19,859 research outputs found

    Robust Exponential Runge-Kutta Embedded Pairs

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    Exponential integrators are explicit methods for solving ordinary differential equations that treat linear behaviour exactly. The stiff-order conditions for exponential integrators derived in a Banach space framework by Hochbruck and Ostermann are solved symbolically by expressing the Runge--Kutta weights as unknown linear combinations of phi functions. Of particular interest are embedded exponential pairs that efficiently generate both a high- and low-order estimate, allowing for dynamic adjustment of the time step. A key requirement is that the pair be robust: if the nonlinear source function has nonzero total time derivatives, the order of the low-order estimate should never exceed its design value. Robust exponential Runge--Kutta (3,2) and (4,3) embedded pairs that are well-suited to initial value problems with a dominant linearity are constructed.Comment: 24 pages, 8 figures. The Mathematica scripts mentioned in the paper can be found in: https://github.com/stiffode/expint

    Nonlocal error bounds for piecewise affine functions

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    The paper is devoted to a detailed analysis of nonlocal error bounds for nonconvex piecewise affine functions. We both improve some existing results on error bounds for such functions and present completely new necessary and/or sufficient conditions for a piecewise affine function to have an error bound on various types of bounded and unbounded sets. In particular, we show that any piecewise affine function has an error bound on an arbitrary bounded set and provide several types of easily verifiable sufficient conditions for such functions to have an error bound on unbounded sets. We also present general necessary and sufficient conditions for a piecewise affine function to have an error bound on a finite union of polyhedral sets (in particular, to have a global error bound), whose derivation reveals a structure of sublevel sets and recession functions of piecewise affine functions

    Dirac Equation for Photons: Origin of Polarisation

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    Spin is a fundamental degree of freedom, whose existence was proven by Dirac for an electron by imposing the relativity to quantum mechanics, leading to the triumph to derive the Dirac equation. Spin of a photon should be linked to polarisation, however, the similar argument for an electron was not applicable to Maxwell equations, which are already Lorentz invariant. Therefore, the origin of polarisation and its relationship with spin are not completely elucidated, yet. Here, we discuss propagation of coherent rays of photons in a graded-index optical fibre, which can be solved exactly using the Laguerre-Gauss or Hermite-Gauss modes in a cylindrical or a Cartesian coordinate. We found that the energy spectrum is massive with the effective mass as a function of the confinement and orbital angular momentum. The propagation is described by the one-dimensional (1D1D) non-relativistic Schr\"odinger equation, which is equivalent to the 2D2D space-time Klein-Gordon equation by a unitary transformation. The probabilistic interpretation and the conservation law require the factorisation of the Klein-Gordon equation, leading to the 2D2D Dirac equation with spin. We applied the Bardeen-Cooper-Schrieffer (BCS)-Bogoliubov theory of superconductivity to a coherent ray from a laser and identified a radiative Nambu-Anderson-Higgs-Goldstone mode for recovering the broken symmetry. The spin expectation value of a photon corresponds to the polarisation state in the Poincar\'e sphere, which is characterised by fixed phases after the onset of lasing due to the broken SU(2)SU(2) symmetry, and it is shown that its azimuthal angle is coming from the phase of the energy gap

    Chiral numerical renormalization group

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    The interplay between the Kondo screening of quantum impurities (by the electronic channels to which they couple) and the interimpurity RKKY interactions (mediated by the same channels) has been extensively studied. However, the effect of unidirectional channels (e.g., chiral or helical edge modes of 2D topological materials) which greatly restrict the mediated interimpurity interactions, has only more recently come under scrutiny, and it can drastically alter the physics. Here we take Wilson's numerical renormalization group (NRG), the most established numerical method for treating quantum impurity models, and extend it to systems consisting of two impurities coupled at different locations to unidirectional channel(s). This is challenging due to the incompatibility of unidirectionality with one of the main ingredients in NRG -- the mapping of the channel(s) to a Wilson chain -- a tight-binding chain with the impurity at one end and hopping amplitudes which decay exponentially with the distance. We bridge this gap by introducing a "Wilson ladder" consisting of two coupled Wilson chains, and demonstrate that this construction successfully captures the unidirectionality of the channel(s), as well as the distance between the two impurities. We use this mapping in order to study two Kondo impurities coupled to a single chiral channel, showing that all local properties and thermodynamic quantities are indifferent to the interimpurity distance, and correspond to two separate single-impurity models. Extensions to more impurities and/or helical channels are possible.Comment: 20 pages, 6 figures - published versio

    Strategies for Early Learners

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    Welcome to learning about how to effectively plan curriculum for young children. This textbook will address: • Developing curriculum through the planning cycle • Theories that inform what we know about how children learn and the best ways for teachers to support learning • The three components of developmentally appropriate practice • Importance and value of play and intentional teaching • Different models of curriculum • Process of lesson planning (documenting planned experiences for children) • Physical, temporal, and social environments that set the stage for children’s learning • Appropriate guidance techniques to support children’s behaviors as the self-regulation abilities mature. • Planning for preschool-aged children in specific domains including o Physical development o Language and literacy o Math o Science o Creative (the visual and performing arts) o Diversity (social science and history) o Health and safety • Making children’s learning visible through documentation and assessmenthttps://scholar.utc.edu/open-textbooks/1001/thumbnail.jp

