Let (0,L)× E_a⊂ℝ^d+1 denote the cylinder of length L, and base E_a⊂ℝ^d, where E_a is an open ellipsoid with semi-axes a=(a_1,...,a_d) . Let w_(0,L)× E_a denote its torsion function. Heat equation tools are used to show that (i) if E_a has small eccentricity and L is sufficiently large, then the maxima of |∇ w_(0,L)× E_a| are located at the centres (0, 0) and (L, 0) respectively, (ii) if E_a has large eccentricity and L is sufficiently large, then the maxima are located on the lateral side of the cylinder.

Upper bounds are obtained for the Newtonian capacity of compact sets in [Formula: see text] in terms of the perimeter of the [Formula: see text]-parallel neighborhood of [Formula: see text]. For compact, convex sets in [Formula: see text] with a [Formula: see text] boundary the Newtonian capacity is bounded from above by [Formula: see text], where [Formula: see text] is the integral of the mean curvature over the boundary of [Formula: see text] with equality if [Formula: see text] is a ball. For compact, convex sets in [Formula: see text] with non-empty interior the Newtonian capacity is bounded from above by [Formula: see text] with equality if [Formula: see text] is a ball. Here, [Formula: see text] is the perimeter of [Formula: see text] and [Formula: see text] is its measure. A quantitative refinement of the latter inequality in terms of the Fraenkel asymmetry is also obtained. An upper bound is obtained for expected Newtonian capacity of the Wiener sausage in [Formula: see text] with radius [Formula: see text] and time length [Formula: see text].

For an open set Ω⊂ℝ^2 let λ (Ω ) denote the bottom of the spectrum of the Dirichlet Laplacian acting in L^2(Ω ) . Let w_Ω be the torsion function for Ω , and let ‖ .‖ _p denote the L^p norm. It is shown that there exist η _1>0,η _2>0 such that (i) ‖ w_Ω‖ _∞λ (Ω )≥ 1+η _1 for any non-empty, open, simply connected set Ω⊂ℝ^2 with λ (Ω ) >0 , (ii) ‖ w_Ω‖ _1λ (Ω )≤(1-η _2)|Ω | for any non-empty, open, simply connected set Ω⊂ℝ^2 with finite measure |Ω | .

Two-sided bounds for the efficiency of the torsion function are obtained in terms of the square of the distance to the boundary function under the hypothesis that the Dirichlet Laplacian satisfies a strong Hardy inequality. Localization properties of the torsion function are obtained under that hypothesis. An example is analyzed in detail.

We investigate extremality properties of shape functionals which are products of Newtonian capacity $\cp(\overline{\Om})$, and powers of the torsional rigidity $T(\Om)$, for an open set $\Om\subset \R^d$ with compact closure $\overline{\Om}$, and prescribed Lebesgue measure. It is shown that if $\Om$ is convex then $\cp(\overline{\Om})T^q(\Om)$ is (i) bounded from above if and only if $q\ge 1$, and (ii) bounded from below and away from $0$ if and only if $q\le \frac{d-2}{2(d-1)}$. Moreover a convex maximiser for the product exists if either $q>1$, or $d=3$ and $q=1$. A convex minimiser exists for $q< \frac{d-2}{2(d-1)}$. If $q\le 0$, then the product is minimised among all bounded sets by a ball of measure $1$.

We consider the torsion function for the Dirichlet Laplacian −Δ, and for the Schrödinger operator −Δ + V on an open set ${\Omega }\subset \mathbb {R}^{m}$ of finite Lebesgue measure $0<|{\Omega }|<\infty $ with a real-valued, non-negative, measurable potential V. We investigate the efficiency and the phenomenon of localisation for the torsion function, and their interplay with the geometry of the first Dirichlet eigenfunction.

Introduction Several countries advocate screening for aneurysms of the abdominal aorta (AAA) in selected patients. In the Netherlands, routine screening is currently under review by the National Health Council. In any screening programme, cost-efficiency and accuracy are key. In this study, we evaluate the Aorta Scan (Verathon, Amsterdam, Netherlands), a cost-effective and easy-to-use screening device based on bladder scan technology, which enables untrained personnel to screen for AAA. Methods We subjected 117 patients to an Aorta Scan and compared the results to the gold standard (abdominal ultrasound). We used statistical analysis to determine sensitivity and specificity of the Aorta Scan, as well as the positive and negative predictive values, accuracy, and inter-test agreement (Kappa). Results Sensitivity and specificity were 0.86 and 0.98, respectively. Positive predictive value was 0.98 and negative predictive value was 0.88. Accuracy was determined at 0.92 and the Kappa value was 0.85. When waist–hip circumferences (WHC) of > 115 cm were excluded, sensitivity raised to 0.96, specificity stayed 0.98, positive and negative predictive value were 0.98 and 0.96, respectively, accuracy to 0.97, and Kappa to 0.94. Conclusion Herein, we show that the Aorta Scan is a cost-effective and very accurate screening tool, especially in patients with WHC below 115 cm, which makes it a suitable candidate for implementation into clinical practice, specifically in the setting of screening selected populations for the presence of AAA.

