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The electricity sector is going through a rapid transition in the energy landscape. The global share of distributed renewable energy sources like solar panels and wind power for energy generation is on the rise. Even though this transition to renewable energy sources is beneficial for reducing global CO2 emissions, the intermittent nature of solar and wind power makes their integration into the electricity grid a highly challenging task. In this thesis we formulate mathematical models and develop computational techniques which facilitate reliability analysis of power grids integrated with wind power and energy storage devices. This thesis focuses on the reliable operation of the power grid which can in general be disrupted by the intermittent nature of wind power. Adverse effects of wind power variability can be mitigated by incorporating energy storage devices into the grid. Hence we investigate how to operate and place energy storage devices into the power grid for reliable operation of the grid. We also develop simple stochastic models for wind power which can be used to study statistical properties of power grids and to estimate grid reliability indices.

The Askey–Wilson polynomials are a four-parameter family of orthogonal symmetric Laurent polynomials Rn[z] that are eigenfunctions of a second-order q-difference operator L, and of a second-order difference operator in the variable n with eigenvalue z + z−1 = 2x. Then, L and multiplication by z + z−1 generate the Askey–Wilson (Zhedanov) algebra. A nice property of the Askey–Wilson polynomials is that the variables z and n occur in the explicit expression in a similar and to some extent exchangeable way. This property is called duality. It returns in the nonsymmetric case and in the underlying algebraic structures: the Askey–Wilson algebra and the double affine Hecke algebra (DAHA). In this paper, we follow the degeneration of the Askey–Wilson polynomials until two arrows down and in four different situations: for the orthogonal polynomials themselves, for the degenerate Askey–Wilson algebras, for the nonsymmetric polynomials, and for the (degenerate) DAHA and its representations.

The unpredictable nature of wind energy makes its integration to the electric grid highly challenging. However, these challenges can be addressed by incorporating storage devices (batteries) in the system. We perform an overall assessment of a single domestic power system with a wind turbine supported by an energy storage device. The aim is to investigate the best operation mode of the storage device such that the occurrence of large power spills can be minimized. For estimating the small probability of large power spills, we use the splitting technique for rare-event simulations. An appropriate Importance Function for splitting is formulated such that it reduces the work-load of the probability estimator as compared to the conventional Crude Monte Carlo probability estimator. Simulation results show that the ramp constraints imposed on the charging/discharging rate of the storage device plays a pivotal role in mitigating large power spills. It is observed that by employing a new charging strategy for the storage device large power spills can be minimized further. There exists a trade-off between reducing the large power spills versus reducing the average power spills.

We consider a fictitious domain formulation of an elliptic partial differential equation and approximate the resulting saddle-point system using a nested inexact preconditioned Uzawa iterative algorithm, which consists of three nested loops. In the outer loop the trial space for the Galerkin approximation of the Lagrange multiplier is enlarged. The intermediate loop solves this Galerkin system by a damped preconditioned Richardson iteration. Each iteration of the latter involves solving an elliptic problem on the fictitious domain whose solution is approximated by an adaptive finite element method in the inner loop. We prove that the overall method converges with the best possible rate and illustrate numerically our theoretical findings.

In this paper, the intersection of bivariate orthogonal polynomials on triangle patches will be investigated. The result is interesting by its own but also has important applications in the theory of a posteriori error estimation for finite element discretizations with $p$-refinement, i.e., if the local polynomial degree of the test and trial functions is increased to improve the accuracy. A triangle patch is a set of disjoint open triangles whose closed union covers a neighborhood of the common triangle vertex. On each triangle we consider the space of orthogonal polynomials of degree n with respect to the weight function which is the product of the barycentric coordinates. We show that the intersection of these polynomial spaces is the null space. The analysis requires the derivation of subtle representations of orthogonal polynomials on triangles. Up to four triangles have to be considered to identify that the intersection is trivial., Comment: 20 pages, 7 figures. v3: minor corrections and additions; accepted by Mathematics of Computation

Motivated by work of Dragt and Abell on accelerator physics, we study the completion of symplectic jets by polynomial maps of low degrees. We use Andersen-Lempert Theory to prove that symplectic completions always exist, and we prove the degree bound conjectured by Dragt and Abell in the physically relevant cases. However, we disprove the degree bound for \(3\)-jets in dimension 4. This follows from the fact that if \(\Sigma \) is the disjoint union of \(r=7\) involutive lines in \(\mathbb P^3\), then \(\Sigma \) is contained in a degree \(d=4\) hypersurface, i.e., the restriction morphism \(\iota :H^0(\mathbb P^3,\mathcal O_{\mathbb P^3}(4))\rightarrow H^0(\Sigma ,\mathcal O_{\Sigma }(4))\) has a nontrivial kernel (Todd). We give two new proofs of this fact, and finally we show that if \((r,d)\ne (7,4)\) then the map \(\iota \) has maximal rank.