    Limit theorems for non-Markovian and fractional processes

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    This thesis examines various non-Markovian and fractional processes---rough volatility models, stochastic Volterra equations, Wiener chaos expansions---through the prism of asymptotic analysis. Stochastic Volterra systems serve as a conducive framework encompassing most rough volatility models used in mathematical finance. In Chapter 2, we provide a unified treatment of pathwise large and moderate deviations principles for a general class of multidimensional stochastic Volterra equations with singular kernels, not necessarily of convolution form. Our methodology is based on the weak convergence approach by Budhiraja, Dupuis and Ellis. This powerful approach also enables us to investigate the pathwise large deviations of families of white noise functionals characterised by their Wiener chaos expansion as~Xε=∑n=0∞εnIn(fnε).X^\varepsilon = \sum_{n=0}^{\infty} \varepsilon^n I_n \big(f_n^{\varepsilon} \big). In Chapter 3, we provide sufficient conditions for the large deviations principle to hold in path space, thereby refreshing a problem left open By Pérez-Abreu (1993). Hinging on analysis on Wiener space, the proof involves describing, controlling and identifying the limit of perturbed multiple stochastic integrals. In Chapter 4, we come back to mathematical finance via the route of Malliavin calculus. We present explicit small-time formulae for the at-the-money implied volatility, skew and curvature in a large class of models, including rough volatility models and their multi-factor versions. Our general setup encompasses both European options on a stock and VIX options. In particular, we develop a detailed analysis of the two-factor rough Bergomi model. Finally, in Chapter 5, we consider the large-time behaviour of affine stochastic Volterra equations, an under-developed area in the absence of Markovianity. We leverage on a measure-valued Markovian lift introduced by Cuchiero and Teichmann and the associated notion of generalised Feller property. This setting allows us to prove the existence of an invariant measure for the lift and hence of a stationary distribution for the affine Volterra process, featuring in the rough Heston model.Open Acces

    INVESTIGATING THE PERCEPTION OF EXPATRIATES TOWARDS IMMIGRATION SERVICE QUALITY IN SHARJAH, UNITED ARAB EMIRATES THROUGH MIXED METHOD APPROACH

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    The public sectors in UAE are under immense pressure to demonstrate that their services are customer-focused and that continuous performance improvement is being delivered. The United Arab Emirates is a favoured destination for expatriates due to its own citizens form a minority of the population and are barely represented in the private sector workforce. These highly unusual demographics confer high importance on the national immigration services. Recently, increased interest in international migration, specifically within the United Arab Emirates, has been shown both by government agencies and by the governments of industrialised countries. Given the importance of the expatriate labour force to economic stability and growth in the Emirates, this research investigates how immigration services are perceived, with the aim of contributing to their improvement, thus ultimately supporting economic growth. It proposes a service quality perception framework to improve understanding within SID of how to raise levels of service delivered to migrants and other persons directly or indirectly affected by SID services. Qualitative data were collected by means of semi-structured interviews and quantitative data by means of a questionnaire survey based on the abovementioned framework. The survey data, on the variables influencing participants’ experiences and perceptions of SID services, were subjected to statistical analysis. The framework was then used to evaluate quality of service in terms of general impressions, delivery, location, response, SID culture and behaviour. Numerical data were analysed using inferential and descriptive statistics. It was found that service quality positively influenced service behaviour and that this relationship was mediated by SID culture. This research makes an original contribution to knowledge as one of the few studies of immigration to the United Arab Emirates. By examining the workings of one immigration department, it adds to the literature on immigration departments and organisational development in developing countries. It illuminates the mechanics of immigration services and demonstrates their increasing importance to the world economy

    Moduli Stabilisation and the Statistics of Low-Energy Physics in the String Landscape

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    In this thesis we present a detailed analysis of the statistical properties of the type IIB flux landscape of string theory. We focus primarily on models constructed via the Large Volume Scenario (LVS) and KKLT and study the distribution of various phenomenologically relevant quantities. First, we compare our considerations with previous results and point out the importance of Kähler moduli stabilisation, which has been neglected in this context so far. We perform different moduli stabilisation procedures and compare the resulting distributions. To this end, we derive the expressions for the gravitino mass, various quantities related to axion physics and other phenomenologically interesting quantities in terms of the fundamental flux dependent quantities gsg_s, W0W_0 and n\mathfrak{n}, the parameter which specifies the nature of the non-perturbative effects. Exploiting our knowledge of the distribution of these fundamental parameters, we can derive a distribution for all the quantities we are interested in. For models that are stabilised via LVS we find a logarithmic distribution, whereas for KKLT and perturbatively stabilised models we find a power-law distribution. We continue by investigating the statistical significance of a newly found class of KKLT vacua and present a search algorithm for such constructions. We conclude by presenting an application of our findings. Given the mild preference for higher scale supersymmetry breaking, we present a model of the early universe, which allows for additional periods of early matter domination and ultimately leads to rather sharp predictions for the dark matter mass in this model. We find the dark matter mass to be in the very heavy range mχ∼1010−1011 GeVm_{\chi}\sim 10^{10}-10^{11}\text{ GeV}
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