The Cauchy problem for general elliptic equations of second order is considered. In a previous paper (Berntsson et al. in Inverse Probl Sci Eng 26(7):1062–1078, 2018), it was suggested that the alternating iterative algorithm suggested by Kozlov and Maz’ya can be convergent, even for large wavenumbers k², in the Helmholtz equation, if the Neumann boundary conditions are replaced by Robin conditions. In this paper, we provide a proof that shows that the Dirichlet–Robin alternating algorithm is indeed convergent for general elliptic operators provided that the parameters in the Robin conditions are chosen appropriately. We also give numerical experiments intended to investigate the precise behaviour of the algorithm for different values of k² in the Helmholtz equation. In particular, we show how the speed of the convergence depends on the choice of Robin parameters.

The variational capacity cap(p) in Euclidean spaces is known to enjoy the density dichotomy at large scales, namely that for every E subset of R-n, infx is an element of R(n)cap(p)(E boolean AND B(x, r), B(x, 2r))/cap(p)(B(x, r), B(x, 2r)) is either zero or tends to 1 as r -amp;gt; infinity. We prove that this property still holds in unbounded complete geodesic metric spaces equipped with a doubling measure supporting a p-Poincare inequality, but that it can fail in nongeodesic metric spaces and also for the Sobolev capacity in R-n. It turns out that the shape of balls impacts the validity of the density dichotomy. Even in more general metric spaces, we construct families of sets, such as John domains, for which the density dichotomy holds. Our arguments include an exact formula for the variational capacity of superlevel sets for capacitary potentials and a quantitative approximation from inside of the variational capacity.

The variational solution to the Zaremba problem for a divergent linear second order elliptic equation with measurable coefficients is considered. The problem is set in a local Lipschitz graph domain. An estimate in L_2+δ, δ > 0, for the gradient of a solution, is proved. An example of the problem with the Dirichlet data supported by a fractal set of zero (n - 1)-dimensional measure and non-zero p-capacity, p > 1 is constructed.

This thesis aims to provide an introduction to the field of potential theory at an undergraduate level, by studying an important mathematical problem in the field, namely the Dirichlet problem. By examining the historical development of different methods for solving the problem in increasingly general contexts, and the mathematical concepts which were established to support these methods, the aim is to provide an overview of various basic techniques in the field of potential theory, as well as a summary of the fundamental results concerning the Dirichlet problem in Euclidean space.

We present an elementary proof of a well-known theorem of Cheeger which states that if a metric-measure space X supports a p-Poincaré inequality, then the N^1,p(X) Sobolev space is reflexive and separable whenever p ∈ (1, ∞). We also prove separability of the space when p=1. Our proof is based on a straightforward construction of an equivalent norm on N^1,p(X), p ∈ [1, ∞), that is uniformly convex when p ∈ (1, ∞). Finally, we explicitly construct a functional that is pointwise comparable to the minimal p-weak upper gradient, when p ∈ (1, ∞).

The linear-drift memristor model, suggested by HP Labs a few years ago, is used in this work together with two window functions. From the equations describing the memristor model, the transfer characteristics of a memristor is formulated and analyzed. A first-order estimation of the cut-off frequency is shown, that illustrates the bandwidth limitation of the memristor and how it varies with some of its physical parameters. The design space is elaborated upon and it is shown that the state speed, the variation of the doped and undoped regions of the memristor, is inversely proportional to the physical length, and depth of the device. The transfer characteristics is simulated for Joglekar-Wolf, and Biolek window functions and the results are analyzed. The Joglekar-Wolf window function causes a distinct behavior in the tranfer characteristics at cut-off frequency. The Biolek window function on the other hand gives a smooth state transfer function, at the cost of loosing the one-to-one mapping between charge and state. We also elaborate on the design constraints derived from the transfer characteristics.

