First, we reconsider the magnetic pseudodifferential calculus and show that for a large class of non-decaying symbols, their corresponding magnetic pseudodifferential operators can be represented, up to a global gauge transform, as generalized Hofstadter-like, bounded matrices. As a by-product, we prove a Calderón-Vaillancourt type result. Second, we make use of this matrix representation and prove sharp results on the spectrum location when the magnetic field strength b varies. Namely, when the operators are self-adjoint, we show that their spectrum (as a set) is at least 1/2-Hölder continuous with respect to b in the Hausdorff distance. Third, when the magnetic perturbation comes from a constant magnetic field we show that their spectral edges are Lipschitz continuous in b. The same Lipschitz continuity holds true for spectral gap edges as long as the gaps do not close.
Let $q\ge2$ be an integer, $\{X_n\}_{n\geq 1}$ a stochastic process with state space $\{0,\ldots,q-1\}$ , and F the cumulative distribution function (CDF) of $\sum_{n=1}^\infty X_n q^{-n}$ . We show that stationarity of $\{X_n\}_{n\geq 1}$ is equivalent to a functional equation obeyed by F, and use this to characterize the characteristic function of X and the structure of F in terms of its Lebesgue decomposition. More precisely, while the absolutely continuous component of F can only be the uniform distribution on the unit interval, its discrete component can only be a countable convex combination of certain explicitly computable CDFs for probability distributions with finite support. We also show that $\mathrm{d} F$ is a Rajchman measure if and only if F is the uniform CDF on [0, 1].
We revisit the problem of constructing one-dimensional acoustic black holes. Instead of considering the Euler–Bernoulli beam theory, we use Timoshenko's approach, which is known to be more realistic at higher frequencies. Our goal is to minimize the reflection coefficient under a constraint imposed on the normalized wavenumber variation. We use the calculus of variations to derive the corresponding Euler–Lagrange equation analytically and then use numerical methods to solve this equation to find the “optimal” height profile for different frequencies. We then compare these profiles to the corresponding ones previously found using the Euler–Bernoulli beam theory and see that in the lower range of the dimensionless frequency Ω (defined using the largest height of the plate), the optimal profiles almost coincide, as expected.
This dissertation deals with three different mathematical subjects: mathematical physics, acoustics and probability theory. The manuscript starts with a brief description of the aforementioned fields, and introduces the notation and terminology needed to understand the content of the published/submitted papers.Paper A deals with the theory of pseudodifferential operators with magnetic fields. Here a generalization of the magnetic Weyl quantization is proposed and then shown that this choice makes it possible to represent a magnetic pseudodifferential operator as a generalized Hofstadter matrix. This generalized matrix structure is then used to show spectral stability when the generating symbol is real, i.e. the corresponding operator is self-adjoint. Specifically, it is shown that the spectrum of the operator is locally Hölder continuous in the strength of the magnetic field, with a Hölder exponent 1/2. Furthermore, if the magnetic field is constant, it is shown that the extreme spectral values, as well as the gap edges (if such gaps exist and they stay open when the strength of the magnetic field is varied) are Lipschitz continuous.Paper B is concerned with the bulk-boundary correspondence for unbounded Dirac operators. First, one shows essential self-adjointness of the edge magnetic Dirac operator defined with infinite mass boundary conditions, along with a detailed analysis of the integral kernel of its resolvent. This is then used to formulate a relativistic bulk-boundary correspondence and a gap labelling theorem, which extends some known results from the Schrödinger setting.Paper C investigates the possibility of constructing acoustic black holes within the Timoshenko beam theory setting. More specifically, the paper deals with determining an optimal height profile of a wedge at the end of thin plate, which minimizes the reflection of waves at the boundary. This is done by considering the partial differential equations describing the beam motion and then derive the Timoshenko dispersion relation. A functional depending on the height profile is then derived, and the associated Euler-Lagrange equation is determined. By solving this equation numerically an optimal profile is determined and compared with the corresponding profile obtained by considering the waves using the Euler-Bernoulli beam theory iii instead.Papers D and E deals with stochastic variables on the unit interval given by a base-q expansion with digits coming from a stationary stochastic process. In Paper D a functional equation for the CDF corresponding to the stationary stochastic process is derived. By applying this functional equation, a characterization of stationarity of the stochastic process in terms of the corresponding CDF is established. In Paper E specific stationery models are characterized, hereunder stationary Markov chains and stationary renewal processes.sammenlignes med højdeprofilen fundet ved at betragte bølger med Euler-Bernoullis bjaelke teori.Artikel D og E omhandler stokastiske variable på enhedsintervallet der er givet v...
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