It is widely known that the sum of the angles of a triangle equals two right angles. Far less known are the answers to similar questions for tetrahedra and higher dimensional simplices. In this paper we review some of these less known results, and look at them from a different point of view. Then we continue to derive tight bounds on the dihedral angle sums for the subclass of nonobtuse simplices. All the dihedral angles of such simplices are less than or equal to right. They have several important applications (Brandts et al., 2009). The main conclusion is that when the spatial dimension n is even, the range of dihedral angle sums of nonobtuse simplices is n times smaller than the corresponding range for arbitrary simplices. When n is odd, it is n-1 times smaller.

Motivated by the discontinuous Petrov-Galerkin method from Demkowicz & Gopalakrishnan [2011, Numer. Methods Partial Differential Equations, 27, 70-105], we study a variational formulation of second-order elliptic equations in mixed form that is obtained by piecewise integrating one of the two equations in the system w.r.t. a partition of the domain into mesh cells. We apply a Petrov-Galerkin discretization with optimal test functions, or equivalently, minimize the residual in the natural norm associated to the variational form. These optimal test functions can be found by solving local problems. Well-posedness, uniformly in the partition, and optimal error estimates are demonstrated.In the second part of the paper, the application to convection-diffusion problems is studied. The available freedom in the variational formulation and in its optimal Petrov-Galerkin discretization is used to construct a method that allows a (smooth) passing to a converging method in the convective limit, being a necessary condition to retain convergence and having a bound on the cost for a vanishing diffusion. The theoretical findings are illustrated by several numerical results.

We provide a comparative study of the Subspace Projected Approximate Matrix method, abbreviated SPAM, which is a fairly recent iterative method of computing a few eigenvalues of a Hermitian matrix A. It falls in the category of inner-outer iteration methods and aims to reduce the costs of matrix-vector products with A within its inner iteration. This is done by choosing an approximation A 0 of A, and then, based on both A and A 0, to define a sequence (A k ) =0 of matrices that increasingly better approximate A as the process progresses. Then the matrix A k is used in the kth inner iteration instead of A. In spite of its main idea being refreshingly new and interesting, SPAM has not yet been studied in detail by the numerical linear algebra community. We would like to change this by explaining the method, and to show that for certain special choices for A 0, SPAM turns out to be mathematically equivalent to known eigenvalue methods. More sophisticated approximations A 0 turn SPAM into a boosted version of Lanczos, whereas it can also be interpreted as an attempt to enhance a certain instance of the preconditioned Jacobi-Davidson method. Numerical experiments are performed that are specifically tailored to illustrate certain aspects of SPAM and its variations. For experiments that test the practical performance of SPAM in comparison with other methods, we refer to other sources. The main conclusion is that SPAM provides a natural transition between the Lanczos method and one-step preconditioned Jacobi-Davidson.

Well-posed space-time variational formulations in fractional order Bochner--Sobolev spaces are proposed for parabolic partial differential equations, and in particular for the instationary Stokes and Navier--Stokes equations on bounded Lipschitz domains. The latter formulations include the pressure variable as a primal unknown and so account for the incompressibility constraint via a Lagrange multiplier. The proposed new variational formulations can be the basis of adaptive numerical solution methods that converge with the best possible rate, which, by exploiting the tensor product structure of a Bochner space, equals the rate of best approximation for the corresponding stationary problem. Unbounded time intervals are admissible in many cases, permitting an optimal adaptive solution of long-term evolution problems.