In this thesis we study various properties of the Lanczos spinor. The results include an algebraic classification scheme for symmetric (3,1)-spinors, a link between Lanczos potentials of the Weyl spinor and the spin coefficients in certain classes of spacetimes, an existence proof for the Lanczos potential of a general Weyl candidate that is much simpler than those previously known and the existence of a symmetric potential H_ABA'B' of an arbitrary symmetric (3,1)-spinor L_ABCA' in Einstein spacetimes according tothe equation L_ABCA' = ∇(_A^B' H_BC)A'B'. In addition we study a large subclass of algebraically special spacetimes and obtain necessary and sufficient conditions for a Lanczos potential of the Weyl spinor to define a metric, curvature-free connection; we also prove existence of such connections. This construction is analogous to a construction of quasi-local momentum in the Kerr spacetime by Bergqvist and Ludvigsen and we therefore obtain an analogue of the Nester-Witten 2-form in these spacetimes.

Human activity and other events can cause environmental changes to the habitat of organisms. The environmental changes effect the vital rates for a population. In order to predict the impact of these environmental changes on populations, we use two different models for population dynamics. One simpler linear model that ignores environmental competition between individuals and another model that does not. Our population models take into consideration the age distribution of the population and thus takes into consideration the impact of demographics. This thesis generalize two theorems, one for each model, developed by Sonja Radosavljevic regarding long term upper and lower bounds of a population with periodic birth rate ; see [6] and [5]. The generalisation consist in including the case where the periodic part of the birth rate can be expressed with a finite Fourier series and also infinite Fourier series under some constraints. The old theorems only considers the case when the periodic part of the birth rate can be expressed with one cosine term. From the theorems we discover a connection between the frequency of oscillation and the effect on population growth. From this derived connection we conclude that periodical changing environments can have both positive and negative effects on the population.

In nature many pathogens and in particular strains of pathogens with negative effects on species coexists. This is for simplicity often ignored in many epidemiological models. It is however still of interest to get a deeper understanding how this coexistence affects the dynamics of the disease. There are several ways at which coexistence can influence the dynamics. Coinfection which is the simultaneous infection of two or more pathogens can cause increased detrimental health effects on the host. Pathogens can also limit each others growth by the effect of cross immunity  as well as promoting isolation. On the contrary one pathogen can also aid another by making the host more vulnerable to as well as more inclined to spread disease.

Spread of disease is dependent on the density of the population. If a pathogen is able to spread or not, is strongly correlated with how many times individuals interact with each other. This in turn depends on how many individuals live in a given area. 

The aim of papers I-III is to provide an understanding how different factors including the carrying capacity of the host population affect the dynamics of two coexisting diseases. 

In papers I-III we investigate how the parameters effects the long term solution in the form of a stable equilibrium point. In particular we want to provide an understanding of how changes in the carrying capacity affects the long term existence of each disease as well as the occurrence of coinfection.

The model that is studied in papers I-III is a generalization of the standard susceptible, infected, recovered (SIR) compartmental model. The SIR model is generalized by the introduction of the second infected compartment as well as the coinfection compartment. We also use a logistic growth term à la Verhulst with associated carrying capacity K. In paper I and II we make the simplifying assumption that a coinfected individual has to, if anything, transmit both of the disease and simply not just one of them. This restriction is relaxed in paper III. In all papers I-III however we do restrict ourselves by letting all transmission rates, that involves scenarios where the newly infected person does not move to same compartment as the infector, to be small. By small we here mean that the results at least hold when the relevant parameters are small enough.

In all paper I-III it turns out that for each set of parameters excluding K there exist a unique branch of mostly stable equilibrium points depending continuously on K. We differentiate the equilibrium points of the branch by which compartments are non-zero which we refer to as the type of the equilibrium. The way that the equilibrium point changes its type with K is made clear with the use of transition diagrams together with graphs for the stable susceptible population over K.

In paper IV we consider a model for a single age-structured population á la Mckendric-von-Foerster with the addition of differing density dependence on the birth and death rates. Each vital rate is a function of age as well as a weighing of the population also referred to as a size. The birth rate influencing size and the death rate influencing size can be weighted differently allowing us to consider different age-groups to influence the birth and death rate in different proportions compared to other age groups. 

It is commonly assumed that an increase of population density is detrimental to the survival of each individual. However, for various reasons, it is know that for some species survival is positively correlated with population density when the population is small. This is called the Allee effect and our model includes this scenario.

It is shown that the trivial equilibrium, which signifies extinction, is locally stable if the basic reproductive rate $R_0$ is less then 1. This implies global stability with certain extinction if no Allee effect is present. However if the Allee effect is present we show that the population can persist even if R₀ < 1.

