A curve shortening flow rule for closed embedded plane curves with a prescribed rate of change in enclosed area

Motivated by a problem from fluid mechanics, we consider a generalization of the standard curve shortening flow problem for a closed embedded plane curve such that the area enclosed by the curve is forced to decrease at a prescribed rate. Using formal asymptotic and numerical techniques, we derive possible extinction shapes as the curve contracts to a point, dependent on the rate of decreasing area; we find there is a wider class of extinction shapes than for standard curve shortening, for which initially simple closed curves are always asymptotically circular. We also provide numerical evidence that self-intersection is possible for non-convex initial conditions, distinguishing between pinch-off and coalescence of the curve interior.

Duration from vine emergence to flowering suggests a long-day or rate of change of photoperiod response in white yam (Dioscorea rotundata Poir.)

The objective of this study was to quantify the effect of photoperiod on the duration from vine (shoot) emergence to flowering in white or Guinea yam (Dioscorea rotundata). The duration from vine emergence to flowering in two clonal varieties of yam (TDr 131 and TDr 99-9) was recorded at 10 different sowing dates/locations in Nigeria. Durations to flowering varied from 40 to > 88 days. Mean daily temperature and photoperiod between vine emergence and flowering varied from 25 to 27 degrees C and 13.1 to 13.4 h day(-1), respectively. Both clones had similar responses to temperature, with base and optimum temperatures of 12 and 25-27 degrees C, respectively. Thermal durations to flowering were strongly related (r(2) > 0.75-0.83) to absolute photoperiod (h) at vine emergence as well as to rate of change of photoperiod (s day(-1)) at vine emergence. The response to absolute photoperiod suggests that white yams are quantitative LDPs, flowering sooner in long than short days. Yams also flowered earlier when the rate of change of photoperiod was positive but small, or was negative. It is suggested that yams may use a combination of photoperiod and rate of change in order to fine tune flowering time. (c) 2006 Elsevier B.V. All rights reserved.

Duration from vine emergence to flowering suggests a long-day or rate of change of photoperiod response in white yam (Dioscorea rotundata Poir.)

The objective of this study was to quantify the effect of photoperiod on the duration from vine (shoot) emergence to flowering in white or Guinea yam (Dioscorea rotundata). The duration from vine emergence to flowering in two clonal varieties of yam (TDr 131 and TDr 99-9) was recorded at 10 different sowing dates/locations in Nigeria. Durations to flowering varied from 40 to > 88 days. Mean daily temperature and photoperiod between vine emergence and flowering varied from 25 to 27 degrees C and 13.1 to 13.4 h day(-1), respectively. Both clones had similar responses to temperature, with base and optimum temperatures of 12 and 25-27 degrees C, respectively. Thermal durations to flowering were strongly related (r(2) > 0.75-0.83) to absolute photoperiod (h) at vine emergence as well as to rate of change of photoperiod (s day(-1)) at vine emergence. The response to absolute photoperiod suggests that white yams are quantitative LDPs, flowering sooner in long than short days. Yams also flowered earlier when the rate of change of photoperiod was positive but small, or was negative. It is suggested that yams may use a combination of photoperiod and rate of change in order to fine tune flowering time. (c) 2006 Elsevier B.V. All rights reserved.

Revealing conceptions of rate of change

Rate of change is an important mathematical concept. Research referring to students’ difficulties with this concept spans more than twenty years. Research suggests that problems experienced by some calculus students are likely a result of pre-existing limited or incorrect conceptions of rate of change. This study investigated 23 Victorian Year 10 students’ understanding of rate as revealed by phenomenographic analysis of interviews. Eight conceptions of rate of change emerged. Four important aspects of the concept were identified and gaps in students’ thinking defined. In addition, the employment of phenomenography, to reveal conceptions of rate, is described in detail.

Potential of technology and a familiar context to enhance students' concept of rate of change

Students' concept image of rate of change may be incomplete or erroneous. This paper reports a pilot study, with secondary school students, which explores the potential of technology (JavaMathWorlds), depicting a familiar context of motion, to develop students' existing schema of informal understandings of rate of change to more formal mathematical representations. Students developed numerous 'models of' rate of change in a motion context which then transferred to serve as a 'model for' rate of change in other contexts.

