Computing the Least Fixed Point of Positive Polynomial Systems

We consider equation systems of the form X_1 = f_1(X_1, ..., X_n), ..., X_n = f_n(X_1, ..., X_n) where f_1, ..., f_n are polynomials with positive real coefficients. In vector form we denote such an equation system by X = f(X) and call f a system of positive polynomials, short SPP. Equation systems of this kind appear naturally in the analysis of stochastic models like stochastic context-free grammars (with numerous applications to natural language processing and computational biology), probabilistic programs with procedures, web-surfing models with back buttons, and branching processes. The least nonnegative solution mu f of an SPP equation X = f(X) is of central interest for these models. Etessami and Yannakakis have suggested a particular version of Newton's method to approximate mu f. We extend a result of Etessami and Yannakakis and show that Newton's method starting at 0 always converges to mu f. We obtain lower bounds on the convergence speed of the method. For so-called strongly connected SPPs we prove the existence of a threshold k_f such that for every i >= 0 the (k_f+i)-th iteration of Newton's method has at least i valid bits of mu f. The proof yields an explicit bound for k_f depending only on syntactic parameters of f. We further show that for arbitrary SPP equations Newton's method still converges linearly: there are k_f>=0 and alpha_f>0 such that for every i>=0 the (k_f+alpha_f i)-th iteration of Newton's method has at least i valid bits of mu f. The proof yields an explicit bound for alpha_f; the bound is exponential in the number of equations, but we also show that it is essentially optimal. Constructing a bound for k_f is still an open problem. Finally, we also provide a geometric interpretation of Newton's method for SPPs.Comment: This is a technical report that goes along with an article to appear in SIAM Journal on Computing

Convergence Thresholds of Newton's Method for Monotone Polynomial Equations

Monotone systems of polynomial equations (MSPEs) are systems of fixed-point equations $X_1 = f_1(X_1, ..., X_n),$ $..., X_n = f_n(X_1, ..., X_n)$ where each $f_i$ is a polynomial with positive real coefficients. The question of computing the least non-negative solution of a given MSPE $\vec X = \vec f(\vec X)$ arises naturally in the analysis of stochastic models such as stochastic context-free grammars, probabilistic pushdown automata, and back-button processes. Etessami and Yannakakis have recently adapted Newton's iterative method to MSPEs. In a previous paper we have proved the existence of a threshold $k_{\vec f}$ for strongly connected MSPEs, such that after $k_{\vec f}$ iterations of Newton's method each new iteration computes at least 1 new bit of the solution. However, the proof was purely existential. In this paper we give an upper bound for $k_{\vec f}$ as a function of the minimal component of the least fixed-point $\mu\vec f$ of $\vec f(\vec X)$. Using this result we show that $k_{\vec f}$ is at most single exponential resp. linear for strongly connected MSPEs derived from probabilistic pushdown automata resp. from back-button processes. Further, we prove the existence of a threshold for arbitrary MSPEs after which each new iteration computes at least $1/w2^h$ new bits of the solution, where $w$ and $h$ are the width and height of the DAG of strongly connected components.Comment: version 2 deposited February 29, after the end of the STACS conference. Two minor mistakes correcte

Bisimilarity of Pushdown Systems is Nonelementary

Given two pushdown systems, the bisimilarity problem asks whether they are bisimilar. While this problem is known to be decidable our main result states that it is nonelementary, improving EXPTIME-hardness, which was the previously best known lower bound for this problem. Our lower bound result holds for normed pushdown systems as well

Tracking Pacific bluefin tuna (Thunnus thynnus orientalis) in the northeastern Pacific with an automated algorithm that estimates latitude by matching sea-surface-temperature data from satellites with temperature data from tags on fish

Data recovered from 11 popup satellite archival tags and 3 surgically implanted archival tags were used to analyze the movement patterns of juvenile northern bluefin tuna (Thunnus thynnus orientalis) in the eastern Pacific. The light sensors on archival and pop-up satellite transmitting archival tags (PSATs) provide data on the time of sunrise and sunset, allowing the calculation of an approximate geographic position of the animal. Light-based estimates of longitude are relatively robust but latitude estimates are prone to large degrees of error, particularly near the times of the equinoxes and when the tag is at low latitudes. Estimating latitude remains a problem for researchers using light-based geolocation algorithms and it has been suggested that sea surface temperature data from satellites may be a useful tool for refining latitude estimates. Tag data from bluefin tuna were subjected to a newly developed algorithm, called “PSAT Tracker,” which automatically matches sea surface temperature data from the tags with sea surface temperatures recorded by satellites. The results of this algorithm compared favorably to the estimates of latitude calculated with the lightbased algorithms and allowed for estimation of fish positions during times of the year when the lightbased algorithms failed. Three near one-year tracks produced by PSAT tracker showed that the fish range from the California−Oregon border to southern Baja California, Mexico, and that the majority of time is spent off the coast of central Baja Mexico. A seasonal movement pattern was evident; the fish spend winter and spring off central Baja California, and summer through fall is spent moving northward to Oregon and returning to Baja California

Exploring Touch Communication Between Coaches and Athletes

In athletics, coaches and athletes share a unique and important relationship. Recently Jowett and her colleagues (Jowett & Cockerill, 2003; Jowett & Meek, 2000; Jowett & Ntoumanis, 2003, 2004; Jowett & Timson-Katchis, 2005) utilized relationship research (focusing on, for example, marital, familial and workplace relationships) from conjoining fields, and in particular social and cognitive psychology, to develop and test a four-component model (4 C’s) that depicts the most influential relational and emotional components (closeness, commitment, complementarity and co-orientation) of coach-athlete relationships. Proceeding from a review of the literature on human touch communication to examine research on the power of touch to exchange relational and emotional messages (Hertenstein et al., 2006), the present study explores coaches’ and athletes’ collective experiences of communicating via touch, utilizing in-depth interviews with eight college coaches and athletes. A phenomenological approach was used to gather, analyze and interpret the data, drawing on Merleau-Ponty’s (1945/1962) philosophical exploration of perception and human experience, which emphasizes the body as a means of communicating with the world. The findings indicate that touch between coaches and athletes increased at major events when emotions and tensions ran high. In addition, touch involved showing appreciation, instructing, comforting and giving attention, and affected perceptions of relationships. The findings also show that touch communication is influenced by societal factors, such as gender, relational stage, and what spectators, parents and other athletes may think. By illustrating how touch is enacted and experienced by a group of college coaches and athletes, the study represents an initial step toward understanding touch communication in the coach-athlete dyad. Indo-Pacific Journal of Phenomenology, Volume 7, Edition 2 September 200

The First Detection of \u3ci\u3eCeratophyllus\u3c/i\u3e Fleas and an Ischnocera Louse on the Great Cormorant \u3ci\u3ePhalacrocorax carbo\u3c/i\u3e in Mongolia

There are summarized data on ectoparasites of Mongolian birds. The Mongolian-German Biological Expeditions found first records for the flea Ceratophyllus vagabundus and the Ischnocera louse Pectinopygus gyricornis at the host Phalacrocorax carbo