Littlewood–Paley Theory for Triangle Buildings

For the natural two-parameter filtration \(\left( {\mathcal {F}_\lambda }: {\lambda \in P}\right) \) on the boundary of a triangle building, we define a maximal function and a square function and show their boundedness on \(L^p(\Omega _0)\) for \(p \in (1, \infty )\). At the end, we consider \(L^p(\Omega _0)\) boundedness of martingale transforms. If the building is of \({\text {GL}}(3, \mathbb {Q}_p)\), then \(\Omega _0\) can be identified with p-adic Heisenberg group.     

Dynamic thermoviscoelastic thermistor problem with contact and nonmonotone friction

Abstract

The paper studies the evolution of the thermomechanical and electric state of a thermoviscoelastic thermistor that is in frictional contact with a reactive foundation. The mechanical process is dynamic, while the electric process is quasistatic. Friction is modeled with a nonmonotone relation between the tangential traction and tangential velocity. Frictional heat generation is taken into account and so is the strong dependence of the electric conductivity on the temperature. The mathematical model for the process is in the form of a system that consists of dynamic hyperbolic subdifferential inclusion for the mechanical state coupled with a nonlinear parabolic equation for the temperature and an elliptic equation for the electric potential. The paper establishes the existence of a weak solution to the problem by using time delays, a priori estimates and a convergence method.     

Limit theorems for random walks

We consider a random walk $S_{\tau }$ which is obtained from the simple random walk $S$ by a discrete time version of Bochner’s subordination. We prove that under certain conditions on the subordinator $\tau$ appropriately scaled random walk $S_{\tau }$ converges in the Skorohod space to the symmetric $\alpha$-stable process $B^{\alpha }$. We also prove asymptotic formula for the transition function of $S_{\tau }$ similar to the Pólya’s asymptotic formula for $B^{\alpha }$.     

Triangle-Free Geometric Intersection Graphs with Large Chromatic Number

Several classical constructions illustrate the fact that the chromatic number of a graph may be arbitrarily large compared to its clique number. However, until very recently no such construction was known for intersection graphs of geometric objects in the plane. We provide a general construction that for any arc-connected compact set $X$ X in $\mathbb{R }^2$ R 2 that is not an axis-aligned rectangle and for any positive integer $k$ k produces a family $\mathcal{F }$ F of sets, each obtained by an independent horizontal and vertical scaling and translation of $X$ X , such that no three sets in $\mathcal{F }$ F pairwise intersect and $\chi (\mathcal{F })>k$ χ ( F ) > k . This provides a negative answer to a question of Gyárfás and Lehel for L-shapes. With extra conditions we also show how to construct a triangle-free family of homothetic (uniformly scaled) copies of a set with arbitrarily large chromatic number. This applies to many common shapes, like circles, square boundaries or equilateral L-shapes. Additionally, we reveal a surprising connection between coloring geometric objects in the plane and on-line coloring of intervals on the line.     

A study of the interaction between perylene and the TiO2(1 1 0)-(1 × 1) surface-based on XPS, UPS and NEXAFS measurements

The interaction between a semi-large aromatic hydrocarbon compound (perylene) and the TiO₂(1 1 0)-(1 × 1) surface under ultra high vacuum conditions has been probed by X-ray photoemission spectroscopy (XPS), ultraviolet photoemission spectroscopy (UPS) and near-edge X-ray absorption fine structure (NEXAFS) methods. UPS measurements of the adsorbate system have been compared with an experimental UPS spectrum of perylene in the gas phase and a calculated spectrum obtained by means of density functional theory (DFT) methods. NEXAFS results of perylene molecules adsorbed on TiO₂(1 1 0)-(1 × 1) were compared with data from an α-phase perylene single crystal. A novel analysis of the valence data has been employed to show that no strong chemical interaction takes place between perylene and the TiO₂(1 1 0)-(1 × 1) surface. Furthermore, angle-dependent NEXAFS measurements and the growth curve results suggest that the perylene molecules are oriented flat down onto the TiO₂ substrate due to weak van der Waals interactions.     

