Understanding Search Trees via Statistical Physics

We study the random m-ary search tree model (where m stands for the number of branches of a search tree), an important problem for data storage in computer science, using a variety of statistical physics techniques that allow us to obtain exact asymptotic results. In particular, we show that the probability distributions of extreme observables associated with a random search tree such as the height and the balanced height of a tree have a traveling front structure. In addition, the variance of the number of nodes needed to store a data string of a given size N is shown to undergo a striking phase transition at a critical value of the branching ratio m_c=26. We identify the mechanism of this phase transition, show that it is generic and occurs in various other problems as well. New results are obtained when each element of the data string is a D-dimensional vector. We show that this problem also has a phase transition at a critical dimension, D_c= \pi/\sin^{-1}(1/\sqrt{8})=8.69363...Comment: 11 pages, 8 .eps figures included. Invited contribution to STATPHYS-22 held at Bangalore (India) in July 2004. To appear in the proceedings of STATPHYS-2

Cutting edges at random in large recursive trees

We comment on old and new results related to the destruction of a random recursive tree (RRT), in which its edges are cut one after the other in a uniform random order. In particular, we study the number of steps needed to isolate or disconnect certain distinguished vertices when the size of the tree tends to infinity. New probabilistic explanations are given in terms of the so-called cut-tree and the tree of component sizes, which both encode different aspects of the destruction process. Finally, we establish the connection to Bernoulli bond percolation on large RRT's and present recent results on the cluster sizes in the supercritical regime.Comment: 29 pages, 3 figure

Long and short paths in uniform random recursive dags

In a uniform random recursive k-dag, there is a root, 0, and each node in turn, from 1 to n, chooses k uniform random parents from among the nodes of smaller index. If S_n is the shortest path distance from node n to the root, then we determine the constant \sigma such that S_n/log(n) tends to \sigma in probability as n tends to infinity. We also show that max_{1 \le i \le n} S_i/log(n) tends to \sigma in probability.Comment: 16 page

Development of two reference materials for all trans-retinol, retinyl palmitate, α- and γ-tocopherol in milk powder and infant formula

AbstractVitamins are important food constituents that can be present in almost every foodstuff. Food quality and safety depends on food surveillance by reliable quantitative analysis enabled by appropriate quality control. Certified matrix reference materials are versatile tools to support quality assurance and control. However, in the case of vitamins, which are important in various foods, there is a lack of matrix reference materials. Two certified reference materials for the determination of all–trans-retinol, retinyl palmitate, and α- and γ-tocopherol in milk powder and infant formula have been developed by the National Institute of Standards, Egypt. This article presents the preparation, characterization, homogeneity, and stability testing as well as statistical treatment of data and certified value assignment. The assignment of the certified values and associated uncertainties in the prepared natural-matrix reference materials were based on the widely used approach of combining data from independent and reliable analytical methods

Phase Transition in a Random Fragmentation Problem with Applications to Computer Science

We study a fragmentation problem where an initial object of size x is broken into m random pieces provided x>x_0 where x_0 is an atomic cut-off. Subsequently the fragmentation process continues for each of those daughter pieces whose sizes are bigger than x_0. The process stops when all the fragments have sizes smaller than x_0. We show that the fluctuation of the total number of splitting events, characterized by the variance, generically undergoes a nontrivial phase transition as one tunes the branching number m through a critical value m=m_c. For m<m_c, the fluctuations are Gaussian where as for m>m_c they are anomalously large and non-Gaussian. We apply this general result to analyze two different search algorithms in computer science.Comment: 5 pages RevTeX, 3 figures (.eps

Dye-sensitized solar cells using dyes extracted from flowers, leaves, parks, and roots of three trees

In this paper, eleven natural dyes were collected from three trees and used as photosensitizers for dye sensitized solar cells (DSSCs). The cells were fabricated using TiO2 as a semiconducting layer deposited on transparent fluorine doped tin oxide (FTO) conductive glass using doctor blade method. The absorption spectra of the extracts were performed in the spectral range from 400 nm to 750 nm. The JV characteristic curves of all fabricated cells were measured and analyzed. The parameters related to the solar cell performance were determined. Moreover, the impedance spectroscopy of the cell with the best performance was investigated

Leadership Statistics in Random Structures

The largest component (``the leader'') in evolving random structures often exhibits universal statistical properties. This phenomenon is demonstrated analytically for two ubiquitous structures: random trees and random graphs. In both cases, lead changes are rare as the average number of lead changes increases quadratically with logarithm of the system size. As a function of time, the number of lead changes is self-similar. Additionally, the probability that no lead change ever occurs decays exponentially with the average number of lead changes.Comment: 5 pages, 3 figure

Entanglement in squeezed two-level atom

In the previous paper, we adopted the method using quantum mutual entropy to measure the degree of entanglement in the time development of the Jaynes-Cummings model. In this paper, we formulate the entanglement in the time development of the Jaynes-Cummings model with squeezed states, and then show that the entanglement can be controlled by means of squeezing.Comment: 6 pages, 5 figures, to be published in J.Phys.