321 research outputs found
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Genus Distributions of Graphs Constructed Through Amalgamations
Graphs are commonly represented as points in space connected by lines. The points in space are the vertices of the graph, and the lines joining them are the edges of the graph. A general definition of a graph is considered here, where multiple edges are allowed between two vertices and an edge is permitted to connect a vertex to itself. It is assumed that graphs are connected, i.e., any vertex in the graph is reachable from another distinct vertex either directly through an edge connecting them or by a path consisting of intermediate vertices and connecting edges. Under this visual representation, graphs can be drawn on various surfaces. The focus of my research is restricted to a class of surfaces that are characterized as compact connected orientable 2-manifolds. The drawings of graphs on surfaces that are of primary interest follow certain prescribed rules. These are called 2-cellular graph embeddings, or simply embeddings. A well-known closed formula makes it easy to enumerate the total number of 2-cellular embeddings for a given graph over all surfaces. A much harder task is to give a surface-wise breakdown of this number as a sequence of numbers that count the number of 2-cellular embeddings of a graph for each orientable surface. This sequence of numbers for a graph is known as the genus distribution of a graph. Prior research on genus distributions of graphs has primarily focused on making calculations of genus distributions for specific families of graphs. These families of graphs have often been contrived, and the methods used for finding their genus distributions have not been general enough to extend to other graph families. The research I have undertaken aims at developing and using a general method that frames the problem of calculating genus distributions of large graphs in terms of a partitioning of the genus distributions of smaller graphs. To this end, I use various operations such as edge-amalgamation, self-edge-amalgamation, and vertex-amalgamation to construct large graphs out of smaller graphs, by coupling their vertices and edges together in certain consistent ways. This method assumes that the partitioned genus distribution of the smaller graphs is known or is easily calculable by computer, for instance, by using the famous Heffter-Edmonds algorithm. As an outcome of the techniques used, I obtain general recurrences and closed-formulas that give genus distributions for infinitely many recursively specifiable graph families. I also give an easily understood method for finding non-trivial examples of distinct graphs having the same genus distribution. In addition to this, I describe an algorithm that computes the genus distributions for a family of graphs known as the 4-regular outerplanar graphs
Mergers and Typical Black Hole Microstates
We use mergers of microstates to obtain the first smooth horizonless
microstate solutions corresponding to a BPS three-charge black hole with a
classically large horizon area. These microstates have very long throats, that
become infinite in the classical limit; nevertheless, their curvature is
everywhere small. Having a classically-infinite throat makes these microstates
very similar to the typical microstates of this black hole. A rough CFT
analysis confirms this intuition, and indicates a possible class of dual CFT
microstates.
We also analyze the properties and the merging of microstates corresponding
to zero-entropy BPS black holes and black rings. We find that these solutions
have the same size as the horizon size of their classical counterparts, and we
examine the changes of internal structure of these microstates during mergers.Comment: 49 pages, 5 figures. v2 references adde
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Methods for Computing Genus Distribution Using Double-Rooted Graphs
This thesis develops general methods for computing the genus distribution of various types of graph families, using the concept of double-rooted graphs, which are defined to be graphs with two vertices designated as roots (the methods developed in this dissertation are limited to the cases where one of the two roots is restricted to be of valence two). I define partials and productions, and I use these as follows: (i) to compute the genus distribution of a graph obtained through the vertex amalgamation of a double-rooted graph with a single-rooted graph, and to show how these can be used to obtain recurrences for the genus distribution of iteratively growing infinite graph families. (ii) to compute the genus distribution of a graph obtained (a) through the operation of self-vertex-amalgamation on a double-rooted graph, and (b) through the operation of edge-addition on a double-rooted graph, and finally (iii) to develop a method to compute the recurrences for the genus distribution of the graph family generated by the Cartesian product of P3 and Pn
Random Embeddings of Graphs: The Expected Number of Faces in Most Graphs is Logarithmic
A random 2-cell embedding of a connected graph in some orientable surface
is obtained by choosing a random local rotation around each vertex. Under this
setup, the number of faces or the genus of the corresponding 2-cell embedding
becomes a random variable. Random embeddings of two particular graph classes --
those of a bouquet of loops and those of parallel edges connecting two
vertices -- have been extensively studied and are well-understood. However,
little is known about more general graphs despite their important connections
with central problems in mainstream mathematics and in theoretical physics (see
[Lando & Zvonkin, Springer 2004]). There are also tight connections with
problems in computing (random generation, approximation algorithms). The
results of this paper, in particular, explain why Monte Carlo methods (see,
e.g., [Gross & Tucker, Ann. NY Acad. Sci 1979] and [Gross & Rieper, JGT 1991])
cannot work for approximating the minimum genus of graphs.
