85 research outputs found
Solving rank-constrained semidefinite programs in exact arithmetic
We consider the problem of minimizing a linear function over an affine
section of the cone of positive semidefinite matrices, with the additional
constraint that the feasible matrix has prescribed rank. When the rank
constraint is active, this is a non-convex optimization problem, otherwise it
is a semidefinite program. Both find numerous applications especially in
systems control theory and combinatorial optimization, but even in more general
contexts such as polynomial optimization or real algebra. While numerical
algorithms exist for solving this problem, such as interior-point or
Newton-like algorithms, in this paper we propose an approach based on symbolic
computation. We design an exact algorithm for solving rank-constrained
semidefinite programs, whose complexity is essentially quadratic on natural
degree bounds associated to the given optimization problem: for subfamilies of
the problem where the size of the feasible matrix is fixed, the complexity is
polynomial in the number of variables. The algorithm works under assumptions on
the input data: we prove that these assumptions are generically satisfied. We
also implement it in Maple and discuss practical experiments.Comment: Published at ISSAC 2016. Extended version submitted to the Journal of
Symbolic Computatio
On the Complexity of Solving Zero-Dimensional Polynomial Systems via Projection
Given a zero-dimensional polynomial system consisting of n integer
polynomials in n variables, we propose a certified and complete method to
compute all complex solutions of the system as well as a corresponding
separating linear form l with coefficients of small bit size. For computing l,
we need to project the solutions into one dimension along O(n) distinct
directions but no further algebraic manipulations. The solutions are then
directly reconstructed from the considered projections. The first step is
deterministic, whereas the second step uses randomization, thus being
Las-Vegas.
The theoretical analysis of our approach shows that the overall cost for the
two problems considered above is dominated by the cost of carrying out the
projections. We also give bounds on the bit complexity of our algorithms that
are exclusively stated in terms of the number of variables, the total degree
and the bitsize of the input polynomials
Parts of Quantum States
It is shown that generic N-party pure quantum states (with equidimensional
subsystems) are uniquely determined by their reduced states of just over half
the parties; in other words, all the information in almost all N-party pure
states is in the set of reduced states of just over half the parties. For N
even, the reduced states in fewer than N/2 parties are shown to be an
insufficient description of almost all states (similar results hold when N is
odd). It is noted that Real Algebraic Geometry is a natural framework for any
analysis of parts of quantum states: two simple polynomials, a quadratic and a
cubic, contain all of their structure. Algorithmic techniques are described
which can provide conditions for sets of reduced states to belong to pure or
mixed states.Comment: 10 pages, 1 figur
Fast Reduction of Bivariate Polynomials with Respect to Sufficiently Regular Gröbner Bases
International audienc
Dissipation in planar resonant planetary systems
Close-in planetary systems detected by the Kepler mission present an excess
of periods ratio that are just slightly larger than some low order resonant
values. This feature occurs naturally when resonant couples undergo dissipation
that damps the eccentricities. However, the resonant angles appear to librate
at the end of the migration process, which is often believed to be an evidence
that the systems remain in resonance.
Here we provide an analytical model for the dissipation in resonant planetary
systems valid for low eccentricities. We confirm that dissipation accounts for
an excess of pairs that lie just aside from the nominal periods ratios, as
observed by the Kepler mission. In addition, by a global analysis of the phase
space of the problem, we demonstrate that these final pairs are non-resonant.
Indeed, the separatrices that exist in the resonant systems disappear with the
dissipation, and remains only a circulation of the orbits around a single
elliptical fixed point. Furthermore, the apparent libration of the resonant
angles can be explained using the classical secular averaging method. We show
that this artifact is only due to the severe damping of the amplitudes of the
eigenmodes in the secular motion.Comment: 18 pages, 20 figures, accepted to A&
Sparse Rational Univariate Representation
International audienceWe present explicit worst case degree and height bounds for the rational univariate representation of the isolated roots of polynomial systems based on mixed volume. We base our estimations on height bounds of resultants and we consider the case of 0-dimensional, positive dimensional, and parametric polynomial systems
The Ks-band Tully-Fisher Relation - A Determination of the Hubble Parameter from 218 ScI Galaxies and 16 Galaxy Clusters
The value of the Hubble Parameter (H0) is determined using the
morphologically type dependent Ks-band Tully-Fisher Relation (K-TFR). The slope
and zero point are determined using 36 calibrator galaxies with ScI morphology.
