also discovered. In the skyrmion phase, dislocations are shown to cause the emergence and switching between domains with different lattice orientations, and the temporal fluctuation of these domains is filmed. These results demonstrate the importance of direct-space and real-time imaging of skyrmion domains for addressing both their long-range topology and stability.

National Academy of Sciences2015

Topological Structures on DMC spaces

Rajai Nasser

2017