text:
One long-established simplifying principle behind the large repertoire and high versatility of human hand movements is the two-thirds power law—an empirical law stating a relationship between local geometry and kinematics of human hand trajectories during planar curved movements. It was further generalized not only to various types of human movements, but also to motion perception and prediction, although it was unsuccessful in explaining unconstrained three-dimensional (3D) movements. Recently, movement obeying the power law was proved to be equivalent to moving with constant planar equi-affine speed. Generalizing such motion to 3D space—i.e., to movement at constant spatial equi-affine speed—predicts the emergence of a new power law, whose utility for describing spatial scribbling movements we have previously demonstrated. In this empirical investigation of the new power law, subjects repetitively traced six different 3D geometrical shapes with their hand. We show that the 3D power law explains the data consistently better than both the two-thirds power law and an additional power law that was previously suggested for spatial hand movements. We also found small yet systematic modifications of the power-law's exponents across the various shapes, which further scrutiny suggested to be correlated with global geometric factors of the traced shape. Nevertheless, averaging over all subjects and shapes, the power-law exponents are generally in accordance with constant spatial equi-affine speed. Taken together, our findings provide evidence for the potential role of non-Euclidean geometry in motion planning and control. Moreover, these results seem to imply a relationship between geometry and kinematics that is more complex than the simple local one stipulated by the two-thirds power law and similar models.
One fundamental problem in the study of motor control is that of trajectory formation—i.e., the selection of the geometrical shape of the path as well as the velocity profile (or speed profile) over that path to accomplish a certain goal. This selection process is highly redundant in many levels (Bernstein 1967), as can be demonstrated by examining the simple act of reaching for a cup of tea. The CNS must choose a specific solution among an infinite number of possible paths leading from the hand's initial position to that of the cup. It must also choose one of an infinite number of speed profiles along that path, specify joint rotations that would realize the chosen hand trajectory, and then choose from all muscle activation patterns that achieve those joint configurations.
Two important classes of models of trajectory formation that have been suggested to solve the redundancy problem for hand movements are those that rely on optimization principles versus those that are based on motion invariants. Because the motor system is a product of evolution, development, learning, and adaptation—all of which are in a sense processes aimed at reaching the most advantageous performance—optimization-based models stress that attempts to understand this system should emphasize the role of optimization. These models thus suggest that planning and control of movement are optimal in some sense—i.e., they minimize some undesired cost or maximize some sought-after benefit.
One successful optimization criterion has been based on the idea that the brain favors smooth movements and thus strives for movements that are maximally smooth (Flash and Hogan 1985). This implies a cost that penalizes the time derivative of either hand acceleration (i.e., jerk) (Flash and Hogan 1985; Todorov and Jordan 1998), joint torques (Nakano et al. 1999; Uno et al. 1989), or muscle forces (Pandy et al. 1995). Another idea is that because the magnitude of motor noise seems proportional to the amount of muscle activation, the sequence of muscle activations is planned so as to minimize the resulting variance of the final hand positions (Harris and Wolpert 1998). Yet another model, based on stochastic optimal feedback control, stresses the role of closed-loop sensorimotor integration, while allowing trial-by-trial variability in task-irrelevant dimensions of the movement (Braun and Wolpert 2007; Guigon et al. 2007; Todorov 2004; Todorov and Jordan 2002).
The second class of models attempts to identify simplifying principles behind the large repertoire and high versatility of human hand movements. Discovery of these laws of motion goes some way to reduce the apparent redundancies of trajectory formation in the motor system. One of many examples is the long-known isochrony principle (Binet and Courtier 1893; Viviani 1986; Viviani and Flash 1995), which suggests that average movement speed scales with the extent of movement, making movement duration only weakly dependent on movement extent. Other invariants are the bell-shaped speed profiles and almost-straight paths typical of planar reaching movements.
Of special importance for us here is another long-established invariant: the inverse relation between instantaneous tangential speed and curvature during planar curvilinear hand trajectories (Binet and Courtier 1893; Jack 1895). More recently it was formally defined as (1) and termed the two-thirds power law, which is often denoted 2/3-PL in the following text. The name "two-thirds power law" stems from the originally used equivalent formulation of this kinematic constraint in terms of angular speed A, where A = ακ2/3 (Lacquaniti et al. 1983). In Eq. 1, the hand's movement speed is (2) and the curvature of the movement path of the hand is (O'Neill 1997; Oprea 1997) (3) where for any function of time ρ(t), the first and second time derivatives are ρ̇ and ρ̈, respectively (i.e., ρ̇ = dρ/dt and ρ̈ = d2ρ/dt2). For the power law of Eq. 1, the velocity (or speed) gain factor α is roughly constant for simple elliptical shapes, but is piecewise constant for more complex shapes (Viviani 1986; Viviani and Cenzato 1985). The power law is therefore a model of trajectory formation, allowing the derivation of local trajectory kinematics from local path geometry1 (up to the constant α).
The power law was shown to describe various types of movement that invoke different effectors and use different muscle groups across the body. Thus besides drawing movements, it was also established to apply to the hand under isometric force conditions (Massey et al. 1992). Smooth-pursuit eye movements were further demonstrated to obey the power law (deSperati and Viviani 1997), as were tongue movements during speech kinematics (Perrier and Fuchs 2008; Tasko and Westbury 2004). Moreover, locomotive trajectories were found to be congruent with the power law (Hicheur et al. 2005, 2007; Pham et al. 2007; Vieilledent et al. 2001). It was also discovered in the perception of motion (Dayan et al. 2007; Levit-Binnun et al. 2006; Viviani and Stucchi 1992), where motion according to the power law was demonstrated to appear more uniform than constant-speed movement. Moreover, prediction of human movement is facilitated by trajectories conforming to the power law (Kandel et al. 2000). It was discovered that this power law is also represented in proprioceptive information from the hand (Viviani et al. 1997) and leg (Albert et al. 2005). The power law was further observed to develop with age in children (Sciaky et al. 1987; Viviani and Schneider 1991), whereas adults find it extremely difficult to produce trajectories that significantly violate this law (Viviani 2002).
Some effort has been invested in associating between different invariants. Thus the isochrony principle was at least partially explained by the two-thirds power law (Viviani and Cenzato 1985; Viviani and Flash 1995; Viviani and Schneider 1991). The ubiquity of many of the invariants motivated attempts at discovering some underlying optimization principles to account for their presence in human movements. Bell-shaped speed profiles and straight paths in planar reaching movements were shown to be an immediate result of the minimum-jerk model (Flash and Hogan 1985). The 2/3-PL was proposed to result from motor-planning strategies of the CNS that favor smoothness maximization (Richardson and Flash 2002; Todorov and Jordan 1998; Viviani and Flash 1995; Wann et al. 1988). The power law was further shown to be a possible consequence of the minimum-variance principle (Harris and Wolpert 1998). It was recently hypothesized that the power law might also stem from correlated noise in the motor system (Maoz et al. 2006).
Others suggested that the power law may stem from more peripheral smoothness effects due to muscle mechanics (Gribble and Ostry 1996) or to the nonlinearities involved in the mapping from joint to hand coordinates and to the smooth oscillatory nature of joint rotations (Dounskaia 2007; Schaal and Sternad 2001). However, single-cell monkey recordings demonstrate that population vector coding in both the motor and the premotor cortical areas obeys the two-thirds power law during motion execution (Polyakov et al. 2001; Schwartz 1994; Schwartz and Moran 1999, 2000). Moreover, a recent functional magnetic resonance imaging (fMRI) study in humans has shown that motion perception seems to reflect this constraint and that the brain's response to this law of motion is much stronger and more widespread than that to other types of movement (Dayan et al. 2007). These studies provide evidence in favor of the origin of the power law in central motion planning and control.
Importantly, it was proved that planar movement that obeys the 2/3-PL is equivalent to moving at constant equi-affine speed (Flash and Handzel 1996, 2007; Handzel and Flash 1999; Pollick and Sapiro 1997). (For a more intuitive explanation of the notion of equi-affine geometry and equi-affine speed, see Supplemental Material.2 ) Formally, for a planar trajectory r(t) = [x(t), y(t)], equi-affine speed is defined as (4) which is the cube root of the signed area of the parallelogram created by the instantaneous first and second time-derivative vectors of the position along the trajectory (see Supplemental Fig. S3). Algebraic manipulations of Eqs. 2, 3, and 4 then yield and when comparing with the power law (Eq. 1) we see that motion with constant equi-affine speed (i.e., with vea = const) is equivalent to motion that obeys the two-thirds power law.
Within the equi-affine framework, moving at constant equi-affine speed (which is equivalent to moving according to the 2/3-PL) is the simplest kind of motion. Thus if the brain tends to generate movement within this framework, the simplest kind of motion would be that which obeys the 2/3-PL. Indeed, analysis of monkey brain recordings during planer scribbling movements suggest's that equi-affine speed might be represented in motor-control neurons (Polyakov 2006, Polyakov et al. 2009). This means that the various types of phenomena that the power law describes may all result from the CNS planning and controlling movement using constant equi-affine speed. This interpretation suggests that the power law may be more than some empirical, and possibly accidental, invariant of motion.
It was recently shown that three-dimensional (3D) movement at constant equi-affine speed entails a novel power law different from the 2/3-PL (Pollick et al. 1997). This is an empirically testable prediction stemming from the equi-affine framework. If movement at constant equi-affine speed is indeed the guiding principle and the 2/3-PL is just a special case, we would expect general 3D movement to conform to 3D constant equi-affine speed instead of to the 2/3-PL. This was in fact demonstrated when it was shown that the 2/3-PL does not sufficiently explain general 3D drawing movements—the less planar the path is, the less the power law is obeyed (Pollick and Ishimura 1996; Schaal and Sternad 2001; Sternad and Schaal 1999). Yet does the novel 3D power law describe 3D movements better than the 2/3-PL?
Following a similar rationale to that used to prove the equivalence of the two-thirds power law with planar movement at constant equi-affine speed, we derive the formula for motion at constant spatial equi-affine speed (for full details see Pollick et al. 2008). Spatial equi-affine transformations preserve the volume (rather than area) enclosed by the shape. We thus define spatial equi-affine speed at any point on a curve in terms of the volume of the parallelepiped defined by the first-, second-, and third-derivative vectors at that point. More formally, we define (5) where r(t) = [x(t), y(t), z(t)] and "|u, v, w|" denotes the scalar triple product of any vectors u, v, w ∈ ℝ3; |u, v, w| = u·(v × w). Here the "·" operator denotes the dot-product between two vectors and "×" denotes the cross-product.
We must now introduce torsion, which intuitively speaking is the local deviation from planarity3 (6) Here for any vector u, ‖u‖ is its vector-norm or magnitude. Algebraic manipulations of Eqs. 2, 3, 5, and 6 yield Therefore motion at constant equi-affine speed entails a new power law (Pollick et al. 2008) (7) which we name the one-sixth power law, designated 1/6-PL in the following text. This law suggests that spatial movement speed (v) is inversely related to curvature (κ) and, to a lesser extent, to torsion (τ). It was demonstrated that for the limited case of 3D scribbling movements, the one-sixth power law explains the data rather well and certainly better than the two-thirds power law (Pollick et al. 2008).
Here we test this new power law on repetitive self-paced manual tracings of six different spatial shapes, to investigate how successful the equi-affine framework is at explaining unconstrained 3D drawing movements. We evaluate it against another 3D hand-movement power law suggested in the literature as well as against the two-thirds power law. As detailed in the following text, the one-sixth power law fits our movement data best among the tested power laws, providing evidence in favor of the CNS planning and executing movement according to the constant equi-affine speed framework. Indeed, when averaged over all subjects and shapes the movement is in general agreement with the 1/6-PL. However, we find small, though systematic, deviations of the power-law exponents from the values predicted from the 1/6-PL across the various shapes. These deviations are found to be in correlation with global geometric characteristics of the shape being traced.
Eight subjects volunteered to participate in the experiment (seven male, ages 28–37 yr; five were right-handed), which was approved by the Ethical Committee of the Weizmann Institute of Science. All gave their full informed consent to participate and none reported any previous arm injuries. They are designated by their initials: A.B., D.L., E.D., F.P., l.Z., J.F., M.O., and S.L.
