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We show how to deduce Rellich inequalities from Hardy inequalities on infinite graphs. Specifically, the obtained Rellich inequality gives an upper bound on a function by the Laplacian of the function in terms of weighted norms. These weights involve the Hardy weight and a function which satisfies an eikonal inequality. The results are proven first for Laplacians and are extended to Schrodinger operators afterwards.
We study graphs whose vertex degree tends to infinity and which are, therefore, called rapidly branching. We prove spectral estimates, discreteness of spectrum, first order eigenvalue and Weyl asymptotics solely in terms of the vertex degree growth. The underlying techniques are estimates on the isoperimetric constant. Furthermore, we give lower volume growth bounds and we provide a new criterion for stochastic incompleteness. (C) 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
We prove a version of the Hopf-Rinow theorem with respect to path metrics on discrete spaces. The novel aspect is that we do not a priori assume local finiteness but isolate a local finiteness type condition, called essentially locally finite, that is indeed necessary. As a side product we identify the maximal weight, called the geodesic weight, generating the path metric in the situation when the space is complete with respect to any of the equivalent notions of completeness proven in the Hopf-Rinow theorem. As an application we characterize the graphs for which the resistance metric is a path metric induced by the graph structure.
We prove Cheeger inequalities for p-Laplacians on finite and infinite weighted graphs. Unlike in previous works, we do not impose boundedness of the vertex degree, nor do we restrict ourselves to the normalized Laplacian and, more generally, we do not impose any boundedness assumption on the geometry. This is achieved by a novel definition of the measure of the boundary which uses the idea of intrinsic metrics. For the non-normalized case, our bounds on the spectral gap of p-Laplacians are already significantly better for finite graphs and for infinite graphs they yield non-trivial bounds even in the case of unbounded vertex degree. We, furthermore, give upper bounds by the Cheeger constant and by the exponential volume growth of distance balls. (C) 2016 Elsevier Ltd. All rights reserved.
In this article we prove upper bounds for the Laplace eigenvalues lambda(k) below the essential spectrum for strictly negatively curved Cartan-Hadamard manifolds. Our bound is given in terms of k(2) and specific geometric data of the manifold. This applies also to the particular case of non-compact manifolds whose sectional curvature tends to -infinity, where no essential spectrum is present due to a theorem of Donnelly/Li. The result stands in clear contrast to Laplacians on graphs where such a bound fails to be true in general.
We present a simple observation showing that the heat kernel on a locally finite graph behaves for short times t roughly like t(d), where d is the combinatorial distance. This is very different from the classical Varadhan-type behavior on manifolds. Moreover, this also gives that short-time behavior and global behavior of the heat kernel are governed by two different metrics whenever the degree of the graph is not uniformly bounded.
It is well-documented that academic achievement is associated with students' self-perceptions of their academic abilities, that is, their academic self-concepts. However, low-achieving students may apply self-protective strategies to maintain a favorable academic self-concept when evaluating their academic abilities. Consequently, the relation between achievement and academic self-concept might not be linear across the entire achievement continuum. Capitalizing on representative data from three large-scale assessments (i.e., TIMSS, PIRLS, PISA; N = 470,804), we conducted an integrative data analysis to address nonlinear trends in the relations between achievement and the corresponding self-concepts in mathematics and the verbal domain across 13 countries and 2 age groups (i.e., elementary and secondary school students). Polynomial and interrupted regression analyses showed nonlinear relations in secondary school students, demonstrating that the relations between achievement and the corresponding self-concepts were weaker for lower achieving students than for higher achieving students. Nonlinear effects were also present in younger students, but the pattern of results was rather heterogeneous. We discuss implications for theory as well as for the assessment and interpretation of self-concept.
Instructional videos are widely used to study teachers' professional vision. A new technological development in video research is mobile eye-tracking (MET). It has the potential to provide fine-grained insights into teachers' professional vision in action, but has yet been scarcely employed. We addressed this research gap by using MET video feedback to examine how expert and novice teachers differed in their noticing and weighing of alternative teaching strategies. Expert and novice teachers' lessons were recorded with MET devices. Then, they commented on what they observe while watching their own teaching videos. Using a mixed methods approach, we found that expert and novice teachers did not differ in the number of classroom events they noticed and alternative teaching strategies they mentioned. However, novice teachers were more critical of their own teaching than expert teachers, particularly when they considered alternative teaching strategies. Practical implications for the field of teacher education are discussed.
The hormone calcitonin (CT) is primarily known for its pharmacologic action as an inhibitor of bone resorption, yet CT-deficient mice display increased bone formation. These findings raised the question about the underlying cellular and molecular mechanism of CT action. Here we show that either ubiquitous or osteoclast-specific inactivation of the murine CT receptor (CTR) causes increased bone formation. CT negatively regulates the osteoclast expression of Spns2 gene, which encodes a transporter for the signalling lipid sphingosine 1-phosphate (S1P). CTR-deficient mice show increased S1P levels, and their skeletal phenotype is normalized by deletion of the S1P receptor S1P(3). Finally, pharmacologic treatment with the nonselective S1P receptor agonist FTY720 causes increased bone formation in wild-type, but not in S1P(3)-deficient mice. This study redefines the role of CT in skeletal biology, confirms that S1P acts as an osteoanabolic molecule in vivo and provides evidence for a pharmacologically exploitable crosstalk between osteoclasts and osteoblasts.