A functional model associated with a generalized Nevanlinna pair

Let $ \mathcal{L} $ be a Hilbert space, and let $ \mathcal{H} $ be a Pontryagin space. For every self-adjoint linear relation $ \tilde{A} $ in $ \mathcal{H} \oplus \mathcal{L} $ , the pair $ \{ I + \lambda \psi (\lambda ),\,\psi (\lambda )\} $ where $ \psi (\lambda ) $ is the compressed resolvent of $ \tilde{A} $ , is a normalized generalized Nevanlinna pair. Conversely, every normalized generalized Nevanlinna pair is shown to be associated with some self-adjoint linear relation $ \tilde{A} $ in the above sense. A functional model for this selfadjoint linear relation $ \tilde{A} $ is constructed.     

Some free iminoxyl radicals in the hydrogenated pyridine series

Summary Five new free radicals of the hydrogenated pyridine series were synthesized and their reaction with phenylhydrazine was studied.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 2, pp. 391–393, February, 1965     

Plasma-induced formation of Ag nanodots for ultra-high-enhancement surface-enhanced Raman scattering substrates

We report here plasma-induced formation of Ag nanostructures for surface-enhanced Raman scattering (SERS) applications. An array of uniform Ag patterned structures of 150 nm diameter was first fabricated on a silicon substrate with imprint lithography; then the substrate was further treated with an oxygen plasma to fracture the patterned structures into clusters of smaller, interconnected, closely packed Ag nanoparticles (20-60 nm) and redeposited Ag nanodots ( approximately 10 nm) between the clusters. The substrate thus formed had a uniform ultrahigh SERS enhancement factor (1010) over the entire substrate for 4-mercaptophenol molecules. By comparison, Au patterned structures fabricated with the same method did not undergo such a morphological change after the plasma treatment and showed no enhancement of Raman scattering.     

Transport numbers and ionic conduction of eutectic methacomposites {(1 ? x )MeWO4 · x WO3}) (Me = Sr, Ba; x = 0?0.55)

Transport numbers of oxygen ions, , in methacomposites (1 − x)MeWO₄ · xWO₃, where Me = Sr and Ba and x = 0−0.55, are determined in the temperature interval 600 to 900°C by a method of the emf of an oxygen-air galvanic cell. It is demonstrated that the region of small contents of the additive (x ≤ 0.2) is predominantly characterized by oxygen-ion conduction ( = 1), which gives way to electronic conduction (t _e = 1) at x > 0.35. It is confirmed once again that subeutectic compositions (1 − x)MeWO₄ · xWO₃ where Me = Sr and Ba and x = 0−0.2 belong in the class of ion-conducting methacomposites. The threshold of percolation of electronic conduction (t _e ≥ 0.5, < 0.5) occurs at x _t ≥ 0.3. Dependences of the transport numbers of the oxygen ions on the volume ratio between components in both composites resemble one another; specifically, the threshold composition contains nearly 20 vol % of WO₃. The dramatic amplification (by 1–1.5 orders of magnitude) of the ionic conductivity in the methacomposites occurs at small contents of tungstic oxide (x ≤ 0.01). A chemical transport removal of excess tungstic oxide, which is segregated in the form of the surface compound MeW-s, from the surface of the MeWO₄ grains destroys MeW-s, leading to a 10–15-fold drop of the ionic conductance. At x ≥ 0.05, the oxygen-ion conductance in the methacomposites is practically independent of their composition. A model for the formation and architecture of the methacomposites is qualitatively modified. The modified model takes into account doubled surface activity and mobility of the MeW-s phase with respect to MeWO₄ and WO₃. Original Russian Text ? N.N. Pestereva, A.Yu. Zhukova, A. Ya. Neiman, 2007, published in Elektrokhimiya, 2007, Vol. 43, No. 11, pp. 1379–1386.     

Advances in metric embedding theory

Metric Embedding plays an important role in a vast range of application areas such as computer vision, computational biology, machine learning, networking, statistics, and mathematical psychology, to name a few. The mathematical theory of metric embedding is well studied in both pure and applied analysis and has more recently been a source of interest for computer scientists as well. Most of this work is focused on the development of bi-Lipschitz mappings between metric spaces. In this paper we present new concepts in metric embeddings as well as new embedding methods for metric spaces. We focus on finite metric spaces, however some of the concepts and methods are applicable in other settings as well.One of the main cornerstones in finite metric embedding theory is a celebrated theorem of Bourgain which states that every finite metric space on n points embeds in Euclidean space with distortion. Bourgain?s result is best possible when considering the worst case distortion over all pairs of points in the metric space. Yet, it is natural to ask: can an embedding do much better in terms of the average distortion? Indeed, in most practical applications of metric embedding the main criteria for the quality of an embedding is its average distortion over all pairs.In this paper we provide an embedding with constant average distortion for arbitrary metric spaces, while maintaining the same worst case bound provided by Bourgain?s theorem. In fact, our embedding possesses a much stronger property. We define the ?q-distortion of a uniformly distributed pair of points. Our embedding achieves the best possible ?q-distortion for all 1?q?∞simultaneously.The results are based on novel embedding methods which improve on previous methods in another important aspect: the dimension of the host space. The dimension of an embedding is of very high importance in particular in applications and much effort has been invested in analyzing it. However, no previous result improved the bound on the dimension which can be derived from Bourgain?s embedding. Our embedding methods achieve better dimension, and in fact, shed new light on another fundamental question in metric embedding, which is: whether the embedding dimension of a metric space is related to its intrinsic dimension? I.e., whether the dimension in which it can be embedded in some real normed space is related to the intrinsic dimension which is reflected by the inherent geometry of the space, measured by the space?s doubling dimension. The existence of such an embedding was conjectured by Assouad,⁴and was later posed as an open problem in several papers. Our embeddings give the first positive result of this type showing any finite metric space obtains a low distortion (and constant average distortion) embedding in Euclidean space in dimension proportional to its doubling dimension.Underlying our results is a novel embedding method. Probabilistic metric decomposition techniques have played a central role in the field of finite metric embedding in recent years. Here we introduce a novel notion of probabilistic metric decompositions which comes particularly natural in the context of embedding. Our new methodology provides a unified approach to all known results on embedding of arbitrary finite metric spaces. Moreover, as described above, with some additional ideas they allow to get far stronger results.The results presented in this paper⁵have been the basis for further developments both within the field of metric embedding and in other areas such as graph theory, distributed computing and algorithms. We present a comprehensive study of the notions and concepts introduced here and provide additional extensions, related results and some examples of algorithmic applications.     

Self-assembly hexanuclear metallacontainer hosting halogenated guest species and sustaining structure of 3D coordination framework

Chloride-centered hexanuclear hydroxopyrazolate reveals potential as a receptor of halomethane and halometallate species and as a molecular building block for coordination polymers.