1,458 research outputs found
Phase transitions in Pareto optimal complex networks
The organization of interactions in complex systems can be described by
networks connecting different units. These graphs are useful representations of
the local and global complexity of the underlying systems. The origin of their
topological structure can be diverse, resulting from different mechanisms
including multiplicative processes and optimization. In spatial networks or in
graphs where cost constraints are at work, as it occurs in a plethora of
situations from power grids to the wiring of neurons in the brain, optimization
plays an important part in shaping their organization. In this paper we study
network designs resulting from a Pareto optimization process, where different
simultaneous constraints are the targets of selection. We analyze three
variations on a problem finding phase transitions of different kinds. Distinct
phases are associated to different arrangements of the connections; but the
need of drastic topological changes does not determine the presence, nor the
nature of the phase transitions encountered. Instead, the functions under
optimization do play a determinant role. This reinforces the view that phase
transitions do not arise from intrinsic properties of a system alone, but from
the interplay of that system with its external constraints.Comment: 14 pages, 7 figure
Modeling the life and death of competing languages from a physical and mathematical perspective
Recent contributions address the problem of language coexistence as that of
two species competing to aggregate speakers, thus focusing on the dynamics of
linguistic traits across populations. They draw inspiration from physics and
biology and share some underlying ideas -- e. g. the search for minimal schemes
to explain complex situations or the notion that languages are extant entities
in a societal context and, accordingly, that objective, mathematical laws
emerge driving the aforementioned dynamics. Different proposals pay attention
to distinct aspects of such systems: Some of them emphasize the distribution of
the population in geographical space, others research exhaustively the role of
bilinguals in idealized situations (e. g. isolated populations), and yet others
rely extremely on equations taken unchanged from physics or biology and whose
parameters bear actual geometrical meaning. Despite the sources of these models
-- so unrelated to linguistics -- sound results begin to surface that establish
conditions and make testable predictions regarding language survival within
populations of speakers, with a decisive role reserved to bilingualism. Here we
review the most recent works and their interesting outcomes stressing their
physical theoretical basis, and discuss the relevance and meaning of the
abstract mathematical findings for real-life situations.Comment: 22 pages, 4 figures. Fifth chapter of the book Bilingualism and
Minority Languages in Europe: Current trends and developments by F. Lauchlan,
M. C. Parafita Couto, ed
A multiobjective optimization approach to statistical mechanics
Optimization problems have been the subject of statistical physics
approximations. A specially relevant and general scenario is provided by
optimization methods considering tradeoffs between cost and efficiency, where
optimal solutions involve a compromise between both. The theory of Pareto (or
multi objective) optimization provides a general framework to explore these
problems and find the space of possible solutions compatible with the
underlying tradeoffs, known as the {\em Pareto front}. Conflicts between
constraints can lead to complex landscapes of Pareto optimal solutions with
interesting implications in economy, engineering, or evolutionary biology.
Despite their disparate nature, here we show how the structure of the Pareto
front uncovers profound universal features that can be understood in the
context of thermodynamics. In particular, our study reveals that different
fronts are connected to different classes of phase transitions, which we can
define robustly, along with critical points and thermodynamic potentials. These
equivalences are illustrated with classic thermodynamic examples.Comment: 14 pages, 8 figure
The Bohr radius of the -dimensional polydisk is equivalent to
We show that the Bohr radius of the polydisk behaves
asymptotically as . Our argument is based on a new
interpolative approach to the Bohnenblust--Hille inequalities which allows us
to prove that the polynomial Bohnenblust--Hille inequality is subexponential.Comment: The introduction was expanded and some misprints correcte
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