In this chapter we review stochastic modelling methods in climate science. First we provide a conceptual framework for stochastic modelling of deterministic dynamical systems based on the Mori-Zwanzig formalism. The Mori-Zwanzig equations contain a Markov term, a memory term and a term suggestive of stochastic noise. Within this framework we express standard model reduction methods such as averaging and homogenization which eliminate the memory term. We further discuss ways to deal with the memory term and how the type of noise depends on the underlying deterministic chaotic system. Secondly, we review current approaches in stochastic data-driven models. We discuss how the drift and diffusion coefficients of models in the form of stochastic differential equations can be estimated from observational data. We pay attention to situations where the data stems from multi scale systems, a relevant topic in the context of data from the climate system. Furthermore, we discuss the use of discrete stochastic processes (Markov chains) for e.g. stochastic subgrid-scale modeling and other topics in climate science.

Shub & Wilkinson and Ruelle & Wilkinson studied a class of volume preserving diffeomorphisms on the three dimensional torus that are stably ergodic. The diffeomorphisms are partially hyperbolic and admit an invariant central foliation of circles. The foliation is not absolutely continuous, in fact, Ruelle & Wilkinson established that the disintegration of volume along central leaves is atomic. We show that in such a class of volume preserving diffeomorphisms the disintegration of volume along central leaves is a single delta measure. We also formulate a general result for conservative three dimensional skew product like diffeomorphisms on circle bundles, providing conditions for delta measures as disintegrations of the smooth invariant measure., 15 pages, accepted by Proc. Amer. Math. Soc

This thesis contains three related articles which are inserted in separate chapters. The chapters are self contained, hence it is possible to read each of them independently. In each of the chapters we discuss iterated function systems, IFSs, generated by finitely many continuous maps f1, . . . , fk on the unit interval I = [0, 1], taken with positive probabilities pi, ∑ki=1 pi =1, randomly at each iterate. We always assume that these maps fix the boundaries of the interval I and study dynamics of the system under some conditions like the sign of the Lyapunov exponent at the boundaries of the interval. The sign of Lyapunov exponent at a boundary determines whether the boundary is, on average, attracting (negative Lyapunov exponent), neutral (zero), or repelling (positive). We discuss occurrence of intriguing dynamical phenomena such as synchronization, intermingled basins and intermittency, in this context.

Fatou components for rational endomorphisms of the Riemann sphere are fully classified and play an important role in our view of one-dimensional dynamics. In higher dimensions, the situation is less satisfactory. In this work we give a nearly complete classification of invariant Fatou components for moderately dissipative Hénon maps. Namely, we prove that any such a component is either an attracting or parabolic basin, or the basin of a rotation domain. More specifically, recurrent Fatou components were classified about 20 years ago (modulo the problem of existence of Herman ring basins), while in this paper we prove that non-recurrent invariant Fatou components are semi-parabolic basins. Most of our methods apply in a more general setting.

We prove that any compact manifold without boundary admits a pair of diffeomorphisms that generates $C^1$ robustly minimal dynamics. We apply the results to the construction of blenders and robustly transitive skew product diffeomorphisms., 16 pages, 2 figures

The instationary Stokes and Navier-Stokes equations are considered in a simultaneously space-time variational saddle point formulation, so involving both velocities u and pressure p. For the instationary Stokes problem, it is shown that the corresponding operator is a boundedly invertible linear mapping between H1 and H'2, both Hilbert spaces H1 and H2 being Cartesian products of (intersections of) Bochner spaces, or duals of those. Based on these results, the operator that corresponds to the Navier−Stokes equations is shown to map H1 into H'2, with a Frechet derivative that, at any (u,p) ∈ H1, is boundedly invertible. These results are essential for the numerical solution of the combined pair of velocities and pressure as function of simultaneously space and time. Such a numerical approach allows for the application of (adaptive) approximation from tensor products of spatial and temporal trial spaces, with which the instationary problem can be solved at a computational complexity that is of the order as for a corresponding stationary problem.

An algorithm is presented that for a local bilinear form evaluates in linear complexity the application of the stiffness matrix w.r.t. a collection of tensor product multiscale basis functions, assuming that this collection has a multi-tree structure. It generalizes an algorithm for sparse-grid index sets [R. Balder, Ch. Zenger, The solution of multidimensional real Helmholtz equations on sparse grids, SIAM J. Sci. Comput. 17 (3) (1996) 631-646] and it finds its application in adaptive tensor product approximation methods.