Video evidence : what gestures tell us about students' understanding of rate of change

This paper reports on insights into students’ understanding of the concept of rate of change, provided by examining the gestures made, by 25 Year 10 students, in videorecorded interviews. Detailed analysis, of both the sound and images, illuminates the meaning of rate-related gestures. Findings indicate that students often use the symbols and metaphors of gesture to complement, supplement, or even contradict verbal descriptions. Many students demonstrated, by the combination of their words and gestures, a sound qualitative understanding of constant rate, with a few attempting to quantify rate. The interpretation of gestures may provide teachers with a better understanding of the progress in their students’ thinking.

An 'emergent model' for rate of change

Does speed provide a ‘model for’ rate of change in other contexts? Does JavaMathWorlds (JMW), animated simulation software, assist in the development of the ‘model for’ rate of change? This project investigates the transference of understandings of rate gained in a motion context to a non-motion context. Students were 27 14–15 year old students at an Australian secondary school. The instructional sequence, utilising JMW, provided rich learning experiences of rate of change in the context of a moving elevator. This context connects to students’ prior knowledge. The data taken from pre- and post-tests and student interviews revealed a wide variation in students’ understanding of rate of change. The variation was mapped on a hypothetical learning trajectory and interpreted in the terms of the ‘emergent models’ theory (Gravemeijer, Math Think Learn 1(2):155–177, 1999) and illustrated by specific examples from the data. The results demonstrate that most students were able to use the ‘model of’ rate of change developed in a vertical motion context as a ‘model for’ rate of change in a horizontal motion context. A smaller majority of students were able to use their, often incomplete, ‘model of’ rate of change as a ‘model for’ reasoning about rate of change in a non-motion context.

Effect of rate of change of slip on the peak braking coefficient of passenger car tires. Technical report.

Mode of access: Internet.

Optimal Controller Designs for Rotating Machines - Penalising the Rate of Change of Control Forcing

In the context of active control of rotating machines, standard optimal controller methods enable a trade-off to be made between (weighted) mean-square vibrations and (weighted) mean-square currents injected into magnetic bearings. One shortcoming of such controllers is that no concern is devoted to the voltages required. In practice, the voltage available imposes a strict limitation on the maximum possible rate of change of control force (force slew rate). This paper removes the aforementioned existing shortcomings of traditional optimal control.

Patterns of coronary heart disease mortality over the 20th century in England and Wales : possible plateaus in the rate of decline

Background
Coronary heart disease (CHD) rates in England and Wales between 1950 and 2005 were high and reasonably steady until the mid 1970s, when they began to fall. Recent work suggests that the rate of change in some groups has begun to decrease and may be starting to plateau or even reverse.

Methods
Data for all deaths between 1931 and 2005 in England and Wales were grouped by year, sex, age at death and contemporaneous ICD code for CHD as cause of death. CHD mortality rates by calendar year and birth cohort were produced for both sexes and rates of change were examined.

Results
The pattern of increased burden of CHD mortality within older age groups has only recently emerged in men, whereas it has been established in women for far longer. CHD mortality rates among younger people showed little variation by birth cohort. For younger women (49 and under), the rate of change in CHD mortality has reversed in the last 20 years, indicating a future plateau and possible reversal of previous improvement in CHD mortality rates. Among younger men the rate of change in CHD mortality has been consistent for the past 15 years indicating that rates in this group have continued to fall steadily.

Conclusion
Although CHD mortality rates continue to drop in older age groups the actual burden of coronary heart disease is increasing due to the ageing of the population. The rate of improvement in CHD mortality appears to be beginning to decline and may even be reversing among younger women.

Change in the effect of constant photoperiods on the rate of sexual maturation in modern genotypes of domestic pullet

1. Data for modern egg-type hybrids reared on constant daylengths show that, as expected, they mature more quickly than earlier genotypes. However, the constant photoperiod which gives earliest sexual maturity has not changed as a result of selection and is 10 h for both early and modern genotypes. 2. Further analysis showed that the rate of delay in sexual maturity for constant photoperiods above 10 h is similar for modern and for early hybrids ( +0.29 d for each incremental one hour of photoperiod), the response of modern hybrids below 10 h ( +4.22 d for each one-hour reduction in photoperiod) is more than double that of early hybrids ( +1.71 d/h).