35,988 research outputs found
Composite Cyclotomic Fourier Transforms with Reduced Complexities
Discrete Fourier transforms~(DFTs) over finite fields have widespread
applications in digital communication and storage systems. Hence, reducing the
computational complexities of DFTs is of great significance. Recently proposed
cyclotomic fast Fourier transforms (CFFTs) are promising due to their low
multiplicative complexities. Unfortunately, there are two issues with CFFTs:
(1) they rely on efficient short cyclic convolution algorithms, which has not
been investigated thoroughly yet, and (2) they have very high additive
complexities when directly implemented. In this paper, we address both issues.
One of the main contributions of this paper is efficient bilinear 11-point
cyclic convolution algorithms, which allow us to construct CFFTs over
GF. The other main contribution of this paper is that we propose
composite cyclotomic Fourier transforms (CCFTs). In comparison to previously
proposed fast Fourier transforms, our CCFTs achieve lower overall complexities
for moderate to long lengths, and the improvement significantly increases as
the length grows. Our 2047-point and 4095-point CCFTs are also first efficient
DFTs of such lengths to the best of our knowledge. Finally, our CCFTs are also
advantageous for hardware implementations due to their regular and modular
structure.Comment: submitted to IEEE trans on Signal Processin
Modular SIMD arithmetic in Mathemagix
Modular integer arithmetic occurs in many algorithms for computer algebra,
cryptography, and error correcting codes. Although recent microprocessors
typically offer a wide range of highly optimized arithmetic functions, modular
integer operations still require dedicated implementations. In this article, we
survey existing algorithms for modular integer arithmetic, and present detailed
vectorized counterparts. We also present several applications, such as fast
modular Fourier transforms and multiplication of integer polynomials and
matrices. The vectorized algorithms have been implemented in C++ inside the
free computer algebra and analysis system Mathemagix. The performance of our
implementation is illustrated by various benchmarks
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Modular pipeline fast Fourier transform
A modular pipeline algorithm and architecture for computing discrete Fourier transforms is described. For an N point transform, two pipeline √{square root over (N)} point fast Fourier transform (FFT) modules are combined with a center element. The center element contains memories, multipliers and control logic. Compared with standard N point pipeline FFTs, the modular pipeline FFT maintains the bandwidth of existing pipeline FFTs with reduced dynamic power consumption and reduced complexity of the overall hardware pipeline.Board of Regents, University of Texas Syste
A multi-variable version of the completed Riemann zeta function and other -functions
We define a generalisation of the completed Riemann zeta function in several
complex variables. It satisfies a functional equation, shuffle product
identities, and has simple poles along finitely many hyperplanes, with a
recursive structure on its residues. The special case of two variables can be
written as a partial Mellin transform of a real analytic Eisenstein series,
which enables us to relate its values at pairs of positive even points to
periods of (simple extensions of symmetric powers of the cohomology of) the CM
elliptic curve corresponding to the Gaussian integers. In general, the totally
even values of these functions are related to new quantities which we call
multiple quadratic sums.
More generally, we cautiously define multiple-variable versions of motivic
-functions and ask whether there is a relation between their special values
and periods of general mixed motives. We show that all periods of mixed Tate
motives over the integers, and all periods of motivic fundamental groups (or
relative completions) of modular groups, are indeed special values of the
multiple motivic -values defined here.Comment: This is the second half of a talk given in honour of Ihara's 80th
birthday, and will appear in the proceedings thereo
On Infrared Universality in Massive Theories. Another Example
The infrared behaviour of the -theory is discussed stressing
analogies with the Witten-Seiberg story about . Though the
microscopic theory is apparently not integrable, the effective theory is shown
to be integrable at classical level, and a general solution of it in terms of
hypergeometric functions is obtained. An effective theory for the multiparticle
soft scattering is sketched.Comment: 9 pages, Latex, ps is available at
http://venus.itep.ru/preprints/1996/96016.ps.g
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