The adaptive wavelet Galerkin method for solving linear, elliptic operator equations introduced by Cohen et al. (Math Comp 70:27---75, 2001) is extended to nonlinear equations and is shown to converge with optimal rates without coarsening. Moreover, when an appropriate scheme is available for the approximate evaluation of residuals, the method is shown to have asymptotically optimal computational complexity. The application of this method to solving least-squares formulations of operator equations $$G(u)=0$$G(u)=0, where $$G:H \rightarrow K'$$G:HźKź, is studied. For formulations of partial differential equations as first-order least-squares systems, a valid approximate residual evaluation is developed that is easy to implement and quantitatively efficient.

The main purpose of this paper is to introduce and study the notion of plurifinely-maximal plurifinely plurisubharmonic functions, which extends the notion of maximal plurisubharmonic functions on a Euclidean domain to a plurifine domain of C^n in a natural way. Our main result is that a finite plurifinely plurisubharmonic function u on a plurifine domain U satisfies (dd^c u)^n=0 if and only if u is plurifinely-locally plurifinely-maximal outside some pluripolar set. In particular, a finite plurifinely-maximal plurisubharmonic function u satisfies (dd^c u)^n=0., 20 pages, manuscript as accepted by publisher. To appear in Potential Analysis

We study the convergence of multigrid schemes for the Helmholtz equation, focusing in particular on the choice of the coarse scale operators. Let G_c denote the number of points per wavelength at the coarse level. If the coarse scale solutions are to approximate the true solutions, then the oscillatory nature of the solutions implies the requirement G_c > 2. However, in examples the requirement is more like G_c >= 10, in a trade-off involving also the amount of damping present and the number of multigrid iterations. We conjecture that this is caused by the difference in phase speeds between the coarse and fine scale operators. Standard 5-point finite differences in 2-D are our first example. A new coarse scale 9-point operator is constructed to match the fine scale phase speeds. We then compare phase speeds and multigrid performance of standard schemes with a scheme using the new operator. The required G_c is reduced from about 10 to about 3.5, with less damping present so that waves propagate over > 100 wavelengths in the new scheme. Next we consider extensions of the method to more general cases. In 3-D comparable results are obtained with standard 7-point differences and optimized 27-point coarse grid operators, leading to an order of magnitude reduction in the number of unknowns for the coarsest scale linear system. Finally we show how to include PML boundary layers, using a regular grid finite element method. Matching coarse scale operators can easily be constructed for other discretizations. The method is therefore potentially useful for a large class of discretized high-frequency Helmholtz equations., Coarse scale operators are simplified and only standard smoothers used in v3; 5 figures, 12 tables

Dit jaar is het vijftig jaar geleden dat de bekende Nederlandse wiskundige L.E.J. Brouwer overleed. Ter gelegenheid van dit feit organiseerden het Koninklijk Wiskundig Genootschap samen met de Koninklijke Hollandsche Maatschappij der Wetenschappen op 2 september een lezingenmiddag in het Hodshon Huis in Haarlem, de plaats waar Brouwer een belangrijk deel van zijn jeugd doorbracht. Dit artikel is een weergave van de lezing die Jaap Korevaar op deze middag hield.

In this work, we construct a well-posed first-order system least squares (FOSLS) simultaneously space-time formulation of parabolic PDEs. Using an adaptive wavelet solver, this problem is solved with the best possible rate in linear complexity. Thanks to the use of a basis that consists of tensor products of wavelets in space and time, this rate is equal to that when solving the corresponding stationary problem. Our findings are illustrated by numerical results.

In this study, we present a novel method for quantifying dependencies in multivariate datasets, based on estimating the Rényi mutual information by minimum spanning trees (MSTs). The extent to which random variables are dependent is an important question, e.g., for uncertainty quantification and sensitivity analysis. The latter is closely related to the question how strongly dependent the output of, e.g., a computer simulation, is on the individual random input variables. To estimate the Rényi mutual information from data, we use a method due to Hero et al. that relies on computing minimum spanning trees (MSTs) of the data and uses the length of the MST in an estimator for the entropy. To reduce the computational cost of constructing the exact MST for large datasets, we explore methods to compute approximations to the exact MST, and find the multilevel approach introduced recently by Zhong et al. (2015) to be the most accurate. Because the MST computation does not require knowledge (or estimation) of the distributions, our methodology is well-suited for situations where only data are available. Furthermore, we show that, in the case where only the ranking of several dependencies is required rather than their exact value, it is not necessary to compute the Rényi divergence, but only an estimator derived from it. The main contributions of this paper are the introduction of this quantifier of dependency, as well as the novel combination of using approximate methods for MSTs with estimating the Rényi mutual information via MSTs. We applied our proposed method to an artificial test case based on the Ishigami function, as well as to a real-world test case involving an El Nino dataset.

