7,846 research outputs found
Gauge sector statistics of intersecting D-brane models
In this article, which is based on the first part of my PhD thesis, I review
the statistics of the open string sector in T^6/(Z_2xZ_2) orientifold
compactifications of the type IIA string. After an introduction to the
orientifold setup, I discuss the two different techniques that have been
developed to analyse the gauge sector statistics, using either a saddle point
approximation or a direct computer based method. The two approaches are
explained and compared by means of eight- and six-dimensional toy models. In
the four-dimensional case the results are presented in detail. Special emphasis
is put on models containing phenomenologically interesting gauge groups and
chiral matter, in particular those containing a standard model or SU(5) part.Comment: 51 pages, 29 figures; v2: ref. added, version to appear in Fortsch.
Phys; v3: ref. adde
Line transversals to disjoint balls
We prove that the set of directions of lines intersecting three disjoint
balls in in a given order is a strictly convex subset of . We then
generalize this result to disjoint balls in . As a consequence, we can
improve upon several old and new results on line transversals to disjoint balls
in arbitrary dimension, such as bounds on the number of connected components
and Helly-type theorems.Comment: 21 pages, includes figure
Complex patterns on the plane: different types of basin fractalization in a two-dimensional mapping
Basins generated by a noninvertible mapping formed by two symmetrically
coupled logistic maps are studied when the only parameter \lambda of the system
is modified. Complex patterns on the plane are visualised as a consequence of
basins' bifurcations. According to the already established nomenclature in the
literature, we present the relevant phenomenology organised in different
scenarios: fractal islands disaggregation, finite disaggregation, infinitely
disconnected basin, infinitely many converging sequences of lakes, countable
self-similar disaggregation and sharp fractal boundary. By use of critical
curves, we determine the influence of zones with different number of first rank
preimages in the mechanisms of basin fractalization.Comment: 19 pages, 11 figure
Some determinants of path generating functions
We evaluate four families of determinants of matrices, where the entries are
sums or differences of generating functions for paths consisting of up-steps,
down-steps and level steps. By specialisation, these determinant evaluations
have numerous corollaries. In particular, they cover numerous determinant
evaluations of combinatorial numbers - most notably of Catalan, ballot, and of
Motzkin numbers - that appeared previously in the literature.Comment: 35 pages, AmS-TeX; minor corrections; final version to appear in Adv.
Appl. Mat
Vicious walkers, friendly walkers and Young tableaux II: With a wall
We derive new results for the number of star and watermelon configurations of
vicious walkers in the presence of an impenetrable wall by showing that these
follow from standard results in the theory of Young tableaux, and combinatorial
descriptions of symmetric functions. For the problem of -friendly walkers,
we derive exact asymptotics for the number of stars and watermelons both in the
absence of a wall and in the presence of a wall.Comment: 35 pages, AmS-LaTeX; Definitions of n-friendly walkers clarified; the
statement of Theorem 4 and its proof were correcte
The Statistics of Supersymmetric D-brane Models
We investigate the statistics of the phenomenologically important D-brane
sector of string compactifications. In particular for the class of intersecting
D-brane models, we generalise methods known from number theory to determine the
asymptotic statistical distribution of solutions to the tadpole cancellation
conditions. Our approach allows us to compute the statistical distribution of
gauge theoretic observables like the rank of the gauge group, the number of
chiral generations or the probability of an SU(N) gauge factor. Concretely, we
study the statistics of intersecting branes on T^2 and T^4/Z_2 and T^6/Z_2 x
Z_2 orientifolds. Intriguingly, we find a statistical correlation between the
rank of the gauge group and the number of chiral generations. Finally, we
combine the statistics of the gauge theory sector with the statistics of the
flux sector and study how distributions of gauge theoretic quantities are
affected.Comment: 62 pages, 31 figures, harvmac; v3: sections 3.2. + 3.7. added, figs.
7,28,29 added, figs. 24,25,26 corrected, refs. added, typos correcte
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