In this paper we develop a compartmental model of SIR type (the abbreviation refers to the number of Susceptible, Infected and Recovered people) that models the population dynamics of two diseases that can coinfect. We discuss how the underlying dynamics depends on the carrying capacity K: from a simple dynamics to a more complex. This can also help in understanding the appearance of more complicated dynamics, for example, chaos and periodic oscillations, for large values of K. It is also presented that pathogens can invade in population and their invasion depends on the carrying capacity K which shows that the progression of disease in population depends on carrying capacity. More specifically, we establish all possible scenarios (the so-called transition diagrams) describing an evolution of an (always unique) locally stable equilibrium state (with only non-negative compartments) for fixed fundamental parameters (density independent transmission and vital rates) as a function of the carrying capacity K. An important implication of our results is the following important observation. Note that one can regard the value of K as the natural ‘size’ (the capacity) of a habitat. From this point of view, an isolation of individuals (the strategy which showed its efficiency for COVID-19 in various countries) into smaller resp. larger groups can be modelled by smaller resp. bigger values of K. Then we conclude that the infection dynamics becomes more complex for larger groups, as it fairly maybe expected for values of the reproduction number R0≈1. We show even more, that for the values R0>1 there are several (in fact four different) distinguished scenarios where the infection complexity (the number of nonzero infected classes) arises with growing K. Our approach is based on a bifurcation analysis which allows to generalize considerably the previous Lotka-Volterra model considered previously in Ghersheen et al. (Math Meth Appl Sci 42(8), 2019).

In this paper we continue the stability analysis of the model for coinfection with density dependent susceptible population introduced in Andersson et al. (Effect of density dependence on coinfection dynamics. arXiv:2008.09987, 2020). We consider the remaining parameter values left out from Andersson et al. (Effect of density dependence on coinfection dynamics. arXiv:2008.09987, 2020). We look for coexistence equilibrium points, their stability and dependence on the carrying capacity K. Two sets of parameter value are determined, each giving rise to different scenarios for the equilibrium branch parametrized by K. In both scenarios the branch includes coexistence points implying that both coinfection and single infection of both diseases can exist together in a stable state. There are no simple explicit expression for these equilibrium points and we will require a more delicate analysis of these points with a new bifurcation technique adapted to such epidemic related problems. The first scenario is described by the branch of stable equilibrium points which includes a continuum of coexistence points starting at a bifurcation equilibrium point with zero single infection strain #1 and finishing at another bifurcation point with zero single infection strain #2. In the second scenario the branch also includes a section of coexistence equilibrium points with the same type of starting point but the branch stays inside the positive cone after this. The coexistence equilibrium points are stable at the start of the section. It stays stable as long as the product of K and the rate γ¯γ¯ of coinfection resulting from two single infections is small but, after this it can reach a Hopf bifurcation and periodic orbits will appear.

This paper introduces a new method for the determination of near-optimal audit strategies with the Erard & Feinstein model. It is demonstrated that the solution method presented by Erard & Feinstein does not lead to optimal audit strategies. The new method simulates taxpayer behaviour for a class of audit functions among which the best function is selected. Using a simple class of functions, the resulting strategies are superior to those obtained with the Erard & Feinstein method.

The simulation method can easily handle variations of the model, for instance with involuntary taxpayer errors as well as with partially discrete true income distributions.

The aim of this project is to study the possibility to apply audit strategies developed for taxation on fraud and involuntary errors in the social benefit sector. The efficiency of different audit strategies is compared using a computer-based optimization algorithm.

Two types of audit strategies are used in this study. One is to adapt the audit intensity to the propensity for errors and fraud in different segments of the group studied. The other type of audit strategy is based on adaptation of behaviour through information concerning the audit intensity. A model for determination of optimal tax audit strategies of the latter type was developed by Erard & Feinstein in 1994.

This study is based on data from a large study of temporary parental benefit performed by the Institute for Evaluation of Labour Market and Education Policy (Institutet för arbetsmarknadspolitisk utvärdering, IFAU) in 2006.

The study has shown that it is possible to apply the Erard & Feinstein model on benefit fraud. However, the solution method developed by Erard & Feinstein has proven to be non-optimal. A new solution method based on simulation has been developed and used in the study.

