The boundary value problem and the Jensen inequality for an entropy functional of a Markov diffusion process

The paper introduces the recent results related to an entropy functional on trajectories of a controlled diffusion process, expressed through an additive functional of the diffusion process, with a Lagrangian, determined by the parameters of a controlled stochastic equation. These results include a minimum condition for the entropy functional and the functional's Jensen inequality, which both are useful for the solution of important mathematical and applied problems.     

Network ensemble clustering using latent roles

We present a clustering method for collections of graphs based on the assumptions that graphs in the same cluster have a similar role structure and that the respective roles can be founded on implicit vertex types. Given a network ensemble (a collection of attributed graphs with some substantive commonality), we start by partitioning the set of all vertices based on attribute similarity. Projection of each graph onto the resulting vertex types yields feature vectors of equal dimensionality, irrespective of the original graph sizes. These feature vectors are then subjected to standard clustering methods. This approach is motivated by social network concepts, and we demonstrate its utility on an ensemble of personal networks of migrants, where we extract structurally similar groups and show their resemblance to predicted acculturation strategies.     

On Sharp Aperture-Weighted Estimates for Square Functions

Let \(S_{\alpha ,\psi }(f)\) be the square function defined by means of the cone in \({\mathbb R}^{n+1}_{+}\) of aperture \(\alpha \) , and a standard kernel \(\psi \) . Let \([w]_{A_p}\) denote the \(A_p\) characteristic of the weight \(w\) . We show that for any \(1<p<\infty \) and \(\alpha \ge 1\) , $$\begin{aligned} \Vert S_{\alpha ,\psi }\Vert _{L^p(w)}\lesssim \alpha ^n[w]_{A_p}^{\max \left( \frac{1}{2},\frac{1}{p-1}\right) }. \end{aligned}$$ For each fixed \(\alpha \) the dependence on \([w]_{A_p}\) is sharp. Also, on all class \(A_p\) the result is sharp in \(\alpha \) . Previously this estimate was proved in the case \(\alpha =1\) using the intrinsic square function. However, that approach does not allow to get the above estimate with sharp dependence on \(\alpha \) . Hence we give a different proof suitable for all \(\alpha \ge 1\) and avoiding the notion of the intrinsic square function.     

Solution to the variation problem for information path functional of a controlled random process

The paper introduces a new approach to dynamic modeling, using the variation principle, applied to a functional on trajectories of a controlled random process, and its connection to the process' information functional. In [V.S. Lerner, Dynamic approximation of a random information functional, J. Math. Anal. Appl. 327 (1) (2007) 494-514, available online 5-24-06], we presented the information path functional with the Lagrangian, determined by the parameters of a controlled stochastic equation. In this paper, the solution to the path functional's variation problem provides both a dynamic model of a random process and the model's optimal control, which allows us to build a two-level information model with a random process at the microlevel and a dynamic process at the macrolevel. A wide class of random objects, modeled by the Markov diffusion process and a common structure of the process' information functional, leads to a universal information structure of the dynamic model, which is specified and identified on a particular object with the applied optimal control functions. The developed mathematical formalism, based on classical methods, is aimed toward the solution of problems identification, combined with an optimal control synthesis, which is practically implemented and also demonstrated in the paper's example.     

Feynman integrals ofp-adic argument in the momentum space. I. Convergence

The regions of convergence of Feynman integrals ofp-adic arguments in the coordinate and momentum spaces are established, and a theorem on the Fourier transformation of Feynman amplitudes is proved.State University, Kazan. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 102, No. 3, pp. 367–377, March, 1995.     

Feynman integrals ofp-adic argument in the momentum space. II. Explicit expressions

A procedure for obtaining explicit expressions for Feynman integrals ofp-adic argument in momentum space is described, and the connection with Feynman amplitudes over a finite field and chromatic polynomials is indicated. Simple explicit expressions are given for the results of integration over the region of large and small momenta.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 3, pp. 371–392, September, 1995.     