Experimental setup
Subjects were seated on a rigid chair with their trunks strapped to its high back from the shoulder of the dominant arm to the contralateral side of the waist by a restraining strap. Sensors were attached to the wrist, upper arm, and the shoulder of the dominant arm as well as to the chest at the collarbone proximal to that arm. The subjects were further fitted with a wrist brace, which prevented wrist motion. The subsequent analysis used data only from the wrist sensor, designating it as the end effector or working point. The positions and orientations of these four sensors were measured using the Polhemus Liberty electromagnetic spatial tracking system (Polhemus Electronics, Colchester, VT). Because the accuracy of this system drops significantly in the presence of metallic objects, the experiments were performed outdoors, far from any metals (the surroundings, including the ground, were carefully tested with a metal detector). The chair and stands used for the experiment were composed entirely of wood and plastic, respectively (the absence of metals was again ascertained with a metal detector). The subjects further removed any metallic items they were carrying before the experiment began.
These strict noise-reduction measures were necessary to make sure that the power law would not originate from measurement noise, following the criticism of Maoz et al. (2006). There it was demonstrated that white and other types of noise can induce power-law compliance in non-power-law signals, if the noise levels after smoothing roughly equal the mean displacement between consecutive samples in the measured data. The Liberty system's specifications suggest a static position accuracy of 0.76 mm. We tested the dynamic accuracy of this system by moving a 1.2-m-long rigid wooden stick, with sensors firmly attached to it, around the working area at approximately the same distance from the magnetic field source as the subjects would move and analyzing the SD of the distance between sensors. The system's measurement error was judged to be about 1 mm before smoothing and at most 0.3 mm after smoothing (see following text). The system's measurement error after smoothing was therefore well below the average displacement between consecutive samples, which is always >1 mm. This effectively ensures that any power laws subsequently computed would not be an artifact of measurement noise.
Data were collected at 240 Hz and stored on an IBM Thinkpad A31p laptop running Windows XP. The software designed to record the movements was developed using the Liberty's software development kit. It notified the experimenter of any data loss or accuracy problems in real time. The very few problematic trials that were encountered were immediately aborted and rerecorded.
Experimental procedure
The experiment was composed of six blocks. A different shape was traced in each block. These shapes, illustrated in Fig. 1, were the double-bent ellipse (DBE), figure eight (FE), large bent ellipse (LBE), left-to-right helix (LRH), spiral helix (SH), and small bent ellipse (SBE). Before each block, the subject was given a wire-frame model of the shape. She or he would then practice tracing the shape in a continuous manner (except for the LRH and SH, in which subjects reached back to the shapes' starting point after completing each trace) until feeling confident in the ability to correctly trace the shape.
The subjects were instructed to draw at a natural pace within their comfortable working area. They were further informed that the overall size and proportions of the shapes were up to them. This was to keep their movements as natural as possible. Before they started moving, the subjects closed their eyes to eliminate visual feedback of their moving arm. They verbally signaled to the experimenter when they were ready for data recording to begin. Recoding then actually began only after the participant reached a stable pace, to eliminate motion-initiation effects, which might result in nonstereotypical trajectories. A recording trial lasted for 20 s, but the subject was requested to stop moving only after ≥25 s to avoid motion-termination effects. Three trials were executed in each block. Participants were encouraged to rest their arms between trials for as long as they felt necessary. Therefore overall the database of movements consists of 8 (subjects) × 6 (blocks or shapes) × 3 (trials per shape) = 144 recording trials.
Data processing
The 3D trajectories were tested for missing data points using the sample index provided by the Liberty system. Missing samples (very rare and at most seven consecutively, i.e., always resulting in time gaps <30 ms) were interpolated with cubic splines. The motion data were approximated with splines using the Matlab (The MathWorks) implementation of the GCVSPL package for generalized, cross-validatory spline smoothing and differentiation (Woltring 1986). This method is suited for nonstationary additive noise and nonequispaced time stamps, both of which are characteristic of our measurement system. The 1-mm measurement error (see earlier text) was used to compute the appropriate tolerance for these splines. The quantities of speed v, curvature κ, and torsion τ could then be calculated by analytically differentiating these splines. Extensive simulations that we performed with the Liberty's characteristic noise suggest that for shape-tracing trajectories such as ours, torsion is calculated to an accuracy of about 10% of the signal's magnitude, whereas speed and curvature are calculated to within 2–3% error.
Processing of the data involved examining whether the predicted relationship between the speed, curvature, and absolute value of torsion in Eq. 7 was obtained. There were two initial considerations in examination of this relationship. The first was whether parts of the movement were planar. In regions where a movement is rather planar, the torsion was approximately zero and the relationship between speed, curvature, and torsion, as presented in Eq. 7, becomes ill-defined. A second consideration is that subjects tend to trace 3D paths whose signed torsion is composed of negative and positive regions. Taking the absolute value of this torsion (as in Eq. 7) thus gives rise to physically unrealizable cusps at locations that are transition regions between positive and negative values for signed torsion. These nonrealizable transition regions must be omitted before any correspondence between speed, curvature, and torsion can be examined.4 To deal with the planar regions and the torsion cusps we placed a threshold of 2 m−1 on the value of absolute torsion for the fitting procedures of power laws described in the following text, effectively using only data that had absolute torsion values of ≥2 m−1 for power-law fitting. This threshold value was the lowest one that enabled us to considerably eliminate not only the planar regions of the movement but also the transition regions of the cusps. Overall, 79% of the data samples had a corresponding torsion value above the threshold and were maintained for the power-law fitting.
The 3D trajectories of all experimental trials were sliced into discrete repetitions using a semiautomatic method, the results of which were examined manually. This slicing procedure eliminated any slow drifts of the drawing pattern in space due to the blindfolded manner of execution. These repetitions are the elemental units of analysis in this study; therefore all power-law fitting (see following text) was carried out globally, encompassing an entire repetition. The perimeter of each repetition was further computed as the sum of the Euclidean distance between consecutive samples of that repetition (i.e., the integral of the path of that repetition). Last, the measure of the nonplanarity of each repetition that we used was the square root of the smallest eigenvalue of the covariance matrix for that repetition, following Schaal and Sternad (2001).
Fitting the power laws to the data
Two power-law comparisons were carried out. In the first, the 2/3-PL and 1/6-PL are compared against each other as well as against the power law (8) This power law, introduced by Gielen et al. (2008), suggests that the speed is proportional to the total curvature, i.e., to , to the power of −1/3. We name it the one-third total-curvature power law and designate it 1/3-Totκ-PL. In this first power-law comparison only the best-fit speed gain factor α (in the least-squares sense) of each of Eqs. 1, 7, and 8 was calculated using linear regression.
In the second power-law comparison, exponents of the power laws were not taken as fixed constants, but rather the least-squares-error speed gain factor and exponents were sought together. In other words, the method was based on simultaneously finding the gain factor and exponents that minimize the squared error between the experimentally measured speed and the speed predicted according to Eqs. 9–12 in the following text (using the curvature and torsion computed from the experimental data).
More formally, instead of the two-thirds power law we have (9) We name this the curvature power law and denote it by the acronym κ-PL. Similarly, instead of the 1/6-PL we have (10) It is termed the curvature–torsion power law and is often denoted κτ-PL in the following text. We also maintain a constrained version of the curvature–torsion power law (11) which is named the constrained curvature–torsion power law because of the constraint between its curvature and torsion values. It is denoted c-κτ-PL in the following text. This constraint was imposed so that we would have a version of the curvature–torsion power law with the same number of free parameters as the rest of the free-exponent power laws, which makes their comparison simpler. Last, instead of the one-third total-curvature power law, there is (12) which we name the total-curvature power law and denote Totκ-PL. These seven power laws are summarized in Table 1.
Fitting the free-exponent power laws required the use of nonlinear regression to find the best-fit exponents. For each of Eqs. 9–12 we did the following: taking the log of both sides of the equations, the approximate least-squares regression values were calculated with linear regression in this log-space. These values then served as starting points for the subspace trust region nonlinear regression method (Coleman and Li 1994, 1996). This resulted in a best-fit solution for the speed gain factor and exponents, which was double-checked using an exhaustive regional search with iteratively increasing resolution.
The R2 goodness-of-fit statistic (Rao 1973) of linear and nonlinear regression was used to compare among the power laws' fits to the data. This statistic measures how successful a fit is in explaining the variance of the data (in our case the movement speed) or what ratio of the data variance is explained by the fit (to one of the power laws). It is also named the square of multiple correlation coefficient and the coefficient of multiple determination. More formally, for some predictor data sequence (independent variable) xi, R2 is the square of the correlation between the response values (dependent variable) yi and the response values predicted by the fit (or the model) ŷi. Quantitatively where ȳ is the mean over all yi. Therefore formally R2 ≤ 1 and can be negative. However, negative R2 scores are actually possible for nonlinear regression only. This arises from the fact that for linear regression ∑i (yi − ŷi)2 ≤ ∑i (yi − ȳ)2 because, by the definition of linear regression, if some trend is found in the data ∑i (yi − ŷi)2 < ∑i (yi − ȳ)2 and thus R2 > 0, whereas if no trend is found in the data ŷi = ȳ for all i, and thus ∑i (yi − ŷi)2 = ∑i (yi − ȳ)2 and R2 = 0.
All subjects were presented with the same set of templates for practice, one for each shape (see Fig. 1). The template was removed before movement onset, leaving the subjects free to trace out their subjective interpretation of the template. Indeed, intersubject variations for each shape were considerable, as can be appreciated from Fig. 2.
Comparing the 2/3-PL, 1/6-PL, and 1/3-Tot-κ-PL
We initially compared the 2/3-PL, 1/6-PL, and 1/3-Totκ-PL (see Eqs. 1, 7, and 8, respectively). Figure 3 illustrates the speed prediction of the various power laws versus the experimentally measured speed. We calculated the predicted tangential hand speed of the 2/3-PL, 1/6-PL, and 1/3-Tot-κ-PL for six sections of movement, one for each shape. As is apparent, the speed profile of the 1/6-PL is usually closest to that of the measured speed and the 1/3-Tot-κ-PL is typically farthest from it.
More rigorous statistical analysis suggests that the trends observed in Fig. 3 are borne out over the entire movement database. Table 2 gives the R2 statistic of the fit between the speeds predicted by each of the 2/3-PL, 1/6-PL, and 1/3-Tot-κ-PL against the experimentally measured speed (Supplemental Tables S1–S3 give the R2 statistic of these power laws for each subject and shape). On average, the 1/6-PL explains 17% more of the experimentally measured speed variance than does the 2/3-PL and 38% more of the variance than does the 1/3-Tot-κ-PL.
When pooled over all subjects (naturally, always with an equal number of randomly selected samples from each subject), the R2 scores of the 1/6-PL are significantly greater than those of the 2/3-PL for all shapes except LRH (one-way ANOVA and multiple-comparison test of means using Tukey's honestly significant difference [HSD] criterion, P < 0.04 for LBE and P < 0.001 for DBE, FE, SH, and SBE) and greater than those of the 1/3-Tot-κ-PL scores for all shapes (one-way ANOVA and multiple-comparison test of means using Tukey's HSD criterion, P < 0.001). The R2 scores of the 2/3-PL pooled over all subjects are significantly greater than those of the 1/3-Tot-κ-PL for all shapes except FE and SH (one-way ANOVA and multiple-comparison test of means using Tukey's HSD criterion, P < 0.01 for DBE and P < 0.001 for LBE, LRH, and SBE).
The average R2 scores of the 1/3-Tot-κ-PL for LRH and SBE in Table 2 are negative. As discussed in methods, this is because for these shapes the speed profile predicted by the 1/3-Tot-κ-PL explains the experimentally measured speed profile even worse than the average of that experimentally measured speed (Rao 1973).
Comparing the κ-PL, κτ-PL, c-κτ-PL, and Tot-κ-PL
We now compare the power laws again when best-fitting both the exponents and the speed gain factors. This tests how well the different power laws describe the movement data without restricting their exponents to the values prescribed in Eqs. 1, 7, and 8. We therefore also calculated the predicted tangential speeds from the κ-PL, κτ-PL, c-κτ-PL, and Tot-κ-PL (see Eqs. 9, 10, 11, and 12, respectively) and compared them to the speeds actually measured in the experiments. Six examples of these speed profiles are depicted in Fig. 4, one for each shape. As is apparent, the speed profiles of the c-κτ-PL and κτ-PL are usually closer to those of the measured speeds than to those predicted by the other power laws.