We answer the question initially posed by Arik Tamir at the Fourth NYU Computational Geometry Day (March, 1987): “Given a collection of compact sets, can one decide in polynomial time whether there exists a convex body whose boundary intersects every set in the collection?”

We prove that when the sets are segments in the plane, deciding existence of the convex stabber is NP-hard. The problem remains NP-hard if the sets are regular polygons. We also show that in 3D the stabbing problem is hard when the sets are balls. On the positive side, we give a polynomial-time algorithm to find a convex transversal of a maximum number of pairwise-disjoint segments in 2D if the vertices of the transversal are restricted to a given set of points. Our algorithm also finds a convex stabber of the maximum number of a set of convex pseudodisks in the plane.

The stabbing problem is related to “convexity” of point sets measured as the minimum distance by which the points must be shifted in order to arrive in convex position; we give a PTAS to find the minimum shift in 2D, and a 2-approximation in any dimension. We also consider stabbing with vertices of a regular polygon – a problem closely related to approximate symmetry detection.

We consider discrete minimal surface algebras (DMSA) as generalized noncommutative analogues of minimal surfaces in higher dimensional spheres. These algebras appear naturally in membrane theory, where sequences of their representations are used as a regularization. After showing that the defining relations of the algebra are consistent, and that one can compute a basis of the enveloping algebra, we give several explicit examples of DMSAs in terms of subsets of sl_n (any semi-simple Lie algebra providing a trivial example by itself). A special class of DMSAs are Yang-Mills algebras. The representation graph is introduced to study representations of DMSAs of dimension d ≤ 4, and properties of representations are related to properties of graphs. The representation graph of a tensor product is (generically) the Cartesian product of the corresponding graphs. We provide explicit examples of irreducible representations and, for coinciding eigenvalues, classify all the unitary representations of the corresponding algebras.

As n-ary operations, generalizing Lie and Poisson algebras, arise in many different physical contexts, it is interesting to study general ways of constructing explicit realizations of such multilinear structures. Generically, they describe the dynamics of a physical system, and there is a need of understanding their quantization. Hom-Nambu-Lie algebras provide a framework that might be an appropriate setting in which n-Lie algebras (n-ary Nambu-Lie algebras) can be deformed, and their quantization studied. We present a procedure to construct (n + 1)-ary Hom-Nambu-Lie algebras from n-ary Hom-Nambu-Lie algebras equipped with a generalized trace function. It turns out that the implications of the compatibility conditions, that are necessary for this construction, can be understood in terms of the kernel of the trace function and the range of the twisting maps. Furthermore, we investigate the possibility of defining (n + k)-Lie algebras from n-Lie algebras and a k-form satisfying certain conditions.

We prove a general interpolation theorem for linear operators acting simultaneously in several approximation spaces which are defined by multiparametric approximation families. As a consequence, we obtain interpolation results for finite families of Besov spaces of various types including those determined by a given set of mixed differences.

For p is an element of [1, infinity], we establish criteria for the one-sided invertibility of binomial discrete difference operators A = aI - bV on the space l(p) = l(p)(Z), where a, b is an element of l(infinity), I is the identity operator and the isometric shift operator V is given on functions f. lp by (Vf)(n) = f (n+ 1) for all n is an element of Z. Applying these criteria, we obtain criteria for the one-sided invertibility of binomial functional operators A = aI - bU(alpha) on the Lebesgue space L-p(R+) for every p is an element of [1, infinity], where a, b is an element of L-infinity (R+), a is an orientation-preserving bi-Lipschitz homeomorphism of [0, +infinity] onto itself with only two fixed points 0 and infinity, and U-alpha is the isometric weighted shift operator on L-p(R+) given by U(alpha)f = (alpha)(1/p)(f circle alpha). Applications of binomial discrete operators to interpolation theory are given.

It is shown that the formula

where and is correct under the restrictions and It is also true if we suppose that and the spaces are functional Banach or quasi-Banach lattices on the same measure space

Let Ω be a rectangle in R². A new algorithm for the construction of a near-minimizer for the couple (L²(Ω), BV(Ω)) is presented. The algorithm is based on the Besicovitch covering theorem and analysis of local approximations of the given function f ∈ L²(Ω).

