Critical power as an endurance index

The relationship between exhaustion time (t(lim)) and the work performed at the end of constant-power exercises can be described by a linear relationship (Wlim = a + b t(lim)) for work involving the whole body (eg cycling) or part of the body (eg knee extensions). The slope b in the equation is termed the critical power and has been proposed as an index of the capacity to perform work over a long period of time. The first objective of the present study was to compare the values of slopes b calculated from whole-body work of short duration, ie maximal and supra-maximal cycling exercises (slope b1), with the values calculated from the same work, the durations of which were between 3.5 and 35 min (slope b3), as in the protocols used by Scherrer and Monod (1960) for body-part work. Slope b1 was significantly higher than slope b3 in 10 subjects who performed 5 cycling exhausting exercises (60, 73, 86, 100 and 120% of maximal aerobic power (MAP) in watts). Exhaustion times corresponding to power outputs equivalent to b1 and b3 were equal to 29.0 +/- 19.1 min and 48.6 +/- 9.8 min respectively. Moreover, the exhaustion times at 60 and 73%,MAP were significantly correlated with slope b3 (expressed in %MAP) but not with slope b1. Consequently, slope b3 should be considered as the critical power instead of slope b1 as in some studies in the literature (Moritani et al, 1981). The second objective was to study the physiological significance of the critical power (slope b3) of whole-body work (cycling). The workload that corresponded to a lactate steady state was not significantly different from b3 (68.8 +/- 6.0 vs 68.7 +/- 6.3% MAP). Nevertheless, slope b3 represents a workload corresponding to a slight but significant drift of heart rate or oxygen uptake. These results probably explain why b3 is a power which can be maintained for a long time but not beyond about l h in an average subject.     

Nested multigrid methods for time-periodic,parabolic optimal control problems

We present a nested multigrid method to optimize time-periodic, parabolic, partial differential equations (PDE). We consider a quadratic tracking objective with a linear parabolic PDE constraint. The first order optimality conditions, given by a coupled system of boundary value problems can be rewritten as an Fredholm integral equation of the second kind, which is solved by a multigrid of the second kind. The evaluation of the integral operator consists of solving sequentially a boundary value problem for respectively the state and the adjoints. Both problems are solved efficiently by a time-periodic space-time multigrid method.     

Benchmarking Least Squares Support Vector Machine Classifiers

In Support Vector Machines (SVMs), the solution of the classification problem is characterized by a (convex) quadratic programming (QP) problem. In a modified version of SVMs, called Least Squares SVM classifiers (LS-SVMs), a least squares cost function is proposed so as to obtain a linear set of equations in the dual space. While the SVM classifier has a large margin interpretation, the LS-SVM formulation is related in this paper to a ridge regression approach for classification with binary targets and to Fisher's linear discriminant analysis in the feature space. Multiclass categorization problems are represented by a set of binary classifiers using different output coding schemes. While regularization is used to control the effective number of parameters of the LS-SVM classifier, the sparseness property of SVMs is lost due to the choice of the 2-norm. Sparseness can be imposed in a second stage by gradually pruning the support value spectrum and optimizing the hyperparameters during the sparse approximation procedure. In this paper, twenty public domain benchmark datasets are used to evaluate the test set performance of LS-SVM classifiers with linear, polynomial and radial basis function (RBF) kernels. Both the SVM and LS-SVM classifier with RBF kernel in combination with standard cross-validation procedures for hyperparameter selection achieve comparable test set performances. These SVM and LS-SVM performances are consistently very good when compared to a variety of methods described in the literature including decision tree based algorithms, statistical algorithms and instance based learning methods. We show on ten UCI datasets that the LS-SVM sparse approximation procedure can be successfully applied.     

Multigrid methods with space–time concurrency

Computing and Visualization in Science - We consider the comparison of multigrid methods for parabolic partial differential equations that allow space–time concurrency. With current trends in...     

Selecting creep models using Bayesian methods

The development of creep prediction models has been a field of extensive research and many different models have already been proposed. This paper presents an evaluation method of the prediction quality of creep models for specific experimental data. Within the scope of this paper, the model according to Bockhold and the model according to Heidolf are examined. First, the parameters of the models are identified with respect to existing experimental data. This is done using a sampling based approach of Bayesian updating developed by Ba?ant and Chern. In extension to the method by Ba?ant and Chern, the uncertainty coming from inaccurate measurement data is taken into account in the definition of the likelihood function within the updating algorithm. The more inaccurate the measurements are, the more uncertain the estimated model parameters and model prognoses become. The identification is performed for different short- and long-term creep tests. The intension is not to validate these models intensively, but to evaluate their prognoses for the individually tested creep behavior. The results show that the identifiability of the models?? parameters is different for both models and consequently the models prognoses differ in their uncertainties. Second, the models are evaluated using two different strategies: the stochastic model selection according to MacKay, Beck and Yuen based on the Ockham factor, and a comparison of the uncertainties taking into account parameter and model uncertainties. The results of the evaluation of the creep models differ for various experimental tests. Model Heidolf is more flexible and gives a better fit to the data, however, it fails to predict reliable long-term creep deformations using only short-term measurements compared to model Bockhold. Comparing the evaluation methods, the analysis of uncertainties of the creep prognosis proofs to be more stable than the evaluation using the stochastic model selection.     

