45,070 research outputs found
Vector space framework for unification of one- and multidimensional filter bank theory
A number of results in filter bank theory can be viewed using vector space notations. This simplifies the proofs of many important results. In this paper, we first introduce the framework of vector space, and then use this framework to derive some known and some new filter bank results as well. For example, the relation among the Hermitian image property, orthonormality, and the perfect reconstruction (PR) property is well-known for the case of one-dimensional (1-D) analysis/synthesis filter banks. We can prove the same result in a more general vector space setting. This vector space framework has the advantage that even the most general filter banks, namely, multidimensional nonuniform filter banks with rational decimation matrices, become a special case. Many results in 1-D filter bank theory are hence extended to the multidimensional case, with some algebraic manipulations of integer matrices. Some examples are: the equivalence of biorthonormality and the PR property, the interchangeability of analysis and synthesis filters, the connection between analysis/synthesis filter banks and synthesis/analysis transmultiplexers, etc. Furthermore, we obtain the subband convolution scheme by starting from the generalized Parseval's relation in vector space. Several theoretical results of wavelet transform can also be derived using this framework. In particular, we derive the wavelet convolution theorem
Reconstructing Rational Functions with
We present the open-source library for the
reconstruction of multivariate rational functions over finite fields. We
discuss the involved algorithms and their implementation. As an application, we
use in the context of integration-by-parts reductions and
compare runtime and memory consumption to a fully algebraic approach with the
program .Comment: 46 pages, 3 figures, 6 tables; v2: matches published versio
Efficient Algorithms for Computing Rational First Integrals and Darboux Polynomials of Planar Polynomial Vector Fields
International audienceWe present fast algorithms for computing rational first integrals with bounded degree of a planar polynomial vector field. Our approach builds upon a method proposed by Ferragut and Giacomini, whose main ingredients are the calculation of a power series solution of a first order differential equation and the reconstruction of a bivariate polynomial annihilating this power series. We provide explicit bounds on the number of terms needed in the power series. This enables us to transform their method into a certified algorithm computing rational first integrals via systems of linear equations. We then significantly improve upon this first algorithm by building a probabilistic algorithm with arithmetic complexity and a deterministic algorithm solving the problem in at most arithmetic operations, where~ denotes the given bound for the degree of the rational first integral, and where is the degree of the vector field, and the exponent of linear algebra. We also provide a fast heuristic variant which computes a rational first integral, or fails, in arithmetic operations. By comparison, the best previous algorithm uses at least arithmetic operations. We then show how to apply a similar method to the computation of Darboux polynomials. The algorithms are implemented in a Maple package RationalFirstIntegrals which is available to interested readers with examples showing its efficiency
Computing Bounds on Network Capacity Regions as a Polytope Reconstruction Problem
We define a notion of network capacity region of networks that generalizes
the notion of network capacity defined by Cannons et al. and prove its notable
properties such as closedness, boundedness and convexity when the finite field
is fixed. We show that the network routing capacity region is a computable
rational polytope and provide exact algorithms and approximation heuristics for
computing the region. We define the semi-network linear coding capacity region,
with respect to a fixed finite field, that inner bounds the corresponding
network linear coding capacity region, show that it is a computable rational
polytope, and provide exact algorithms and approximation heuristics. We show
connections between computing these regions and a polytope reconstruction
problem and some combinatorial optimization problems, such as the minimum cost
directed Steiner tree problem. We provide an example to illustrate our results.
The algorithms are not necessarily polynomial-time.Comment: Appeared in the 2011 IEEE International Symposium on Information
Theory, 5 pages, 1 figur
Complete algebraic vector fields on affine surfaces
Let \AAutH (X) be the subgroup of the group \AutH (X) of holomorphic
automorphisms of a normal affine algebraic surface generated by elements of
flows associated with complete algebraic vector fields. Our main result is a
classification of all normal affine algebraic surfaces quasi-homogeneous
under \AAutH (X) in terms of the dual graphs of the boundaries \bX \setminus
X of their SNC-completions \bX.Comment: 44 page
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