14,369 research outputs found
Algebraic Signal Processing Theory: Cooley-Tukey Type Algorithms for DCTs and DSTs
This paper presents a systematic methodology based on the algebraic theory of
signal processing to classify and derive fast algorithms for linear transforms.
Instead of manipulating the entries of transform matrices, our approach derives
the algorithms by stepwise decomposition of the associated signal models, or
polynomial algebras. This decomposition is based on two generic methods or
algebraic principles that generalize the well-known Cooley-Tukey FFT and make
the algorithms' derivations concise and transparent. Application to the 16
discrete cosine and sine transforms yields a large class of fast algorithms,
many of which have not been found before.Comment: 31 pages, more information at http://www.ece.cmu.edu/~smar
Igusa-type functions associated to finite formed spaces and their functional equations
We study symmetries enjoyed by the polynomials enumerating non-degenerate
flags in finite vector spaces, equipped with a non-degenerate alternating
bilinear, hermitian or quadratic form. To this end we introduce Igusa-type
rational functions encoding these polynomials and prove that they satisfy
certain functional equations. Some of our results are achieved by expressing
the polynomials in question in terms of what we call parabolic length functions
on Coxeter groups of type . While our treatment of the orthogonal case
exploits combinatorial properties of integer compositions and their
refinements, we formulate a precise conjecture how in this situation, too, the
polynomials may be described in terms of parabolic length functions.Comment: Slightly revised version, to appear in Trans. Amer. Math. Soc
On computing Belyi maps
We survey methods to compute three-point branched covers of the projective
line, also known as Belyi maps. These methods include a direct approach,
involving the solution of a system of polynomial equations, as well as complex
analytic methods, modular forms methods, and p-adic methods. Along the way, we
pose several questions and provide numerous examples.Comment: 57 pages, 3 figures, extensive bibliography; English and French
abstract; revised according to referee's suggestion
Entanglement of four-qubit systems: a geometric atlas with polynomial compass II (the tame world)
We propose a new approach to the geometry of the four-qubit entanglement
classes depending on parameters. More precisely, we use invariant theory and
algebraic geometry to describe various stratifications of the Hilbert space by
SLOCC invariant algebraic varieties. The normal forms of the four-qubit
classification of Verstraete {\em et al.} are interpreted as dense subsets of
components of the dual variety of the set of separable states and an algorithm
based on the invariants/covariants of the four-qubit quantum states is proposed
to identify a state with a SLOCC equivalent normal form (up to qubits
permutation).Comment: 49 pages, 16 figure
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