10,880 research outputs found
Non-Schlesinger Isomonodromic Deformations of Fuchsian Systems and Middle Convolution
The paper is devoted to non-Schlesinger isomonodromic deformations for
resonant Fuchsian systems. There are very few explicit examples of such
deformations in the literature. In this paper we construct a new example of the
non-Schlesinger isomonodromic deformation for a resonant Fuchsian system of
order 5 by using middle convolution for a resonant Fuchsian system of order 2.
Moreover, it is known that middle convolution is an operation that preserves
Schlesinger's deformation equations for non-resonant Fuchsian systems. In this
paper we show that Bolibruch's non-Schlesinger deformations of resonant
Fuchsian systems are, in general, not preserved by middle convolution
Design of First-Order Optimization Algorithms via Sum-of-Squares Programming
In this paper, we propose a framework based on sum-of-squares programming to
design iterative first-order optimization algorithms for smooth and strongly
convex problems. Our starting point is to develop a polynomial matrix
inequality as a sufficient condition for exponential convergence of the
algorithm. The entries of this matrix are polynomial functions of the unknown
parameters (exponential decay rate, stepsize, momentum coefficient, etc.). We
then formulate a polynomial optimization, in which the objective is to optimize
the exponential decay rate over the parameters of the algorithm. Finally, we
use sum-of-squares programming as a tractable relaxation of the proposed
polynomial optimization problem. We illustrate the utility of the proposed
framework by designing a first-order algorithm that shares the same structure
as Nesterov's accelerated gradient method
A survey on signature-based Gr\"obner basis computations
This paper is a survey on the area of signature-based Gr\"obner basis
algorithms that was initiated by Faug\`ere's F5 algorithm in 2002. We explain
the general ideas behind the usage of signatures. We show how to classify the
various known variants by 3 different orderings. For this we give translations
between different notations and show that besides notations many approaches are
just the same. Moreover, we give a general description of how the idea of
signatures is quite natural when performing the reduction process using linear
algebra. This survey shall help to outline this field of active research.Comment: 53 pages, 8 figures, 11 table
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