Once again, more rigorous statistical analysis suggests that the trends apparent in Fig. 4 are manifested over the entire movement database. Table 3 portrays the R2 statistic of the fit between the speeds predicted by each of the κ-PL, κτ-PL, c-κτ-PL, and Tot-κ-PL against the experimentally measured speed for every shape, when pooled over all subjects (Supplemental Tables S4–S7 give the R2 statistic of these power laws for each subject and shape). The κτ-PL explains 3% more of the experimentally measured speed variance than does the c-κτ-PL, 11% more than does the κ-PL, and 19% more of the variance than does the Tot-κ-PL, averaged over all subjects and shapes.
When pooled over all subjects (again, with an equal number of randomly selected samples from each subject), the R2 scores of the κτ-PL are not significantly different from those of the c-κτ-PL for all shapes except for SBE (two-way ANOVA and one-way ANOVA and multiple-comparison test of means using Tukey's HSD criterion, respectively). Moreover, the R2 scores of the c-κτ-PL are significantly greater than those of the κ-PL for four of the six shapes (DBE, FE, LBE, and SH; one-way ANOVA and multiple-comparison test of means using Tukey's HSD criterion, P < 0.01) and not significantly different for the other two shapes (two-way ANOVA) as well as significantly greater than those of the Tot-κ-PL for four of the six shapes (DBE, FE, LBE, and SBE; one-way ANOVA and multiple-comparison test of means using Tukey's HSD criterion, P < 10−5) and not significantly different for the other two shapes (two-way ANOVA). Last, the R2 scores of the κ-PL are significantly greater than those of the Tot-κ-PL for four of the six shapes (DBE, FE, LBE, and SBE; one-way ANOVA and multiple-comparison test of means using Tukey's HSD criterion, P < 0.001), significantly smaller than the Tot-κ-PL for the SH (one-way ANOVA and multiple-comparison test of means using Tukey's HSD criterion, P < 0.01), and not significantly different for the LRH (two-way ANOVA).
Pooling over all subjects and shapes (the last row of Table 3), we find that the R2 scores of the κτ-PL once more are not significantly different from those of the c-κτ-PL, whereas the scores of both these power laws are greater than the κ-PL and Tot-κ-PL (one-way ANOVA and multiple-comparison test of means using Tukey's HSD criterion, P < 10−10). The R2 scores of the κ-PL are also significantly greater than those of the Tot-κ-PL (one-way ANOVA and multiple-comparison test of means using Tukey's HSD criterion, P < 10−4).
Summing up, when averaging not only over all subjects and shapes, but also for each shape separately, the κτ-PL and the c-κτ-PL share similar explanatory powers over our data set of movements, even though the first has an extra free parameter over the second. The c-κτ-PL explains the data just as well as, or significantly better than, the κ-PL and the Tot-κ-PL for all shapes separately as well as overall on average. Last, the Tot-κ-PL has the overall worst explanatory power on our movement data set.
Further investigation of speed–shape relations
At this point we focus our investigation on the κτ-PL. We already showed that the constrained version of the κτ-PL, the c-κτ-PL, which has the same number of free parameters as the 2/3-PL and the Tot-κ-PL, explains the data better than these two power laws. Because the κτ-PL is a generalization of both the c-κτ-PL and the 2/3-PL, it is the most flexible among these three power laws. Also, by definition as in practice, it fits the data just as well as, or better than, these other two. The Tot-κ-PL, with its relatively poor fit to the data, is no longer pursued.
Table 4 presents the mean and SD of the best-fit β and γ exponent distributions for the κτ-PL when pooled over all subjects (Supplemental Tables S8 and S9, respectively, give the best-fit β and γ values of the κτ-PL for each subject and shape individually). Average β and γ values among the shapes in Table 4 range between −0.39 and −0.28 and −0.15 and −0.08, respectively. The mean values of β and γ over all subjects and shapes are −0.32 and −0.12, respectively. These values are not very far from the values of −0.333 and −0.167, which are congruent with movement at constant equi-affine speed.
Figure 5 presents the results presented in Table 4 in more detail, depicting the joint β–γ distribution histograms for all shapes. For each shape the β and γ values are pooled over all subjects, with an equal number of randomly chosen movement repetitions from each subject. (Empirical distributions can be derived from these histograms via normalization and vice versa, given the overall number of repetitions. The terms histogram and distribution will therefore be used interchangeably.)
Taken together, these results suggest that, when grand-averaging over all subjects and shapes, it may be appropriate to describe the speed–shape relations in our data as movement at constant equi-affine speed. Nevertheless, we now seek to delve deeper into the details and examine similarities and differences between the best-fit exponents of the κτ-PL for each shape separately.
Rigorous statistical testing suggests that when comparing the β–γ distributions of the various shapes, almost all distributions are significantly different in pairs. [Thirteen of the 15 pairs (87%) or all pairs, except FE–LBE and LRH–SBE, are significantly different. Hotelling's multivariate extension to Student's t-test was combined with one-way ANOVA and multiple-comparison test of means using Tukey's HSD criterion on each variable separately, after a Bonferroni correction for multiple comparisons at P < 0.05. Performing the same test when raising the significance level to P < 0.001 results in just one more pair, DBE–SH, becoming nonsignificantly different.]
Further statistical testing suggests that whereas the intersubject variations in the power-law's exponent values are small enough to be insignificant, the intershape variations of these exponents are greater and thus significant (mixed-model ANOVA suggests that whereas the means of the exponent values across the different shapes are not all similar, P = 0.009 for β and P = 4 × 10−6 for γ, the means of the exponent values across the different subjects are likely the same, P > 0.9 for β and P > 0.29 for γ; a further F-test suggests that the variance of shapes is significantly greater than that of the subjects, P = 4 × 10−4 for β and P = 0.03 for γ). These differences suggest that the empirical β–γ joint distributions for each shape originate from different parent distributions. Put less technically, there tends to be a distinct β–γ distribution for each shape.
The above-cited results suggest the following: >60% of the variance in spatial movement speed can be explained by the path being traversed at 3D constant equi-affine speed in 3D (i.e., according to the 1/6-PL; Table 2). Moreover, almost three quarters of spatial movement speed variance can be explained by relaxing the constant equi-affine speed constraint on the exponents of the κτ-PL (Table 3), with the over-all-shapes and subjects exponent averages varying around values close to those predicted by spatial constant equi-affine speed movement. Nevertheless, this small variance of the exponent values among the different shapes is still significant and systematic.
As noted in the introduction, a few studies demonstrated that for the planar case, the exponent of the κ-PL may change systematically with the path (Sternad and Schaal 1999; Wann et al. 1988). Of special interest is the study by Schaal and Sternad (2001), which investigates the 2/3-PL for unconstrained ellipse tracing in 3D space. They find that the fit to the planar κ-PL decreases with the increase in the global geometric parameters of perimeter and nonplanarity, both in terms of the exponent value deviating from −1/3 and in terms of the R2 score decreasing. They attribute this to nonlinearities inherent in mapping from joint to hand coordinates and to the smooth oscillatory nature of joint rotations. Yet we suspect that their observed deviations are due, at least in part, to their attempt to explain movement data that are inherently 3D with a power law resulting from planar constant equi-affine speed. Indeed, because their ellipses became less planar the divergence they measured from the two-thirds power law (i.e., from planar constant equi-affine speed) increased.
The fact that the spatial 1/6-PL and κτ-PL both explain spatial movement data significantly better than the planar 2/3-PL and κ-PL, respectively, seems to lend evidence in favor of our suspicion that Schaal and Sternad (2001) used the inappropriate planar equi-affine framework to account for their data, instead of a more suitable explanation in terms of spatial equi-affine speed. However, the small, yet systematic, variability of the power-law exponents with shape demonstrated in Table 4 suggests that the global geometric parameters of perimeter and nonplanarity might affect the κτ-PL as well. We therefore decided to test the effect of the global-shape parameters of perimeter and nonplanarity on the fit of the κτ-PL to the data in terms of exponent values and R2 scores.
The correlation between the mean R2 value of the power-law fit for each shape, given in Table 3, and the corresponding mean perimeter and nonplanarity values of these shapes, in Table 5, is insignificant (0.48 between the R2 scores and the perimeter and −0.11 between the R2 scores and the degree of nonplanarity, with corresponding t-test P values of 0.34 and 0.84, respectively). This suggests that the R2 score of the fit of the κτ-PL to the data does not decrease when the perimeter or nonplanarity increase, unlike the fit of the κ-PL to the data analyzed by Schaal and Sternad (2001).
We further wanted to test whether shapes with similar average perimeter and nonplanarity values (Table 5) tend to obtain similar average values of the exponents of the κτ-PL (Table 4). We therefore defined a distance measure for the perimeter and nonplanarity values between all shape pairs as well as a distance measure for the exponents of the κτ-PL between all shape pairs. We then tested whether these distances tended to covary—whether shapes separated by small perimeter–nonplanarity distances (i.e., shapes with similar average perimeter and nonplanarity values) tend to also be separated by small β–γ distances (i.e., tend to have similar average β and γ values).
We defined the distance measure between the mean β and γ exponents of every pair of shapes as the L2 distance between corresponding pair of points in the β–γ plane. For example, if the mean β and γ exponent values for a pair of shapes are β1 and γ1, respectively, for the first shape and β2 and γ2, respectively, for the second shape, we define (β2 − β1)2 + (γ2 − γ1)2 to be the β–γ distance between that pair of shapes. The distance measure between the perimeter and nonplanarity values of every pair of shapes was similarly defined as the L2 distance between the corresponding pair of points in the perimeter–nonplanarity plane. For example, if the mean perimeter and nonplanarity of a pair of shapes are p1 and n1, respectively, for the first shape and p2 and n2, respectively, for the second shape, then (p2 − p1)2 + (n2 − n1)2 is the perimeter–nonplanarity distance between that pair of shapes.
Because there are six shapes, there are 15 [(6 × 5)/2] shape pairs. The correlation between the β–γ distances and the perimeter–nonplanarity distances was found to be significant (0.74, with the corresponding t-test P value of 0.002), suggesting that shapes with similar average perimeter and nonplanarity values indeed tend to have similar average κτ-PL exponents. We similarly examined whether shapes with perimeter and nonplanarity values alike have R2 scores for the power-law fit that are alike. This time, however, we found no significant correlation (the correlation between the perimeter–nonplanarity distance and the R2 distance is 0.44, with the corresponding t-test P value of 0.1, and for the κτ-PL and 0.43, with the corresponding t-test P value of 0.11, for the 1/6-PL). This indicates that shapes with similar mean perimeter and nonplanarity values do not necessarily have R2 scores that are alike and vice versa.
Under the hypothesis that the brain might use non-Euclidean geometry to plan and control movements, this study empirically examines the idea that the hand may move at constant equi-affine speed when repetitively tracing 3D shapes. Whereas for the planar case constant equi-affine speed results in the two-thirds power law between drawing speed and curvature, constant spatial equi-affine speed entails a new power law, which we name the one-sixth power law between drawing speed, curvature, and torsion.
We have previously demonstrated that this new power law explains 3D scribbling movements rather well (Pollick et al. 2008). The results presented here suggest that the power law also explains 3D shape-tracing movements just as well and certainly significantly better than the two-thirds power law and another power law previously proposed for this purpose. If we allow the power-law's exponents to take their best-fit values over each repetition of each shape, the power law again explains the data significantly better than the other power laws, with the average exponent values over all subjects and shapes not far from those consistent with movement at constant spatial equi-affine speed.
The inherent noisiness of the human motor system makes variability in end-effector motion inescapable, especially during unconstrained self-paced 3D movement. Moreover, our subjects traced their own subjective interpretation of complex spatial shapes (there was no visual feedback while drawing due to the blindfolded manner of execution),5 which resulted in considerable additional variability among the trajectories traced for each shape (see Fig. 2). Therefore given the differences among subjects for each shape as well as between consecutive repetitions of every shape for the same subject, the fact that a deterministic power law can capture almost three quarters of the variability of motion over the duration of an entire repetition of a complex shape (lasting ≥5 s) is noteworthy.