In his breakthrough work ([Stahl, JCTB 1991] and a series of other papers),
Stahl developed the foundation of "random topological graph theory". Most of
his results have been unsurpassed until today. In our work, we analyze the
expected number of faces of random embeddings (equivalently, the average genus)
of a graph . It was very recently shown [Campion Loth & Mohar, arXiv 2022]
that for any graph , the expected number of faces is at most linear. We show
that the actual expected number of faces is usually much smaller. In
particular, we prove the following results:
1) , for
sufficiently large. This greatly improves Stahl's upper bound for
this case.
2) For random models containing only graphs, whose maximum
degree is at most , we show that the expected number of faces is
.Comment: 44 pages, 6 figure
Renormalization in tensor field theory and the melonic fixed point
This thesis focuses on renormalization of tensor field theories. Its first
part considers a quartic tensor model with symmetry and long-range
propagator. The existence of a non-perturbative fixed point in any at large
is established. We found four lines of fixed points parametrized by the
so-called tetrahedral coupling. One of them is infrared attractive, strongly
interacting and gives rise to a new kind of CFT, called melonic CFTs which are
then studied in more details. We first compute dimensions of bilinears and OPE
coefficients at the fixed point which are consistent with a unitary CFT at
large . We then compute corrections. At next-to-leading order, the
line of fixed points collapses to one fixed point. However, the corrections are
complex and unitarity is broken at NLO. Finally, we show that this model
respects the -theorem. The next part of the thesis investigates sextic
tensor field theories in rank and . In rank , we found two IR stable
real fixed points in short range and a line of IR stable real fixed points in
long range. Surprisingly, the only fixed point in rank is the Gaussian one.
For the rank model, in the short-range case, we still find two IR stable
fixed points at NLO. However, in the long-range case, the corrections to the
fixed points are non-perturbative and hence unreliable: we found no precursor
of the large fixed point. The last part of the thesis investigates the
class of model exhibiting a melonic large limit. We prove that models with
tensors in an irreducible representation of or in rank
indeed admit a large limit. This generalization relies on recursive bounds
derived from a detailed combinatorial analysis of Feynman graphs involved in
the perturbative expansion of our model.Comment: PhD thesis, 277 pages. Based on papers: arXiv:1903.03578,
arXiv:1909.07767, arXiv:1912.06641, arXiv:2007.04603, arXiv:2011.11276,
arXiv:2104.03665, arXiv:2109.08034, arXiv:2111.1179
Black Holes as Effective Geometries
Gravitational entropy arises in string theory via coarse graining over an
underlying space of microstates. In this review we would like to address the
question of how the classical black hole geometry itself arises as an effective
or approximate description of a pure state, in a closed string theory, which
semiclassical observers are unable to distinguish from the "naive" geometry. In
cases with enough supersymmetry it has been possible to explicitly construct
these microstates in spacetime, and understand how coarse-graining of
non-singular, horizon-free objects can lead to an effective description as an
extremal black hole. We discuss how these results arise for examples in Type II
string theory on AdS_5 x S^5 and on AdS_3 x S^3 x T^4 that preserve 16 and 8
supercharges respectively. For such a picture of black holes as effective
geometries to extend to cases with finite horizon area the scale of quantum
effects in gravity would have to extend well beyond the vicinity of the
singularities in the effective theory. By studying examples in M-theory on
AdS_3 x S^2 x CY that preserve 4 supersymmetries we show how this can happen.Comment: Review based on lectures of JdB at CERN RTN Winter School and of VB
at PIMS Summer School. 68 pages. Added reference
The Polytope Formalism: isomerism and associated unimolecular isomerisation
This thesis concerns the ontology of isomerism, this encompassing the conceptual frameworks and relationships that comprise the subject matter; the necessary formal definitions, nomenclature, and representations that have impacts reaching into unexpected areas such as drug registration and patent specifications; the requisite controlled and precise vocabulary that facilitates nuanced communication; and the digital/computational formalisms that underpin the chemistry software and database tools that empower chemists to perform much of their work.