We give an explicit construction of families of Dm-equivariant polynomial vector fields in possessing a codimension one homoclinic cycle. The homoclinic cycle consists of m homoclinic trajectories all connected to the equilibrium at the origin. The constructed vector fields can provide a setting for a (numerical) bifurcation study of these homoclinic cycles, in particular for m equal to a multiple of 4, where the bifurcations form an open problem. © 2013 © 2013 Taylor & Francis.

We will define the Monge-Ampere operator on finite (weakly) plurifinely plurisubharmonic functions in plurifinely open sets U ⊂ ℂ n and show that it defines a positive measure. Ingredients of the proof include a direct proof for bounded strongly plurifinely plurisubharmonic functions, which is based on the fact that such functions can plurifinely locally be written as difference of ordinary plurisubharmonic functions, and an approximation result stating that in the Dirichlet norm weakly plurifinely plurisubharmonic functions are locally limits of plurisubharmonic functions. As a consequence of the latter, weakly plurifinely plurisubharmonic functions are strongly plurifinely plurisubharmonic outside of a pluripolar set.

A new domain decomposition method is introduced for the heterogeneous 2-D and 3-D Helmholtz equations. Transmission conditions based on the perfectly matched layer (PML) are derived that avoid artificial reflections and match incoming and outgoing waves at the subdomain interfaces. We focus on a subdivision of the rectangular domain into many thin subdomains along one of the axes, in combination with a certain ordering for solving the subdomain problems and a GMRES outer iteration. When combined with multifrontal methods, the solver has near-linear cost in examples, due to very small iteration numbers that are essentially independent of problem size and number of subdomains. It is to our knowledge only the second method with this property next to the moving PML sweeping method., Comment: 16 pages, 3 figures, 6 tables - v2 accepted for publication in the Journal of Computational Physics

A wide class of well-posed operator equations can be solved in optimal computational complexity by adaptive wavelet methods. A quantitative bottleneck is the approximate evaluation of the arising residuals that steer the adaptive refinements. In this paper, we consider multi-tree approximations from tensor product wavelet bases for solving linear PDE’s. In this setting, we develop a new efficient approximate residual evaluation. Other than the commonly applied method, that uses the so-called APPLY routine, our approximate residual depends affinely on the current approximation of the solution. Our findings are illustrated by numerical results that show a considerable speed-up.

Triangulations of the cube into a minimal number of simplices without additional vertices have been studied by several authors over the past decades. For $3\leq n\leq 7$ this so-called simplexity of the unit cube $I^n$ is now known to be $5,16,67,308,1493$, respectively. In this paper, we study triangulations of $I^n$ with simplices that only have nonobtuse dihedral angles. A trivial example is the standard triangulation into $n!$ simplices. In this paper we show that, surprisingly, for each $n\geq 3$ there is essentially only one other nonobtuse triangulation of $I^n$, and give its explicit construction. The number of nonobtuse simplices in this triangulation is equal to the smallest integer larger than $n!({\rm e}-2)$., 17 pages, 7 figures

Following [Studia Math., 76(2) (1983), pp. 1-58 and 95-136] by Z. Ciesielski and T. Figiel and [SIAM J. Math. Anal., 31 (1999), pp. 184-230] by W. Dahmen and R. Schneider, by the application of extension operators we construct a basis for a range of Sobolev spaces on a domain $ \Omega $ from corresponding bases on subdomains that form a non-overlapping decomposition. As subdomains, we take hypercubes, or smooth parametric images of those, and equip them with tensor product wavelet bases. We prove approximation rates from the resulting piecewise tensor product basis that are independent of the spatial dimension of $ \Omega $. For two- and three-dimensional polytopes we show that the solution of Poisson type problems satisfies the required regularity condition. The dimension independent rates will be realized numerically in linear complexity by the application of the adaptive wavelet-Galerkin scheme.