Most research on power laws of hand motion for shape-tracing is based on data containing at most two or three shapes; moreover, even studies that do cover a larger number of paths do not directly examine differences between the best-fit exponents over different shapes. Any such differences tend to be attributed to noise in the motor system or to other transitory irregularities in execution. Our data, which consist of hand movements along six different complex paths, enable us to study the differences in the curvature–torsion power-law's exponents over this relatively large number of different shapes. We therefore decided to investigate these intershape differences more thoroughly, seeking to find some explanation for the observed differences.
The best-fit exponents of the curvature–torsion power law across the various shapes were shown to have small, though significant, differences among almost all shape pairs. This suggests that the amount of local influence of geometrical shape (characterized by curvature and torsion) on instantaneous movement speed slightly varies among shapes. Further analysis suggests that the combination of global geometrical parameters of shape perimeter and nonplanarity is significantly correlated with the power-law exponents of the different shapes—similar global geometric form translates into similar power-law exponents and vice versa.
Importantly, unlike the power-law exponents, the goodness of fit of the power law to the data (i.e., the R2 score) is not influenced by global geometric parameters, such as the extent of the perimeter and the degree of nonplanarity. This indicates that the curvature–torsion power law, as it was formulated in Eq. 10, is broad enough to capture the shape-induced variability in speed–shape relations. Moreover, its grand-averaged exponent values over all shapes, as well as those for scribbling (Pollick et al. 2008), are in general accordance with constant spatial equi-affine speed. Nevertheless, origins of the deviations from constant equi-affine speed remain unclear and may well be due to anything from centrally controlled to epiphenomenal behavior. Further research is therefore required to give a fuller and more structured account of the observed results.
Some implications of this work
Evidence is mounting that movements at roughly planar constant equi-affine speed or, equivalently, movements that approximately obey the two-thirds power law originate from central (i.e., CNS-based) control and planning mechanisms. Single-cell monkey recordings demonstrate that population vector coding in the motor and premotor cortex obeys the two-thirds power law during motion execution (Schwartz 1994; Schwartz and Moran 1999, 2000). Moreover, a recent fMRI study in humans shows that motion perception seems to reflect this constraint and the brain's response to this law of motion is much stronger and more widespread than that to other types of movement (Dayan et al. 2007). It therefore seems plausible to hypothesize that movements at roughly constant 3D equi-affine speed—like motion according to the curvature–torsion power law with its slightly different exponents for different shapes—may also stem from central mechanisms.
Under this assumption, our current results seem to suggest that when planning and controlling the execution of complex shapes, a major objective of the motor system might be to generate motion that is constant under the equi-affine framework. Moreover, this constant equi-affine speed motion seems to undergo fine-tuning of its local speed–shape relation, possibly according to global characteristics of the path, such as the perimeter and nonplanarity of the traced shape. Viewing scribbling movements as sequential temporal concatenation of various 3D segments, its planning, and execution may follow some average case shape-based exponent values, which our data suggest to be not very far from those compatible with constant equi-affine speed (see Table 4). This may explain why scribbling movements tend to conform to constant spatial equi-affine speed (Pollick et al. 2008).
The fact that speed varies with torsion as well as curvature may further explain, at least partially, the results of Schaal and Sternad (2001). There it was demonstrated that for ellipses traced freely in 3D space, the two-thirds power law is systematically violated with increased pattern size, which is also consistent with increased nonplanarity. This violation is manifested both in exponent values that depart from −1/3 and in a deterioration in the goodness of fit of the two-thirds power law to the data. Moreover, as planarity decreases torsion becomes more prominent. This caused their data to be increasingly incompatible with movement at planar constant equi-affine speed movement (i.e., to be incompatible with the 2/3-PL). By contrast, the goodness of fit of our movement data to the spatial constant equi-affine speed model does not deteriorate with the increase in perimeter or in nonplanarity. This provides further evidence for our claim that the corresponding increasing divergence of the data of Schaal and Sternad (2001) from the two-thirds power law is a result of their attempt to explain their 3D data with a model that is suitable only for the planar case, i.e., with constant planar equi-affine speed.
Future research directions
As detailed in methods, our subjects repetitively traced the various shapes of Fig. 1. These 20-s-long movement trajectories were then sliced into discrete repetitions of the shape (see Fig. 2 for some examples of repetitions). Each repetition was then fit with the various power laws. It should thus be borne in mind that for the above-cited analyses we used only one value for the speed gain factor (α in Eqs. 7 and 10) per movement repetition, which lasted ≥5 s. However, it was demonstrated that the speed gain factor of the two-thirds power law (α in Eq. 1) is merely piecewise constant when tracing complex planar shapes, with the shifts in the otherwise roughly constant speed gain factor occurring in the transitions between consecutive segments of the drawing movement (Viviani 1986; Viviani and Cenzato 1985; Viviani and Flash 1995). This means that such paths are traced with only piecewise constant equi-affine speed (Flash and Handzel 2007; Handzel and Flash 1999). Thus consecutive segments may invoke the use of different equi-affine speeds.
Nevertheless, additional claims that this segmentation may reflect central segmented control (Viviani 1986; Viviani and Cenzato 1985) were contrasted with results suggesting this segmentation to be epiphenomenal to continuous smooth minimum-jerk control at the hand level (Richardson and Flash 2002) or to nonlinear transformations of the forward kinematics of human arms that perform smooth multijoint rotations, which are governed by continuous control (Dounskaia 2007; Sternad and Schaal 1999). It will therefore be interesting to test what type of segmentation is implied by the one-sixth or curvature–torsion power laws and how much more of the variance in the speed profile might be explained by geometry under the assumption of movement segmentation. Furthermore, it would be intriguing to investigate the relation between any movement segmentation and the slight dependence of the curvature–torsion power-law exponents on shape.
The two-thirds power law was shown to possibly result from various planning strategies that favor smooth trajectories in hand- or joint space (Gribble and Ostry 1996; Richardson and Flash 2002; Schaal and Sternad 2001; Todorov and Jordan 1998; Viviani and Flash 1995; Wann et al. 1988). The minimum variance principle was also demonstrated to predict this power law, at least for simple elliptical trajectories (Harris and Wolpert 1998). It would therefore be important to test whether the curvature–torsion power law and any segmentation scheme it may imply can be derived from smoothness (or smoothness-compatible) criteria in joint- or hand-space. Moreover, it was recently suggested that the two-thirds power law might be at least partially due to correlated noise in the motor system (Maoz et al. 2006). It would thus be interesting to test whether similar results might be extended to the one-sixth or curvature–torsion power laws.
We have demonstrated that the global shape characteristics of perimeter and nonplanarity modulate the exponents of the curvature–torsion power law. However, more research is needed to establish to what extent these modulations in the speed–shape relations reflect central planning and to what extent they are a by-product of peripheral factors, such as muscle dynamics. Further research is also required to investigate which shapes and which other characteristics of movement, if any, systematically modulate the predicted spatial speed–shape relation derived from the assumption of constant spatial equi-affine speed. Other geometrical features that may possibly also influence the speed profile include, for example, orientation and ratios between the lengths of various segments in different directions. Average speed, bounds on maximum acceleration, and other kinematic parameters may also play a role, as may joint rotations, muscle flexion, gravity, and other dynamic factors. Optimal feedback control and the choice of task-relevant and task-irrelevant parameters (Todorov and Jordan 2002) may also modify speed–shape relations.
Last, we found a small dependence of the speed–shape relation on the global shape of the drawing, on the one hand, combined with an over-all-shapes general compliance of this relation with constant equi-affine speed, on the other hand. This dual behavior might indicate that although planar or spatial motion at constant equi-affine may be the primary governing principle for the corresponding drawing movements, it may not give a full and final account of the finer-grain intershape modulations in speed–shape relations. From the perspective of non-Euclidean geometry, the constant equi-affine framework might need refinement because it is possibly not broad enough to completely explain the intricacies of the functional organization of human sensorimotor space for the aforementioned drawing task.
Herein we have proposed and experimentally tested a model of instantaneous speed–shape covariation for hand movement, which stems from the notion of motion at constant spatial equi-affine speed. With a database composed of shape-tracing movements of six different geometrical paths, our results suggest that the model's predictions fit the data rather well, with unconstrained spatial drawing movements slowing down not only in regions of high curvature but also in regions of high torsion (although to a lesser extent, as the model predicts). This new model results in a power law that describes the relationship between spatial speed and geometry significantly better than the two-thirds and total-curvature power laws, enabling a more accurate speed profile prediction for a given path. We further demonstrated small systematic modulations of the power-law's exponents, which were shown to depend, at least partially, on global geometric characteristics of the path. However, these exponents are in general accordance with constant spatial equi-affine speed when averaged over all shapes. This suggests that although local path geometry is the primary determining factor for instantaneous hand speed, via the constant equi-affine framework, this speed is also dependent, to a much lesser extent, on global geometric factors.
U. Maoz was partially supported by the Dean Fellowship of the Faculty of Mathematics and Computer Science of the Weizmann Institute of Science and by a fellowship from the Van Leer Jerusalem Institute. This research was also supported in part by Human Frontier of Science Program Grants HFSP-RGP0054/2004-C and EU Grant FP6-2005-NEST-Path Imp 043403, both to T. Flash and A. Berthoz. T. Flash is an incumbent of the Dr. Hymie Moross professorial chair.
We thank the anonymous reviewers of this manuscript for important comments and suggestions, J. Friedman and F. Polyakov for meaningful discussions and advice, and R. Fuchs for many fruitful discussions and several crucial corrections and suggestions concerning the manuscript.
↵1 Curvature is the inverse of the radius of curvature and is a purely geometric property of the path, irrespective of speed along the path. Moreover, curvature at every point along a planar path completely determines it up to Euclidian transformations (which include translations, rotations, and reflections) (Oprea 1997).
↵2 The online version of this article contains supplemental data.
↵3 Torsion, like curvature, is a purely geometric property of the path, irrespective of the speed profile over the path. However, whereas curvature at every point along a planar path is sufficient to completely determine it up to Euclidian transformations, only curvature and torsion together determine a 3D curve up to Euclidian transformations (Oprea 1997).
↵4 Similar singularities and cusps in power-law relationships have been noted for the planar two-thirds power law at inflection points (e.g. Viviani and Flash 1995). There, such regions were also discarded.
↵5 Todorov and Jordan (1998) reported that when they let their subjects hold and observe the wire-frame models of 3D shapes while they were tracing them, they could not well replicate the specified shapes and did not converge to a stable movement path.
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
- Copyright © 2009 the American Physiological Society
- Albert et al. 2005.↵
- Bernstein 1967.↵
- Binet and Courtier 1893.↵
- Braun and Wolpert 2007.↵
- Coleman and Li 1994.↵
- Coleman and Li 1996.↵
- Dayan et al. 2007.↵
- deSperati and Viviani 1997.↵
- Dounskaia 2007.↵
- Flash and Handzel 1996.↵
- Flash and Handzel 2007.↵
- Flash and Hogan 1985.↵
- Gielen et al. 1996.↵
- Gribble and Ostry 1996.↵
- Guigon et al. 2007.↵
- Handzel and Flash 1999.↵
- Harris and Wolpert 1998.↵
- Hicheur et al. 2007.↵
- Hicheur et al. 2005.↵
- Jack 1895.↵
- Kandel et al. 2000.↵
- Lacquaniti et al. 1983.↵
- Levit-Binnun et al. 2006.↵
- Maoz et al. 2006.↵
- Massey et al. 1992.↵
- Nakano et al. 1999.↵
- O'Neill 1997.↵
- Oprea 1997.↵
- Pandy et al. 1995.↵
- Perrier and Fuchs 2008.↵
- Pham et al. 2007.↵
- Pollick et al. 1997.
- Pollick and Ishimura 1996.↵
- Pollick et al. 1997.↵
- Pollick and Sapiro 1997.↵
- Polyakov 2006.↵
- Polyakov et al. 2001.↵
- Polyakov et al. 2009.↵
- Rao 1973.↵
- Richardson and Flash 2002.↵
- Schaal and Sternad 2001.↵
- Schwartz 1994.↵
- Schwartz and Moran 1999.↵
- Schwartz and Moran 2000.↵
- Sciaky et al. 1987.↵
- Sternad and Schaal 1999.↵
- Tasko and Westbury 2004.↵
- Todorov 2004.↵
- Todorov and Jordan 1998.↵
- Todorov and Jordan 2002.↵
- Uno et al. 1989.↵
- Vieilledent et al. 2001.↵
- Viviani 1986.↵
- Viviani 2002.↵
- Viviani et al. 1997.↵
- Viviani and Cenzato 1985.↵
- Viviani and Flash 1995.↵
- Viviani and Schneider 1991.↵
- Viviani and Stucchi 1992.↵
- Wann et al. 1988.↵
- Woltring 1986.↵


