Conditioned Brownian trees

We consider a Brownian tree consisting of a collection of one-dimensional Brownian paths started from the origin, whose genealogical structure is given by the Continuum Random Tree (CRT). This Brownian tree may be generated from the Brownian snake driven by a normalized Brownian excursion, and thus yields a convenient representation of the so-called Integrated Super-Brownian Excursion (ISE), which can be viewed as the uniform probability measure on the tree of paths. We discuss different approaches that lead to the definition of the Brownian tree conditioned to stay on the positive half-line. We also establish a Verwaat-like theorem showing that this conditioned Brownian tree can be obtained by re-rooting the unconditioned one at the vertex corresponding to the minimal spatial position. In terms of ISE, this theorem yields the following fact: Conditioning ISE to put no mass on $]-\infty,-\epsilon[$ and letting $\epsilon$ go to 0 is equivalent to shifting the unconditioned ISE to the right so that the left-most point of its support becomes the origin. We derive a number of explicit estimates and formulas for our conditioned Brownian trees. In particular, the probability that ISE puts no mass on $]-\infty,-\epsilon[$ is shown to behave like $2\epsilon^4/21$ when $\epsilon$ goes to 0. Finally, for the conditioned Brownian tree with a fixed height $h$, we obtain a decomposition involving a spine whose distribution is absolutely continuous with respect to that of a nine-dimensional Bessel process on the time interval $[0,h]$, and Poisson processes of subtrees originating from this spine.Comment: 42 page

A new path probing strategy for inter-domain multicast routing

Many already in use applications require the provision of QoS services from the underlying network infra-structure. This is particularly true for multicast, since it involves many participants at very sparse locations usually aiming to receive or send multimedia real-time streams. One way to provide QoS is through routing, since QoS aware multicast routing protocols can find feasible multicast trees. At large scale, scalability issues make the QoS multicast routing task a lot more difficult, since it is not possible to have a complete knowledge of the network topology and its path QoS metrics in a clear up-to-date way. The most promising proposals are therefore based on path probing strategies that evaluate a subset of the available connecting paths. Assuming that each member can express its requirements as a combination of QoS metrics like available bandwidth, end-to-end loss probability and delay, it is up to QoS routing multicast protocols to build distribution trees connecting members through paths that can satisfy those requirements. In this paper an inter-domain QoS multicast routing protocol is presented, specifically designed for the hierarchical inter-domain scenario, where requirements like intra-domain independency and policy awareness should be met. Emphasis is given to the path probing mechanism used to connect new members to the multicast tree, stressing how it differs from others. Simulation results show that despite using a less aggressive and simplified probing mechanism - more suitable for inter-domain scenarios - the proposed routing strategy can build multicast trees with metrics similar to those build by more aggressive technics, with considerable less effort

A Fast Algorithm for Computing Geodesic Distances in Tree Space

Comparing and computing distances between phylogenetic trees are important biological problems, especially for models where edge lengths play an important role. The geodesic distance measure between two phylogenetic trees with edge lengths is the length of the shortest path between them in the continuous tree space introduced by Billera, Holmes, and Vogtmann. This tree space provides a powerful tool for studying and comparing phylogenetic trees, both in exhibiting a natural distance measure and in providing a Euclidean-like structure for solving optimization problems on trees. An important open problem is to find a polynomial time algorithm for finding geodesics in tree space. This paper gives such an algorithm, which starts with a simple initial path and moves through a series of successively shorter paths until the geodesic is attained

Symbolic and Visual Retrieval of Mathematical Notation using Formula Graph Symbol Pair Matching and Structural Alignment

Large data collections containing millions of math formulae in different formats are available on-line. Retrieving math expressions from these collections is challenging. We propose a framework for retrieval of mathematical notation using symbol pairs extracted from visual and semantic representations of mathematical expressions on the symbolic domain for retrieval of text documents. We further adapt our model for retrieval of mathematical notation on images and lecture videos. Graph-based representations are used on each modality to describe math formulas. For symbolic formula retrieval, where the structure is known, we use symbol layout trees and operator trees. For image-based formula retrieval, since the structure is unknown we use a more general Line of Sight graph representation. Paths of these graphs define symbol pairs tuples that are used as the entries for our inverted index of mathematical notation. Our retrieval framework uses a three-stage approach with a fast selection of candidates as the first layer, a more detailed matching algorithm with similarity metric computation in the second stage, and finally when relevance assessments are available, we use an optional third layer with linear regression for estimation of relevance using multiple similarity scores for final re-ranking. Our model has been evaluated using large collections of documents, and preliminary results are presented for videos and cross-modal search. The proposed framework can be adapted for other domains like chemistry or technical diagrams where two visually similar elements from a collection are usually related to each other

Exploring the Use of Free Bioinformatics Modules in an Introductory Biochemistry Course

Although bioinformatics, the use of computational science to study biology, has become imperative in many areas of the biological sciences and related career paths, introductory biochemistry courses may disregard practical knowledge on bioinformatics. For this reason, we merged a hands-on activity module into an undergraduate biochemistry course in two ways. First, we incorporated bioinformatics modules for building phylogenetic trees by aligning the active sites of 10 chosen related α-amylase enzymes using freely available data. Secondly, we chose three of those 10 α-amylase enzymes to compare the 3D structure of their active sites. This module gives the students an opportunity to understand how to access biological information from public databases such as GenBank, and analyze the information using software like MEGA, ClustalW, and RasMol. Overall, our module should provide instructors with ideas on how to develop similar modules and encourage students to develop further independence in the use of bioinformatics tools