Several countries advocate screening for aneurysms of the abdominal aorta (AAA) in selected patients. In the Netherlands, routine screening is currently under review by the National Health Council. In any screening programme, cost-efficiency and accuracy are key. In this study, we evaluate the Aorta Scan (Verathon, Amsterdam, Netherlands), a cost-effective and easy-to-use screening device based on bladder scan technology, which enables untrained personnel to screen for AAA.We subjected 117 patients to an Aorta Scan and compared the results to the gold standard (abdominal ultrasound). We used statistical analysis to determine sensitivity and specificity of the Aorta Scan, as well as the positive and negative predictive values, accuracy, and inter-test agreement (Kappa).Sensitivity and specificity were 0.86 and 0.98, respectively. Positive predictive value was 0.98 and negative predictive value was 0.88. Accuracy was determined at 0.92 and the Kappa value was 0.85. When waist-hip circumferences (WHC) of > 115 cm were excluded, sensitivity raised to 0.96, specificity stayed 0.98, positive and negative predictive value were 0.98 and 0.96, respectively, accuracy to 0.97, and Kappa to 0.94.Herein, we show that the Aorta Scan is a cost-effective and very accurate screening tool, especially in patients with WHC below 115 cm, which makes it a suitable candidate for implementation into clinical practice, specifically in the setting of screening selected populations for the presence of AAA.

Introduction: Several countries advocate screening for aneurysms of the abdominal aorta (AAA) in selected patients. In the Netherlands, routine screening is currently under review by the National Health Council. In any screening programme, cost-efficiency and accuracy are key. In this study, we evaluate the Aorta Scan (Verathon, Amsterdam, Netherlands), a cost-effective and easy-to-use screening device based on bladder scan technology, which enables untrained personnel to screen for AAA. Methods: We subjected 117 patients to an Aorta Scan and compared the results to the gold standard (abdominal ultrasound). We used statistical analysis to determine sensitivity and specificity of the Aorta Scan, as well as the positive and negative predictive values, accuracy, and inter-test agreement (Kappa). Results: Sensitivity and specificity were 0.86 and 0.98, respectively. Positive predictive value was 0.98 and negative predictive value was 0.88. Accuracy was determined at 0.92 and the Kappa value was 0.85. When waist-hip circumferences (WHC) of > 115 cm were excluded, sensitivity raised to 0.96, specificity stayed 0.98, positive and negative predictive value were 0.98 and 0.96, respectively, accuracy to 0.97, and Kappa to 0.94. Conclusion: Herein, we show that the Aorta Scan is a cost-effective and very accurate screening tool, especially in patients with WHC below 115 cm, which makes it a suitable candidate for implementation into clinical practice, specifically in the setting of screening selected populations for the presence of AAA.

Let $D$ be a non-empty open subset of $\R^m,\,m\ge 2$, with boundary $\partial D$, with finite Lebesgue measure $|D|$, and which satisfies a parabolic Harnack principle. Let $K$ be a compact, non-polar subset of $D$. We obtain the leading asymptotic behaviour as $\varepsilon\downarrow 0$ of the $L^{\infty}$ norm of the torsion function with a Neumann boundary condition on $\partial D$, and a Dirichlet boundary condition on $\partial (\varepsilon K)$, in terms of the first eigenvalue of the Laplacian with corresponding boundary conditions. These estimates quantify those of Burdzy, Chen and Marshall who showed that $D\setminus K$ is a non-trap domain.

Let omega be an open, possibly unbounded, set in Euclidean spaceRmwith boundary partial differential omega, letAbe a measurable subset of omega with measure|A|and let gamma is an element of(0,1). We investigate whether the solutionv omega,A,gamma of-Delta v=gamma 1 omega set minus A-(1-gamma)1Awithv=0on partial differential omega changes sign. Bounds are obtained for|A|in terms of geometric characteristics of omega(bottom of the spectrum of the Dirichlet Laplacian, torsion, measure orR-smoothness of the boundary) such thatessinfv omega,A,gamma > 0. We show thatessinfv omega,A,gamma<0for any measurable setA, provided|A|>gamma|omega|. This value is sharp. We also study the shape optimisation problem of the optimal location ofA(with prescribed measure) which minimises the essential infimum ofv omega,A,gamma. Surprisingly, if omega is a ball, a symmetry breaking phenomenon occurs.

We study the heat flow from an open, bounded set D in $\mathbb {R}^{2}$ with a polygonal boundary ∂D. The initial condition is the indicator function of D. A Dirichlet 0 boundary condition has been imposed on some but not all of the edges of ∂D. We calculate the heat content of D in $\mathbb {R}^{2}$ at t up to an exponentially small remainder as t ↓ 0.

Bounds are obtained for the \(L^p\) norm of the torsion function \(v_{\varOmega }\), i.e. the solution of \(-\varDelta v=1,\, v\in H_0^1(\varOmega ),\) in terms of the Lebesgue measure of \(\varOmega \) and the principal eigenvalue \(\lambda _1(\varOmega )\) of the Dirichlet Laplacian acting in \(L^2(\varOmega )\). We show that these bounds are sharp for \(1\le p\le 2\).

