7,639 research outputs found
One-way permutations, computational asymmetry and distortion
Computational asymmetry, i.e., the discrepancy between the complexity of
transformations and the complexity of their inverses, is at the core of one-way
transformations. We introduce a computational asymmetry function that measures
the amount of one-wayness of permutations. We also introduce the word-length
asymmetry function for groups, which is an algebraic analogue of computational
asymmetry. We relate boolean circuits to words in a Thompson monoid, over a
fixed generating set, in such a way that circuit size is equal to word-length.
Moreover, boolean circuits have a representation in terms of elements of a
Thompson group, in such a way that circuit size is polynomially equivalent to
word-length. We show that circuits built with gates that are not constrained to
have fixed-length inputs and outputs, are at most quadratically more compact
than circuits built from traditional gates (with fixed-length inputs and
outputs). Finally, we show that the computational asymmetry function is closely
related to certain distortion functions: The computational asymmetry function
is polynomially equivalent to the distortion of the path length in Schreier
graphs of certain Thompson groups, compared to the path length in Cayley graphs
of certain Thompson monoids. We also show that the results of Razborov and
others on monotone circuit complexity lead to exponential lower bounds on
certain distortions.Comment: 33 page
Bernoulli measure on strings, and Thompson-Higman monoids
The Bernoulli measure on strings is used to define height functions for the
dense R- and L-orders of the Thompson-Higman monoids M_{k,1}. The measure can
also be used to characterize the D-relation of certain submonoids of M_{k,1}.
The computational complexity of computing the Bernoulli measure of certain
sets, and in particular, of computing the R- and L-height of an element of
M_{k,1} is investigated.Comment: 27 pages
Deflagration to detonation transition by amplification of acoustic waves in type Ia supernovae
We study a new mechanism for deflagration to detonation transition in
thermonuclear supernovae (SNe Ia), based on the formation of shocks by
amplification of sound waves in the steep density gradients of white dwarfs
envelopes. Given a large enough jump in density a small pressure and velocity
perturbation, produced by the turbulent deflagration, turns into a shock down
of the gradient, where it will dissipate and heat up the media. With the right
frequency and amplitude the heating can be enough to initiate a detonation,
which can propagate backward and up the density gradient. We studied planar and
spherical geometry. In the planar case we made a parametric study of the
frequency and amplitude. We found it possible to obtain a detonation for
perturbations down to Mach number M=0.003. In the spherical case, geometrical
damping makes it harder to initiate a detonation, but considering a small He
atmosphere (<0.01 Msol) makes it possible again to obtain a detonation down to
small perturbation (M=0.002). In the context of thermonuclear supernovae, this
could be a mean to turn a turbulent flame producing sound waves to a
detonation.Comment: accepted for publication in Astronomy and Astrophysic
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