text:
I know this presidential election gets people emotional, but that's no excuse for bad grammar, especially if your name is Madonna. Take a look at Madge's recent Facebook post promoting Hillary:
I love that the Material Girl has no shame, but her first sentence makes me squirm. Here's a breakdown of her misdeeds, according to the fourth edition of The Elements of Style by William Strunk Jr. and E.B. White: 1) Bitches is a name or title that deserves parenthetic treatment when used in direct address. The Blonde One is talking to we bitches as if we were right there beside her. Therefore, "bitches" needs a comma. Ex. Bitches, are you in my gang??? 2) Because her statement is a sentence and not a title, Madonna really should ease up on the capitalization. If she wanted to pen a book, like The Catcher in the Rye or A Tale of Two Cities, she would have to italicize and use lower case letters for prepositions like "in." Ex. Bitches, Are You in My Gang??? 3) Finally, no need for three question marks when one would do. With the way Madge zips through time zones, I recommend simplicity, especially when thumb typing on tiny keys.
As you can see, 29,000 people liked her shabbily written post. More than 2,000 fans actually shared it, which means Madonna is a person of influence. We have free speech in this country. I support her right to be offensive. She can continue to offer free fellatio to Hillary voters, but first, she has to polish her communication skills.
Bitches, are you in my gang?