There are given necessary and sufficient conditions on a measure dμ(x)=w(x)dx under which the key estimates for the distribution and rearrangement of the maximal function due to Riesz, Wiener, Herz and Stein are valid. As a consequence, we obtain the equivalence of the Riesz and Wiener inequalities which seems to be new even for the Lebesgue measure. Our main tools are estimates of the distribution of the averaging function f** and a modified version of the Calderón-Zygmund decomposition. Analogous methods allow us to obtain K-functional formulas in terms of the maximal function for couples of weighted $L_p$-spaces.

The main aim of this paper is to prove novel results on stability of the semi-Fredholm property of operators on interpolation spaces generated by interpolation functors. The methods are based on some general ideas we develop in the paper. This allows us to extend some previous work in literature to the abstract setting. We show an application to interpolation methods introduced by Cwikel-Kalton-Milman-Rochberg which includes, as special cases, the real and complex methods up to equivalence of norms and also some other well known methods of interpolation. A by-product of these results get the stability of isomorphisms on Calderon products of Banach function lattices. We also study the important characteristics in operator Banach space theory, the so-called modules of injection and surjection, and we prove interpolation estimates of these modules of operators on scales of the Calderon complex interpolation spaces. (C) 2020 Elsevier Inc. All rights reserved.

A complete description of the real interpolation space L=(Lp0(ω0),…,Lpn(ωn))θ⃗ ,q is given. An interesting feature of the result is that the whole measure space (Ω,μ) can be divided into disjoint pieces Ωi (i∈I) such that L is an lq sum of the restrictions of L to Ωi, and L on each Ωi is a result of interpolation of just two weighted Lp spaces. The proof is based on a generalization of some recent results of the first two authors concerning real interpolation of vector-valued spaces.

We present, discuss and apply two reiteration theorems for triples of quasi-Banach function lattices. Some interpolation results for block-Lorentz spaces and triples of weighted Lp-spaces are proved. By using these results and a wavelet theory approach we calculate (θ,q)-spaces for triples of smooth function spaces (such as Besov spaces, Sobolev spaces, etc.). In contrast to the case of couples, for which even the scale of Besov spaces is not stable under interpolation, for triples we obtain stability in the frame of Besov spaces based on Lorentz spaces. Moreover, by using the results and ideas of this paper, we can extend the Stein–Weiss interpolation theorem known for Lp(μ)-spaces with change of measures to Lorentz spaces with change of measures. In particular, the results obtained show that for some problems in analysis the three-space real interpolation approach is really more useful than the usual real interpolation between couples.

We continue the development, by reduction to a first-order system for the conormal gradient, of L2a priori estimates and solvability for boundary value problems of Dirichlet, regularity, Neumann type for divergence-form second-order complex elliptic systems. We work here on the unit ball and more generally its bi-Lipschitz images, assuming a Carleson condition as introduced by Dahlberg which measures the discrepancy of the coefficients to their boundary trace near the boundary. We sharpen our estimates by proving a general result concerning a priori almost everywhere nontangential convergence at the boundary. Also, compactness of the boundary yields more solvability results using Fredholm theory. Comparison between classes of solutions and uniqueness issues are discussed. As a consequence, we are able to solve a long standing regularity problem for real equations, which may not be true on the upper half-space, justifying a posteriori a separate work on bounded domains.

A sharp pointwise differential inequality for vectorial second-order partial differential operators, with Uhlenbeck structure, is offered. As a consequence, optimal second-order regularity properties of solutions to nonlinear elliptic systems in domains in R-n are derived. Both local and global estimates are established. Minimal assumptions on the boundary of the domain are required for the latter. In the special case of the p-Laplace system, our conclusions broaden the range of the admissible values of the exponent p previously known.

