663 research outputs found

    The Kato Square Root Problem for Divergence Form Operators with Potential

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    The Kato square root problem for divergence form elliptic operators with potential V:RnCV : \mathbb{R}^{n} \rightarrow \mathbb{C} is the equivalence statement (L+V)12u2u2+V12u2\left\Vert (L + V)^{\frac{1}{2}} u\right\Vert_{2} \simeq \left\Vert \nabla u \right\Vert_{2} + \left\Vert V^{\frac{1}{2}} u \right\Vert_{2}, where L+V:=divA+VL + V := - \mathrm{div} A \nabla + V and the perturbation AA is an LL^{\infty} complex matrix-valued function satisfying an accretivity condition. This relation is proved for any potential with range contained in some positive sector and satisfying Vα2u2+(Δ)α22(VΔ)α2u2\left\Vert |V|^{\frac{\alpha}{2}} u\right\Vert_{2} + \left\Vert (-\Delta)^{\frac{\alpha}{2}} \right\Vert_{2} \lesssim \left\Vert ( |V| - \Delta)^{\frac{\alpha}{2}}u \right\Vert_{2} for all uD(VΔ)u \in D(|V| -\Delta) and some α(1,2]\alpha \in (1,2]. The class of potentials that will satisfy such a condition is known to contain the reverse H\"{o}lder class RH2RH_{2} and Ln2(Rn)L^{\frac{n}{2}}(\mathbb{R}^{n}) in dimension n>4n > 4. To prove the Kato estimate with potential, a non-homogeneous version of the framework introduced by A. Axelsson, S. Keith and A. McIntosh for proving quadratic estimates is developed. In addition to applying this non-homogeneous framework to the scalar Kato problem with zero-order potential, it will also be applied to the Kato problem for systems of equations with zero-order potential.Comment: arXiv admin note: text overlap with arXiv:1902.0110

    High School Student Athletes: If Stress is the “Lock” is Communication the “Key”

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    The focus of this Capstone Project was meant to analyze the possible issues high school student-athletes may face academically as well as mentally while in school. This issue is important to note because there is an increase in students participating in sports and the offered school support should reflect the growing population. An argument that was found was the possible overlook of the mental well-being of the student-athletes if they are academically succeeding or are accomplished athletes. The primary stakeholder\u27s perspectives chosen were current high school student-athletes because they can provide insight into the current issues that student-athletes face. Three themes emerged from an analysis of the data: 1) Providing student-athletes with tutors, study programs, and other resources for academic support. 2) Seasonal mental health and time management training for teachers. 3) The creation of an online platform where teachers and coaches can keep track of the student\u27s grades, homework, tests, and schedules. Three action options have suggested the creation of an online platform for the streamlining of communication between teachers, coaches, and students. is argued to be the most effective way to achieve the goals of supporting high school student-athletes

    “Faith, Hope, and Charity”: The Role of Good Faith in Construction – a Common Law Perspective

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    Los conflictos respecto a la “buena fe” en el desarrollo de contratos de construcción e ingeniería son comunes. Más aún, los conflictos surgen en la mayoría de jurisdicciones del mundo, tanto de common law como civil law. Este artículo aborda el enfoque asumido por el common law británico respecto al concepto legal de “buena fe”. Toma en cuenta las consecuencias de que el Derecho inglés no acepte el concepto de “buena fe” en el Derecho Comercial en general, incluso en relación con los contratos de construcción e ingeniería, y algunas de las doctrinas desarrolladas por el Derecho inglés para aminorar las potenciales graves consecuencias de que una parte ejerza sus derechos contractuales de manera abusiva. También se aborda el New Engineering Contract y otras formas contractuales que requieren expresamente a las partes actuar de buena fe.Issues of “good faith” in the performance of construction and engineering contracts are common. Moreover, “good faith” issues arise in most jurisdictions of the world, including common law and civil law jurisdictions. This paper considers the approach taken by English common law to the legal concept of “good faith”. It considers the consequences of English law not embracing “good faith” in commercial law in any general way, including in relation to construction and engineering contracts, and some of the doctrines developed by English law to ameliorate the potentially harsh consequences of a party exercising its contractual rights in an unfair manner. Consideration is also given to the New Engineering Contract and other forms of contract which expressly require parties to act in good faith

    Schrodinger Operators and the Kato Square Root Problem

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    The general theme of this thesis is the harmonic analysis of Schrodinger operators and its applications. We will focus on two distinct but related open problems in this field. The first problem is the construction of potential dependent averaging operators and will be primarily considered in the first part of this thesis. Here, a Hardy-Littlewood type maximal operator adapted to the Schrodinger operator L:=Δ+x2\mathcal{L} :=-\Delta + |x|^{2} and acting on L2(Rn)L^{2}(\mathbb{R}^{n}) is constructed. This is achieved through the use of the Gaussian grid Δ0γ\Delta^{\gamma}_{0}, constructed by J. Maas, J. van Neerven and P. Portal with the Ornstein-Uhlenbeck operator in mind. At the scale of this grid, the maximal operator will resemble the classical Hardy-Littlewood operator. At a larger scale, the constituent averaging operators of the maximal function are decomposed over the cubes from Δ0γ\Delta^{\gamma}_{0} and weighted appropriately. Through this maximal function, a new class of weights is defined, Ap+A_{p}^{+}, with the property that for any wAp+w \in A_{p}^{+} the heat maximal operator associated with L\mathcal{L} is bounded from Lp(w)L^{p}(w) to itself. This class contains any other known class that possesses this property and contains weights of exponential growth. In particular, it is strictly larger than ApA_{p}. The second problem that we consider is the Kato square root problem for divergence form elliptic operators with potential V:RnCV : \mathbb{R}^{n} \rightarrow \mathbb{C}. This is the equivalence statement (L+V)1/2uu+V1/2u\left\Vert (L + V)^{1/2} u \right\Vert \simeq \left\Vert \nabla u \right\Vert + \left\Vert V^{1/2} u \right\Vert, where L+V:=div(A)+VL + V := - div (A \nabla) + V and the perturbation AA is an LL^{\infty} complex matrix-valued function satisfying an ellipticity condition. One possible path to a solution for this problem is by proving square function estimates for perturbations of associated non-homogeneous Dirac-type operators. At present, there is no general method to obtain such square function estimates other than for potentials bounded both from above and below. We develop such a method by adapting the homogeneous framework introduced by A. Axelsson, S. Keith and A. McIntosh. Two distinct approaches will be considered when adapting this framework. The second such approach will yield a satisfying solution to the potential dependent Kato problem for a large class of potentials with range contained in the right-half of the complex plane

    Behavioural simulation of biological neuron systems using VHDL and VHDL-AMS

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    The investigation of neuron structures is an incredibly difficult and complex task that yields relatively low rewards in terms of information from biological forms (either animals or tissue). The structures and connectivity of even the simplest invertebrates are almost impossible to establish with standard laboratory techniques, and even when this is possible it is generally time consuming, complex and expensive. Recent work has shown how a simplified behavioural approach to modelling neurons can allow “virtual” experiments to be carried out that map the behaviour of a simulated structure onto a hypothetical biological one, with correlation of behaviour rather than underlying connectivity. The problems with such approaches are numerous. The first is the difficulty of simulating realistic aggregates efficiently, the second is making sense of the results and finally, it would be helpful to have an implementation that could be synthesised to hardware for acceleration. In this paper we present a VHDL implementation of Neuron models that allow large aggregates to be simulated. The models are demonstrated using a system level VHDL and VHDL-AMS model of the C. Elegans locomotory system