The Cut Metric,Random Graphs,and Branching Processes

In this paper we study the component structure of random graphs with independence between the edges. Under mild assumptions, we determine whether there is a giant component, and find its asymptotic size when it exists. We assume that the sequence of matrices of edge probabilities converges to an appropriate limit object (a kernel), but only in a very weak sense, namely in the cut metric. Our results thus generalize previous results on the phase transition in the already very general inhomogeneous random graph model introduced by the present authors in Random Struct. Algorithms 31:3–122 (2007), as well as related results of Bollobás, Borgs, Chayes and Riordan (Ann. Probab. 38:150–183, 2010), all of which involve considerably stronger assumptions. We also prove corresponding results for random hypergraphs; these generalize our results on the phase transition in inhomogeneous random graphs with clustering (Random Struct. Algorithms, 2010, to appear).     

Sesqui-type branching processes

We consider branching processes consisting of particles (individuals) of two types (type $L$ and type $S$) in which only particles of type $L$ have offspring, proving estimates for the survival probability and the (tail of) the distribution of the total number of particles. Such processes are in some sense closer to single- than to multi-type branching processes. Nonetheless, the second, barren, type complicates the analysis significantly. The results proved here (about point and survival probabilities) are a key ingredient in the analysis of bounded-size Achlioptas processes in a recent paper by the last two authors.     

Precise Logarithmic Asymptotics for the Right Tails of Some Limit Random Variables for Random Trees

For certain random variables that arise as limits of functionals of random finite trees, we obtain precise asymptotics for the logarithm of the right-hand tail. Our results are based on the facts (i) that the random variables we study can be represented as functionals of a Brownian excursion and (ii) that a large deviation principle with good rate function is known explicitly for Brownian excursion. Examples include limit distributions of the total path length and of the Wiener index in conditioned Galton-Watson trees (also known as simply generated trees). In the case of Wiener index (where we recover results proved by Svante Janson and Philippe Chassaing by a different method) and for some other examples, a key constant is expressed as the solution to a certain optimization problem, but the constant’s precise value remains unknown. Research supported by NSF grants DMS-0104167 and DMS-0406104 and by The Johns Hopkins University’s Acheson J. Duncan Fund for the Advancement of Research in Statistics.     

Sorting Using Complete Subintervals and the Maximum Number of Runs in a Randomly Evolving Sequence

We study the space requirements of a sorting algorithm where only items that at the end will be adjacent are kept together. This is equivalent to the following combinatorial problem: Consider a string of fixed length n that starts as a string of 0’s, and then evolves by changing each 0 to 1, with the n changes done in random order. What is the maximal number of runs of 1’s? We give asymptotic results for the distribution and mean. It turns out that, as in many problems involving a maximum, the maximum is asymptotically normal, with fluctuations of order n ^1/2, and to the first order well approximated by the number of runs at the instance when the expectation is maximized, in this case when half the elements have changed to 1; there is also a second order term of order n ^1/3. We also treat some variations, including priority queues and sock-sorting. The proofs use methods originally developed for random graphs.     

A new approach to the giant component problem

We study the largest component of a random (multi)graph on n vertices with a given degree sequence. We let n → ∞. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the degree sequence that imply that with high probability all the components are small, and other conditions that imply that with high probability there is a giant component and the sizes of its vertex and edge sets satisfy a law of large numbers; under suitable assumptions these are the only two possibilities. In particular, we recover the results by Molloy and Reed on the size of the largest component in a random graph with a given degree sequence. We further obtain a new sharp result for the giant component just above the threshold, generalizing the case of G(n,p) with np = 1 + ω(n)n^?1/3, where ω(n) → ∞ arbitrarily slowly. Our method is based on the properties of empirical distributions of independent random variables, and leads to simple proofs. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Simultaneous analysis of vocal fold vibration and transglottal airflow: exploring a new experimental setup

The purpose of this study was to develop an analysis system for studying the relationship between vocal fold vibration and the associated transglottal airflow. Recordings of airflow, electroglottography (EGG), oral air pressure, and acoustic signals were performed simultaneously with high-speed imaging at a rate of approximately 1900 frames/s. Inverse filtered airflow is compared with the simultaneous glottal area extracted from the high-speed image sequence. The accuracy of the synchronization between the camera images and the foot pedal synchronization pulse was examined, showing that potential synchronization errors increase with time distance to the synchronization pulse. Therefore, analysis was limited to material near the synchronization pulse. Results corroborate previous predictions that air flow lags behind area, but also they reveal that relationships between these two entities may be complex and apparently varying with phonation mode.     

Comparing closed quotient in children singers' voices as measured by high-speed-imaging, electroglottography, and inverse filtering

The closed quotient, i.e., the ratio between the closed phase and the period, is commonly studied in voice research. However, the term may refer to measures derived from different methods, such as inverse filtering, electroglottography or high-speed digital imaging (HSDI). This investigation compares closed quotient data measured by these three methods in two boy singers. Each singer produced sustained tones on two different pitches and a glissando. Audio, electroglottographic signal (EGG), and HSDI were recorded simultaneously. The audio signal was inverse filtered by means of the decap program; the closed phase was defined as the flat minimum portion of the flow glottogram. Glottal area was automatically measured in the high speed images by the built-in camera software, and the closed phase was defined as the flat minimum portion of the area-signal. The EGG-signal was analyzed in four different ways using the matlab open quotient interface. The closed quotient data taken from the EGG were found to be considerably higher than those obtained from inverse filtering. Also, substantial differences were found between the closed quotient derived from HSDI and those derived from inverse filtering. The findings illustrate the importance of distinguishing between these quotients.     

Big Hankel operators of higher weight

Known results concerning the smoothness and boundedness of «big» Hankel operators (Hankel operators in the sense of Axler) are generalized to the case of higher weight (in the sense of representation theory). The key result is a certain estimate for thes-numbers of a particular such operator, involving a combinatorial sum.     

Preservation of convexity of solutions to parabolic equations

In the present paper we find necessary and sufficient conditions on the coefficients of a parabolic equation for convexity to be preserved. A parabolic equation is said to preserve convexity if given a convex initial condition, any solution of moderate growth remains a convex function of the spatial variables for each fixed time.