Capillary surface discontinuities above reentrant corners

A particular configuration of a vertical capillary tube for which S is the equilibrium interface between two fluids in the presence of a downward pointing gravitational field was investigated. S is the graph a function u whose domain is the (horizontal) cross section gamma of the tube. The mean curvature of S is proportional to its height above a fixed reference plane and lambda is a prescribed constant and may be taken between zero and pi/2. Domains gamma for which us is a bounded function but does not extend continuously to d gamma are sought. Simple domains are found and the behavior of u in those domains is studied. An important comparison principle that has been used in the literature to derive many of the results in capillarity is reviewed. It allows one to deduce the approximate shape of a capillary surface by constructing comparison surfaces with mean curvature and contact angle close to those of the (unknown) solution surface. In the context of nonparametric problems the comparison principle leads to height estimates above and below for the function u. An example from the literature where these height estimates have been used successfully is described. The promised domains for which the bounded u does not extend continuously to the boundary are constructed. The point on the boundary at which u has a jump discontinuity will be the vertext of a re-entrant corner having any interior angle theta pi. Using the comparison principle the behavior of u near this point is studied

Average prime-pair counting formula

Taking $r>0$, let $\pi_{2r}(x)$ denote the number of prime pairs $(p, p+2r)$ with $p\le x$. The prime-pair conjecture of Hardy and Littlewood (1923) asserts that $\pi_{2r}(x)\sim 2C_{2r} {\rm li}_2(x)$ with an explicit constant $C_{2r}>0$. There seems to be no good conjecture for the remainders $\omega_{2r}(x)=\pi_{2r}(x)- 2C_{2r} {\rm li}_2(x)$ that corresponds to Riemann's formula for $\pi(x)-{\rm li}(x)$. However, there is a heuristic approximate formula for averages of the remainders $\omega_{2r}(x)$ which is supported by numerical results.Comment: 26 pages, 6 figure

On the nondegeneracy of constant mean curvature surfaces

We prove that many complete, noncompact, constant mean curvature (CMC) surfaces $f:\Sigma \to \R^3$ are nondegenerate; that is, the Jacobi operator $\Delta_f + |A_f|^2$ has no $L^2$ kernel. In fact, if $\Sigma$ has genus zero and $f(\Sigma)$ is contained in a half-space, then we find an explicit upper bound for the dimension of the $L^2$ jernel in terms of the number of non-cylindrical ends. Our main tool is a conjugation operation on Jacobi fields which linearizes the conjugate cousin construction. Consequences include partial regularity for CMC moduli space, a larger class of CMC surfaces to use in gluing constructions, and a surprising characterization of CMC surfaces via spinning spheres.Comment: v2: substantial revisions, to appear in Geom. Funct. Anal.; three figure

Refined asymptotics for constant scalar curvature metrics with isolated singularities

We consider the asymptotic behaviour of positive solutions u of the conformal scalar curvature equation, \Delta u + n(n-2)/4 u^{(n+2)(n-2) = 0, in the neighbourhood of isolated singularities in the standard Euclidean ball. Although asymptotic radial symmetry for such solutions was proved some time ago, by Caffarelli, Gidas and Spruck, we present a much simpler and more geometric derivation of this fact. We also discuss a refinement, showing that any such solution is asymptotic to one of the deformed radial singular solutions. Finally we give some applications of these refined asymptotics, first to computing the global Pohozaev invariants of solutions on the sphere with isolated singularities, and then to the regularity of the moduli space of all such solutions.Comment: To appear, Inventiones Mathematica

Supported Education for youth with psychiatric disabilities

Higher and advanced vocational education prepare young adults for a career and enhance their life goals.The onset of mental illness generally occurs between 17 and 25 years. For young adults with psychiatric disabilities, educational resources are largely unavailable

Roots under convolution of sequences

AbstractThe convolution a * b of the sequences a = a0, a1, a2, ⋯ ∼ and b is the sequence with elements ∑0n akbn − k. One sets 1, 1, 1, ⋯ equal to σ. Given that a * a with a ≥ 0 is close to σ * σ, how close is a to σ? More generally, one asks how close a is to σ if the p-th convolution power, a*P with a ≥ 0, is close to σ*P. Power series and complex analysis form a natural tool to estimate the ‘summed deviation’ ρ = σ * (a — σ) in terms of b = a * a — σ * σ or b = a*P − σ*P. Optimal estimates are found under the condition ∑k=0n bk2 = %plane1D;512;(n2β + 1) whenever −½ < β < p − 1. It is not known what the optimal estimates are for the special case bn = %plane1D;512;(nβ)

