Knots And Crosses Epub 12

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The eight knots in this section are the most basic knots - the building blocks of knot tying. They illustrate the fundamental principles of knot tying. Many are also components of other knots or they provide the underlying structure. The Square Knot (Reef Knot) and Sheet Bend are the two basic methods of joining two ropes; and the Figure 8 underlies many other important knots.

The terms Overhand Knot, Half Hitch, and Half Knot are often confused and frequently used as though they are interchangeable. Similarly with the Slip Knot and Noose. Their importance and their differences are explained for these five knots and cross-links are provided with each animation to facilitate quick comparison.

7 Figure 7:The coronal inversion recovery-weighted image visualizes the skin marker directly over the intersection of the extensor pollicis longus (EPL) and extensor carpi radialis brevis (ECRB). Tenosynovitis is again seen, with fluid intensity signal distending the paratenons and internal synovial hypertrophy. Note the angulation of the extensor pollicis longus tendon as it crosses Lister's tubercle (asterisk), which acts as a mechanical pulley for the tendon.

A sagittal fat suppressed fast spin-echo T2-weighted image through the midfoot reveals peritendinous edema and fluid around the flexor hallucis longus at the level of the navicular and medial cuneiform. In this region, the flexor hallucis longus tendon (FHL) crosses deep to the adjacent flexor digitorum longus tendon (FDL). This intersection point is called the master knot of Henry.

The flexor hallucis longus (FHL) muscle originates from the lower fibula and interosseous membrane, lateral to the flexor digitorum longus (FDL) muscle which originates from the inferior tibia. The FDL tendon crosses superficially over the FHL in the plantar midfoot at the knot of Henry, which is usually located over the navicular and medial cuneiform in most patients. This anatomic relationship creates a mechanically disadvantageous site similar to the intersection syndromes of the upper extremity.26,27,28 A fibrous slip connects the FHL and FDL at the knot of Henry, which tends to maintain their anatomic relationship at this location.29 The tendon sheaths of the FHL and FDL usually communicate, which allows inflammation of one structure to spread to the adjacent tendon.27

Another tendon intersection site has been identified in the lower leg as the FDL crosses superficially over the posterior tibial tendon, called the chiasma crurale. However, there is no reported clinical impingement syndrome at this site, and histologic examination of cadaveric tendons at this location did not reveal significant tendon degeneration.30

We used Gaussian regression to estimate the association between the dietary share of ultra-processed foods and the four component factor scores. To relax the linearity assumption of the association, the dietary contribution of ultra-processed foods variable was transformed using restricted cubic splines with five knots. The model was also fit using z-standardized scores. The factor scores were then regressed on the quintiles of the dietary share of ultra-processed foods. Finally, factor scores were categorized into tertiles to express low, middle, and high adherence to the dietary pattern in order to examine the category distribution across quintiles of the dietary share of ultra-processed foods.

Diaphragmatic paralysis after coronary artery by pass grafting in adult patients is commonly attributed to topical cooling [16, 17]. However, topical cooling is not currently used, which decreased the frequency of diaphragm paralysis. One of the possible causes of diaphragm paralysis after coronary artery by pass grafting is harvest of internal mammary artery. It was shown that phrenic nerve crosses over internal mammary artery in anterior thoracic wall in 54% of patients and in posterior thoracic wall in 14% of patients [18]. Furthermore, pericardiophrenic artery originates from internal mammary artery in 89% of cases [19, 20]. In case of thermal injury of internal mammary artery by electroknife, phrenic nerve may become ischemic. In addition to surgical technique, diabetes and older age have been considered as potential risk factors for diaphragm paralysis [20, 21].

We provide a stepwise approach that builds from simple to complex models, and account for the intrinsic complexity of the data. We start with standard cubic splines regression models and build up to a model that includes subject-specific random intercepts and slopes and residual autocorrelation. We then compared cubic regression splines vis-à-vis linear piecewise splines, and with varying number of knots and positions. Statistical code is provided to ensure reproducibility and improve dissemination of methods. Models are applied to longitudinal height measurements in a cohort of 215 Peruvian children followed from birth until their fourth year of life.

