On Bubble Generators in Directed Graphs

International audienceBubbles are pairs of internally vertex-disjoint (s, t)-paths with applications in the processing of DNA and RNA data. For example, enumerating alternative splicing events in a reference-free context can be done by enumerating all bubbles in a de Bruijn graph built from RNA-seq reads [16]. However, listing and analysing all bubbles in a given graph is usually unfeasible in practice, due to the exponential number of bubbles present in real data graphs. In this paper, we propose a notion of a bubble generator set, i.e. a polynomial-sized subset of bubbles from which all the others can be obtained through the application of a specific symmetric difference operator. This set provides a compact representation of the bubble space of a graph, which can be useful in practice since some pertinent information about all the bubbles can be more conveniently extracted from this compact set. Furthermore, we provide a polynomial-time algorithm to decompose any bubble of a graph into the bubbles of such a generator in a tree-like fashion

A family of tree-based generators for bubbles in directed graphs

6sìopenBubbles are pairs of internally vertex-disjoint (s, t)-paths in a directed graph. In de Bruijn graphs built from reads of RNA and DNA data, bubbles represent interesting biological events, such as alternative splicing (AS) and allelic differences (SNPs and indels). However, the set of all bubbles in a de Bruijn graph built from real data is usually too large to be efficiently enumerated and analysed in practice. In particular, despite significant research done in this area, listing bubbles still remains the main bottleneck for tools that detect AS events in a reference-free context. Recently, in the concept of a bubble generator was introduced as a way for obtaining a compact representation of the bubble space of a graph. Although this bubble generator was quite effective in finding AS events, preliminary experiments showed that it is about 5 times slower than state-of-art methods. In this paper we propose a new family of bubble generators which improve substantially on previous work: bubble generators in this new family are about two orders of magnitude faster and are still able to achieve similar precision in identifying AS events. To highlight the practical value of our new bubble generators, we also report some experimental results on real datasets.openAcuña, Vicente; Soares de Lima, Leandro Ishi; Italiano, Giuseppe F.; Pepè Sciarria, Luca; Sagot, Marie-France; Sinaimeri, BlerinaAcuña, Vicente; Soares de Lima, Leandro Ishi; Italiano, Giuseppe F.; Pepè Sciarria, Luca; Sagot, Marie-France; Sinaimeri, Blerin

Finite Type Invariants of w-Knotted Objects II: Tangles, Foams and the Kashiwara-Vergne Problem

This is the second in a series of papers dedicated to studying w-knots, and more generally, w-knotted objects (w-braids, w-tangles, etc.). These are classes of knotted objects that are wider but weaker than their "usual" counterparts. To get (say) w-knots from usual knots (or u-knots), one has to allow non-planar "virtual" knot diagrams, hence enlarging the the base set of knots. But then one imposes a new relation beyond the ordinary collection of Reidemeister moves, called the "overcrossings commute" relation, making w-knotted objects a bit weaker once again. Satoh studied several classes of w-knotted objects (under the name "weakly-virtual") and has shown them to be closely related to certain classes of knotted surfaces in R4. In this article we study finite type invariants of w-tangles and w-trivalent graphs (also referred to as w-tangled foams). Much as the spaces A of chord diagrams for ordinary knotted objects are related to metrized Lie algebras, the spaces Aw of "arrow diagrams" for w-knotted objects are related to not-necessarily-metrized Lie algebras. Many questions concerning w-knotted objects turn out to be equivalent to questions about Lie algebras. Most notably we find that a homomorphic universal finite type invariant of w-foams is essentially the same as a solution of the Kashiwara-Vergne conjecture and much of the Alekseev-Torossian work on Drinfel'd associators and Kashiwara-Vergne can be re-interpreted as a study of w-foams.Comment: 57 pages. Improvements to the exposition following a referee repor