Ikehara-type theorem involving boundedness

Consider any Dirichlet series sum a_n/n^z with nonnegative coefficients a_n and finite sum function f(z)=f(x+iy) when x is greater than 1. Denoting the partial sum a_1+...+a_N by s_N, the paper gives the following necessary and sufficient condition in order that (s_N)/N remain bounded as N goes to infinity. For x tending to 1 from above, the quotient q(x+iy)=f(x+iy)/(x+iy) must converge to a pseudomeasure q(1+iy), the distributional Fourier transform of a bounded function. The paper also gives an optimal estimate for (s_N)/N under the "real condition" that (1-x)f(x) remain bounded as x tends to 1 from above.Comment: 6 page

The impact of computer interface design on Saudi students’ performance on a L2 reading test

A thesis submitted to the University of Bedfordshire in partial fulfillment of the requirements for the degree of Doctor of PhilosophyThis study investigates the effect of testing mode on lower-level Saudi Arabian test-takers’ performance and cognitive processes when taking an L2 reading test on computer compared to its paper-based counterpart from an interface design perspective. An interface was developed and implemented into the computer-based version of the L2 reading test in this study, which was administered to 102 Saudi Arabian University students for quantitative analyses and to an additional eighteen for qualitative analyses. All participants were assessed on the same L2 reading test in two modes on two separate occasions in a within-subject design. Statistical tests such as correlations, group comparisons, and item analyses were employed to investigate test-mode effect on test-takers’ performance whereas test-takers’ concurrent verbalizations were recorded when taking the reading test to investigate their cognitive processes. Strategies found in both modes were compared through their frequency of occurrence. In addition, a qualitative illustration of test-takers cognitive behavior was given to describe the processes when taking a lower-level L2 reading test. A mixed-method approach was adhered to when collecting data consisting of questionnaires think-aloud protocols, and post-experimental interviews as main data collection instruments. Results on test-takers’ performance showed that there was no significant difference between the two modes of testing on overall reading performance, however, item level analyses discovered significant differences on two of the test’s items. Further qualitative investigation into possible interface design related causes for these differences showed no identifiable relationship between test-takers’ performance and the computer-based testing mode. Results of the cognitive processes analyses showed significant differences in three out of the total number of cognitive processes employed by test-takers indicating that test-takers had more difficulties in processing text in the paper-based test than in the computer-based test. Both product and process analyses carried out further provided convincing supporting evidence for the cognitive validity, content validity, and context validity contributing to the construct validity of the computer-based test used in this study

The Changing role of agriculture in Dutch society

Dutch agriculture has undergone significant changes in the past century, similar to many countries in the European Union. Due to economies of scale and in order to remain economically profitable, it became necessary for farmers to increase farm size, efficiency and external inputs, while minimizing labour use per hectare. The latter has resulted in fewer people working in the agricultural sector. Consequently, Dutch society gradually lost its connection to agricultural production. This divergence resulted in a poor image for the agricultural sector, because of environmental pollution, homogenization of the landscape, outbreaks of contagious animal diseases and reduced animal welfare. Although the general attitude towards agriculture seems to have improved slightly in recent years, there is still a long way to go in regaining this trust. In order to keep the Dutch countryside viable, farmers are considered indispensable. However, their methods of production should match the demands of society in terms of sustainability. This applies both to farming systems that are used in a monofunctional way (production only) and to multifunctional farming systems. For researchers involved in development of these farming systems, this requires new capabilities; contrary to the situation in the past, citizens and stakeholder groups now demand involvement in the design of farming systems. In the current paper, it is suggested that, besides traditional mainstream agriculture, other alternative farming systems should be developed and implemented. Hence, Dutch agricultural research should remain focused on the cutting edge of economy and society. Despite all efforts, not all of these newly developed systems will acquire a position within the agricultural spectrum. However, some of the successful ones may prove extremely valuable