Distinguishing sceneries by observing the scenery along a random walk path

LetG be an infinite connected graph with vertex setV. Ascenery onG is a map ξ :V → 0, 1 (equivalently, an assignment of zeroes and ones to the vertices ofG). LetS _{n n≥0} be a simple random walk onG, starting at some distinguished vertex v₀. Now let ξ and η be twoknown sceneries and assume that we observe one of the two sequences ξ(S _n) _n≥0 or {η(S _n)} _n≥0 but we do not know which of the two sequences is observed. Can we decide, with a zero probability of error, which of the two sequences is observed? We show that ifG = Z orG = Z², then the answer is “yes” for each fixed ξ and “almost all” η. We also give some examples of graphsG for which almost all pairs (ξ, η) are not distinguishable, and discuss some variants of this problem.     

Survival of the weak in hyperbolic spaces, a remark on competition and geometry

A simple competition model is presented. While in Euclidean spaces the weak will ``die out', in the presence of hyperbolicity, coexistence take place.

     

A theoretical study of CHx chemisorption on the Ni(100) and Ni(111) surfaces

Cluster model calculations have been performed for CH_x, x = 0−3, chemisorbed on Ni(100) and Ni(111). The predicted chemisorption energies, at the present level of theory, based on bond-prepared clusters for Ni(100) are for carbon 150 kcal/mol, for CH 136 kcal/mol, for CH₂ 91 kcal/mol and for CH₃ 46 kcal/mol. The corresponding energies for Ni(111) are for CH 120 kcal/mol, for CH₂ 88 cal/mol and for CH₃ 49 kcal/mol. These chemisorption energies lead to similar stabilities for all CH_x fragments on both Ni(100) and Ni(111). Large basis sets and multi-reference correlation treatments are found to be very important in particular for the multiply bonded species. The vibrational C-H stretching frequencies predicted for CH_x on Ni(111) are for CH 3054 cm⁻¹ (2980 cm⁻¹), for CH₂ 3204 cm⁻¹ and for CH₃ 2709 cm⁻¹ (2680 cm⁻¹), where the available experimental values are given in parent The predicted ionization spectra of adsorbed CH_x are also in general agreement with experimental findings.     

Inclusion complex formation in a particular geometry by a water-soluble paracyclophane in aqueous solution — NMR Studies —

NMR studies (¹H and ¹³C) demonstrating an inclusion complex formation in a particular geometry between a water-soluble paracyclophane, CP44, and hydrophobic substrates in acidic aqueous solution are described.     

Formulation design of an oral, fast-disintegrating dosage form containing taste-masked particles of famotidine

A fast-disintegrating dosage form has been developed as a user-friendly formulation that disintegrates in the mouth immediately. Patients can take it without water like a liquid formulation. In this study famotidine taste-masking technology was applied to the new fast-disintegrating tablet in an attempt to produce a novel, taste-masked, fast-disintegrating tablet. Partial granulation was found to be an effective and practical way to address content uniformity, however, oral disintegration time tended to become longer as content uniformity improved. The disintegration time was improved considerably by controlling ambient humidity during the compression process (>50% RH). Furthermore, since the new fast-disintegrating technology made it possible to use low compression force, there was no change in the structure or dissolution rate of the taste-masked particles after compression. Therefore, this system can produce a taste-masked fast-disintegrating tablet with satisfactory attributes.     

Bi-Lipschitz bijection between the Boolean cube and the Hamming ball

We construct a bi-Lipschitz bijection from the Boolean cube to the Hamming ball of equal volume. More precisely, we show that for all even n ∈ N there exists an explicit bijection ψ: {0, 1}ⁿ → {x ∈ {0, 1}ⁿ⁺¹ : |x| > n/2} such that for every x ≠ y ∈ {0, 1}ⁿ, \(\frac{1}{5} \leqslant \frac{{dis\tan ce\left( {\psi \left( x \right),\psi \left( y \right)} \right)}}{{dis\tan ce\left( {x,y} \right)}} \leqslant 4,\) where distance(·, ·) denotes the Hamming distance. In particular, this implies that the Hamming ball is bi-Lipschitz transitive.     

On a minimization problem with a mass constraint involving a potential vanishing on two curves

We study a singular perturbation type minimization problem with a mass constraint over a domain in ℝ ^N , involving a potential vanishing on two curves in the plane. We demonstrate how the behavior of the minimizers depends on the geometry of the domain and, more precisely, on its isoperimetric profile.     

Geodesics and almost geodesic cycles in random regular graphs

A geodesic in a graph G is a shortest path between two vertices of G. For a specific function e(n) of n, we define an almost geodesic cycle C in G to be a cycle in which for every two vertices u and v in C, the distance d_G(u, v) is at least d_C(u, v)?e(n). Let ω(n) be any function tending to infinity with n. We consider a random d‐regular graph on n vertices. We show that almost all pairs of vertices belong to an almost geodesic cycle C with e(n) = log_d?1log_d?1n+ ω(n) and |C| = 2log_d?1n+ O(ω(n)). Along the way, we obtain results on near‐geodesic paths. We also give the limiting distribution of the number of geodesics between two random vertices in this random graph. Copyright © 2010 John Wiley & Sons, Ltd. J Graph Theory 66:115‐136, 2011     

Simple random walk on the uniform infinite planar quadrangulation: subdiffusivity via pioneer points

We study the pioneer points of the simple random walk on the uniform infinite planar quadrangulation (UIPQ) using an adaptation of the peeling procedure of Angel (Geom Funct Anal 13:935–974, 2003) to the quadrangulation case. Our main result is that, up to polylogarithmic factors, n ³ pioneer points have been discovered before the walk exits the ball of radius n in the UIPQ. As a result we verify the KPZ relation Knizhnik et al. (Modern Phys Lett A 3:819–826, 1988) in the particular case of the pioneer exponent and prove that the walk is subdiffusive with exponent less than 1/3. Along the way, new geometric controls on the UIPQ are established.