On Borel equivalence relations related to self-adjoint operators

In a recent work, the authors studied various Borel equivalence relations defined on the Polish space ${\rm{SA}}(H)$ of all (not necessarily bounded) self-adjoint operators on a separable infinite-dimensional Hilbert space $H$. In this paper we study the domain equivalence relation $E_{\rm{dom}}^{{\rm{SA}}(H)}$ given by $AE_{\rm{dom}}^{{\rm{SA}}(H)}B\Leftrightarrow {\rm{dom}}{A}={\rm{dom}}{B}$ and determine its exact Borel complexity: $E_{\rm{dom}}^{{\rm{SA}}(H)}$ is an $F_{\sigma}$ (but not $K_{\sigma}$) equivalence relation which is continuously bireducible with the orbit equivalence relation $E_{\ell^{\infty}}^{\mathbb{R}^{\mathbb{N}}}$ of the standard Borel group $\ell^{\infty}=\ell^{\infty}(\mathbb{N},\mathbb{R})$ on $\mathbb{R}^{\mathbb{N}}$. This, by Rosendal's Theorem, shows that $E_{\rm{dom}}^{{\rm{SA}}(H)}$ is universal for $K_{\sigma}$ equivalence relations. Moreover, we show that generic self-adjoint operators have purely singular continuous spectrum equal to $\mathbb{R}$.Comment: 10 pages, added more detail of the proof of Proposition 3.8 after the referee's suggestio

The Crystal Structure of Monovalent Streptavidin.

The strong interaction between streptavidin (SA) and biotin is widely utilized in biotechnological applications. A SA variant, monovalent SA, was developed with a single and high affinity biotin-binding site within the intact tetramer. However, its structural characterization remains undetermined. Here, we seek to determine the crystal structure of monovalent SA at 1.7-Å resolution. We show that, in contrast to its 'close-state' in the only wild-type subunit, the L3,4 loops of three Dead SA subunits are free from crystal packing and remain in an 'open state', stabilized by a consistent H-bonding network involving S52. This H-bonding network also applies to the previously reported open state of the wild-type apo-SA. These results suggest that specific substitutions (N23A/S27D/S45A) at biotin-binding sites stabilize the open state of SA L3,4 loop, thereby further reducing biotin-binding affinity. The general features of the 'open state' SA among different SA variants may facilitate its rational design. The structural information of monovalent SA will be valuable for its applications across a wide range of biotechnological areas

Expression kinetics and innate immune response after electroporation and LNP-mediated delivery of a self-amplifying mRNA in the skin

In this work, we studied the expression kinetics and innate immune response of a self-amplifying mRNA (sa-RNA) after electroporation and lipid-nanoparticle (LNP)-mediated delivery in the skin of mice. Intradermal electroporation of the sa-RNA resulted in a plateau-shaped expression, with the plateau between day 3 and day 10. The overall protein expression of sa-RNA was significantly higher than that obtained after electroporation of plasmid DNA (pDNA) or non-replication mRNAs. Moreover, using IFN-beta reporter mice, we elucidated that intradermal electroporation of sa-RNA induced a short-lived moderate innate immune response, which did not affect the expression of the sa-RNA. A completely different expression profile and innate immune response were observed when LNPs were used. The expression peaked 24 h after intradermal injection of sa-RNA-LNPs and subsequently showed a sharp drop. This drop might be explained by a translational blockage caused by the strong innate immune response that we observed in IFN-beta reporter mice shortly (4 h) after intradermal injection of sa-RNA-LNPs. A final interesting observation was the capacity of sa-RNA-LNPs to transfect the draining lymph nodes after intradermal injection

