268 research outputs found
Combinatorics and Geometry of Transportation Polytopes: An Update
A transportation polytope consists of all multidimensional arrays or tables
of non-negative real numbers that satisfy certain sum conditions on subsets of
the entries. They arise naturally in optimization and statistics, and also have
interest for discrete mathematics because permutation matrices, latin squares,
and magic squares appear naturally as lattice points of these polytopes.
In this paper we survey advances on the understanding of the combinatorics
and geometry of these polyhedra and include some recent unpublished results on
the diameter of graphs of these polytopes. In particular, this is a thirty-year
update on the status of a list of open questions last visited in the 1984 book
by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.Comment: 35 pages, 13 figure
Asymptotic analysis of noisy fitness maximization, applied to metabolism and growth
We consider a population dynamics model coupling cell growth to a diffusion
in the space of metabolic phenotypes as it can be obtained from realistic
constraints-based modelling. In the asymptotic regime of slow diffusion, that
coincides with the relevant experimental range, the resulting non-linear
Fokker-Planck equation is solved for the steady state in the WKB approximation
that maps it into the ground state of a quantum particle in an Airy potential
plus a centrifugal term. We retrieve scaling laws for growth rate fluctuations
and time response with respect to the distance from the maximum growth rate
suggesting that suboptimal populations can have a faster response to
perturbations.Comment: 24 pages, 6 figure
Quantifying multipartite nonlocality via the size of the resource
The generation of (Bell-)nonlocal correlations, i.e., correlations leading to
the violation of a Bell-like inequality, requires the usage of a nonlocal
resource, such as an entangled state. When given a correlation (a collection of
conditional probability distributions) from an experiment or from a theory, it
is desirable to determine the extent to which the participating parties would
need to collaborate nonlocally for its (re)production. Here, we propose to
achieve this via the minimal group size (MGS) of the resource, i.e., the
smallest number of parties that need to share a given type of nonlocal resource
for the above-mentioned purpose. In addition, we provide a general recipe ---
based on the lifting of Bell-like inequalities --- to construct MGS witnesses
for non-signaling resources starting from any given ones. En route to
illustrating the applicability of this recipe, we also show that when
restricted to the space of full-correlation functions, non-signaling resources
are as powerful as unconstrained signaling resources. Explicit examples of
correlations where their MGS can be determined using this recipe and other
numerical techniques are provided.Comment: 8+3 pages, 2 figures, 2 theorems + 1 corollary; comments very
welcomed
An analytic approximation of the feasible space of metabolic networks
Assuming a steady-state condition within a cell, metabolic fluxes satisfy an
under-determined linear system of stoichiometric equations. Characterizing the
space of fluxes that satisfy such equations along with given bounds (and
possibly additional relevant constraints) is considered of utmost importance
for the understanding of cellular metabolism. Extreme values for each
individual flux can be computed with Linear Programming (as Flux Balance
Analysis), and their marginal distributions can be approximately computed with
Monte-Carlo sampling. Here we present an approximate analytic method for the
latter task based on Expectation Propagation equations that does not involve
sampling and can achieve much better predictions than other existing analytic
methods. The method is iterative, and its computation time is dominated by one
matrix inversion per iteration. With respect to sampling, we show through
extensive simulation that it has some advantages including computation time,
and the ability to efficiently fix empirically estimated distributions of
fluxes
Methods for Structural Pattern Recognition: Complexity and Applications
Katedra kybernetik
Genuine quantum correlations in quantum many-body systems: a review of recent progress
Quantum information theory has considerably helped in the understanding of
quantum many-body systems. The role of quantum correlations and in particular,
bipartite entanglement, has become crucial to characterise, classify and
simulate quantum many body systems. Furthermore, the scaling of entanglement
has inspired modifications to numerical techniques for the simulation of
many-body systems leading to the, now established, area of tensor networks.
However, the notions and methods brought by quantum information do not end with
bipartite entanglement. There are other forms of correlations embedded in the
ground, excited and thermal states of quantum many-body systems that also need
to be explored and might be utilised as potential resources for quantum
technologies. The aim of this work is to review the most recent developments
regarding correlations in quantum many-body systems focussing on multipartite
entanglement, quantum nonlocality, quantum discord, mutual information but also
other non classical measures of correlations based on quantum coherence.
Moreover, we also discuss applications of quantum metrology in quantum
many-body systems.Comment: Review. Close to published version. Comments are welcome! Please
write an email to g.dechiara[(at)]qub.ac.u
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