One‐Step Synthesis of a [20]Silafullerane with an Endohedral Chloride Ion

Silicon analogues of the most prominent carbon nanostructures, namely, hollow spheroidals such as C₆₀ and the fullerene family, have been unknown to date. Herein we show that discrete Si₂₀ dodecahedra, stabilized by an endohedral guest and valence saturation, are accessible in preparative yields through a chloride‐induced disproportionation reaction of hexachlorodisilane in the presence of tri(n‐butyl)amine. X‐ray crystallography revealed that each silicon dodecahedron contains an endohedral chloride ion that imparts a net negative charge. Eight chloro substituents and twelve trichlorosilyl groups are attached to the surface of each cluster in a strictly regioregular arrangement, a thermodynamically preferred substitution pattern according to quantum‐chemical assessment. Our results demonstrate that the wet‐chemical self‐assembly of a complex, monodisperse Si nanostructure is possible under mild conditions starting from simple Si₂ building blocks.     

Boron‐Containing Polycyclic Aromatic Hydrocarbons: Facile Synthesis of Stable,Redox‐Active Luminophores

Herein we show that replacing the two meso carbon atoms of the polycyclic aromatic hydrocarbon (PAH) bisanthene by boron atoms transforms a near‐infrared dye into an efficient blue luminophore. This observation impressively illustrates the impact of boron doping on the frontier orbitals of PAHs. To take full advantage of this tool for the targeted design of organic electronic materials, the underlying structure–property relationships need to be further elucidated. We therefore developed a modular synthesis sequence based on a Peterson olefination, a stilbene‐type photocyclization, and an Si–B exchange reaction to substantially broaden the palette of accessible polycyclic aromatic organoboranes and to permit a direct comparison with their PAH congeners.     

Unsymmetrically substituted 9,10-dihydro-9,10-diboraanthracenes as versatile building blocks for boron-doped π-conjugated systems

The targeted hydrolysis of the 9,10-dihydro-9,10-diboraanthracene adduct (Me(2)S)HB(C(6)H(4))(2)BH(SMe(2)) (1) with 0.5 equiv of H(2)O leads to formation of the borinic acid anhydride [(Me(2)S)HB(C(6)H(4))(2)B](2)O (2) and thereby provides access to the field of unsymmetrically substituted 9,10-dihydro-9,10-diboraanthracenes. Compound 2 reacts with tBuC≡CH to give the corresponding vinyl derivative in an essentially quantitative conversion. Subsequent cleavage of the B-O-B bridge by LiAlH(4) with formation of hydridoborate functionalities is possible but is accompanied by partial B-C(vinyl) bond degradation. This situation changes when the related mesityl derivative [MesB(C(6)H(4))(2)B](2)O (7) is employed, which can be synthesized from BrB(C(6)H(4))(2)BBr (6) by treatment with 1 equiv of MesMgBr and subsequent hydrolysis. The reaction of 7 with LiAlH(4) in tetrahydrofuran (THF) furnishes Li[MesB(C(6)H(4))(2)BH(2)] (8); hydride elimination with Me(3)SiCl leads to formation of the THF adduct MesB(C(6)H(4))(2)BH(THF) (9·THF). Alternatively, 7 can be transformed into the bromoborane MesB(C(6)H(4))(2)BBr (10) by treatment with BBr(3). A Br/H-exchange reaction between 10 and Et(3)SiH yields the donor-free borane MesB(C(6)H(4))(2)BH (9), which forms B-H-B bridged dimers (9)(2) in the solid state. The vinyl borane MesB(C(6)H(4))(2)BC(H)=C(H)Mes (14) is accessible from MesC≡CH and either 9·THF or 9. Compared with the related compound Mes(2)BC(H)=C(H)Mes, the electronic absorption and emission spectra of 14 reveal bathochromic shifts of Δλ(abs)=17 nm and Δλ(em)=74 nm, which can be attributed to the rigid, fully delocalized π framework of the [MesB(C(6)H(4))(2)B] chromophore.