9,886 research outputs found
Upper Bounds on the Length of Minimal Solutions to Certain Quadratic Word Equations
It is a long standing conjecture that the problem of deciding whether a quadratic word equation has a solution is in NP. It has also been conjectured that the length of a minimal solution to a quadratic equation is at most exponential in the length of the equation, with the latter conjecture implying the former. We show that both conjectures hold for some natural subclasses of quadratic equations, namely the classes of regular-reversed, k-ordered, and variable-sparse quadratic equations. We also discuss a connection of our techniques to the topic of unavoidable patterns, and the possibility of exploiting this connection to produce further similar results
The Hardness of Solving Simple Word Equations
We investigate the class of regular-ordered word equations. In such equations, each variable occurs at most once in each side and the order of the variables occurring in both left and right hand sides is preserved (the variables can be, however, separated by potentially distinct constant factors). Surprisingly, we obtain that solving such simple equations, even when the sides contain exactly the same variables, is NP-hard. By considerations regarding the combinatorial structure of the minimal solutions of the more general quadratic equations we obtain that the satisfiability problem for regular-ordered equations is in NP. The complexity of solving such word equations under regular constraints is also settled. Finally, we show that a related class of simple word equations, that generalises one-variable equations, is in P
A lower bound for Garsia's entropy for certain Bernoulli convolutions
Let be a Pisot number and let denote Garsia's
entropy for the Bernoulli convolution associated with . Garsia, in 1963
showed that for any Pisot . For the Pisot numbers which
satisfy (with ) Garsia's entropy has been
evaluated with high precision by Alexander and Zagier and later improved by
Grabner, Kirschenhofer and Tichy, and it proves to be close to 1. No other
numerical values for are known.
In the present paper we show that for all Pisot , and
improve this lower bound for certain ranges of . Our method is
computational in nature.Comment: 16 pages, 4 figure
Sub-quadratic Decoding of One-point Hermitian Codes
We present the first two sub-quadratic complexity decoding algorithms for
one-point Hermitian codes. The first is based on a fast realisation of the
Guruswami-Sudan algorithm by using state-of-the-art algorithms from computer
algebra for polynomial-ring matrix minimisation. The second is a Power decoding
algorithm: an extension of classical key equation decoding which gives a
probabilistic decoding algorithm up to the Sudan radius. We show how the
resulting key equations can be solved by the same methods from computer
algebra, yielding similar asymptotic complexities.Comment: New version includes simulation results, improves some complexity
results, as well as a number of reviewer corrections. 20 page
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