130,253 research outputs found
Generating and Searching Families of FFT Algorithms
A fundamental question of longstanding theoretical interest is to prove the
lowest exact count of real additions and multiplications required to compute a
power-of-two discrete Fourier transform (DFT). For 35 years the split-radix
algorithm held the record by requiring just 4n log n - 6n + 8 arithmetic
operations on real numbers for a size-n DFT, and was widely believed to be the
best possible. Recent work by Van Buskirk et al. demonstrated improvements to
the split-radix operation count by using multiplier coefficients or "twiddle
factors" that are not n-th roots of unity for a size-n DFT. This paper presents
a Boolean Satisfiability-based proof of the lowest operation count for certain
classes of DFT algorithms. First, we present a novel way to choose new yet
valid twiddle factors for the nodes in flowgraphs generated by common
power-of-two fast Fourier transform algorithms, FFTs. With this new technique,
we can generate a large family of FFTs realizable by a fixed flowgraph. This
solution space of FFTs is cast as a Boolean Satisfiability problem, and a
modern Satisfiability Modulo Theory solver is applied to search for FFTs
requiring the fewest arithmetic operations. Surprisingly, we find that there
are FFTs requiring fewer operations than the split-radix even when all twiddle
factors are n-th roots of unity.Comment: Preprint submitted on March 28, 2011, to the Journal on
Satisfiability, Boolean Modeling and Computatio
A bit of tropical geometry
This friendly introduction to tropical geometry is meant to be accessible to
first year students in mathematics. The topics discussed here are basic
tropical algebra, tropical plane curves, some tropical intersections, and
Viro's patchworking. Each definition is explained with concrete examples and
illustrations. To a great exten, this text is an updated of a translation from
a french text by the first author. There is also a newly added section
highlighting new developments and perspectives on tropical geometry. In
addition, the final section provides an extensive list of references on the
subject.Comment: 27 pages, 19 figure
Anomalies and Invertible Field Theories
We give a modern geometric viewpoint on anomalies in quantum field theory and
illustrate it in a 1-dimensional theory: supersymmetric quantum mechanics. This
is background for the resolution of worldsheet anomalies in orientifold
superstring theory.Comment: 21 pages, based talk at String-Math 2013; small corrections in v
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