Modeling tubulin at interfaces. Immobilization of microtubules on self-assembled monolayers

A theoretical study of protein docking to self-assembled monolayers using a new approach is presented. Docking experiments based on space complementarity implemented in FTDock software were performed for three different proteins: tubulin dimer, cytochrome c, and lysozyme. The proteins were adsorbed on alkanethiol surfaces with different terminating groups and 50,000 best orientations of each protein were analyzed. For all systems three filters based on different chemical and biological approaches were applied. Correctly docked proteins for the cytochrome c and lysozyme systems were found in a list of the first 12 results after applying the geometrical and grouping filter and in a list of the first 3 results after applying the biological filter. We have found that alkanethiol monolayers with odd and even numbers of -CH2- groups have similar properties in terms of interactions with the two proteins. Docking of the tubulin dimer revealed that the orientation favored from the applicational point of view can be found in a list of the first 14 results for monolayers with different terminating groups and that there may be a noticeable difference in tubulin dimer interactions with alkanethiol chains of various length. The results for tubulin dimer docking combined with microtubules ability of reversible assembly suggest that these biological structures may become good candidates to serve as templates for fabrication of nanowires and other nanoscale electronic devices. The new method of theoretical docking presented may be used as a fast and reliable tool complementing other theoretical and experimental techniques of exploring other protein-surface interfaces.     

Capacities of quantum channels for massive bosons and fermions

We consider the capacity of classical information transfer for noiseless quantum channels carrying a finite average number of massive bosons and fermions. The maximum capacity is attained by transferring the Fock states generated from the grand-canonical ensemble. Interestingly, the channel capacity for a Bose gas indicates the onset of Bose-Einstein condensation, by changing its qualitative behavior at the criticality, while for a channel carrying weakly attractive fermions, it exhibits the signatures of Bardeen-Cooper-Schrieffer transition. We also show that, for noninteracting particles, fermions are better carriers of information than bosons.     

Phenol adsorption on closed carbon nanotubes

We present the results of systematic studies of phenol adsorption on closed commercially available, unmodified carbon nanotubes. Phenol adsorption is determined by the value of tube-specific surface area, the presence of small amount of surface groups influence adsorption only in very small amount. Phenol can be applied as a probe molecule for comparative analysis of tube surface areas. Tube curvature influences adsorption from solution, i.e., we observe increasing adsorption energy (and slower desorption process) with the decrease in tube curvature. This is in full accordance with molecular simulation results.     

Great expectations: can artificial molecular machines deliver on their promise?

The development and fabrication of mechanical devices powered by artificial molecular machines is one of the contemporary goals of nanoscience. Before this goal can be realized, however, we must learn how to control the coupling/uncoupling to the environment of individual switchable molecules, and also how to integrate these bistable molecules into organized, hierarchical assemblies that can perform significant work on their immediate environment at nano-, micro- and macroscopic levels. In this tutorial review, we seek to draw an all-important distinction between artificial molecular switches which are now ten a penny-or a dime a dozen-in the chemical literature and artificial molecular machines which are few and far between despite the ubiquitous presence of their naturally occurring counterparts in living systems. At the single molecule level, a prevailing perspective as to how machine-like characteristics may be achieved focuses on harnessing, rather than competing with, the ineluctable effects of thermal noise. At the macroscopic level, one of the major challenges inherent to the construction of machine-like assemblies lies in our ability to control the spatial ordering of switchable molecules-e.g., into linear chains and then into muscle-like bundles-and to influence the cross-talk between their switching kinetics. In this regard, situations where all the bistable molecules switch synchronously appear desirable for maximizing mechanical power generated. On the other hand, when the bistable molecules switch "out of phase," the assemblies could develop intricate spatial or spatiotemporal patterns. Assembling and controlling synergistically artificial molecular machines housed in highly interactive and robust architectural domains heralds a game-changer for chemical synthesis and a defining moment for nanofabrication.