Non-Liouville groups with return probability exponent at most 1/2

We construct a finitely generated group $G$ without the Liouville property such that the return probability of a random walk satisfies $p_{2n}(e,e) \gtrsim e^{-n^{1/2 + o(1)}}$. Recent results suggest that $1/2$ is indeed the smallest possible return probability exponent for non-Liouville groups. Our construction is based on permutational wreath products over tree-like Schreier graphs and the analysis of large deviations of inverted orbits on such graphs.Comment: 15 pages, 1 figure; v2: minor correction

Continuum spin foam model for 3d gravity

An example illustrating a continuum spin foam framework is presented. This covariant framework induces the kinematics of canonical loop quantization, and its dynamics is generated by a {\em renormalized} sum over colored polyhedra. Physically the example corresponds to 3d gravity with cosmological constant. Starting from a kinematical structure that accommodates local degrees of freedom and does not involve the choice of any background structure (e. g. triangulation), the dynamics reduces the field theory to have only global degrees of freedom. The result is {\em projectively} equivalent to the Turaev-Viro model.Comment: 12 pages, 3 figure

A Family of Tree-Based Generators for Bubbles in Directed Graphs

International audienceBubbles are pairs of internally vertex-disjoint (s, t)-paths in a directed graph. In de Bruijn graphs built from reads of RNA and DNA data, bubbles represent interesting biological events, such as alternative splicing (AS) and allelic differences (SNPs and indels). However, the set of all bubbles in a de Bruijn graph built from real data is usually too large to be efficiently enumerated and analysed in practice. In particular, despite significant research done in this area, listing bubbles still remains the main bottleneck for tools that detect AS events in a reference-free context. Recently, in [1] the concept of a bubble generator was introduced as a way for obtaining a compact representation of the bubble space of a graph. Although this generator was quite effective in finding AS events, preliminary experiments showed that it is about 5 times slower than state-of-art methods. In this paper we propose a new family of bubble generators which improve substantially on the previous generator: generators in this new family are about two orders of magnitude faster and are still able to achieve similar precision in identifying AS events. To highlight the practical value of our new generators, we also report some experimental results on a real dataset

On Lorentzian causality with continuous metrics

We present a systematic study of causality theory on Lorentzian manifolds with continuous metrics. Examples are given which show that some standard facts in smooth Lorentzian geometry, such as light-cones being hypersurfaces, are wrong when metrics which are merely continuous are considered. We show that existence of time functions remains true on domains of dependence with continuous metrics, and that $C^{0,1}$ differentiability of the metric suffices for many key results of the smooth causality theory.Comment: Minor changes. Version to appear in Classical and Quantum Gravit

Partial duality of hypermaps

We introduce a collection of new operations on hypermaps, partial duality, which include the classical Euler-Poincar\'e dualities as particular cases. These operations generalize the partial duality for maps, or ribbon graphs, recently discovered in a connection with knot theory. Partial duality is different from previous studied operations of S. Wilson, G. Jones, L. James, and A. Vince. Combinatorially hypermaps may be described in one of three ways: as three involutions on the set of flags ($\tau$-model), or as three permutations on the set of half-edges ($\sigma$-model in orientable case), or as edge 3-colored graphs. We express partial duality in each of these models.Comment: 19 pages, 16 figure

Ocean foam generation and modeling

A laboratory investigation was conducted to determine the physical and microwave properties of ocean foam. Special foam generators were designed and fabricated, using porous glass sheets, known as glass frits, as the principal element. The glass frit was sealed into a water-tight vertical box, a few centimeters from the bottom. Compressed air, applied to the lower chamber, created ocean foam from sea water lying on the frit. Foam heights of 30 cm were readily achieved, with relatively low air pressures. Special photographic techniques and analytical procedures were employed to determine foam bubble size distributions. In addition, the percentage water content of ocean foam was determined with the aid of a particulate sampling procedure. A glass frit foam generator, with pore diameters in the range 70 - 100 micrometers, produced foam with bubble distributions very similar to those found on the surface of natural ocean foam patches