Let 𝕋 m be the m-dimensional unit torus, m ∈ ℕ. The torsional rigidity of an open set Ω ⊂ 𝕋 m is the integral with respect to Lebesgue measure over all starting points x ∈ Ω of the expected lifetime in Ω of a Brownian motion starting at x. In this paper we consider Ω = 𝕋 m \β[0, t], the complement of the path ß[0, t] of an independent Brownian motion up to time t. We compute the leading order asymptotic behaviour of the expectation of the torsional rigidity in the limit as t → ∞. For m = 2 the main contribution comes from the components in 𝕋2\β0, t] whose inradius is comparable to the largest inradius, while for m = 3 most of 𝕋3\β[0, t] contributes. A similar result holds for m ≥ 4 after the Brownian path is replaced by a shrinking Wiener sausage W r(t)[0, t] of radius r(t) = o(t -1/(m-2)), provided the shrinking is slow enough to ensure that the torsional rigidity tends to zero. Asymptotic properties of the capacity of ß[0, t] in ℝ3 and W 1[0, t] in ℝ m , m ≥ 4, play a central role throughout the paper. Our results contribute to a better understanding of the geometry of the complement of Brownian motion on 𝕋 m , which has received a lot of attention in the literature in past years.

Let Ω be an open set in a complete, smooth, non-compact, m -dimensional Riemannian manifold M without boundary, where M satisfies a two-sided Li-Yau gaussian heat kernel bound. It is shown that if Ω has infinite measure, and if Ω has finite heat content H_Ω(T) for some T>0 , then H_Ω(t)<∞ for all t>0 . Comparable two-sided bounds for H_Ω(t) are obtained for such Ω .

We consider the minimisation of Dirichlet eigenvalues $\lambda_k$, $k \in \N$, of the Laplacian on cuboids of unit measure in $\R^3$. We prove that any sequence of optimal cuboids in $\R^3$ converges to a cube of unit measure in the sense of Hausdorff as $k \rightarrow \infty$.

Let $\Omega$ be an open set in Euclidean space $\R^m,\, m=2,3,...$, and let $v_{\Omega}$ denote the torsion function for $\Omega$. It is known that $v_{\Omega}$ is bounded if and only if the bottom of the spectrum of the Dirichlet Laplacian acting in $\Leb^2(\Omega)$, denoted by $\lambda(\Omega)$, is bounded away from $0$. It is shown that the previously obtained bound $\|v_{\Omega}\|_{\Leb^{\infty}(\Omega)}\lambda(\Omega)\ge 1$ is sharp: for $m\in\{2,3,...\}$, and any $\epsilon>0$ we construct an open, bounded and connected set $\Omega_{\epsilon}\subset \R^m$ such that $\|v_{\Omega_{\epsilon}}\|_{\Leb^{\infty}(\Omega_{\epsilon})} \lambda(\Omega_{\epsilon})<1+\epsilon$. An upper bound for $v_{\Omega}$ is obtained for planar, convex sets in Euclidean space $M=\R^2$, which is sharp in the limit of elongation. For a complete, non-compact, $m$-dimensional Riemannian manifold $M$ with non-negative Ricci curvature, and without boundary it is shown that $v_{\Omega}$ is bounded if and only if the bottom of the spectrum of the Dirichlet-Laplace-Beltrami operator acting in $\Leb^2(\Omega)$ is bounded away from $0$.

In the last two decades Dutch primary school students scored below expectation in international mathematics tests. An explanation for this may be that teachers fail to adequately assess their students' understanding of learning goals and provide timely feedback. To improve the teachers' formative assessment practice, researchers, curriculum experts and teachers worked together to develop a model for classroom formative assessment (CFA). In three pilot studies, six teachers from three different schools implemented the CFA-model and evaluated its feasibility together with the researchers by means of checklists. The CFA-model was primarily changed with regard to the assessment techniques. Teachers indicated that classroom management and preparation time were preconditions for an optimal implementation. Analysis of covariance was used to explore students' learning outcomes. The results showed that a correct implementation of the CFA-model might result in the enhancement of students' mathematical performance. The implications of the three pilots for the implementation of the CFA-model on a larger scale are discussed.

Let Ω be an open set in Euclidean space with finite Lebesgue measure |Ω| . We obtain some properties of the set function F:Ω↦ℝ^+ defined by F(Ω )=T(Ω )λ _1(Ω )/|Ω| , where T(Ω ) and λ _1(Ω ) are the torsional rigidity and the first eigenvalue of the Dirichlet Laplacian respectively. We improve the classical Pólya bound F(Ω )≤ 1, and show that F(Ω )≤ 1- ν _m T(Ω )|Ω |^-1-2/m, where ν _m depends only on m . For any m=2,3,… and ϵ∈ (0,1) we construct an open set Ω _ϵ⊂ℝ^m such that F(Ω _ϵ)≥ 1-ϵ .