subset: commoncrawl


text:
- Glenn W. Burton,
- J. E. Jackson and
- R. H. Hart2
Annual forage yields increased as the interval between cuttings increased to 6 weeks, leveled off as the interval increased to 12 weeks and dropped sharply in plots cut only once. Increasing the interval decreased crude protein and vitamin A equivalent content, leaf percent, and in uitro digestibility and increased stem length and fiber content. Increasing N applications increased yields of dry matter and crude protein and the content of crude protein and vitamin A equivalent but had no effect on the percent fiber in the grass.Please view the pdf by using the Full Text (PDF) link under 'View' to the left.
Copyright © . .


text:
GraphicRiver Multipurpose Business Flyers Templates 11544908
Category: Print Template | Comment: ( 0 )
GraphicRiver Multipurpose Business Flyers Templates 11544908
Print Template \ Flyers \ Corporate
Flyer \ Business \ Marketing \ Product \ Print \ Multipurpose \ Promotion \ Ready
Print Template - Multipurpose Business Flyers Templates 11544908 by afjamaal @ GraphicRiver
Print Template \ Flyers \ Corporate
Layered: Yes; Graphics Files Included: Photoshop PSD; Minimum Adobe CS Version: CS4; Print Dimensions: 210x297.
Multipurpose Business Flyers Templates.
This flyer is made in photoshop the files included are help file and photoshop psd's. All psd's are very well organized proper name groups and proper name layers. Very easy to edit very easy to insert images very easy to change text. Image placeholders are smartObjects to make easy for you to add image. And edit file.
We are always available to help you and you can contact us from our profile or comments on this product 24/7. We can also change text and images for you. NO CUSTOMIZATION!
We do not provide fonts, or images you are seeing in previews. But we can give you links for fonts and images.
Font http://www.fontsquirrel.com/fonts/lato.
Font http://www.fontsquirrel.com/fonts/roboto.
Font http://www.behance.net/gallery/ALEO-Free-Font-Family/8018673.
Do not forget to rate our product if you like work please provide 5 stars. We wish you good luck in your project as well.
Letterhead Print Templates; Corporate Business Flyer Template; Product Promotion Flyer Bundle; Business Flyer Template; Corporate Business Flyer Bundle; Corporate Business Flyer Template; Corporate Business Flyer Bundle.
Keywords: agency, ai, blue, business, business flyer, consulting, corporate flyer, creative flyer, digital, editable, flyer, graphic, marketing, marketing flyer, modern, multipurpose flyer, print, product, promotion, psd, ready, template, yellow.


text:
There's the need for balance – to understand both our beauty and depravity. When you look around the world, it's easy to get downright depressed, discouraged, and overwhelmed. I certainly feel that way sometimes. But in the midst of all this junk [including my personal junk], I'm also freshly reminded that when God created the universe and all that which it encompassed – including humanity – He saw and acknowledged that it was 'good.' That may be the most incredible part of God's narrative story – right next to the Resurrection.
So, while we acknowledge the story of brokenness and sin, we can also acknowledge that God is at work – all around the earth – through the power of the Holy Spirit – to restore, reconcile, and redeem.
Locally, nationally, and globally, there have been [and always will be] a list of heavy, painful, and tragic things. The same week that the tragedy of Virginia Tech came to pass, I received an email from a Karen Christian brother I had met during my fall visit to Station 1o1 in Burma.
How are you? you are remember me. i am xxxxxxx. i would like to tell you last two year you will come to mae sot i sending you team to 101 area church. you are remember when you visit to 101 church you are sharing the message. in this time 101 area church is S.P.D.C attack and occupy. many people flee…please pray. [SPDC is the military ruling regime – State Peace and Development Council]
Heavy and painful stuff. The above isn't just news to me; they are people I visited. Looked into their eyes; encouraged them to have hope; taught the scriptures in that church that is not occupied by the Burmese military.
Locally in the Quest family, as it has and will continue to be, there are plenty of stories. Why? Because that's life. There are beautiful stories indeed but there are also painful stories. Tense relationships, marriages in crisis, cancer, sickness, addictions, and so forth. Painful stuff but this is where God's work of restoration and redemption can take place.
I stumbled onto this guy's blog who lost his dear brother to cancer this past year. His recent letter written to his brother gives me another glimpse of hope and beauty in a broken world.
Who shall separate us from the love of Christ? Shall trouble or hardship or persecution or famine or nakedness or danger or sword? As it is written: "For your sake we face death all day long; we are considered as sheep to be slaughtered." No, in all these things we are more than conquerors through him who loved us. For I am convinced that neither death nor life, neither angels nor demons, neither the present nor the future, nor any powers, neither height nor depth, nor anything else in all creation, will be able to separate us from the love of God that is in Christ Jesus our Lord. [Romans 8:35-39]
Love Wins.
Filed under: christianity, quest church


text:
The last words
Before walking the walk, James DeGale and George Groves talk the talk for one last time in our quickfire Q&A...
By Tim Hobbs
Last Updated: 21/05/11 10:07am
Tonight at the O2 Arena James DeGale and George Groves will walk the walk.
They have of course, more than talked the talk. Over the past few weeks the insults and boasts have been flying back and forth as years of animosity have come pouring out in press conferences and interviews from both.
skysports.com has been with both men from the very start and although we couldn't get them to sit down together for the same interview, we did get them to take part in a quickfire Q&A.
Over the past month we have put the same questions to DeGale and Groves, to get to the heart of their bitter feud and find out what they really think bout each other, their fued and the fight itself.
With fight night hours away, here are the famous last words from two young men with so much to win - and a whole lot more to lose...
Describe your relationship as amatuers...
GROVES: He was a fat light-heavyweight and I was probably a light-welterweight and we used to do a bit of sparring together. We were tolerable of each other, I guess. Oh, and he was always the one at the back when we used to go running.
DeGALE: We never fell out because we were never close enough to fall in. We were never friends, we were associates, we trained together but we never went out. We sparred a lot. I used to mash him up, school him. Every time.
What do you dislike most about him now?
GROVES: I feel neither way about him... it's hard to say what I dislike - but it's much harder to say anything that I like.
DeGALE: Come on! Look at him, look at him his team, the way he acts. He's very smug. I think he wants everyone to feel sorry for him. He's very softly-spoken so he comes across well but he's not like that. He's smug and he's horrible.
When was the last time you were nice to each other?
GROVES: I genuinely can't remember. If he did come up and wish me well after I beat him in the ABA's, I'm pretty sure it would have been through gritted teeth
DeGALE: It might have been then, but I can't remember. I probably wished him well, but I really can't remember.
Was this fight always unavoidable?
GROVES: Of course. As long as he stayed unbeaten, as I plan to do, this fight was inevitable - that's what we'd have called it if we'd promoted it, Inevitable.
DeGALE: It had to happen now otherwise it wouldn't have happened because he's ready to get beat and I want to be the first one to beat him. He's gonna get beat by the Golden Boy James DeGale, so he can come again.
Did you expect it to happen so early?
GROVES: I did, I genuinely did. As soon as I realised he wanted to fight, I knew it was gonna happen. That's why I strategically went down against Kenny Anderson, just to grease the wheels. They were peachy-keen after that!
DeGALE: I've always been telling Frank to get it on and Frank wanted to make it as well. It was just all the politics, them going back and forth with Frank, that delayed it. That was all bull**** but it does't matter now... the fight's on.
Have you followed each other's pro careers closely?
GROVES: Not closely. I've kept an eye on him because he's a super-middleweight and therefore a rival; you keep an eye on them make sure they're all coming along. Once he started stepping up he was not performing well, like against Nathan King, or they were getting blown over in a couple of minutes like Carl Dilks. His defining fight was against Smith - that's probably the fight I've seen the most.
DeGALE: Obviously I watch Saturday Fight Night every week so you get to see everyone. I've watched him - and I'm not impressed to be honest. He's got more balls than brains this guy - but I still see fear.
What's been different about this camp?
GROVES: Apart from the fact I got some sunshine when I started? We've kicked it up a notch. It's been intense, really intense. We've done a lot of sparring because it's been a long time since I got in with a southpaw.
DeGALE: We ain't changed nothing. It's just been the hard graft as normal, a lot of sparring and a lot of hard graft.
What would defeat mean to you?
GROVES: I literally don't have a clue because I've not thought about it; as a boxer you never think about getting beat. I think about situations I might be in during the fight because it's no good when you're doing your long jogs and you're thinking about bashing someone up in the corner. I think about if I get trapped, what am I going to do? When I get caught with this shot, what am I going to do? But think about losing? It's never entered my head.
DeGALE: I've not even thought about it because it ain't gonna happen. All I've been thinking about is how I'm going to win, how I'm going to knock him out. I can't wait to knock him out.
What would victory mean to you?
GROVES: I'm going in with one and coming out with two - that was one of my friend's lines. I've always wanted the British title, it's a lovely belt, the history, the heritage. But I haven't thought about where I would go from there, I haven't mentioned it to Adam. We're happy where we'll be, my profile will have been raised and there'll be more big, big fights in 2012.
DeGALE: I need to get this out of the way. This is just an itch on my shoulder and once I've got rid of the itch I can move on. And once I've got rid of that itch, I don't want to hear his name agaiin. Everyone keeps asking me about this ugly ginger kid; who is he? Can you deal with him? He keeps running his mouth.
Give us your prediction...
GROVES: It won't go the distance. He'll get stopped; he'll either get knocked out or I genuinely believe Jim McDonnell's going to throw the towel in. And he won't be able to speak to poor Jim again.
DeGALE: Ugly Kid will get knocked out in four rounds if he comes to fight; if he tries to make it technical it might last a couple more rounds. But it doesn't matter, I'm going to get the 'w'. I just want to knock him out early - and look good doing it.
And finally... will you make up afterwards?
GROVES: I'm not sure. It depends... he's definitely not coming to the family barbecue! It would be nice to think we'll shake hands at the end, as you always do. I don't hate anyone I box, I don't hate James DeGale - but I definitely don't like him. It would mean nothing if we did shake hands.
DeGALE: I will probably shake his hand once I've knocked him out, give him a pat on the head and say 'don't worry son, you'll be back'.