We study a novel degenerate and singular elliptic operator Δ˜(τ,χ) defined by Δ˜(τ,χ)u=τ(x,Du)(|Du|Δ1u+χ(x,Du)Δ∞u), where the singular weights τ(x,s)>0 and χ(x,s)≥0 are continuous functions on Ω×Rn∖{0}. The operator Δ˜(τ,χ) is an extension of Δ(p,q)u=|Du|qΔ1u+(p−1)|Du|p−2Δ∞u,p≥1,q≥0, introduced by the authors in Baravdishet al. (2020), which in turn is an extension of the p-Laplace operator Δp. We establish the well-posedness of the Neumann boundary value problem for the parabolic equation ut=Δ˜(τ,χ)u in the framework of viscosity solutions. For the solution u, the weight χ controls the evolution along the tangential and the normal directions, respectively, on the level surface of u. The weight τ controls the total speed of the evolution of u. We also prove the consistency and the convergence of the numerical scheme for the finite differences method of the parabolic equation above. Numerical simulations show that our novel nonlinear operator Δ˜(τ,χ) gives better results than both the Perona–Malik (Perona and Malik, 1990) and total variation (TV) methods (Chan and Shen, 2005) when applied to image enhancement.

In this work we introduce a novel operator $$\displaystyle \varDelta _(p,q)$$ as an extended family of operators that generalize the p-Laplace operator. The operator is derived with an emphasis on image processing applications, and particularly, with a focus on image denoising applications. We propose a non-linear transition function, coupling p and q, which yields a non-linear filtering scheme analogous to adaptive spatially dependent total variation and linear filtering. Well-posedness of the final parabolic PDE is established via pertubation theory and connection to classical results in functional analysis. Numerical results demonstrates the applicability of the novel operator $$\displaystyle \varDelta _(p,q)$$ .

We have in a previous study introduced a novel elliptic operator Δ(p,q)u=|∇u|^qΔ₁u+(p−1)|∇u|^p−2Δ_∞u, p≥1, q≥0, as a generalization of the p-Laplace operator. In this paper, we establish the well-posedness of the parabolic equation u_t=|∇u|^1−qΔ(_1+q,q), where q=q(|∇u|) is continuous and has range in [0,1],in the framework of viscosity solutions. We prove the consistency and convergence of the numerical scheme of finite differences of this parabolic equation. Numerical simulations shows the advantage of this operator applied to image enhancement.

Evolution of brain tumours backwards in time is studied using well-established brain tumour growth models being semilinear parabolic equations of reaction-diffusion type. To run the models backwards, the tumour cell density data at a fixed (final) time is used, rendering an inverse ill-posed problem. This problem is recast as the minimisation of a cost functional matching the data against the solution at a final time of a forward parabolic model having the initial cell density as a control function. Regularisation is incorporated via penalising terms involving Sobolev norms. Mathematical properties of the semilinear parabolic equations are shown in Sobolev-Bochner spaces including uniqueness of a solution to the inverse problem. Differentiability of the control-to-state map is established rendering a sensitivity problem. The derivative of the cost functional is calculated and the adjoint state is derived via the Lagrange formalism. A non-linear conjugate gradient method (NCG) is presented for the minimisation. Numerical realisation of the minimisation on the BraTS'20 dataset is included using a standard finite difference discretisation of the space and time derivatives, showing that tumour evolution backwards in time can be accomplished and that the initial tumour cell density can be reconstructed. Comparison is done with a non-linear Landweber method.

In this study, we investigate the backward p(x)-parabolic equation as a new methodology to enhance images. We propose a novel iterative regularization procedure for the backward p(x)-parabolic equation based on the nonlinear Landweber method for inverse problems. The proposed scheme can also be extended to the family of iterative regularization methods involving the nonlinear Landweber method. We also investigate the connection between the variable exponent p(x) in the proposed energy functional and the diffusivity function in the corresponding Euler-Lagrange equation. It is well known that the forward problems converges to a constant solution destroying the image. The purpose of the approach of the backward problems is twofold. First, solving the backward problem by a sequence of forward problems we obtain a smooth image which is denoised. Second, by choosing the initial data properly we try to reduce the blurriness of the image. The numerical results for denoising appear to give improvement over standard methods as shown by preliminary results.

Eigenvalue problems for the p-Laplace operator in domains with finite volume, on noncompact Riemannian manifolds, are considered. If the domain does not coincide with the whole manifold, Neumann boundary conditions are imposed. Sharp assumptions ensuring L-q- or L-infinity-bounds for eigenfunctions are offered either in terms of the isoperimetric function or of the isocapacitary function of the domain.