The primary aim of our analysis was to model height and height velocity. We included the following predictors in our models: age in months, an indicator variable for greater than 24 months to account for unit differences in length vs. height measurement methods, and sex. To model the non-linear relationship between age and height over time, we used smooth, flexible functions known as cubic regression splines. While there are several forms of regression splines that can be used to model non-linear relationships between a predictor (i.e., age) and an outcome (i.e., height), we chose to use cubic regression splines because they are simple to construct and understand [21]. We purposely varied the number and positions of the interval knots in several of our examples to demonstrate that our models are not affected by these changes. As mentioned, derivation of different types of splines to calculate height velocity is straightforward (Table 1). Since other investigators [6] have proposed the use of linear splines to model child growth because of their ease of use, in this paper, we compared adequacy of both estimation and prediction between cubic and linear regression splines with variations in the number and positions of the knots. We compared cubic and linear regression spline models using an in-sample (i.e., estimation) mean square error (MSE) and out-of-sample (i.e., prediction) MSE using standard methods. For out-of-sample prediction, we used 80 % of the data for training and 20 % of the data for validation. For each individual growth curve, we randomly sampled 80 % of the data values and used them to construct the model fit. The validation data consisting of 20 % of the data was then used to generate predicted height values. The predicted height and observed height were used to compute subject-specific prediction MSEs.

Cubic regression splines were superior to linear regression splines in both estimation and prediction at each modeling step, and even when varying the number and position of knots. In Table 4, we report the values of AIC and BIC for OLS, LME, and LME with CAR(1) models even when the number of knots and their positions are varied. All results indicate that mixed effects models with cubic splines by far outperform the other models considered and that cubic splines outperform linear splines for every level of model complexity. AIC and BIC results show that cubic splines outperform linear splines even when cubic splines use three knots and linear splines use five knots. This is probably because the growth curvature is better captured by a cubic function than by multiple piecewise linear splines. Equally importantly, however, is the important reduction in AIC and BIC noted when a CAR(1) error structure was incorporated into the regression model.

An interesting characteristic of the height velocity and acceleration curves from a cubic regression model is that they are continuous and make biological sense. In contrast, a linear regression spline would assume that growth velocity is piecewise constant within knots but discontinuous across knots. Moreover, growth acceleration curve is zero. Both these assumptions of linear spline approaches contradict scientific knowledge about growth trajectories and the data in our application. For example, in Fig. 5 we estimated growth velocity (top panels) and acceleration (bottom panels) for three subjects in the study using linear (left panels) and cubic (right panels) splines with three knots (at 3, 10, and 29 months.) A different number of knots and knot locations would result in slightly different plots, but with the same qualitative interpretation. The linear spline plot assumes that growth velocity is piecewise constant between the knots, an assumption that once highlighted is very difficult to accept from a scientific perspective. In contrast, the cubic spline estimate of the velocity curve is much more aligned with the current knowledge of human growth with the velocity curve being continuous and smooth. Even more striking, the acceleration curve estimated using linear splines is always zero. In contrast, cubic regression splines estimate a negative acceleration, which is especially large in absolute terms in the first part of the curve. This corresponds to the obvious patterns we observe in growth curves: they have a slightly concave shape, with concavity much more pronounced immediately after birth (indicating deceleration of growth). Interestingly, the cubic spline estimator gets much closer to zero after month 10, but continues to be negative, which may indicate continuous concavity of the function, though much subtler.

Estimated growth velocity (top panels) and acceleration (bottom panels) for three subjects in the study using linear (left panels) and cubic (right panels) splines with three knots (at 3, 10, and 29 months). A different number of knots and knot locations would result in slightly different plots, but with the same qualitative interpretation 2b1af7f3a8