Using conceptual tools taken from Combinatorics, and Graph Theory, means are presented to provide a unified description of isomerism and associated unimolecular isomerisation spanning both constitutional isomerism and stereoisomerism called the Polytope Formalism. This includes unification of the varying approaches historically taken to describe and understand stereoisomerism in organic and inorganic compounds.
Work for this Thesis began with the synthesis, isolation, and characterisation of compounds not adequately describable using existing IUPAC recommendations. Generalisation of the polytopal-rearrangements model of stereoisomerisation used for inorganic chemistry led to the prescriptions that could deal with the synthesised compounds, revealing an unrecognised fundamental form of isomerism called akamptisomerism.
Following on, this Thesis describes how in attempting to place akamptisomerism within the context of existing stereoisomerism reveals significant systematic deficiencies in the IUPAC recommendations. These shortcomings have limited the conceptualisation of broad classes of compounds and hindered development of molecules for medicinal and technological applications.
It is shown how the Polytope Formalism can be applied to the description of constitutional isomerism in a practical manner. Finally, a radically different medicinal chemistry design strategy with broad application, based upon the principles, is describe
Commemorative Issue in Honor of Professor Karlheinz Schwarz on the Occasion of His 80th Birthday
A collection of 18 scientific papers written in honor of Professor Karlheinz Schwarz's 80th birthday. The main topics include spectroscopy, excited states, DFT developments, results analysis, solid states, and surfaces
TURBOMOLE: Modular program suite for ab initio quantum-chemical and condensed-matter simulations
TURBOMOLE is a collaborative, multi-national software development project aiming to provide highly efficient and stable computational tools for quantum chemical simulations of molecules, clusters, periodic systems, and solutions. The TURBOMOLE software suite is optimized for widely available, inexpensive, and resource-efficient hardware such as multi-core workstations and small computer clusters. TURBOMOLE specializes in electronic structure methods with outstanding accuracy–cost ratio, such as density functional theory including local hybrids and the random phase approximation (RPA), GW-Bethe–Salpeter methods, second-order Møller–Plesset theory, and explicitly correlated coupled-cluster methods. TURBOMOLE is based on Gaussian basis sets and has been pivotal for the development of many fast and low-scaling algorithms in the past three decades, such as integral-direct methods, fast multipole methods, the resolution-of-the-identity approximation, imaginary frequency integration, Laplace transform, and pair natural orbital methods. This review focuses on recent additions to TURBOMOLE’s functionality, including excited-state methods, RPA and Green’s function methods, relativistic approaches, high-order molecular properties, solvation effects, and periodic systems. A variety of illustrative applications along with accuracy and timing data are discussed. Moreover, available interfaces to users as well as other software are summarized. TURBOMOLE’s current licensing, distribution, and support model are discussed, and an overview of TURBOMOLE’s development workflow is provided. Challenges such as communication and outreach, software infrastructure, and funding are highlighted
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