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Register a new userThe Ramsey properties for Grassmannians over $mathbb R$, $mathbb C$
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In this note we study and obtain factorization theorems for colorings of matrices and Grassmannians over $mathbb{R}$ and ${mathbb{C}}$, which can be considered metr
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Let $p(cdot): mathbb R^nto(0,infty)$ be a variable exponent function satisfying the globally log-Holder continuous condition. In this article, the authors first obtain a decomposition for any distribution of the variable weak Hardy space into good and bad parts and then prove the following real interpolation theorem between the variable Hardy space $H^{p(cdot)}(mathbb R^n)$ and the space $L^{infty}(mathbb R^n)$: begin{equation*} (H^{p(cdot)}(mathbb R^n),L^{infty}(mathbb R^n))_{theta,infty} =W!H^{p(cdot)/(1-theta)}(mathbb R^n),quad thetain(0,1), end{equation*} where $W!H^{p(cdot)/(1-theta)}(mathbb R^n)$ denotes the variable weak Hardy space. As an application, the variable weak Hardy space $W!H^{p(cdot)}(mathbb R^n)$ with $p_-:=mathopmathrm{ess,inf}_{xinrn}p(x)in(1,infty)$ is proved to coincide with the variable Lebesgue space $W!L^{p(cdot)}(mathbb R^n)$.
This survey paper examines the effective model theory obtained with the BSS model of real number computation. It treats the following topics: computable ordinals, satisfaction of computable infinitary formulas, forcing as a construction technique, effective categoricity, effective topology, and relations with other models for the effective theory of uncountable structures.
In this paper, we prove two improv
Quantum Hall Effects (QHEs) on the complex Grassmann manifolds $mathbf{Gr}_2(mathbb{C}^N)$ are formulated. We set up the Landau problem in $mathbf{Gr}_2(mathbb{C}^N)$ and solve it using group theoretical techniques and provide the energy spectrum and the eigenstates in terms of the $SU(N)$ Wigner ${cal D}$-functions for charged particles on $mathbf{Gr}_2(mathbb{C}^N)$ under the influence of abelian and non-abelian background magnetic monopoles or a combination of these thereof. In particular, for the simplest case of $mathbf{Gr}_2(mathbb{C}^4)$ we explicitly write down the $U(1)$ background gauge field as well as the single and many-particle eigenstates by introducing the Pl{u}cker coordinates and show by calculating the two-point correlation function that the Lowest Landau Level (LLL) at filling factor $ u =1$ forms an incompressible fluid. Our results are in agreement with the previous results in the literature for QHE on ${mathbb C}P^N$ and generalize them to all $mathbf{Gr}_2(mathbb{C}^N)$ in a suitable manner. Finally, we heuristically identify a relation between the $U(1)$ Hall effect on $mathbf{Gr}_2(mathbb{C}^4)$ and the Hall effect on the odd sphere $S^5$, which is yet to be investigated in detail, by appealing to the already known analogous relations between the Hall effects on ${mathbb C}P^3$ and ${mathbb C}P^7$ and those on the spheres $S^4$ and $S^8$, respectively.
We study the barycenter of the Hellinger--Kantorovich metric over non-negative measures on compact, convex subsets of $mathbb{R}^d$. The article establishes existence, uniqueness (under suitable assumptions) and equivalence between a coupled-two-marginal and a multi-marginal formulation. We analyze the HK barycenter between Dirac measures in detail, and find that it differs substantially from the Wasserstein barycenter by exhibiting a local `clustering behaviour, depending on the length scale of the input measures. In applications it makes sense to simultaneously consider all choices of this scale, leading to a 1-parameter family of barycenters. We demonstrate the usefulness of this family by analyzing point clouds sampled from a mixture of Gaussians and inferring the number and location of the underlying Gaussians.
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