We study the convergence of the so-called entangled ergodic averages $\frac{1}{N^k}\sum_{n_1,...,n_k=1}^{N}T_m^{n_{\alpha(m)}}A_{m-1}T_{m-1}^{n_{\alpha(m-1)}}A_{m-2}...A_1T_1^{n_{\alpha(1)}},$ where $k\leq m$ and $\alpha:\{1,...,m\}\to\{1,...,k\}$ is a surjective map. We show that, on general Banach spaces and without any restriction on the partition $\alpha$, the above averages converge strongly as $N\to \infty$ under some quite weak compactness assumptions on the operators $T_j$ and $A_j$. A formula for the limit based on the spectral analysis of the operators $T_j$ and the continuous version of the result are presented as well., Comment: 14 pages, submitted; minor corrections, references updated

Let ${F}_{BC} (\lambda , k; t)$ be the Heckman–Opdam hypergeometric function of type BC with multiplicities $k= ({k}_{1} , {k}_{2} , {k}_{3} )$ and weighted half-sum $\rho (k)$ of positive roots. We prove that ${F}_{BC} (\lambda + \rho (k), k; t)$ converges as ${k}_{1} + {k}_{2} \rightarrow \infty $ and ${k}_{1} / {k}_{2} \rightarrow \infty $ to a function of type A for $t\in { \mathbb{R} }^{n} $ and $\lambda \in { \mathbb{C} }^{n} $. This limit is obtained from a corresponding result for Jacobi polynomials of type BC, which is proven for a slightly more general limit behavior of the multiplicities, using an explicit representation of Jacobi polynomials in terms of Jack polynomials. Our limits include limit transitions for the spherical functions of non-compact Grassmann manifolds over one of the fields $ \mathbb{F} = \mathbb{R} , \mathbb{C} , \mathbb{H} $ when the rank is fixed and the dimension tends to infinity. The limit functions turn out to be exactly the spherical functions of the corresponding infinite-dimensional Grassmann manifold in the sense of Olshanski.

We prove a uniform extension of the Wiener-Wintner theorem for nilsequences due to Host and Kra and a nilsequence extension of the topological Wiener-Wintner theorem due to Assani. Our argument is based on (vertical) Fourier analysis and a Sobolev embedding theorem., Comment: v3: 18 p., proof that the cube construction produces compact homogeneous spaces added, measurability issues in the proof of Theorem 1.5 addressed. We thank the anonymous referees for pointing out these gaps in v2

There are parallels between de Bruijn’s early work in analysis and that of the author. However, Dick’s work soon became much broader and deeper. While the present paper reviews several topics of common interest, its main content is a short version of Dick’s important article related to Riemann’s Hypothesis, entitled ‘The roots of trigonometric integrals’ [N.G. de Bruijn, The roots of trigonometric integrals, Duke Math. J. 17 (1950) 197–226]. The associated ‘de Bruijn–Newman constant’ is also discussed.

We review recent results from the theory of random differential equations with bounded noise. Assuming the noise to be “sufficiently robust in its effects” we discuss the feature that any stationary measure of the system is supported on a “Minimal Forward Invariant” (MFI) set. We review basic properties of the MFI sets, including their relationship to attractors in systems where the noise is small. In the main part of the paper we discuss how MFI sets can undergo discontinuous changes that we have called hard bifurcations. We characterize such bifurcations for systems in one and two dimensions and we give an example of the effects of bounded noise in the context of a Hopf–Andronov bifurcation.

In this paper we introduce a conceptual model for vegetation patterns that generalizes the Klausmeier model for semi-arid ecosystems on a sloped terrain (Klausmeier in Science 284:1826-1828, 1999). Our model not only incorporates downhill flow, but also linear or nonlinear diffusion for the water component.To relate the model to observations and simulations in ecology, we first consider the onset of pattern formation through a Turing or a Turing-Hopf bifurcation. We perform a Ginzburg-Landau analysis to study the weakly nonlinear evolution of small amplitude patterns and we show that the Turing/Turing-Hopf bifurcation is supercritical under realistic circumstances.In the second part we numerically construct Busse balloons to further follow the family of stable spatially periodic (vegetation) patterns. We find that destabilization (and thus desertification) can be caused by three different mechanisms: fold, Hopf and sideband instability, and show that the Hopf instability can no longer occur when the gradient of the domain is above a certain threshold. We encounter a number of intriguing phenomena, such as a 'Hopf dance' and a fine structure of sideband instabilities. Finally, we conclude that there exists no decisive qualitative difference between the Busse balloons for the model with standard diffusion and the Busse balloons for the model with nonlinear diffusion.