The existence of solutions to a class of quasilinear elliptic problems on noncompact Riemannian manifolds, with finite volume, is investigated. Boundary value problems, with homogeneous Neumann conditions, in possibly irregular Euclidean domains are included as a special instance. A nontrivial solution is shown to exist under an unconventional growth condition on the right-hand side, which depends on the geometry of the underlying manifold. The identification of the critical growth is a crucial step in our analysis, and entails the use of the isocapacitary function of the manifold. A condition involving its isoperimetric function is also provided. (C) 2017 Elsevier Inc. All rights reserved.

We show that a stochastic (Markov) operator S acting on a Schatten class C-1 satisfies the Noether condition (i.e. S' (A) = A and S' (A(2)) = A(2), where A is an element of C-infinity is a Hermitian and bounded operator on a fixed separable and complex Hilbert space (H, <.,.>)), if and only if S(E-A(G)XEA(G)) = E-A (G)S(X)E-A (G) for any state X is an element of C-1 and all Borel sets G subset of R, where E-A (G) denotes the orthogonal projection coming from the spectral resolution A = integral(sigma(A)) zE(A)(dz). Similar results are obtained for stochastic one-parameter continuous semigroups.

In this work, we consider a class of centred kernel quadratic stochastic operators. We prove that in this class a centred kernel quadratic stochastic operator convergences almost surely and in L 2 with an exponential L 2-rate to its limit distribution. We propose an approximation scheme for this class of quadratic stochastic operators and describe three algorithms for simulating them. We consider in detail an example where the kernel is a Guassian one.

Quadratic stochastic operators can exhibit a wide variety of asymptotic behaviours andthese have been introduced and studied recently in the l1 space. It turns out that inprinciple most of the results can be carried over to the L1 space. However, due to topologicalproperties of this space one has to restrict in some situations to kernel quadratic stochasticoperators. In this article we study the uniform and strong asymptotic stability of quadratic stochastic operators acting on the L1 space in terms of convergence of the associated (linear)nonhomogeneous Markov chains.

The problem of linearization for third order evolution equations is considered. Criteria for testing equations for linearity are presented. A class of linearizable equations depending on arbitrary functions is obtained by requiring presence of an infinite-dimensional symmetry group. Linearizing transformations for this class are found using symmetry structure and local conservation laws. A number of special cases as examples are discussed. Their transformation to equations within the same class by differential substitutions and connection with KdV and mKdV equations is also reviewed in this framework. Published by AIP Publishing.

When moving into a more connected world together with machines, a mutual understanding will be very important. With the increased availability in wear- able sensors, a better understanding of human needs is suggested. The Dart- mouth Research study at the Psychiatric Research Center has examined the viability of detecting and further on predicting human behaviour and complex tasks. The field of smoking detection was challenged by using the Q-sensor by Affectiva as a prototype. Further more, this study implemented a framework for future research on the basis for developing a low cost, connected, device with Thayer Engineering School at Dartmouth College. With 3 days of data from 10 subjects smoking sessions was detected with just under 90% accuracy using the Conditional Random Field algorithm. However, predicting smoking with Electrodermal Momentary Assessment (EMA) remains an unanswered ques- tion. Hopefully a tool has been provided as a platform for better understanding of habits and behaviour. 

We establish sharp global regularity of a class of multilinear oscillatory integral operators that are associated to nonlinear dispersive equations with both Banach and quasi-Banach target spaces. As a consequence we also prove the (local in time) continuous dependence on the initial data for solutions of a large class of coupled systems of dispersive partial differential equations.

