On embeddings of proper and equicontinuous actions in zero-dimensional compactifications

We provide a tool for studying properly discontinuous actions of non-compact groups on locally compact, connected and paracompact spaces, by embedding such an action in a suitable zero-dimensional compactification of the underlying space with pleasant properties. Precisely, given such an action we construct a zero-dimensional compactification of with the properties: (a) there exists an extension of the action on , (b) if is the set of the limit points of the orbits of the initial action in , then the restricted action remains properly discontinuous, is indivisible and equicontinuous with respect to the uniformity induced on by that of , and (c) is the maximal among the zero-dimensional compactifications of with these properties. Proper actions are usually embedded in the endpoint compactification of , in order to obtain topological invariants concerning the cardinality of the space of the ends of , provided that has an additional ``nice" property of rather local character (``property Z", i.e., every compact subset of is contained in a compact and connected one). If the considered space has this property, our new compactification coincides with the endpoint one. On the other hand, we give an example of a space not having the ``property Z" for which our compactification is different from the endpoint compactification. As an application, we show that the invariant concerning the cardinality of the ends of holds also for a class of actions strictly containing the properly discontinuous ones and for spaces not necessarily having ``property Z".

     

Characterization of functions as radiation patterns in linear elasticity

In this paper a necessary and sufficient condition for a pair of vector functions to be radiation patterns is presented. More precisely, it is proved that two vector functions, the first in the radial direction and the second in the tangential one, are radiation patterns if and only if there are two entire harmonic vector functions whose radial and tangential projections, respectively, are identical with the previous functions on the unit sphere and whose L²-norm over a sphere of radius R is a function of exponential type in the variable R.     

A new graph parameter related to bounded rank positive semidefinite matrix completions

The Gram dimension $\mathrm{gd}(G)$ of a graph $G$ is the smallest integer $k\ge 1$ such that any partial real symmetric matrix, whose entries are specified on the diagonal and at the off-diagonal positions corresponding to edges of $G$ , can be completed to a positive semidefinite matrix of rank at most $k$ (assuming a positive semidefinite completion exists). For any fixed $k$ the class of graphs satisfying $\mathrm{gd}(G) \le k$ is minor closed, hence it can be characterized by a finite list of forbidden minors. We show that the only minimal forbidden minor is $K_{k+1}$ for $k\le 3$ and that there are two minimal forbidden minors: $K_5$ and $K_{2,2,2}$ for $k=4$ . We also show some close connections to Euclidean realizations of graphs and to the graph parameter $\nu ^=(G)$ of van der Holst (Combinatorica 23(4):633–651, 2003). In particular, our characterization of the graphs with $\mathrm{gd}(G)\le 4$ implies the forbidden minor characterization of the 3-realizable graphs of Belk (Discret Comput Geom 37:139–162, 2007) and Belk and Connelly (Discret Comput Geom 37:125–137, 2007) and of the graphs with $\nu ^=(G) \le 4$ of van der Holst (Combinatorica 23(4):633–651, 2003).     

The group of isometries of a locally compact metric space with one end

In this note we study the dynamics of the natural evaluation action of the group of isometries G of a locally compact metric space (X,d) with one end. Using the notion of pseudo-components introduced by S. Gao and A.S. Kechris we show that X has only finitely many pseudo-components exactly one of which is not compact and G acts properly on this pseudo-component. The complement of the non-compact component is a compact subset of X and G may fail to act properly on it.     

Identification of Polyphenol Derivatives as Novel SARS-CoV-2 and DENV Non-Nucleoside RdRp Inhibitors

The Coronavirus Disease 2019 (COVID-19) and dengue fever (DF) pandemics both remain to be significant public health concerns in the foreseeable future. Anti-SARS-CoV-2 drugs and vaccines are both indispensable to eliminate the epidemic situation. Here, two piperazine-based polyphenol derivatives DF-47 and DF-51 were identified as potential inhibitors directly blocking the active site of SARS-CoV-2 and DENV RdRp. Data through RdRp inhibition screening of an in-house library and in vitro antiviral study selected DF-47 and DF-51 as effective inhibitors of SARS-CoV-2/DENV polymerase. Moreover, in silico simulation revealed stable binding modes between the DF-47/DF-51 and SARS-CoV-2/DENV RdRp, respectively, including chelating with Mg²⁺ near polymerase active site. This work discovered the inhibitory effect of two polyphenols on distinct viral RdRp, which are expected to be developed into broad-spectrum, non-nucleoside RdRp inhibitors with new scaffold.     

Formation and plasmonic response of self-assembled layers of colloidal gold nanorods and branched gold nanoparticles

The plasmonic properties of self-assembled layers of rod- and branched-shaped gold nanoparticles were investigated using optical techniques. Nanoparticles were synthesized by a surfactant-guided, seed-mediated growth method. The layers were obtained by gradual assembly of nanoparticles at the interface between a polar and a nonpolar solvent and were transferred to a glass slide. Polarization and angle-dependent extinction measurements showed that the layers made of gold nanorods were governed by an effective medium response. The response of the layers made by branched gold particles was characterized by random light scattering. Microscopic mapping of the spatial mode structure demonstrates a uniform optical response of the nanoparticle layers down to a submicrometer length scale.     

A lower bound for sums of eigenvalues of the Laplacian

Let be the th Dirichlet eigenvalue of a bounded domain in . According to Weyl's asymptotic formula we have

The optimal in view of this asymptotic relation lower estimate for the sums has been proven by P.Li and S.T.Yau (Comm. Math. Phys. 88 (1983), 309-318). Here we will improve this estimate by adding to its right-hand side a term of the order of that depends on the ratio of the volume to the moment of inertia of .

     

Induced cosmology on a regularized brane in six-dimensional flux compactification

We consider a six-dimensional Einstein–Maxwell system compactified in an axisymmetric two-dimensional space with one capped regularized conical brane of codimension one. We study the cosmological evolution which is induced on the regularized brane as it moves in between known static   bulk and cap solutions. Looking at the resulting Friedmann equation, we see that the brane cosmology at high energies is dominated by a five-dimensional ρ²${\rho }^2$ energy density term. At low energies, we obtain a Friedmann equation with a term linear to the energy density with, however, negative coefficient in the small four-brane radius limit (i.e., with negative effective Newton's constant). We discuss ways out of this problem.     

Local Lower Norm Estimates for Dyadic Maximal Operators and Related Bellman Functions

We provide lower \(L^{q}\) and weak \(L^{q}\)-bounds for the localized dyadic maximal operator on \(\mathbb {R}^{n}\), when the local \(L^{1}\) and the local \(L^{p}\) norms of the function are given. We actually do that in the more general context of homogeneous trees in probability spaces.