Numerical study of two ill-posed one-phase Stefan problems

We treat two related moving boundary problems. The first is the ill-posed Stefan problem for melting a superheated solid in one Cartesian coordinate. Mathematically, this is the same problem as that for freezing a supercooled liquid, with applications to crystal growth. By applying a front-fixing technique with finite differences, we reproduce existing numerical results in the literature, concentrating on solutions that break down in finite time. This sort of finite-time blow-up is characterised by the speed of the moving boundary becoming unbounded in the blow-up limit. The second problem, which is an extension of the first, is proposed to simulate aspects of a particular two-phase Stefan problem with surface tension. We study this novel moving boundary problem numerically, and provide results that support the hypothesis that it exhibits a similar type of finite-time blow-up as the more complicated two-phase problem. The results are unusual in the sense that it appears the addition of surface tension transforms a well-posed problem into an ill-posed one.

Outcomes and implications of one teacher's approach to context-based science in the middle years

In Australia, there is a crisis in science education with students becoming disengaged with canonical science in the middle years of schooling. One recent initiative that aims to improve student interest and motivation without diminishing conceptual understanding is the context-based approach. Contextual units that connect the canonical science with the students’ real world of their local community have been used in the senior years but are new in the middle years. This ethnographic study explored the learning transactions that occurred in one 9th grade science class studying a context-based Environmental Science unit for 11 weeks. Outcomes of the study and implications are discussed in this paper.

The pursuit of thinness : an outcome study of anorexia nervosa

INTRODUCTION: Anorexia nervosa (AN) is a growing problem among young female Singaporeans. We studied the demographics and follow-up data of AN patients referred to dietitians for nutritional intervention. METHODS: A retrospective nutritional notes review was done on 94 patients seen from 1992 to 2004. All patients were given nutritional intervention, which included individualised counselling for weight gain, personalised diet plan, correction of poor dietary intake and correction of perception towards healthy eating. We collected data on body mass index (BMI), patient demographics and outcome. RESULTS: 96 percent of the patients were female and 86.2 percent were Chinese. The median BMI at initial consultation was 14.7 kilogramme per square metre (range, 8.6-18.8 kilogramme per square metre). 76 percent were between 13 and 20 years old. 83 percent of the patients came back for follow-up appointments with the dietitians in addition to consultation with the psychiatrist. Overall, there was significant improvement in weight and BMI from an average 37 kg to 41 kg and 14.7 kilogramme per square metre to 16.4 kilogramme per square metre, respectively, between the fi rst and fi nal consultations (p-value is less than 0.001). The average duration of followup was about eight months. Among the patients on follow-up, 68 percent showed improvement with an average weight gain of 6 kg. Patients that improved had more outpatient follow-up sessions with the dietitians (4.2 consultations versus 1.6 consultations; p-value is less than 0.05), lower BMI at presentation (14.2 kilogramme per square metre versus 15.7 kilogramme per square metre; p-value is less than 0.01) and shorter duration of disease at presentation (one year versus three years; p-value is less than 0.05) compared with those who did not improve. Seven patients with the disease for more than two years did not show improvement with follow-up. CONCLUSION: We gained valuable understanding of the AN patients referred to our tertiary hospital for treatment, two-thirds of whom improved with adequate follow-up treatment. Patients that had suffered AN longer before seeking help appeared more resistant to improvement.

Risk of falls, injurious falls, and other injuries resulting from visual impairment among older adults with age-related macular degeneration

Purpose: Age-related macular degeneration (AMD) is the leading cause of irreversible visual impairment among older adults. This study explored the relationship between AMD, falls risk and other injuries and identified visual risk factors for these adverse events. Methods: Participants included 76 community-dwelling individuals with a range of severity of AMD (mean age, 77.0±6.9 years). Baseline assessment included binocular visual acuity, contrast sensitivity and merged visual fields. Participants completed monthly falls and injury diaries for one year following the baseline assessment. Results: Overall, 74% of participants reported having either a fall, injurious fall or other injury. Fifty-four percent of participants reported a fall and 30% reported more than one fall; of the 102 falls reported, 63% resulted in an injury. Most occurred outdoors (52%), between late morning and late afternoon (61%) and when navigating on level ground (62%). The most common non-fall injuries were lacerations (36%) and collisions with an object (35%). Reduced contrast sensitivity and visual acuity were associated with increased fall rate, after controlling for age, gender, cognitive function, cataract severity and self-reported physical function. Reduced contrast sensitivity was the only significant predictor of falls and other injuries. Conclusion: Among older adults with AMD, increased visual impairment was significantly associated with an increased incidence of falls and other injuries. Reduced contrast sensitivity was significantly associated with increased rates of falls, injurious falls and injuries, while reduced visual acuity was only associated with increased falls risk. These findings have important implications for the assessment of visually impaired older adults.