Efficient computer analysis of ideal switched-capacitor circuits using matrix compaction techniques

In this paper we describe how three simple observations can be used in order to obtain an efficient algorithm for the computer analysis of ideal switched-capacitor circuits. the resulting algorithm is linear in the number of phases. the first observation uses the structure of the r-domain MNA-matrix to come up with a new LU-decomposition scheme which is gradual per timeslot. the second observation allows a great reduction in size of the z-domain MNA-matrix by a matrix compaction algorithm which also operates gradually per timeslot and which can be interleaved with the first gradual LU-decomposition process. This leads to a small matrix which can then be used for a time- and direct frequency-domain analysis. Third, the computations of transfer functions, aliasing functions and sensitivities can be optimized by applying appropriate excitations and making appropriate combinations of the terms in the expressions. These algorithms have been implemented in the SC-analysis program DIANA.SC. the usefulness and efficiency of the program is then illustrated with some examples.     

Bayesian framework for least-squares support vector machine classifiers, gaussian processes, and kernel Fisher discriminant analysis

The Bayesian evidence framework has been successfully applied to the design of multilayer perceptrons (MLPs) in the work of MacKay. Nevertheless, the training of MLPs suffers from drawbacks like the nonconvex optimization problem and the choice of the number of hidden units. In support vector machines (SVMs) for classification, as introduced by Vapnik, a nonlinear decision boundary is obtained by mapping the input vector first in a nonlinear way to a high-dimensional kernel-induced feature space in which a linear large margin classifier is constructed. Practical expressions are formulated in the dual space in terms of the related kernel function, and the solution follows from a (convex) quadratic programming (QP) problem. In least-squares SVMs (LS-SVMs), the SVM problem formulation is modified by introducing a least-squares cost function and equality instead of inequality constraints, and the solution follows from a linear system in the dual space. Implicitly, the least-squares formulation corresponds to a regression formulation and is also related to kernel Fisher discriminant analysis. The least-squares regression formulation has advantages for deriving analytic expressions in a Bayesian evidence framework, in contrast to the classification formulations used, for example, in gaussian processes (GPs). The LS-SVM formulation has clear primal-dual interpretations, and without the bias term, one explicitly constructs a model that yields the same expressions as have been obtained with GPs for regression. In this article, the Bayesian evidence framework is combined with the LS-SVM classifier formulation. Starting from the feature space formulation, analytic expressions are obtained in the dual space on the different levels of Bayesian inference, while posterior class probabilities are obtained by marginalizing over the model parameters. Empirical results obtained on 10 public domain data sets show that the LS-SVM classifier designed within the Bayesian evidence framework consistently yields good generalization performances.     

Ratings of perceived exertion (RPE) during cycling exercises at constant power output

The purpose of the present investigation was to study the overall rating of perceived exertion (RPEov) according to the 6-20 scale proposed by Borg (1970) and muscular RPE (RPEmu) in exercises at constant load. The relationship between RPE and heart rate for three different loads was studied during exhausting exercises in 10 participants. Whether the drift of RPE during a 20 min exercise at constant load could be an index of the endurance time during long-lasting exercises at constant load was also investigated. At 1-week intervals, the participants performed cycling exercises up to exhaustion at 60, 73, and 86% maximal aerobic power (MAP) measured during an incremental test. Heart rate, RPEov, RPEmu and exhaustion time (tlim) were measured. The upward shift of the HRmax-RPE regressions was significant between 86, 73 and 60% MAP (p < 0.001) for RPEov and RPEmu. This result suggests that the equation HR = 10 x RPE proposed by Borg (1973) for incremental exercise is not valid for long-lasting exercise at constant load until exhaustion because the heart rate corresponding to a given RPE depends on load and time. Mean RPE increased linearly with time up to exhaustion. Unexpectedly, the relationships between RPEmu or RPEov and percentage of exhaustion time were similar for exercises at 60 and 73% MAP although the exhaustion times were very different (79.40 +/- 30.64 min versus 36.19 +/- 15.99 min, respectively) (p < 0.001). Consequently, it is likely that RPE was a subjective estimation of the hardness of exercise rather than the intensity of exercise. The RPE pattern at the beginning of long-lasting exercises at constant load (60 and 73% MAP) cannot be considered as a sensitive predictor of the point of self-imposed exhaustion for individuals. Indeed, the errors in the estimation of exhaustion time from extrapolation of RPE at the beginning of exercise were very large. Moreover, at 60% MAP, a steady-state in RPE was observed during 20 min in five subjects whose tlim were not longer than tlim of the other subjects. In addition, the data of the present study indicate that RPEmu could be more useful than RPEov in cycling.     

A unified theory of symmetry for nonlinear multiport and multiterminal resistors

Simple group-theoretic concepts are used to develop a rigorous and comprehensive theory of symmetry for nonlinear multiport and multiterminal resistors which do not rely on geometrical arguments or other ad hoc techniques normally invoked in such studies. This theory unifies all forms of symmetry, including rotation, reflection and complementary symmetry, into a single framework. It also includes all known nonlinear symmetry principles as special cases. Moreover, a general method for identifying all symmetry characteristics possessed by a nonlinear multiport and multiterminal element is given. The main results of this paper:

• 1 Several algorithms for synthesizing a nonlinear multiport or multiterminal element having any prescribed form of symmetry are presented. In particular, various examples are given which illustrate how these algorithms can be used to derive well-known symmetrical nonlinear circuit modules such as push-pull amplifiers, complementary-symmetric amplifiers, rectifiers, modulators, etc.;
• 2 A reduction algorithm is presented which allows a complicated symmetric element to be analysed by a much simpler reduced element;
• 3 A general principle is derived for applying symmetry to achieve frequency separation in nonlinear communication circuits where the even harmonic components are separated from the odd harmonic components.