305 research outputs found
On the Second-Order Wiener Ratios of Iterated Line Graphs
The Wiener index W(G) of a graph G is the sum of distances between all
unordered pairs of its vertices. Dobrynin and Mel'nikov [in: Distance in
Molecular Graphs - Theory, 2012, p. 85-121] propose the study of estimates for
extremal values of the ratio R_k(G) = W(L^k(G))/W(G) where L^k(G) denotes the
k-th iterated line graph of G. Hri\v{n}\'akov\'a, Knor and \v{S}krekovski [Art
Discrete Appl. Math. 1 (2018) #P1.09] prove that for each k>2, the path P_n has
the smallest value of the ratio R_k among all trees of large order n, and they
conjecture that the same holds for the case k=2. We give a counterexample of
every order n>21 to this conjecture
Laplacian energy of graphs and digraphs.
Spectral graph theory (Algebraic graph theory) which emerged in 1950s and 1960s is the study of properties of a graph in relationship to the characteristic polynomial, eigenvalues and eigenvectors of matrices associated to the graph. The major source of research in spectral graph theory has been the study of relationship between the structural and spectral properties of graphs. Another source has research in quantum chemistry. Just as astronomers study stellar spectra to determine the make-up of distant stars, one of the main goals in spectral graph theory is to deduce the principal properties and structure of a graph from its graph spectrum (or from a short list of easily computable invariants). The spectral approach for general graphs is a step in this direction.Digital copy of Thesis.University of Kashmir
Amoeba Techniques for Shape and Texture Analysis
Morphological amoebas are image-adaptive structuring elements for
morphological and other local image filters introduced by Lerallut et al. Their
construction is based on combining spatial distance with contrast information
into an image-dependent metric. Amoeba filters show interesting parallels to
image filtering methods based on partial differential equations (PDEs), which
can be confirmed by asymptotic equivalence results. In computing amoebas, graph
structures are generated that hold information about local image texture. This
paper reviews and summarises the work of the author and his coauthors on
morphological amoebas, particularly their relations to PDE filters and texture
analysis. It presents some extensions and points out directions for future
investigation on the subject.Comment: 38 pages, 19 figures v2: minor corrections and rephrasing, Section 5
(pre-smoothing) extende
From constructive field theory to fractional stochastic calculus. (I) An introduction: rough path theory and perturbative heuristics
Let be a -dimensional fractional Brownian motion
with Hurst index , or more generally a Gaussian process whose
paths have the same local regularity. Defining properly iterated integrals of
is a difficult task because of the low H\"older regularity index of its
paths. Yet rough path theory shows it is the key to the construction of a
stochastic calculus with respect to , or to solving differential equations
driven by . We intend to show in a forthcoming series of papers how to
desingularize iterated integrals by a weak singular non-Gaussian perturbation
of the Gaussian measure defined by a limit in law procedure.
Convergence is proved by using "standard" tools of constructive field theory,
in particular cluster expansions and renormalization. These powerful tools
allow optimal estimates of the moments and call for an extension of the
Gaussian tools such as for instance the Malliavin calculus. This first paper
aims to be both a presentation of the basics of rough path theory to
physicists, and of perturbative field theory to probabilists; it is only
heuristic, in particular because the desingularization of iterated integrals is
really a {\em non-perturbative} effect. It is also meant to be a general
motivating introduction to the subject, with some insights into quantum field
theory and stochastic calculus. The interested reader should read in a second
time the companion article \cite{MagUnt2} (or a preliminary version
arXiv:1006.1255) for the constructive proofs
From Physics to Number Theory via Noncommutative Geometry, Part II: Renormalization, the Riemann-Hilbert correspondence, and motivic Galois theory
We establish a precise relation between Galois theory in its motivic form
with the mathematical theory of perturbative renormalization (in the minimal
subtraction scheme with dimensional regularization). We identify, through a
Riemann-Hilbert correspondence based on the Birkhoff decomposition and the
t'Hooft relations, a universal symmetry group (the "cosmic Galois group"
suggested by Cartier), which contains the renormalization group and acts on the
set of physical theories. This group is closely related to motivic Galois
theory. We construct a universal singular frame of geometric nature, in which
all divergences disappear. The paper includes a detailed overview of the work
of Connes-Kreimer and background material on the main quantum field theoretic
and algebro-geometric notions involved. We give a complete account of our
results announced in math.NT/0409306.Comment: 97 pages LaTeX, 17 eps figure
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