An identity by Chaundy and Bullard writes 1/(1-x)^n (n=1,2,...) as a sum of two truncated binomial series. In a paper which appeared in 2008 in Indag. Math. the authors surveyed many aspects of this identity. In the present paper we discuss much earlier occurrences of this identity in works by Hering (1868), de Moivre (1738) and de Montmort (1713). A relationship with Krawtchouk polynomials in work by Greville (1966) is also discussed., 8 pages, in v2 slightly more material added

Conditional Markov chain (CMC) models have proven to be promising building blocks for stochastic convection parameterizations. In this paper, it is demonstrated how two different CMC models can be used as mass flux closures in convection parameterizations. More specifically, the CMC models provide a stochastic estimate of the convective area fraction that is directly proportional to the cloud-base mass flux. Since, in one of the models, the number of CMCs decreases with increasing resolution, this approach makes convection parameterizations scale aware and introduces stochastic fluctuations that increase with resolution in a realistic way. Both CMC models are implemented in a GCM of intermediate complexity. It is shown that with the CMC models, trained with observational data, it is possible to improve both the subgrid-scale variability and the autocorrelation function of the cloud-base mass flux as well as the distribution of the daily accumulated precipitation in the tropics. Hovmöller diagrams and wavenumber–frequency diagrams of the equatorial precipitation indicate that, in this specific GCM, convectively coupled equatorial waves are more sensitive to the mean cloud-base mass flux than to stochastic fluctuations. A smaller mean mass flux tends to increase the power of the simulated MJO and to diminish equatorial Kelvin waves.

In this thesis, Discontinuous Petrov-Galerkin (DPG) finite element methods are developed for convection-diffusion equations. In particular, this thesis focuses on the use of optimal test spaces. A convection-diffusion equation is a singularly perturbed problem. That is, the nature of the problem changes when the diffusion term vanishes, which makes it challenging to solve numerically for small diffusion values, i.e. when convection dominates. Standard finite element methods give very unsatisfactory results, producing approximations that exhibit spurious oscillations and other nonphysical behavior. Recently, a class of finite element methods has been developed, in which optimal test spaces are used. These spaces guarantee that one gets the best approximation from the trial space in which the solution is sought. The methods are examples of least-squares methods, with the special property that one can choose the norm in which the residual is minimized. This freedom of choice allows us to control the norm in which the best approximation is obtained. The new approach in this thesis is that the variational formulation associated with the convection-diffusion problem also gives a well-posed variational formulation of the limit convection problem if the diffusion term vanishes. This is necessary in order to retain stability, and to make sure that the computational cost does not grow, when the diffusion term decreases. Special attention is paid to the transport problem which, besides being the limit problem for vanishing diffusion, also has other applications. A new method is introduced that outperforms existing methods in convergence rates, but also in reducing the smearing of discontinuities of solutions. The theory developed in this thesis is illustrated by various numerical results.

We show that there exist polynomial endomorphisms of C^2, possessing a wandering Fatou component. These mappings are polynomial skew-products, and can be chosen to extend holomorphically of P^2(C). We also find real examples with wandering domains in R^2. The proof is based on parabolic implosion techniques, and is based on an original idea of M. Lyubich., Comment: Revised introduction

Until recently, little was known about the existence of wandering Fatou components for rational maps in more than one complex variables. In 2014, examples of wandering Fatou components were constructed in Astorg et al. [1] for polynomial skew-products with an invariant parabolic fiber. In 2004 Lilov already proved the non-existence of wandering Fatou components for polynomial skew-products in the basin of an invariant super-attracting fiber. In the current work we investigate in how far his methods carry over to the geometrically attracting case. Lilov proved a stronger statement, namely that the orbit of any horizontal disk in the super-attracting basin must eventually intersect a fattened Fatou components. Here we give explicit constructions to show that this stronger result is false in the geometrically attracting case. However, we also prove that the constructed disks do not give rise to wandering Fatou components. Our construction therefore leaves open the existence of wandering Fatou components in the geometrically attracting case, showing however that the situation is significantly more complicated than in the super-attracting case.

