58,412 research outputs found
On Sequences, Rational Functions and Decomposition
Our overall goal is to unify and extend some results in the literature
related to the approximation of generating functions of finite and infinite
sequences over a field by rational functions. In our approach, numerators play
a significant role. We revisit a theorem of Niederreiter on (i) linear
complexities and (ii) ' minimal polynomials' of an infinite sequence,
proved using partial quotients. We prove (i) and its converse from first
principles and generalise (ii) to rational functions where the denominator need
not have minimal degree. We prove (ii) in two parts: firstly for geometric
sequences and then for sequences with a jump in linear complexity. The basic
idea is to decompose the denominator as a sum of polynomial multiples of two
polynomials of minimal degree; there is a similar decomposition for the
numerators. The decomposition is unique when the denominator has degree at most
the length of the sequence. The proof also applies to rational functions
related to finite sequences, generalising a result of Massey. We give a number
of applications to rational functions associated to sequences.Comment: Several more typos corrected. To appear in J. Applied Algebra in
Engineering, Communication and Computing. The final publication version is
available at Springer via http://dx.doi.org/10.1007/s00200-015-0256-
Minimal Polynomial Algorithms for Finite Sequences
We show that a straightforward rewrite of a known minimal polynomial
algorithm yields a simpler version of a recent algorithm of A. Salagean.Comment: Section 2 added, remarks and references expanded. To appear in IEEE
Transactions on Information Theory
Drift-Diffusion in Mangled Worlds Quantum Mechanics
In Everett's many worlds interpretation, where quantum measurements are seen
as decoherence events, inexact decoherence may let large worlds mangle the
memories of observers in small worlds, creating a cutoff in observable world
size. I solve a growth-drift-diffusion-absorption model of such a mangled
worlds scenario, and show that it reproduces the Born probability rule closely,
though not exactly. Thus deviations from exact decoherence can allow the Born
rule to be derived in a many worlds approach via world counting, using a finite
number of worlds and no new fundamental physics
Vacuum polarization energy of the Shifman-Voloshin soliton
We compute the vacuum polarization energy of soliton configurations in a
model with two scalar fields in one space dimension using spectral methods. The
second field represents an extension of the conventional kink soliton
model. We find that the vacuum polarization energy destabilizes the soliton
except when the fields have identical masses. In that case the model is
equivalent to two independent models.Comment: nine pape
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