35,293 research outputs found
Decimation-in-Frequency Fast Fourier Transforms for the Symmetric Group
In this thesis, we present a new class of algorithms that determine fast Fourier transforms for a given finite group G. These algorithms use eigenspace projections determined by a chain of subgroups of G, and rely on a path algebraic approach to the representation theory of finite groups developed by Ram (26). Applying this framework to the symmetric group, Sn, yields a class of fast Fourier transforms that we conjecture to run in O(n2n!) time. We also discuss several future directions for this research
Multidimensional CooleyâTukey Algorithms Revisited
AbstractThe representation theory of Abelian groups is used to obtain an algebraic divide-and-conquer algorithm for computing the finite Fourier transform. The algorithm computes the Fourier transform of a finite Abelian group in terms of the Fourier transforms of an arbitrary subgroup and its quotient. From this algebraic algorithm a procedure is derived for obtaining concrete factorizations of the Fourier transform matrix in terms of smaller Fourier transform matrices, diagonal multiplications, and permutations. For cyclic groups this gives as special cases the CooleyâTukey and GoodâThomas algorithms. For groups with several generators, the procedure gives a variety of multidimensional CooleyâTukey type algorithms. This method of designing multidimensional fast Fourier transform algorithms gives different data flow patterns from the standard ârowâcolumnâ approaches. We present some experimental evidence that suggests that in hierarchical memory environments these data flows are more efficient
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
Fourier Mukai Transforms and Applications to String Theory
We give an introductory review of Fourier-Mukai transforms and their
application to various aspects of moduli problems, string theory and mirror
symmetry. We develop the necessary mathematical background for Fourier-Mukai
transforms such as aspects of derived categories and integral functors as well
as their relative version which becomes important for making precise the notion
of fiberwise T-duality on elliptic Calabi-Yau threefolds. We discuss various
applications of the Fourier-Mukai transform to D-branes on Calabi-Yau manifolds
as well as homological mirror symmetry and the construction of vector bundles
for heterotic string theory.Comment: 52 pages. To appear in Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A
Mat. Minor changes, reference of conjecture in section 7.5 changed,
references update
Algebraic Signal Processing Theory: Cooley-Tukey Type Algorithms for Polynomial Transforms Based on Induction
A polynomial transform is the multiplication of an input vector x\in\C^n by
a matrix \PT_{b,\alpha}\in\C^{n\times n}, whose -th element is
defined as for polynomials p_\ell(x)\in\C[x] from a list
and sample points \alpha_k\in\C from a list
. Such transforms find applications in
the areas of signal processing, data compression, and function interpolation.
Important examples include the discrete Fourier and cosine transforms. In this
paper we introduce a novel technique to derive fast algorithms for polynomial
transforms. The technique uses the relationship between polynomial transforms
and the representation theory of polynomial algebras. Specifically, we derive
algorithms by decomposing the regular modules of these algebras as a stepwise
induction. As an application, we derive novel general-radix
algorithms for the discrete Fourier transform and the discrete cosine transform
of type 4.Comment: 19 pages. Submitted to SIAM Journal on Matrix Analysis and
Application
Fast Erasure-and-Error Decoding and Systematic Encoding of a Class of Affine Variety Codes
In this paper, a lemma in algebraic coding theory is established, which is
frequently appeared in the encoding and decoding for algebraic codes such as
Reed-Solomon codes and algebraic geometry codes. This lemma states that two
vector spaces, one corresponds to information symbols and the other is indexed
by the support of Grobner basis, are canonically isomorphic, and moreover, the
isomorphism is given by the extension through linear feedback shift registers
from Grobner basis and discrete Fourier transforms. Next, the lemma is applied
to fast unified system of encoding and decoding erasures and errors in a
certain class of affine variety codes.Comment: 6 pages, 2 columns, presented at The 34th Symposium on Information
Theory and Its Applications (SITA2011
Conservative descent for semi-orthogonal decompositions
Motivated by the local flavor of several well-known semi-orthogonal
decompositions in algebraic geometry, we introduce a technique called
conservative descent, which shows that it is enough to establish these
decompositions locally. The decompositions we have in mind are those for
projectivized vector bundles and blow-ups, due to Orlov, and root stacks, due
to Ishii and Ueda. Our technique simplifies the proofs of these decompositions
and establishes them in greater generality for arbitrary algebraic stacks.Comment: Final versio
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