Generalized Stirling permutations, families of increasing trees and urn models

Bóna (2007) [6] studied the distribution of ascents, plateaux and descents in the class of Stirling permutations, introduced by Gessel and Stanley (1978) [13]. Recently, Janson (2008) [17] showed the connection between Stirling permutations and plane recursive trees and proved a joint normal law for the parameters considered by Bóna. Here we will consider generalized Stirling permutations extending the earlier results of Bóna (2007) [6] and Janson (2008) [17], and relate them with certain families of generalized plane recursive trees, and also (k+1)-ary increasing trees. We also give two different bijections between certain families of increasing trees, which both give as a special case a bijection between ternary increasing trees and plane recursive trees. In order to describe the (asymptotic) behaviour of the parameters of interests, we study three (generalized) Pólya urn models using various methods.     

Analysis of a generalized Friedman’s urn with multiple drawings

We study a generalized Friedman’s urn model with multiple drawings of white and blue balls. After a drawing, the replacement follows a policy of opposite reinforcement. We give the exact expected value and variance of the number of white balls after a number of draws, and determine the structure of the moments. Moreover, we obtain a strong law of large numbers, and a central limit theorem for the number of white balls. Interestingly, the central limit theorem is obtained combinatorially via the method of moments and probabilistically via martingales. We briefly discuss the merits of each approach. The connection to a few other related urn models is briefly sketched.     

Nucleotide-Bearing Benzylidene-Tetrahydroxanthylium Near-IR Fluorophore for Sensing DNA Replication,Secondary Structures and Interactions

Thymidine triphosphate bearing benzylidene-tetrahydroxanthylium near-IR fluorophore linked to the 5-methyl group via triazole was synthesized through the CuAAC reaction and was used for polymerase synthesis of labelled DNA probes. The fluorophore lights up upon incorporation to DNA (up to 348-times) presumably due to interactions in major groove and the fluorescence further increases in the single-stranded oligonucleotide. The labelled dsDNA senses binding of small molecules and proteins by a strong decrease of fluorescence. The nucleotide was used as a light-up building block in real-time PCR for detection of SARS-CoV-2 virus.     

On Rounding Error Sums Related to the Circle Problem. ii

For a positive real parameter t, real numbers , , and , we consider sums , where is the rounding error function, i.e.\ . Generalizing and improving the main result of Part I of the paper we show that there exists an absolute constant such that for all , and all . Further, we give applications concerning the circle problem with linear, polynomial, and general weight.     

Zur Verteilung der Quadrate ganzer Zahlen in rationalen Quaternionenalgebren

For γ ∈ ?letQ 〈γ〉 = ?[i]+?[i]j. where j. is a hypercomplex number withj2 = γ, and define addition and multiplication formally with respect to $zj = j\overline z $ for all z ∈ ?[i], so thatQ〈γ〉 becomes a quaternion algebra over the rationals. Further fix γ s.t.Q 〈γ 〉 is a division algebra and define for real X ≥ 1 where |Re(α)|, |Im(α)|, |Re(β)|, |Im(ö)|≤ X and Generalizing former results concerning Hamilton’s quaternions (i.e. the case γ =- 1) we show that, as X → ∞, when γ < 0, when γ > 0, when γ < 0, wheny γ 0. Thereby δ(t) is any upper bound of the error term in Dirichlet’s divisor problem, e.g. δ(t) =t0.315, C_γ, D_γ > 0 are numerical constants, and c, d are given by c := π(1 + log 2 - 2η) and d := π2(1 + 4 log 4 - 4π)/8, where π = 0.577 … is Euler’s constant.     

Molecular dynamics simulations of Ar gas ejection from a ruptured subsurface bubble in Cu(100) induced by impact of 200 eV Ar atoms

We present a molecular dynamics simulation of the rupture of a subsurface Ar bubble in Cu(100) and the ejection of Ar atoms and Ar₂ dimers. A cavity in the subsurface region was pressurised by inserting Ar atoms until a stability condition was met and the bubble rupture was initiated by the impact of a 200 eV Ar atom. We calculate local temperature and pressure profiles inside the Ar bubble, the angular and energy distributions of emitted Ar atoms and Ar₂ dimers and compare with experiments.     

On edge-weighted recursive trees and inversions in random permutations

We introduce random recursive trees, where deterministically weights are attached to the edges according to the labeling of the trees. We will give a bijection between recursive trees and permutations, which relates the arising edge-weights in recursive trees with inversions of the corresponding permutations. Using this bijection we obtain exact and limiting distribution results for the number of permutations of size n, where exactly m elements have j inversions. Furthermore we analyze the distribution of the sum of labels of the elements, which have exactly j inversions, where we can identify Dickman's infinitely divisible distribution as the limit law. Moreover we give a distributional analysis of weighted depths and weighted distances in edge-weighted recursive trees.