SUPPORT OF MAXIMIZING MEASURES FOR TYPICAL C(0) DYNAMICS ON COMPACT MANIFOLDS

Given a compact manifold X, a continuous function g : X -> IR, and a map T : X -> X, we study properties of the T-invariant Borel probability measures that maximize the integral of g. We show that if X is a n-dimensional connected Riemaniann manifold, with n >= 2, then the set of homeomorphisms for which there is a maximizing measure supported on a periodic orbit is meager. We also show that, if X is the circle, then the ""topological size"" of the set of endomorphisms for which there are g maximizing measures with support on a periodic orbit depends on properties of the function g. In particular, if g is C(1), it has interior points.

Maximizing measures for endomorphisms of the circle

We study a given fixed continuous function phi : S(1) -> R and an endomorphism f : S(1)-> S(1), whose f-invariant probability measures maximize integral phi d mu. We prove that the set of endomorphisms having a f maximizing invariant measure supported on a periodic orbit is C(0) dense.

On maximizing measures of homeomorphisms on compact manifolds

We prove that given a compact n-dimensional connected Riemannian manifold X and a continuous function g : X -> R, there exists a dense subset of the space of homeomorphisms of X such that for all T in this subset, the integral integral(X) g d mu, considered as a function on the space of all T-invariant Borel probability measures mu, attains its maximum on a measure supported on a periodic orbit.

COMPARISON BETWEEN LINEAR AND DAILY UNDULATING PERIODIZED RESISTANCE TRAINING TO INCREASE STRENGTH

Prestes, J, Frollini, AB, De Lima, C, Donatto, FF, Foschini, D, de Marqueti, RC, Figueira Jr, A, and Fleck, SJ. Comparison between linear and daily undulating periodized resistance training to increase strength. J Strength Cond Res 23(9): 2437-2442, 2009-To determine the most effective periodization model for strength and hypertrophy is an important step for strength and conditioning professionals. The aim of this study was to compare the effects of linear (LP) and daily undulating periodized (DUP) resistance training on body composition and maximal strength levels. Forty men aged 21.5 +/- 8.3 and with a minimum 1-year strength training experience were assigned to an LP (n = 20) or DUP group (n = 20). Subjects were tested for maximal strength in bench press, leg press 45 degrees, and arm curl (1 repetition maximum [RM]) at baseline (T1), after 8 weeks (T2), and after 12 weeks of training (T3). Increases of 18.2 and 25.08% in bench press 1 RM were observed for LP and DUP groups in T3 compared with T1, respectively (p <= 0.05). In leg press 45 degrees, LP group exhibited an increase of 24.71% and DUP of 40.61% at T3 compared with T1. Additionally, DUP showed an increase of 12.23% at T2 compared with T1 and 25.48% at T3 compared with T2. For the arm curl exercise, LP group increased 14.15% and DUP 23.53% at T3 when compared with T1. An increase of 20% was also found at T2 when compared with T1, for DUP. Although the DUP group increased strength the most in all exercises, no statistical differences were found between groups. In conclusion, undulating periodized strength training induced higher increases in maximal strength than the linear model in strength-trained men. For maximizing strength increases, daily intensity and volume variations were more effective than weekly variations.

Approximating a class of combinatorial problems with rational objective function

In the late seventies, Megiddo proposed a way to use an algorithm for the problem of minimizing a linear function a(0) + a(1)x(1) + ... + a(n)x(n) subject to certain constraints to solve the problem of minimizing a rational function of the form (a(0) + a(1)x(1) + ... + a(n)x(n))/(b(0) + b(1)x(1) + ... + b(n)x(n)) subject to the same set of constraints, assuming that the denominator is always positive. Using a rather strong assumption, Hashizume et al. extended Megiddo`s result to include approximation algorithms. Their assumption essentially asks for the existence of good approximation algorithms for optimization problems with possibly negative coefficients in the (linear) objective function, which is rather unusual for most combinatorial problems. In this paper, we present an alternative extension of Megiddo`s result for approximations that avoids this issue and applies to a large class of optimization problems. Specifically, we show that, if there is an alpha-approximation for the problem of minimizing a nonnegative linear function subject to constraints satisfying a certain increasing property then there is an alpha-approximation (1 1/alpha-approximation) for the problem of minimizing (maximizing) a nonnegative rational function subject to the same constraints. Our framework applies to covering problems and network design problems, among others.