Path-pairability of infinite planar grids

For a given integer $$k\ge 1$$ k ≥ 1 , a graph G with at least 2k vertices is called k-path-pairable, if for any set of k disjoint pairs of vertices, $$s_i,t_i$$ s i , t i , $$1\le i\le k$$ 1 ≤ i ≤ k , there exist pairwise edge-disjoint $$s_i,t_i$$ s i , t i -paths in G. The path-pairability numberis the largest k such that G is k-path-pairable. Bounds on the path-pairability number are given here if G is the graph of infinite integer grids in the Euclidean plane. We prove that the path-pairability number of the integer quadrant is 4, and we show that the integer half-plane is 6-path-pairable and at most 7-path-pairable.

Quantum-Mechanical Assessment of the Energetics of Silver Decahedron Nanoparticles

We present a quantum-mechanical study of silver decahedral nanoclusters and nanoparticles containing from 1 to 181 atoms in their static atomic configurations corresponding to the minimum of the ab initio computed total energies. Our thermodynamic analysis compares T = 0 K excess energies (without any excitations) obtained from a phenomenological approach, which mostly uses bulk-related properties, with excess energies from ab initio calculations of actual nanoclusters/nanoparticles. The phenomenological thermodynamic modeling employs (i) the bulk reference energy, (ii) surface energies obtained for infinite planar (bulk-related) surfaces and (iii) the bulk atomic volume. We show that it can predict the excess energy (per atom) of nanoclusters/nanoparticles containing as few as 7 atoms with the error lower than 3%. The only information related to the nanoclusters/nanoparticles of interest, which enters the phenomenological modeling, is the number of atoms in the nanocluster/nanoparticle, the shape and the crystallographic orientation(s) of facets. The agreement between both approaches is conditioned by computing the bulk-related properties with the same computational parameters as in the case of the nanoclusters/nanoparticles but, importantly, the phenomenological approach is much less computationally demanding. Our work thus indicates that it is possible to substantially reduce computational demands when computing excess energies of nanoclusters and nanoparticles by ab initio methods.

Approximation Model for Radiation Impedance of a Square Piston Source on an Infinite Planar Baffle

Approximation models based on a finite sum of Bessel functions of the first kind and a pair of simple rational transfer functions are proposed for radiation resistance and reactance of a square piston source mounted on an infinite planar baffle. Model accuracy is better than 1.6% for reactance and 0.5% for resistance within a very wide range of dimensionless frequency k√S (0.1–100). The very low and high frequency behaviors of radiation impedance are incorporated into the models' closed-form expressions so that the approximation error outside the specified frequency range tends to zero.

A new biologically active molecular scaffold: crystal structure of 7-(3-hydroxyphenyl)-4-methyl-2H-[1,2,4]triazolo[3,2-c][1,2,4]triazole and selective antiproliferative activity of three isomeric triazolo–triazoles

A study of three isomeric compounds containing a phenolic moiety attached to the nitrogen-rich triazolo–triazole bicycle is presented. In the three isomers, the phenolic OH group is in the ortho, meta and para positions. The crystal structure analysis of the meta isomer (C10H9N5O) shows that the 2H-tautomer is present in the crystal and that the molecule adopts a substantially planar geometry. However, the conformation found in the crystal is different compared to the monoprotonated cation of the same compound previously investigated in several salts. The packing of the meta isomer is driven by the formation of strong hydrogen bonds and shows the formation of infinite planar ribbons, parallel to a, formed around 21 crystallographic axes. The three isomers were tested against some cancer cell lines and also against normal cell lines. The ortho isomer shows a weak antiproliferative activity, the meta isomer shows significant antiproliferative activity against some cancer lines and no activity against healthy cell lines, and the para isomer is active against all the tested cell lines.

Exotic Patterns of Synchrony in Planar Lattice Networks

Patterns of dynamical synchrony that can occur robustly in networks of coupled dynamical systems are associated with balanced colorings of the nodes of the network. In symmetric networks, the orbits of any group of symmetries automatically determine a balanced orbit coloring. Balanced colorings not of this kind are said to be exotic. Exotic colorings occur in infinite planar lattices, both square and hexagonal, with various short-range couplings. In some cases, a balanced two-coloring remains balanced when colors are swapped along suitable diagonals, giving rise to uncountably many distinct exotic colorings. We explain this phenomenon in terms of iterated orbit colorings, in which the quotient of the lattice by an orbit coloring has extra symmetries, allowing new orbit colorings on the quotient, which then lift back to the lattice. We apply the same construction to several other exotic lattice colorings. Two appendices discuss how to modify the notion of balance for networks with diffusive coupling, and how to formalize the differential equations in infinitely many variables that arise for lattices.

Volumes in the Uniform Infinite Planar Triangulation: From Skeletons to Generating Functions

We develop a method to compute the generating function of the number of vertices inside certain regions of the Uniform Infinite Planar Triangulation (UIPT). The computations are mostly combinatorial in flavour and the main tool is the decomposition of the UIPT into layers, called the skeleton decomposition, introduced by Krikun [20]. In particular, we get explicit formulas for the generating functions of the number of vertices inside hulls (or completed metric balls) centred around the root, and the number of vertices inside geodesic slices of these hulls. We also recover known results about the scaling limit of the volume of hulls previously obtained by Curien and Le Gall by studying the peeling process of the UIPT in [17].

Mullins' Self-Similar Grooving Solution Revisited

In W.W. Mullins' classical 1957 paper on thermal grooving, motion by surface diffusion was proposed to describe the development of a thermal groove separating two grains in a simple semi-infinite planar geometry. After making a small slope approximation which is often realistic, Mullins' sought self-similar solutions, and obtained an explicit time series solution for the groove depth. In the years since, Mullins' grooving solution has become a standard tool; however it has yet to be rigorously demonstrated that self-similar solutions exist when the small slope approximation is not applicable. Here we demonstrate that reformulation of Mullins' nonlinear problem in arc-length variables yields a particularly simple fully nonlinear formulation, which is useful for verifying large slope grooving properties and which should aid in proving existence.