Accurately pinpointing and understanding how and where profits arise within companies is ofgreat importance for understanding how achieved results should be interpreted. For multi-national organizations, foreign exchange and interest rate risks are a particularly large andunpredictable source of impact on what the profit actually turns out to be. Previous attemptsusing a performance attribution model that can break down profits, changes in daily net presentvalue, has been shown successful. The used performance attribution model incorporates factorssuch as interest rate risk and foreign exchange risk in addition to classic factors such as marginsin explaining where the profit comes from in a way that is numerically exact.In this thesis, the aim is to further develop previous attempts in collaboration with the Swedishmining equipment company Epiroc. This is done by dividing one of their facilities into businessunits and extracting all available data from a four-year period for these respectively, and usingthe data to develop a mathematical model that can link the performance attribution frameworkto an industrial company. Conducting performance attribution on the individual business unitsillustrates how the model can be used to evaluate and interpret where profits originate from.To answer the purpose of the thesis, literature studies on performance attribution and internalpricing methods are carried out. The studies are used to develop an understanding of how acompany can be decomposed into business functions and how internal prices are set for businessunit specific profits. Studies on performance attribution are essential to gain an understandingof how the implemented framework should be used to capture activities of an industrial companyin the context of financial assets and instruments. The literature studies are also performed toensure that the chosen framework is the most appropriate for this study.The idea and aim of the work is based on previous master theses made in the years prior tothis. However, the implementation and mathematical interpretation of an industrial companyin the context of a financial performance attribution framework is entirely new and developedfor this thesis. This to as accurately as possible describe the real world through mathematicsand incorporate extensively larger data sets than in previous attempts.The results of the thesis show that it is possible to conduct performance attribution on specificbusiness units, decomposing and explaining profits in detail on a daily basis. This is demon-strated though applying the implementation of the performance attribution and mathematicalmodel without introducing any significantly large error terms when comparing the results toactual data. Performance attribution is executed on as much data as possible, as well as onsubsets of data to demonstrate the possibilities of use in an industrial context. Showing resultsthat could be of great interest for companies acting on a global market.

The eigenvalue problem for linear differential operators is important since eigenvalues correspond to the possible energy levels of a physical system. It is also important to have good estimates of the error in the computed eigenvalues. In this work, we use spline interpolation to construct approximate eigenfunctions of a linear operator using the corresponding eigenvectors of a discretized approximation of the operator. We show that an error estimate for the approximate eigenvalues can be obtained by evaluating the residual for an approximate eigenpair. The interpolation scheme is selected in such a way that the residual can be evaluated analytically. To demonstrate that the method gives useful error bounds, we apply it to a problem originating from the study of graphene quantum dots where the goal was to investigate the change in the spectrum from incorporating electron–electron interactions in the potential.

In this paper, we study cluster sets and essential cluster sets for Sobolev functions and quasiharmonic functions (i.e., continuous quasiminimizers). We develop their basic theory with a particular emphasis on when they coincide and when they are connected. As a main result, we obtain that if a Sobolev function u on an open set has boundary values f in Sobolev sense and f |∂is continuous at x0 ∈ ∂, then the essential cluster set C(u, x0, Ω) is connected. We characterize precisely in which metric spaces this result holds. Further, we provide some new boundary regularity results for quasiharmonic functions. Most of the results are new also in the Euclidean case.

We fill in a gap in the proofs of Theorems 1.1-1.4 in The Kellogg property and boundary regularity for p-harmonic functions with respect to the Mazurkiewicz boundary and other compactifications, to appear in Complex Var. Elliptic Equ., doi:10.1080/17476933.2017.1410799.

It is well known that sets of p-capacity zero are removable for bounded p-harmonic functions, but on metric spaces there are examples of removable sets of positive capacity. In this paper, we show that this can happen even on unweighted when n > p, although only in very special cases. A complete characterization of removable singularities for bounded A-harmonic functions on weighted , is also given, where the weight is p-admissible. The same characterization is also shown to hold for bounded quasiharmonic functions on weighted , as well as on unweighted . For bounded A-superharmonic functions and bounded quasisuperharmonic functions on weighted , we show that relatively closed sets are removable if and only if they have zero capacity.

In this paper, boundary regularity for p-harmonic functions is studied with respect to the Mazurkiewicz boundary and other compactifications. In particular, the Kellogg property (which says that the set of irregular boundary points has capacity zero) is obtained for a large class of compactifications, but also two examples when it fails are given. This study is done for complete metric spaces equipped with doubling measures supporting a p-Poincare inequality, but the results are new also in unweighted Euclidean spaces.

We show that every Sobolev function in W-loc(1, p) (U) on a p-quasiopen set U subset of R-n with a.e.-vanishing p-fine gradient is a.e.-constant if and only if U is p-quasiconnected. To prove this we use the theory of Newtonian Sobolev spaces on metric measure spaces, and obtain the corresponding equivalence also for complete metric spaces equipped with a doubling measure supporting a p-Poincare inequality. On unweighted R-n, we also obtain the corresponding result for p-finely open sets in terms of p-fine connectedness, using a deep result by Latvala.