359 research outputs found
Random trees with superexponential branching weights
We study rooted planar random trees with a probability distribution which is
proportional to a product of weight factors associated to the vertices of
the tree and depending only on their individual degrees . We focus on the
case when grows faster than exponentially with . In this case the
measures on trees of finite size converge weakly as tends to infinity
to a measure which is concentrated on a single tree with one vertex of infinite
degree. For explicit weight factors of the form with
we obtain more refined results about the approach to the infinite
volume limit.Comment: 19 page
Extremes of geometric variables with applications to branching processes
We obtain limit theorems for the row extrema of a triangular array of
zero-modified geometric random variables. Some of this is used to obtain limit
theorems for the maximum family size within a generation of a simple branching
process with varying geometric offspring laws.Comment: 12 pages, some proofs are added to the published versio
Splitting fields and general differential Galois theory
An algebraic technique is presented that does not use results of model theory
and makes it possible to construct a general Galois theory of arbitrary
nonlinear systems of partial differential equations. The algebraic technique is
based on the search for prime differential ideals of special form in tensor
products of differential rings. The main results demonstrating the work of the
technique obtained are the theorem on the constructedness of the differential
closure and the general theorem on the Galois correspondence for normal
extensions..Comment: 33 pages, this version coincides with the published on
Composition-Diamond lemma for -differential associative algebras with multiple operators
In this paper, we establish the Composition-Diamond lemma for
-differential associative algebras over a field with multiple
operators. As applications, we obtain Gr\"{o}bner-Shirshov bases of free
-differential Rota-Baxter algebras. In particular, linear bases of
free -differential Rota-Baxter algebras are obtained and consequently,
the free -differential Rota-Baxter algebras are constructed by words
Random Networks Tossing Biased Coins
In statistical mechanical investigations on complex networks, it is useful to
employ random graphs ensembles as null models, to compare with experimental
realizations. Motivated by transcription networks, we present here a simple way
to generate an ensemble of random directed graphs with, asymptotically,
scale-free outdegree and compact indegree. Entries in each row of the adjacency
matrix are set to be zero or one according to the toss of a biased coin, with a
chosen probability distribution for the biases. This defines a quick and simple
algorithm, which yields good results already for graphs of size n ~ 100.
Perhaps more importantly, many of the relevant observables are accessible
analytically, improving upon previous estimates for similar graphs
The differential-algebraic and bi-Hamiltonian integrability analysis of the Riemann type hierarchy revisited
A differential-algebraic approach to studying the Lax type integrability of
the generalized Riemann type hydrodynamic hierarchy is revisited, its new Lax
type representation and Poisson structures constructed in exact form. The
related bi-Hamiltonian integrability and compatible Poissonian structures of
the generalized Riemann type hierarchy are also discussed.Comment: 18 page
Gravitational detection of a low-mass dark satellite at cosmological distance
The mass-function of dwarf satellite galaxies that are observed around Local
Group galaxies substantially differs from simulations based on cold dark
matter: the simulations predict many more dwarf galaxies than are seen. The
Local Group, however, may be anomalous in this regard. A massive dark satellite
in an early-type lens galaxy at z = 0.222 was recently found using a new method
based on gravitational lensing, suggesting that the mass fraction contained in
substructure could be higher than is predicted from simulations. The lack of
very low mass detections, however, prohibited any constraint on their mass
function. Here we report the presence of a 1.9 +/- 0.1 x 10^8 M_sun dark
satellite in the Einstein-ring system JVAS B1938+666 at z = 0.881, where M_sun
denotes solar mass. This satellite galaxy has a mass similar to the Sagittarius
galaxy, which is a satellite of the Milky Way. We determine the logarithmic
slope of the mass function for substructure beyond the local Universe to be
alpha = 1.1^+0.6_-0.4, with an average mass-fraction of f = 3.3^+3.6_-1.8 %, by
combining data on both of these recently discovered galaxies. Our results are
consistent with the predictions from cold dark matter simulations at the 95 per
cent confidence level, and therefore agree with the view that galaxies formed
hierarchically in a Universe composed of cold dark matter.Comment: 25 pages, 7 figures, accepted for publication in Nature (19 January
2012
Zero Order Estimates for Analytic Functions
The primary goal of this paper is to provide a general multiplicity estimate.
Our main theorem allows to reduce a proof of multiplicity lemma to the study of
ideals stable under some appropriate transformation of a polynomial ring. In
particular, this result leads to a new link between the theory of polarized
algebraic dynamical systems and transcendental number theory. On the other
hand, it allows to establish an improvement of Nesterenko's conditional result
on solutions of systems of differential equations. We also deduce, under some
condition on stable varieties, the optimal multiplicity estimate in the case of
generalized Mahler's functional equations, previously studied by Mahler,
Nishioka, Topfer and others. Further, analyzing stable ideals we prove the
unconditional optimal result in the case of linear functional systems of
generalized Mahler's type. The latter result generalizes a famous theorem of
Nishioka (1986) previously conjectured by Mahler (1969), and simultaneously it
gives a counterpart in the case of functional systems for an important
unconditional result of Nesterenko (1977) concerning linear differential
systems. In summary, we provide a new universal tool for transcendental number
theory, applicable with fields of any characteristic. It opens the way to new
results on algebraic independence, as shown in Zorin (2010).Comment: 42 page
A Bose-Einstein Approach to the Random Partitioning of an Integer
Consider N equally-spaced points on a circle of circumference N. Choose at
random n points out of on this circle and append clockwise an arc of
integral length k to each such point. The resulting random set is made of a
random number of connected components. Questions such as the evaluation of the
probability of random covering and parking configurations, number and length of
the gaps are addressed. They are the discrete versions of similar problems
raised in the continuum. For each value of k, asymptotic results are presented
when n,N both go to infinity according to two different regimes. This model may
equivalently be viewed as a random partitioning problem of N items into n
recipients. A grand-canonical balls in boxes approach is also supplied, giving
some insight into the multiplicities of the box filling amounts or spacings.
The latter model is a k-nearest neighbor random graph with N vertices and kn
edges. We shall also briefly consider the covering problem in the context of a
random graph model with N vertices and n (out-degree 1) edges whose endpoints
are no more bound to be neighbors
The place and role of leisure in the Moscow studentsâ everyday life in the context of the information society development
The proposed article discusses issues related to the daily life of Moscow students and their leisure, the relevance of which in the context of the development of the information society is associated with increased attention of the state to the educational component of modern educational systems. Moreover, in the conditions of modern realities of the development of the information society, and, consequently, information culture, the format of leisure becomes a significant factor in the socialisation of the individual.  As a goal, the authors of the article identified the analysis of various forms of leisure activities of Moscow students through their questionnaires, which was designed to outline vectors for further research of the place and role of study, family relations and forms of spending free time in a studentâs life.  Special attention is paid to modern forms of leisure for students, such as: spending time on the Internet, visiting experimental theaters and cinemas, watching talk shows and entertainment programs, visiting immersive shows, participating in modern youth organisations, etc. Summarising the results of the study, it was concluded that at the moment no more than 20 % of the surveyed Moscow students choose âmodernâ forms of leisure activities. The conducted research revealed the studentsâ clear preferences in spending their free time, as well as their value orientations
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