Comparison between the forces and moments applied on the residuum of above-knee amputees during daily life activities and walking

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The regularity of the boundary of a multidimensional aggregation patch

Let $d \geq 2$ and let $N(y)$ be the fundamental solution of the Laplace equation in $R^d$ We consider the aggregation equation $$\frac{\partial \rho}{\partial t} + \operatorname{div}(\rho v) =0, v = -\nabla N * \rho$$ with initial data $\rho(x,0) = \chi_{D_0}$, where $\chi_{D_0}$ is the indicator function of a bounded domain $D_0 \subset R^d.$ We now fix $0 < \gamma < 1$ and take $D_0$ to be a bounded $C^{1+\gamma}$ domain (a domain with smooth boundary of class $C^{1+\gamma}$). Then we have Theorem: If $D_0$ is a $C^{1 + \gamma}$ domain, then the initial value problem above has a solution given by $$\rho(x,t) = \frac{1}{1 -t} \chi_{D_t}(x), \quad x \in R^d, \quad 0 \le t < 1$$ where $D_t$ is a $C^{1 + \gamma}$ domain for all $0 \leq t < 1$

Characterization of radially symmetric finite time blowup in multidimensional aggregation equations,

This paper studies the transport of a mass $\mu$ in $\real^d, d \geq 2,$ by a flow field $v= -\nabla K*\mu$. We focus on kernels $K=|x|^\alpha/ \alpha$ for $2-d\leq \alpha<2$ for which the smooth densities are known to develop singularities in finite time. For this range This paper studies the transport of a mass $\mu$ in $\real^d, d \geq 2,$ by a flow field $v= -\nabla K*\mu$. We focus on kernels $K=|x|^\alpha/ \alpha$ for $2-d\leq \alpha<2$ for which the smooth densities are known to develop singularities in finite time. For this range we prove the existence for all time of radially symmetric measure solutions that are monotone decreasing as a function of the radius, thus allowing for continuation of the solution past the blowup time. The monotone constraint on the data is consistent with the typical blowup profiles observed in recent numerical studies of these singularities. We prove monotonicity is preserved for all time, even after blowup, in contrast to the case $\alpha >2$ where radially symmetric solutions are known to lose monotonicity. In the case of the Newtonian potential ($\alpha=2-d$), under the assumption of radial symmetry the equation can be transformed into the inviscid Burgers equation on a half line. This enables us to prove preservation of monotonicity using the classical theory of conservation laws. In the case $2 -d < \alpha < 2$ and at the critical exponent $p$ we exhibit initial data in $L^p$ for which the solution immediately develops a Dirac mass singularity. This extends recent work on the local ill-posedness of solutions at the critical exponent.Comment: 30 page

Diablo Canyon power plant site ecological study Annual Report July 1, 1976 - June 30, 1977 and Quarterly Report no. 16 April 1, 1977 - June 30, 1977

More stations were surveyed during this report period than during any previous period. A total of 818 mandays was spent surveying 18 permanent and 67 random subtidal stations, 19 permanent and 50 random intertidal stations, and 64 sportfish catch-per-unit-of-effort stations, as well as conducting corollary laboratory work. In the subtidal areas, Laminaria dentigera and Pterygophora californica, both important subsurface kelp species, increased in their combined total numbers from the 1976 survey. However, Nereocystis luetkeana, the bull kelp, declined substantially in all study areas. Population trends of many of the dominant subtidal macro-invertebrates have varied depending on the species and on the type of method utilized. Regression analyses have been performed on selected species for numbers versus depth. Red algal abundance and diversity appear to have increased in both subtidal study areas. Of the three observation areas, sea otters were observed least frequently in Diablo Cove. However, there continues to be fresh evidence of otter foraging within Diablo Cove. Several seasons' data for intertidal algae and invertebrates have been summarized for one study area. While the algae biomass shows a fairly clear seasonality of abundance, the trends in numbers of the six invertebrate species considered are not as well defined. Because of high variability in the data, the sportfish catch-per-unit-of-effort and hook-and-line study was cancelled in December. Populations of intertidal red and black abalones, Haliotis rufescens and H. cracherodii, respectively, appear to have remained fairly stable during the 1976-77 period. Two other ancillary studies were also terminated during this period: interviews of commercial abalone and urchin fishermen, and observations of foam in Diablo Cove. (107pp.

Diablo Canyon power plant site ecological study Quarterly Report no. 22: October 1 - December 31, 1978

Field work for this quarter consisted of completion of random 1/4-m2 subtidal stations from the summer sampling season. In addition, one permanent subtidal station was surveyed. No intertidal stations were surveyed. Sea otters, Enhydra lutris, remained scarce in the vicinity of the power plant. Lab work was comprised of processing subtidal algae samples and sorting and identifying invertebrates from intertidal samples. Analysis of preoperational data for final report proceeded apace. (7pp.

Diablo Canyon power plant site ecological study Quarterly Report no. 19: January 1 - March 31, 1978

Field work was limited to monitoring permanent intertidal and subtidal stations. Only a few stations were surveyed due to a long series of winter storms which produced large seas. Office and laboratory work was devoted to completing computer data sheets for keypunching, completing the 1976/77 annual report and several quarterly reports, and processing the remaining algae samples collected at subtidal and intertidal stations in 1977. Sea otter activity increased in South Cove and Diablo Cove where sea otters were observed on five out of 20 observation days. (11pp.

Diablo Canyon power plant site ecological study Quarterly Report no. 18; October 1 - December 31, 1977

Large storm-generated swells prevented us from completing the random stations in the North Control. The project has shifted to a low level of field work to allow for analysis of all data and preparation of the final report. Field work during the next year will be confined to monitoring permanent stations. Four permanent subtidal stations were surveyed during the quarter, including a new station located at the entrance of Diablo Cove. Nine random 30m2 stations and 32 - 1/4m2 quadrats were also completed. It appears, from this year's subtidal studies, that there has been a decline in the abundance of lingcod, Ophiodon elongatus, and kelp greenling, Hexagrammos decagrammus, in the Diablo Canyon area since our studies began in 1974. A few sea otters continue to raft and forage in Lion Rock Cove. (12pp.

Diablo Canyon power plant site ecological study Quarterly Report No. 21: July 1 - September 30, 1978

Rough seas and the hiring freeze further delayed completion of our field and laboratory work. All random 1/4-m2 stations were completed in Diablo Cove but ten stations remain in the North Control. Nine random fish species counts were conducted in Diablo Cove. Very few sea otters, Enhydra lutris, were observed during this quarter, none in Diablo Cove. (10pp.