Calibration distances are adopted from direct Cepheid distances, and group or
companion distances derived with the Surface Brightness Fluctuation Method or
Type Ia Supernova. Distances are determined to 16 galaxy clusters and 218 ScI
galaxies with minimum distances of 40.0 Mpc. From the 16 galaxy clusters a
weighted mean Hubble Parameter of H0=84.2 +/-6 km s-1 Mpc-1 is found. From the
218 ScI galaxies a Hubble Parameter of H0=83.4 +/-8 km s-1 Mpc-1 is found. When
the zero point of the K-TFR is corrected to account for recent results that
find a Large Magellanic Cloud distance modulus of 18.39 +/-0.05 a Hubble
Parameter of 88.0 +/-6 km s-1 Mpc-1 is found. A comparison with the results of
the Hubble Key Project (Freedman et al 2001) is made and discrepancies between
the K-TFR distances and the HKP I-TFR distances are discussed. Implications for
Lamda-CDM cosmology are considered with H0=84 km s-1 Mpc-1. (Abridged)Comment: 37 pages including 12 tables and 7 figures. Final version accepted
for publication in the Journal of Astrophysics & Astronom
Semidefinite Characterization and Computation of Real Radical Ideals
For an ideal given by a set of generators, a new
semidefinite characterization of its real radical is
presented, provided it is zero-dimensional (even if is not). Moreover we
propose an algorithm using numerical linear algebra and semidefinite
optimization techniques, to compute all (finitely many) points of the real
variety as well as a set of generators of the real radical
ideal. The latter is obtained in the form of a border or Gr\"obner basis. The
algorithm is based on moment relaxations and, in contrast to other existing
methods, it exploits the real algebraic nature of the problem right from the
beginning and avoids the computation of complex components.Comment: 41 page
Numeric and Certified Isolation of the Singularities of the Projection of a Smooth Space Curve
International audienceLet CP ∩Q be a smooth real analytic curve embedded in R 3 , defined as the solutions of real analytic equations of the form P (x, y, z) = Q(x, y, z) = 0 or P (x, y, z) = ∂P ∂z = 0. Our main objective is to describe its projection C onto the (x, y)-plane. In general, the curve C is not a regular submanifold of R 2 and describing it requires to isolate the points of its singularity locus Σ. After describing the types of singularities that can arise under some assumptions on P and Q, we present a new method to isolate the points of Σ. We experimented our method on pairs of independent random polynomials (P, Q) and on pairs of random polynomials of the form (P, ∂P ∂z) and got promising results
The antimalarial MMV688533 provides potential for single-dose cures with a high barrier to
The emergence and spread of Plasmodium falciparum resistance to first-line antimalarials creates an imperative to identify and develop potent preclinical candidates with distinct modes of action. Here, we report the identification of MMV688533, an acylguanidine that was developed following a whole-cell screen with compounds known to hit high-value targets in human cells. MMV688533 displays fast parasite clearance in vitro and is not cross-resistant with known antimalarials. In a P. falciparum NSG mouse model, MMV688533 displays a long-lasting pharmacokinetic profile and excellent safety. Selection studies reveal a low propensity for resistance, with modest loss of potency mediated by point mutations in PfACG1 and PfEHD. These proteins are implicated in intracellular trafficking, lipid utilization, and endocytosis, suggesting interference with these pathways as a potential mode of action. This preclinical candidate may offer the potential for a single low-dose cure for malaria
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