text:
Guide In Purchasing A Compression Garment
It would be important to see that for so many athletes out there that are training the hard way they could, even the smallest of change that they are experiencing with respect to their training can certainly make a very big difference along the way. Apart from all the vitamins, and supplements, there are also some accessories that athletes need to equip themselves with so that it can somehow help them with the performance. There are so many athletes out there that are making use of compression garments so that it can help sustain them in their gameplay. There is the need for you to get familiar with what a compression garments is all about, and that it is important that you will see that it is a garment that is made up from spandex materials and that it is stretchy for that matter so that it can somehow help you with the blood flow that is going through your body.
With such a garment then you will see that you can actually recover fast and that you will be able to last long in the kind of training that you are doing and that you will be certainly enjoying the activities that you do. The use of the compression garments will actually allow you to get the necessary stabilization and that you will be happy that it is something that will somehow protect you from any forms of injury that might actually happen to you. You will certainly be enjoying the winning streaks that you are going to see as you are having the best performance as aided with the use of the garment that will help you perform you best so that you can always see yourself as someone that is really above and ahead of others. There are now a lot of technologies that are being added on the compression garments that you are wearing and that it is essential that you will know what is the one that can fully help you out.
There are so many innovative concepts that you must know, and that is why, before you even begin on purchasing the compression garments, you must first do proper research to what you think is ultimately the right one for you. There is a need for you to check on what is the kind of garment that will really help you along the way so that you will see to it that you are getting the most out from what you are seeing.
Always see to it that you get the garment that will ensure that you will be supporting you in all the things that you do, and that you must make sure that you get only the best and high quality ones.
Source: Related Site


text:
Critically appreciate the essay 'A Piece of Chalk' by G.K.Chesterton?
- print Print
- list Cite
The essay 'A Piece of Chalk' by G.K. Chesterton was typical of those writings of his which reflected on life's deeper purpose, meaning and facilitation which were brought on by the simplest of life's little daily events - in this case forgetting a white color and picking up a piece of soft rock (chalk) to write with instead. He was not the first to write like this - similar famous reflections include those by St Augustine and Thomas Aquinas (to hold all the world in 'a grain of sand.')
Chesterton was staying in the south of England and went out the chalk downs to draw. He brought all his chalks with him except the most important one - white! He had to pick up a chalky pebble from nearby to finish his work and got to thinking - that England was a piece of chalk.
Some things you might want to crtically appreciate - is that staement true or simplistic? Did he twist the concept to fit his reflections? (England has chalk cliffs in the south-east corner - it also has coal and very many igneous rocks,granite moors and headlands in the west country,slate,shale,London Beds and shale!)
Look for paradox (white can be a color - if it's used on a dark background) is that true? Did Chesterton did have our benefit of science/light technology education?
Look for other 'ordinary' things in the essay (brown paper) and comment on those in the same way.


text:
My first video ever - when I started swinging
Tags: homemade mom wife
Today it was already quite late when I started my hunt.
Tags: blowjobs gays reality twink
Gay movie of I started by wiggling him a bit and when he
Tags: amateur gays twink
Free medical porn de and naked gay medical exams full length I started out with the
Tags: amateur gays medical nude student twink
getting eatin out sitting on his face when i cum
Tags: amateur ass booty cum face sitting fingering latin
I started touching it and stroking it and it felt harder than a rock and he started moaning quietly
Tags: gays moaning
Masha was surprised when she saw that I took out the camera
Tags: amateur cumshots teens
Riley certainly is shy when it comes to the power of black dick. She's used to getting fucked by white boys. When I say "fuck" I mean i
Tags: amazing black dick hardcore shy white
I thought gym would be a great location for picking up sexy girls. Really fit and sporty girls who like to work out... I love that :) But when Edik an
Tags: amateur gym hardcore pretty sporty
Jayden is eager to get the fun started when he starts
Tags: blowjobs funny gays group sex
i started sucking up and down as fast as I could go
Tags: gays sucking twink
I picked up Veronika on the street a sunny day and brought her to a nearby grass field where we started to have sex in public. Veronika turned out to
Tags: ass asshole public sunny day tight
Nika was at home all alone when she started to feel horny. S
Tags: babes brunettes european homemade masturbation solo
When I first met this teen cutie I never thought I'd be
Tags: blowjobs brunettes hardcore russian small tits teens
I must anal masturbate when i ejaculate am i gay and twinks using condoms
Tags: anal emo gays masturbation shaved solo toys
Finally finishing what I started...
Asian ass-whore Lucy Lee was at the top of our Rear POV list when we started
Tags: anal asian lucy lee pov
I haven't even started yet
Tags: asian bondage brunettes couple domination femdom fetish
Kylie Ireland started out as a standard regulation porn whore
Tags: group sex kylie ireland orgy whore
I started watching porn with this woman in 1988
Tags: interracial matures pornstars
I know that he loves when I do this
Tags: blowjobs deep throat gagging handjobs sucking
I bet that she loves when I do this
I know that she loves when I do this
Tags: amateur dildo dyke kiss lesbians licking masturbation
When I think about you I touch myself
Tags: erotic
TrafficHolder.com - Buy and Sell Adult Traffic|Trafficshop.com - Let your business grow|Contact Us|Webmaster
All content, including images, text, graphics, video and audio content contained in the website is protected under the laws of copyright, owned or under license to company or its designees,and represents proprietary and valuable intellectual property.
Contact Webmaster 18 U.S.C. 2257 Record-Keeping Requirements Compliance Statement
Powered by Stream Rotator


text:
Cinque Terre: 5 pretty villages on Italy's west coast
Cinque Terre (pronounced CHEEN-kway TEHR-reh) is "five lands" perched upon a stretch of lush, terraced hills on Italy's western coastline halfway between Genoa and Pisa. It is a place of unrivaled beauty, a national park, a United Nations World Heritage Site.
On the southbound train that brings visitors from Genoa, the shimmering Ligurian Sea is on one side while the five towns - the "lands" of Cinque Terre - appear one after another as pastel jumbles dotting the rolling slopes of green.
Monterosso, Vernazza, Corniglia, Manarola, Riomaggiore - each punctuated by the railroad tunnels in between.
Roads as we know them don't exist here, and even the few residents with vehicles have to park their cars and motorbikes behind gates that separate the villages from the outside world. There aren't many museums here, either, but why should there be?
Nearly all the buildings stand five stories tall, with green wood shutters and well-worn pastel façades of pink, yellow, purple and orange. Time has faded the paint, leaving a rustic warmth behind.
Peak tourism season in Cinque Terre falls during the summer months. Expect sold-out guest rooms, shoulder-to-shoulder crowds (even on some parts of the hiking trails) and sticky humidity. Go between September and early April and you'll likely have the place all to yourself.
The five villages are living, breathing museums of the Italian essence. No matter what village you're in, you'll walk the narrow pedestrian ways among apartments, trattorias and olive oil purveyors that could be no place else. You can base yourself anywhere and move among them as you wish by train, boat or hiking trails - only 7 miles separate the five villages. You just can't drive beyond Monterosso. Like five sisters, each village in the Cinque Terre has its own personality:
IT'S LIKE . . . the big sister.
YOU'LL FIND It's the closest thing to a resort community, with a sandy beach, several hotels and the most modern amenities.
GOOD TO KNOW The largest of all the villages, Monterosso is known for its good trattorias. A short tunnel separates one side of the village from the other.
IT'S LIKE . . . the pretty prom queen.
YOU'LL FIND With its town square and natural harbor, it's the most picturesque and the favorite Cinque Terre town among tourists.
GOOD TO KNOW People-watching is a favorite pastime. Visitors can grab a picnic lunch and settle themselves anywhere on the long dock that juts out into the harbor to watch the power boats come in, or gaze up at hikers trekking the long trail that winds down the side of the hill.
IT'S LIKE . . . the quiet sister who keeps to herself.
YOU'LL FIND Perched high above the water, it doesn't feel like a tourist trap (it's a grueling 365-step stair climb from the train station to town center, or ride the convenient $2 shuttle bus).
GOOD TO KNOW Located in the middle of Cinque Terre, Corniglia is the only village that isn't situated directly on the sea - rather, it's perched high up on a hill with vineyards and lush terraces that step down to the water.
Manarola and Riomaggiore
THEY'RE LIKE . . . fraternal twins.
YOU'LL FIND They're similar in layout and feel, with a workaday rhythm and intimate, homey charm.
GOOD TO KNOW The two villages are connected by a paved path, Via dell'Amore (the road of love) - it's a leisurely 20-minute stroll.
Milan is the best international gateway to Cinque Terre. From Milan's Central Station (Milano Centrale), it's a three-hour train ride to Monterosso for about $30. Add roughly another $10 for first-class comfort. More information at trenitalia.com.
Consider Genoa, halfway between Milan and Cinque Terre, a good rest stop if three hours on a train is too much after a long flight.
Buy a Cinque Terre Train Card at the train station the minute you arrive. A two-day pass costs about $20 (one day is $11), which covers unlimited train rides between the five villages and access to the hiking trails and museums (by "museum," they mean small rooms with glossy photos on the walls). Revenue from the card helps maintain the national park. The card does not cover boat rides, which run between the villages and beyond during spring and summer; one-way tickets begin at $16 (navigazion egolfodeipoeti.it).
There are plenty of guest room-locating services - to find one, step off the train and throw a rock in any direction. You'll also find locals soliciting their private rooms at the train station to arriving visitors. Look for ones advertising "camere con bagno" - a room with private bathroom. Know that many places accept only cash. And if you can't find a room (especially during the summer months), consider nearby La Spezia home base, 10 minutes by train from Riomaggiore. Levanto, five minutes by train north of Monterosso, is a prettier option.
In Vernazza, Gambero Rosso is a sit-down restaurant by the harbor serving top-notch (and fairly expensive) Ligurian cuisine (the address is Piazza Marconi 7).
In Monterosso, Ristoranti Il Gabbiano is just steps away from the train station and serves simple, inexpensive pastas, such as the ubiquitous spaghetti al frutti di mare, the tomatoey seafood pasta, for $9 (Via Fegina 78).
The spaghetti al cartoccio at Riomaggiore's Ristorante la Lampara -just like spaghetti al frutti di mare, but steam-baked in foil - was especially enjoyable (Via Malborghetto 10).


