272 research outputs found
Random induced subgraphs of Cayley graphs induced by transpositions
In this paper we study random induced subgraphs of Cayley graphs of the
symmetric group induced by an arbitrary minimal generating set of
transpositions. A random induced subgraph of this Cayley graph is obtained by
selecting permutations with independent probability, . Our main
result is that for any minimal generating set of transpositions, for
probabilities where , a random induced subgraph has a.s. a unique
largest component of size , where
is the survival probability of a specific branching process.Comment: 18 pages, 1 figur
The Cheshire Cap
A key role in black hole dynamics is played by the inner horizon; most of the
entropy of a slightly nonextremal charged or rotating black hole is carried
there, and the covariant entropy bound suggests that the rest lies in the
region between the inner and outer horizon. An attempt to match this onto
results of the microstate geometries program suggests that a `Higgs branch' of
underlying long string states of the configuration space realizes the degrees
of freedom on the inner horizon, while the `Coulomb branch' describes the
inter-horizon region and beyond. Support for this proposal comes from an
analysis of the way singularities develop in microstate geometries, and their
close analogy to corresponding structures in fivebrane dynamics. These
singularities signal the opening up of the long string degrees of freedom of
the theory, which are partly visible from the geometry side. A conjectural
picture of the black hole interior is proposed, wherein the long string degrees
of freedom resolve the geometrical singularity on the inner horizon, yet are
sufficiently nonlocal to communicate information to the outer horizon and
beyond.Comment: 64 pages, 8 figures. Version 2: References added, together with
substantial elaborations and clarification
The generalized 4-connectivity of burnt pancake graphs
The generalized -connectivity of a graph , denoted by , is
the minimum number of internally edge disjoint -trees for any and . The generalized -connectivity is a natural extension of
the classical connectivity and plays a key role in applications related to the
modern interconnection networks. An -dimensional burnt pancake graph
is a Cayley graph which posses many desirable properties. In this paper, we try
to evaluate the reliability of by investigating its generalized
4-connectivity. By introducing the notation of inclusive tree and by studying
structural properties of , we show that for , that is, for any four vertices in , there exist () internally
edge disjoint trees connecting them in
Fermion condensation and super pivotal categories
We study fermionic topological phases using the technique of fermion
condensation. We give a prescription for performing fermion condensation in
bosonic topological phases which contain a fermion. Our approach to fermion
condensation can roughly be understood as coupling the parent bosonic
topological phase to a phase of physical fermions, and condensing pairs of
physical and emergent fermions. There are two distinct types of objects in
fermionic theories, which we call "m-type" and "q-type" particles. The
endomorphism algebras of q-type particles are complex Clifford algebras, and
they have no analogues in bosonic theories. We construct a fermionic
generalization of the tube category, which allows us to compute the
quasiparticle excitations in fermionic topological phases. We then prove a
series of results relating data in condensed theories to data in their parent
theories; for example, if is a modular tensor category containing
a fermion, then the tube category of the condensed theory satisfies
.
We also study how modular transformations, fusion rules, and coherence
relations are modified in the fermionic setting, prove a fermionic version of
the Verlinde dimension formula, construct a commuting projector lattice
Hamiltonian for fermionic theories, and write down a fermionic version of the
Turaev-Viro-Barrett-Westbury state sum. A large portion of this work is devoted
to three detailed examples of performing fermion condensation to produce
fermionic topological phases: we condense fermions in the Ising theory, the
theory, and the theory, and compute the
quasiparticle excitation spectrum in each of these examples.Comment: 161 pages; v2: corrected typos (including 18 instances of "the the")
and added some reference
Critical manifold of the kagome-lattice Potts model
Any two-dimensional infinite regular lattice G can be produced by tiling the
plane with a finite subgraph B of G; we call B a basis of G. We introduce a
two-parameter graph polynomial P_B(q,v) that depends on B and its embedding in
G. The algebraic curve P_B(q,v) = 0 is shown to provide an approximation to the
critical manifold of the q-state Potts model, with coupling v = exp(K)-1,
defined on G. This curve predicts the phase diagram both in the ferromagnetic
(v>0) and antiferromagnetic (v<0) regions. For larger bases B the
approximations become increasingly accurate, and we conjecture that P_B(q,v) =
0 provides the exact critical manifold in the limit of infinite B. Furthermore,
for some lattices G, or for the Ising model (q=2) on any G, P_B(q,v) factorises
for any choice of B: the zero set of the recurrent factor then provides the
exact critical manifold. In this sense, the computation of P_B(q,v) can be used
to detect exact solvability of the Potts model on G.
We illustrate the method for the square lattice, where the Potts model has
been exactly solved, and the kagome lattice, where it has not. For the square
lattice we correctly reproduce the known phase diagram, including the
antiferromagnetic transition and the singularities in the Berker-Kadanoff
phase. For the kagome lattice, taking the smallest basis with six edges we
recover a well-known (but now refuted) conjecture of F.Y. Wu. Larger bases
provide successive improvements on this formula, giving a natural extension of
Wu's approach. The polynomial predictions are in excellent agreement with
numerical computations. For v>0 the accuracy of the predicted critical coupling
v_c is of the order 10^{-4} or 10^{-5} for the 6-edge basis, and improves to
10^{-6} or 10^{-7} for the largest basis studied (with 36 edges).Comment: 31 pages, 12 figure
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
Novel graph based algorithms for transcriptome sequence analysis
RNA-sequencing (RNA-seq) is one of the most-widely used techniques in molecular biology. A key bioinformatics task in any RNA-seq workflow is the assembling the reads. As the size of transcriptomics data sets is constantly increasing, scalable and accurate assembly approaches have to be developed.Here, we propose several approaches to improve assembling of RNA-seq data generated by second-generation sequencing technologies. We demonstrated that the systematic removal of irrelevant reads from a high coverage dataset prior to assembly, reduces runtime and improves the quality of the assembly. Further, we propose a novel RNA-seq assembly work- flow comprised of read error correction, normalization, assembly with informed parameter selection and transcript-level expression computation.
In recent years, the popularity of third-generation sequencing technologies in- creased as long reads allow for accurate isoform quantification and gene-fusion detection, which is essential for biomedical research. We present a sequence-to-graph alignment method to detect and to quantify transcripts for third-generation sequencing data. Also, we propose the first gene-fusion prediction tool which is specifically tailored towards long-read data and hence achieves accurate expression estimation even on complex data sets. Moreover, our method predicted experimentally verified fusion events along with some novel events, which can be validated in the future
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