John Machacek

Squares in Partial Words

Abstract We investigate the number of positions that do not start a square, the number of square occurrences, and the number of distinct squares in partial words, i.e., sequences that may have undefined positions called holes. We show that the limit of the ratio of the maximum number of positions not starting a square in a binary partial word with h holes over its length n is 15/31 and the limit of the ratio of the minimum number of square occurrences in a binary partial word with h holes over its length n is 103/187, provided the limit of h / n is 0. Both limits turn out to match with the known limits for binary full words (those without holes). We prove another surprising result that the maximal proportion of defined positions that are square-free to the number of defined positions in a binary partial word with h holes of length n is 1/2, provided the limit of h / n is in the interval [ 1 / 11 , 1 ) . We also give a 2 k h tight bound on the number of rightmost occurrences of squares per position in a k -ary partial word with h holes. In addition, we provide a more detailed analysis than earlier ones for the maximum number of distinct squares in a one-hole partial word of length n over an alphabet of size k , bound that is independent of k .

Squares in binary partial words

In this paper, we investigate the number of positions that do not start a square, the number of square occurrences, and the number of distinct squares in binary partial words. Letting σh(n) be the maximum number of positions not starting a square for binary partial words with h holes of length n, we show that limσh(n)/n=15/31 provided the limit of h/n is zero. Letting γh(n) be the minimum number of square occurrences in a binary partial word of length n with h holes, we show, under some condition on h, that limγh(n)/n=103/187. Both limits turn out to match with the known limits for binary full words. We also bound the difference between the maximum number of distinct squares in a binary partial word and that of a binary full word by (2h−1)(n+2), where n is the length and h is the number of holes. This allows us to find a simple proof of the known 3n upper bound in a one-hole binary partial word using the completions of such a partial word.

Log-canonical coordinates for Poisson brackets and rational changes of coordinates

Abstract Goodearl and Launois have shown in Goodearl and Launois (2011) that for a log-canonical Poisson bracket on affine space there is no rational change of coordinates for which the Poisson bracket is constant. Our main result is a proof of a conjecture of Michael Shapiro which states that if affine space is given a log-canonical Poisson bracket, then there does not exist any rational change of coordinates for which the Poisson bracket is linear. Hence, log-canonical coordinates can be thought of as the simplest possible algebraic coordinates for affine space with a log-canonical coordinate system. In proving this conjecture we find certain invariants of log-canonical Poisson brackets on affine space which linear Poisson brackets do not have.

An Interior Point Method for Solving Semidefinite Programs Using Cutting Planes and Weighted Analytic Centers

We investigate solving semidefinite programs (SDPs) with an interior point method called SDP-CUT, which utilizes weighted analytic centers and cutting plane constraints. SDP-CUT iteratively refines the feasible region to achieve the optimal solution. The algorithm uses Newton’s method to compute the weighted analytic center. We investigate different stepsize determining techniques. We found that using Newtons method with exact line search is generally the best implementation of the algorithm. We have also compared our algorithm to the SDPT3 method and found that SDP-CUT initially gets into the neighborhood of the optimal solution in less iterations on all our test problems. SDP-CUT also took less iterations to reach optimality on many of the problems. However, SDPT3 required less iterations on most of the test problems and less time on all the problems. Some theoretical properties of the convergence of SDP-CUT are also discussed.

Deleting Powers in Words.

We consider the language consisting of all words such that it is possible to obtain the empty word by iteratively deleting powers. It turns out that in the case of deleting squares in binary words this language is regular, and in the case of deleting squares in words over a larger alphabet the language is not regular. However, for deleting squares over any alphabet we find that this language can be generated by a linear index grammar which is a mildly context sensitive grammar formalism. In the general case we show that this language is generated by an indexed grammar.