text:
Advanced Search |
Search Tips
Everyone's favorite! Don't be caught w/out green this year...get a 1/2 pound of our tasty Velvet Mints.
We promise to never spam you, and just use your email address to identify you as a valid customer.
This product hasn't received any reviews yet. Be the first to review this product!
Click the button below to add the St. Patrick's Day - Velvet Mints to your wish list.
All prices are in USD
Copyright 2017 Weiser Classic Candy.
Web Design by J&S Design House
Sitemap |


text:
FREE SHIPPING on orders of $85 or more
An excerpt from www.HouseOfNames.com archives copyright © 2000 - 2017
occupational name for an armorer.
Penser Early Origins
Westphalia, where the name emerged in mediaeval times as one of the notable families in the western region. From the 13th century onwards the surname was identified with the great social and economic evolution which made this territory a landmark contributor to the development of the nation.
Penser Spelling Variations
Penser Spelling Variations
Spelling variations of this family name include: Panser, Panse, Pansern, Pansen, Pansey, Panzer, Pantzer, Pantzner, Panzner, Panzen and many more.
Penser Early History
Penser Early History
This web page shows only a small excerpt of our Penser research. Another 145 words (10 lines of text) covering the years 1654, 1634, 1633 and 1703 are included under the topic Early Penser History in all our PDF Extended History products and printed products wherever possible.
Penser Early Notables (pre 1700)
Penser Early Notables (pre 1700)
More information is included under the topic Early Penser Notables in all our PDF Extended History products and printed products wherever possible.
The Great Migration
The Great Migration
Some of the first settlers of this family name or some of its variants were: Friedrich Panzer, who came to America in 1783; Joseph Panser, who arrived in New York, NY in 1824; Charles Panzer, who settled in New York in 1838; Eduard Panzer, who came to New York, NY in 1853.
Penser Family Crest Products
Penser Family Crest Products
See Also
See Also
Other References
The Penser Family Crest was acquired from the Houseofnames.com archives. The Penser Family Crest was drawn according to heraldic standards based on published blazons. We generally include the oldest published family crest once associated with each surname.
This page was last modified on 28 November 2015 at 08:05.
Sign Up
FREE SHIPPING on orders of $85 or more


text:
Original paper
Phalacroma gibbosum sp. nov. (Dinophyceae) from the southern Gulf of Mexico
Parra-Toriz, Dulce; Hernández-Becerril, David Uriel; Esqueda-Lara, Karina
Nova Hedwigia Band 99 Heft 1-2 (2014), p. 83 - 96
published: Aug 1, 2014
Within the marine dinoflagellates, the order Dinophysales includes members with very distinctive morphology and ecology. During a survey of the phytoplankton from the southern Gulf of Mexico, three specimens of an unknown dinophysoid species were found in formalin-fixed net samples, and were studied by light and scanning electron microscopy. A new species is proposed and formally described: Phalacroma gibbosum sp. nov. Its most characteristic features are as follows: (1) a distinctive trapezoid shape, (2) epitheca low with an elevation in the dorsal part of the epitheca (hump), (3) cingular lists fairly developed, with strong but discontinuous ribs, (4) an elongation of the posterior end of the hypotheca, (5) a mamilliform structure in the ventral margin of the hypotheca, and (6) theca covered by deep, circular areolae, with small pores, one per 3–5 areolae. This new species is morphologically similar and presumably closely related to Phalacroma favus and P. rapa. Molecular phylogenetic studies grouped these two species together with true Phalacroma species (including the type species), and this is regarded as an indication that Phalacroma gibbosum is also a member of that genus.
dinoflagellates • dinophysales • gulf of mexico • new species • phytoplankton


text:
Home Posts Tagged "Scenic QC"
Scenic Queen Creek, Sunlight Over the Mountains
Feb 5th, 2017 · Comments:
Queen Creek photographer April Haroldsen photographed sunlight peeking through the storm clouds over the Superstition Mountains in November. The Queen Creek Independent encourages residents to submit their favorite scenic or wildlife photos taken within the community, with a short description of the shot, for publication with photo credit. Do not use a filter on submitted […]


text:
Introduction to Fluid Mechanics is a mathematically efficient introductory text for a basal course in mechanical engineering.
An outgrowth of the development of the one-semester course in fluid mechanics that is required of all mechanical engineering students at MIT, this text emphasizes the important principles of the conservation of mass, momentum, and energy as they apply to a fluid. At each stage of the exposition, practical engineering problems are solved, principally in engineering systems such as dams, pumps, turbines, pipe flows, propellers, and jets, but with occasional illustrations from physiological and meteorological flows. Annotation c. Book News, Inc., Portland, OR (booknews.com)


text:
Basidomycota. Maggie Blackmon, Carolyn Poliseo , Jaime Swank. Hyphae. Septate hyphae Septa divide hyphae into cells Pores allow nuclei, ribosomes, and mitochondria to travel. Spore formation. Sexual or asexual Formed from hyphae, basida , yeast, and other basidiospores Cool video.
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
Maggie Blackmon, Carolyn Poliseo, Jaime Swank


text:
WITH blond tussled hair and an electronic guitar in hand, the tempestuous Mozart who now struts the stage in Shanghai is hardly the legendary 18th-century virtuoso whom most people imagine.
Composing at the age of five, writing symphonies by nine, and completing his first opera at 11, Mozart is regarded with saint-like reverence in the classical music world. Yet, as playwright Michael Kunze sees it, Mozart's genius led him to tragedy rather than happiness. He died alone at only 36.
"I believe that Mozart was eager to devote himself to a life filled with happiness and love, which can be discovered in his memoirs; but he could not do it. He saw it as his responsibility to make full use of his talents," says Kunze, "He was torn between this conflict throughout his life. It's been estimated that it would take seven years to copy all of his scores by hand. You can imagine how he'd been working constantly."
The Austrian musical "Mozart," staged at SAIC Shanghai Culture Square until January 15, is meant to help audiences understand the tumultuous inner world of the great composer. Since its premier at Theater an der Wien in Vienna, the musical has toured seven counties and been seen by more than 1.9 million people.
"Everybody knows about Mozart as a musical genius with a short life. But hardly anyone knows him as a real person. How did he die? How did he move from Salzburg to Paris and Vienna? How did he get along with his father, mother and wife?" asks Andrea Friedrich, producer of the musical.
In the German-language show, Mozart is more than just a gifted musician, but also a young man with a rebellious heart and an intense longing for freedom. To win his self-respect, he broke with the archbishop who had long sponsored his work. He went to Vienna to build his career but failed to win his father's approval. He married a loving woman yet lost her due to his relentless devotion to music.
On stage, Mozart's talent is represented by a boy in a red court costume and traditional wig, while the grown Mozart is depicted much like a rock star. The boy is named Amade and he carries a mysterious box throughout the show.
"Amade is part of Mozart, while the mysterious box symbolizes his gift, the source of his inspiration and possibly the whole musical world. Without his gift, Mozart would not have been the famous Mozart, but it was also this gift that ruined his life," says Kunze.
As show reaches its tragic climax, Amade starts to compose with Mozart's own blood, and in the end stabs his pen into the adult composer's heart to get the final drops out of him.
According to Oedo Kuipers who plays Mozart in the musical, it is quite a demanding task to enter the role every night.
"Mozart is a dramatic character with rich emotions like joy, sorrow, love, jealousy and fear. I am experiencing his rise and fall every night," he says. "And it's very difficult to balance between acting and singing. I cannot roar while singing even if the emotion leads me there."
The music in the show was composed by Sylvester Levay, who infuses the musical with pop, jazz and rock. Excerpts from Mozart's composition are also used throughout the play.
Though the story is set in 18th-century Austria, the costumes for the characters vary from traditional Baroque dress to ripped jeans in an attempt to connect the real-life historical drama with modern sensibilities.
Mozart appears throughout the show in a pure white suit. His outfit sets him apart from the colorful cast, just as his gifts and passions made him a misfit during his short life.
Click HERE to buy tickets.
Date: Through January 15, 2pm, 7:15pm
Venue: SAIC Shanghai Culture Square, 597 Fuxing Rd M.
Tickets: 80-1280 yuan
Tel: 021-5292 0164
Shanghai Culture Square
From ¥280
From ¥35
沪公网安备 31010602000209号


text:
I drew the image and made the word backing paper by writing with different pens and a white pencil
I am in the process of making some more cards with other images I have created.
I know lots of teenage girls who would love a card like that! Great design!
Gorgeous as usual Nicola. Lovely colours :)
Love the image! Great card!
Oooh, I commented on this on CB forum, but I didn't realise you drew the image yourself!
Wow, I bow down at your talent!
Jules x
Fantastic image and background. You are very talented!!!!


text:
Sign In
Sign Out | Hello, null
Browse more:
These tools are used for fine sculpting, detailing, and trimming when working with clay, greenware, plaster, wax, bread dough, or any other modeling medium.
Set of 6 tools. Each tool is approximately 5½" long.
(based on 4 reviews)
Ratings Distribution
5 Stars
4 Stars
3 Stars
2 Stars
1 Stars
of respondents would recommend this to a friend.
Reviewed by 4 customers
Displaying reviews 1-4
Back to top
Dick Blick Art Supplies Are Excellent!
By Jackie
from Rockford, Michigan
Comments about Mini Ribbon Clay Sculpting Set:
I use this product sculpting clay and creating small clay pottery.
Bottom Line Yes, I would recommend this to a friend
Was this review helpful? Yes / No
- You may also flag this review
(1 of 1 customers found this review helpful)
not the same as a kemper
By claygirl
from annapolis,MD
blades are not as sharp as the Kemper set.
Bottom Line No, I would not recommend this to a friend
(2 of 2 customers found this review helpful)
Works well with Porceilan
By Magnus
from The Jersey Shore, NJ, USA
These were great! They worked well with porceilan. The size is perfect and the fact they only have a "tool" on one end. The non-slip handle is great also because you can get close to the end of the tool for more control. also they are well made. I bought these for myself they were not a gift.
Sturdy and well made
By Laurie
from Juneau, AK
Great for precise carving. The different tips are all well made.
Loading item data...
Please allow ADDITIONAL DELIVERY TIME for factory-shipped items.
Click here to view expected shipping dates, or call our Order Dept (800)-828-4548.
Copyright © 2015 Dick Blick Art Materials. All rights reserved.
Your cart is currently empty.
Your cart currently contains N item.
Dick Blick Art Materials · P.O. Box 1267 · Galesburg, IL 61402-1267 · Toll-free Phone (800) 828-4548 International Phone +1-309-343-6181 ext. 5402 · Fax (800) 621-8293
Dick Blick Art Materials®, Blick®, Blick Studio®, and Artists Pick Blick® are registered trademarks ofDick Blick Holdings Inc. © Copyright 1999-2017 Dick Blick Holdings Inc. All rights reserved.


text:
As the online world grows so does the voices of experience across all spectrums of knowledge.
In this latest issue of 4Ward! Magazine, MIND POWER will be focussing on how you can find personal growth as an entrepreneur and learn more from online learning and sharing from people who have been doing business for years. One of the more popular channels of communication are podcasts - a recorded sound or video medium in which the author shares knowledge, talks about a particular subject, and conducts interviews amongst many other things.
So what is out there for you to learn more from the likes of Patt Flynn, John Lee Dumas and Chris Ducker? Find out more in Online learning for Young Entrepreneurs