Shifting : one-inclusion mistake bounds and sample compression

We present new expected risk bounds for binary and multiclass prediction, and resolve several recent conjectures on sample compressibility due to Kuzmin and Warmuth. By exploiting the combinatorial structure of concept class F, Haussler et al. achieved a VC(F)/n bound for the natural one-inclusion prediction strategy. The key step in their proof is a d = VC(F) bound on the graph density of a subgraph of the hypercube—oneinclusion graph. The first main result of this paper is a density bound of n [n−1 <=d-1]/[n <=d] < d, which positively resolves a conjecture of Kuzmin and Warmuth relating to their unlabeled Peeling compression scheme and also leads to an improved one-inclusion mistake bound. The proof uses a new form of VC-invariant shifting and a group-theoretic symmetrization. Our second main result is an algebraic topological property of maximum classes of VC-dimension d as being d contractible simplicial complexes, extending the well-known characterization that d = 1 maximum classes are trees. We negatively resolve a minimum degree conjecture of Kuzmin and Warmuth—the second part to a conjectured proof of correctness for Peeling—that every class has one-inclusion minimum degree at most its VCdimension. Our final main result is a k-class analogue of the d/n mistake bound, replacing the VC-dimension by the Pollard pseudo-dimension and the one-inclusion strategy by its natural hypergraph generalization. This result improves on known PAC-based expected risk bounds by a factor of O(logn) and is shown to be optimal up to an O(logk) factor. The combinatorial technique of shifting takes a central role in understanding the one-inclusion (hyper)graph and is a running theme throughout.

Corrigendum to “Shifting : one-inclusion mistake bounds and sample compression” [J. Comput. System Sci. 75 (1) (2009) 37–59]

H. Simon and B. Szörényi have found an error in the proof of Theorem 52 of “Shifting: One-inclusion mistake bounds and sample compression”, Rubinstein et al. (2009). In this note we provide a corrected proof of a slightly weakened version of this theorem. Our new bound on the density of one-inclusion hypergraphs is again in terms of the capacity of the multilabel concept class. Simon and Szörényi have recently proved an alternate result in Simon and Szörényi (2009).

Shifting : one-inclusion mistake bounds and sample compression

We present new expected risk bounds for binary and multiclass prediction, and resolve several recent conjectures on sample compressibility due to Kuzmin and Warmuth. By exploiting the combinatorial structure of concept class F, Haussler et al. achieved a VC(F)/n bound for the natural one-inclusion prediction strategy. The key step in their proof is a d=VC(F) bound on the graph density of a subgraph of the hypercube—one-inclusion graph. The first main result of this report is a density bound of n∙choose(n-1,≤d-1)/choose(n,≤d) < d, which positively resolves a conjecture of Kuzmin and Warmuth relating to their unlabeled Peeling compression scheme and also leads to an improved one-inclusion mistake bound. The proof uses a new form of VC-invariant shifting and a group-theoretic symmetrization. Our second main result is an algebraic topological property of maximum classes of VC-dimension d as being d-contractible simplicial complexes, extending the well-known characterization that d=1 maximum classes are trees. We negatively resolve a minimum degree conjecture of Kuzmin and Warmuth—the second part to a conjectured proof of correctness for Peeling—that every class has one-inclusion minimum degree at most its VC-dimension. Our final main result is a k-class analogue of the d/n mistake bound, replacing the VC-dimension by the Pollard pseudo-dimension and the one-inclusion strategy by its natural hypergraph generalization. This result improves on known PAC-based expected risk bounds by a factor of O(log n) and is shown to be optimal up to a O(log k) factor. The combinatorial technique of shifting takes a central role in understanding the one-inclusion (hyper)graph and is a running theme throughout