We extend a recent result of Avelin, Hed, and Persson about approximation of functions $u$ that are plurisubharmonic on a domain $\Omega$ and continuous on $\bar\Omega$, with functions that are plurisubharmonic on (shrinking) neighborhoods of $\bar\Omega$. We show that such approximation is possible if the boundary of $\Omega$ is $C^0$ outside a countable exceptional set $E\subset\partial \Omega$. In particular, approximation is possible on the Hartogs triangle. For H\"older continuous $u$, approximation is possible under less restrictive conditions on $E$. We next give examples of domains where this kind of approximation is not possible, even when approximation in the H\"older continuous case is possible., Comment: 11 pages

For the Poisson problem in two dimensions, we consider the standard adaptive finite element loop solve, estimate, mark, refine, with estimate being implemented using the p-robust equilibrated flux estimator, and, mark being Dörfler marking. As a refinement strategy we employ p-refinement. We investigate the question by which amount the local polynomial degree on any marked patch has to be incremented in order to achieve a p-independent error reduction. We show that the analysis can be transferred from the patches to a reference triangle, and therein we provide clear-cut computational evidence that any increment proportional to the polynomial degree (for any fixed proportionality constant) yields the desired reduction. The resulting adaptive method can be turned into an instance optimal hp-adaptive method by the addition of a coarsening routine.

On the n-dimensional hypercube, for given \(k\in {\mathbb {N}}\), wavelet Riesz bases are constructed for the subspace of divergence-free vector fields of the Sobolev space \(H^k((0,1)^n)^n\) with general homogeneous Dirichlet boundary conditions, including slip or no-slip boundary conditions. Both primal and suitable dual wavelets can be constructed to be locally supported. The construction of the isotropic wavelet bases is restricted to the square, but that of the anisotropic wavelet bases applies for any space dimension n.

This thesis focuses on the study of certain special classes of n-simplices that occur in the context of numerical analysis, linear algebra, abstract algebra, geometry, and combinatorics. The type of simplex that is of central interest is the nonobtuse simplex, a simplex without any obtuse dihedral angles. Nonobtuse simplices without right dihedral angles are called acute. Special attention will be paid to acute and nonobtuse simplices whose vertices are vertices of the unit n-cube, the so-called 0/1-simplices. Several qualitative properties of finite element approximations of PDEs do not allow simplices in the triangulation of the physical domain to have obtuse or even right dihedral angles. This motivates to investigate whether such triangulations of a given computational domain into nonobtuse or acute simplices actually exist. As a possible tool to tackle related restricted triangulation problems, we prove sharp upper and lower bounds on the sum of all the dihedral angles of a nonobtuse n-simplex. Recognizing a nonnegative matrix as the inverse of an M-matrix, without actually performing the inversion, is an open problem. M-matrices figure in iterative methods for linear systems, are used in numerical linear algebra to yield eigenvalue bounds, and appear in finite Markov chains. In this thesis, we associate the set of inverses of symmetric diagonally dominant M-matrices with nonobtuse simplices. This enables to study the inverse M-matrix problem from a geometric viewpoint, and we prove several results, in particular for simplices whose facets are all nonobtuse. To facilitate the study of acute and nonobtuse 0/1-simplices, we enumerate all of them modulo the symmetries of the unit n-cube, for some larger values of n. Our interest in this type of simplices comes from Hadamard’s maximal determinant conjecture which is equivalent to showing that there exists a regular 0/1-simplex in the unit n-cube when n − 3 is a multiple of 4. In particular, we give a full description of acute 0/1-simplices that can be represented by an irreducible upper Hessenberg matrix.

Persson and Strang (2003) evaluated the integral over [-1,1] of a squared odd degree Legendre polynomial divided by x^2 as being equal to 2. We consider a similar integral for orthogonal polynomials with respect to a general even orthogonality measure, with Gegenbauer and Hermite polynomials as explicit special cases. Next, after a quadratic transformation, we are led to the general nonsymmetric case, with Jacobi and Laguerre polynomials as explicit special cases. Examples of indefinite summation also occur in this context. The paper concludes with a generalization of the earlier results for Hahn polynomials. There some adaptations have to be made in order to arrive at relatively nice explicit evaluations., Comment: 14 pages; minor corrections and reference added; J. Math. Anal. Appl. (2011)