Analytical and computational estimation of patellofemoral forces in the knee under squatting and isometric motion

This study presents an intermediate step in prosthesis design, by introducing a newly developed two-dimensional mathematical, and a three-dimensional computational knee model. The analytical model is derived from Newton’s law with respect to the equilibrium equations, thus based on theoretical assumptions, and experimentally obtained parameter. The numeric model is built from an existing prosthesis, involving three parts as patella, femur and tibia, and currently it is under development. The models are capable to predict – with their standard deviation – the patellofemoral (numerically tibiofemoral as well) forces in the knee joint during squatting motion. The reason why the squatting is investigated is due to its relative simplicity and the fact, that during the movement the forces reach extremity in the knee joint. The obtained forces – as a function of flexion angle – are used firstly as fundaments to the knee design method, and secondly to extend the results related to the existing isometric kinetics, where one of the newly obtained functions appears as an essential – and so far missing – input function. Most results are compared and validated to the ones found in the relevant literature and put into a dimensionless form in order to have more general meaning

Influence of Surface Conditions on Wear and Friction of Hardmetals

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On the transitivity of the comonotonic and countermonotonic comparison of random variables

AbstractA recently proposed method for the pairwise comparison of arbitrary independent random variables results in a probabilistic relation. When restricted to discrete random variables uniformly distributed on finite multisets of numbers, this probabilistic relation expresses the winning probabilities between pairs of hypothetical dice that carry these numbers and exhibits a particular type of transitivity called dice-transitivity. In case these multisets have equal cardinality, two alternative methods for statistically comparing the ordered lists of the numbers on the faces of the dice have been studied recently: the comonotonic method based upon the comparison of the numbers of the same rank when the lists are in increasing order, and the countermonotonic method, also based upon the comparison of only numbers of the same rank but with the lists in opposite order. In terms of the discrete random variables associated to these lists, these methods each turn out to be related to a particular copula that joins the marginal cumulative distribution functions into a bivariate cumulative distribution function. The transitivity of the generated probabilistic relation has been completely characterized. In this paper, the list comparison methods are generalized for the purpose of comparing arbitrary random variables. The transitivity properties derived in the case of discrete uniform random variables are shown to be generic. Additionally, it is shown that for a collection of normal random variables, both comparison methods lead to a probabilistic relation that is at least moderately stochastic transitive

Characterizations of quasitrivial symmetric nondecreasing associative operations

We provide a description of the class of n-ary operations on an arbitrary chain that are quasitrivial, symmetric, nondecreasing, and associative. We also prove that associativity can be replaced with bisymmetry in the definition of this class. Finally we investigate the special situation where the chain is finite