Current Status of Radio Source Databases

We review the history and present status of radio-source catalogue archiving and on-line retrieval of radio source data. Large efforts were spent by the first author in collecting and restoring electronic versions of new and old source catalogues. Some 67 catalogues with ~520,000 entries were searchable via the "Einstein On-line Service" (EOLS). When EOLS lost maintenance support in 1994 a group at SAO (Russia) started building software tools to search and cross-identify objects between the major radio catalogues, maintained as the "CATalog supporting System" (CATS) at the Special Astrophysical Observatory (SAO, Russia). The independent efforts in east and west have recently been joined. Almost 400 different source lists with ~2,000,000 entries have been archived (and partly prepared) by us. All 5C and Penticton "P"-surveys and many of the published WSRT survey lists are now available. CATS has been developed by O. Verkhodanov, S. Trushkin, V. Chernenkov at SAO primarily to support RATAN-600 radio observations. CATS runs under LINUX and can process requests on the basis of various net protocols and via email. Almost 70 well-known radio source catalogues and tables with about 1.3 Mrecords are now available via ftp from CATS, as well as their documentation files. Twenty of the larger tables may be searched simultaneously for objects in rectangular boxes of coordinates. New routines for cross-matching are in progress. More and more catalogues are being folded into CATS. CATS is supported by RFBR grant 96-07-89075.Comment: 2 pages, no figures; to appear in Proc. "Observational Cosmology with the New Radio Surveys", eds. M. Bremer, N. Jackson & I. Perez-Fournon, Kluwer Acad. Pres

Packing Topological Minors Half-Integrally

The packing problem and the covering problem are two of the most general questions in graph theory. The Erd\H{o}s-P\'{o}sa property characterizes the cases when the optimal solutions of these two problems are bounded by functions of each other. Robertson and Seymour proved that when packing and covering $H$-minors for any fixed graph $H$, the planarity of $H$ is equivalent with the Erd\H{o}s-P\'{o}sa property. Thomas conjectured that the planarity is no longer required if the solution of the packing problem is allowed to be half-integral. In this paper, we prove that this half-integral version of Erd\H{o}s-P\'{o}sa property holds with respect to the topological minor containment, which easily implies Thomas' conjecture. Indeed, we prove an even stronger statement in which those subdivisions are rooted at any choice of prescribed subsets of vertices. Precisely, we prove that for every graph $H$, there exists a function $f$ such that for every graph $G$, every sequence $(R_v: v \in V(H))$ of subsets of $V(G)$ and every integer $k$, either there exist $k$ subgraphs $G_1,G_2,...,G_k$ of $G$ such that every vertex of $G$ belongs to at most two of $G_1,...,G_k$ and each $G_i$ is isomorphic to a subdivision of $H$ whose branch vertex corresponding to $v$ belongs to $R_v$ for each $v \in V(H)$, or there exists a set $Z \subseteq V(G)$ with size at most $f(k)$ intersecting all subgraphs of $G$ isomorphic to a subdivision of $H$ whose branch vertex corresponding to $v$ belongs to $R_v$ for each $v \in V(H)$. Applications of this theorem include generalizations of algorithmic meta-theorems and structure theorems for $H$-topological minor free (or $H$-minor free) graphs to graphs that do not half-integrally pack many $H$-topological minors (or $H$-minors)

Steric Hindrance as a Mechanistic Probe for Olefin Reactivity: Variability of the Hydrogenic Canopy over the Isomeric Adamantylideneadamantane/Sesquihomoadamantene Pair (A Combined Experimental and Theoretical Study)

Access to each CC face of adamantylideneadamantane (AA) and sesquihomoadamantene (SA) is hindered by the hydrogenic canopy consisting of four β-hydrogens; otherwise, these olefins have quite normal environments. X-ray crystallography and density functional (DFT) calculations show a 0.5 Å larger annular opening in the protective cover of AA than that in SA. This contributes to the remarkable differences in reactivity toward various reagents, not only by limiting access to the olefin site in SA but also by inhibiting reactions which force these hydrogens closer together. Thus, AA is subject to typical olefin-addition reactions with bromine, sulfuryl chloride, m-chloroperbenzoic acid, dioxygen, and so forth, albeit sometimes at attenuated rates. On the other hand, SA is singularly unreactive under identical reaction conditions, except for the notable exceptions that include Brønsted (protonic) acids, a nitrosonium cation, and dichlorine. The exceptions are characterized as three sterically limited (electrophilic) reagents whose unique reactivity patterns are shown to be strongly influenced by steric access to the CC center. As such, the different degrees of steric encumbrance in the isomeric donors AA and SA shed considerable light on the diverse nature of olefinic reactions. In particular, they evoke mechanistic features in electrophilic addition versus electron transfer, which are otherwise not readily discernible with other less hindered olefinic donors. Transient structures of the olefinic-reaction intermediates such as the protonated carbocations AA−H+ and SA−H+ as well as the cation radicals AA•+ and SA•+ are probed by the combination of X-ray crystallographic analyses and density functional theoretical computations