1,099 research outputs found

    Global results for a Cauchy problem related to biharmonic wave maps

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    We prove global existence of a derivative bi-harmonic wave equation with a non-generic quadratic nonlinearity and small initial data in the scaling critical space B˙d22,1(Rd)×B˙d222,1(Rd)\dot{B}^{2,1}_{\frac{d}{2}}(\mathbb{R}^d) \times \dot{B}^{2,1}_{\frac{d}{2}-2}(\mathbb{R}^d) for d3 d \geq 3 . Since the solution persists higher regularity of the initial data, we obtain a small data global regularity result for the biharmonic wave maps equation for a certain class of target manifolds including the sphere

    Energy bounds for a fourth-order equation in low dimensions related to wave maps

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    For compact, isometrically embedded Riemannian manifolds NRL N \hookrightarrow \mathbb{R}^L, we introduce a fourth-order version of the wave map equation. By energy estimates, we prove an a priori\textit{a priori} estimate for smooth local solutions in the energy subcritical dimension n=1,2 n = 1,2. The estimate excludes blow-up of a Sobolev norm in finite existence times. In particular, combining this with recent work of local well-posedness of the Cauchy problem, it follows that for smooth initial data with compact support, there exists a (smooth) unique global solution in dimension n=1,2n = 1,2. We also give a proof of the uniqueness of solutions that are bounded in these Sobolev norms.Comment: v2: typos fixed, introductory section updated and title changed according to request of referee. To appear Proc. Amer. Math. So

    Blow up dynamics for the 3D energy-critical Nonlinear Schr\"odinger equation

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    We construct a two-parameter continuum of type II blow up solutions for the energy-critical focusing NLS in dimension d=3 d = 3. The solutions collapse to a single energy bubble in finite time, precisely they have the form u(t,x)=eiα(t)λ(t)12W(λ(t)x)+η(t,x) u(t,x) = e^{i \alpha(t)}\lambda(t)^{\frac{1}{2}}W(\lambda(t) x) + \eta(t, x ), t[0,T) t \in[0, T), xR3 x \in \mathbb{R}^3, where W(x)=(1+x23)12 W( x) = \big( 1 + \frac{|x|^2}{3}\big)^{-\frac{1}{2}} is the ground state solution, λ(t)=(Tt)12ν\lambda(t) = (T-t)^{- \frac12 - \nu} for suitable ν>0 \nu > 0 , α(t)=α0log(Tt) \alpha(t) = \alpha_0 \log(T - t) and T=T(ν,α0)>0 T= T(\nu, \alpha_0) > 0 . Further η(t)ηTH˙1H˙2=o(1) \|\eta(t) - \eta_T\|_{\dot{H}^1 \cap \dot{H}^2} = o(1) as tT t \to T^- for some ηTH˙1 H˙2 \eta_T \in \dot{H}^{1} \cap~ \dot{H}^2.Comment: 123 pages, typos correcte

    Local wellposedness and global regularity results for biharmonic wave maps

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    This thesis is concerned with biharmonic wave maps, i.e. a bi-harmonic version of the wave maps equation, which is a Hamiltonian equation for a higher order energy functional and arises variationally from an elastic action functional for a manifold valued map.\\[1pt] In the first part we present local and global results from energy estimates for biharmonic wave maps into compact, embedded target manifolds. This includes local wellposedness in high regularity and global regularity in subcritical dimension n=1,2n = 1, 2. The results rely on the use of careful a priori energy estimates, compactness arguments in weak topologies and sharp Sobolev embeddings combined with energy conservation in the proof of global regularity.\\[1pt] In part two, we extend these results to global regularity in dimension n3 n \geq 3 for biharmonic wave maps into spheres and initial data of small size in a scale invariant Besov norm. This follows from a small data global wellposedness and persistence of regularity result for more general systems of biharmonic wave equations with non-generic nonlinearity. In contrast to part one, the arguments in part two of the thesis rely on the analysis of bilinear frequency interactions based on Fourier restriction methods and Strichartz estimates.\\[1pt] The results in both parts of the thesis fundamentally depend on the non-generic form of the nonlinearity that is introduced by our biharmonic model problem

    Energy bounds for biharmonic wave maps in low dimensions

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    For compact, isometrically embedded Riemannian manifolds NRLN\hookrightarrow \mathbb{R}^L, we introduce a fourth-order version of the wave map equation. By energy estimates, we prove an priori estimate for smooth local solutions in the energy subcritical dimension n=1,2n = 1, 2. The estimate excludes blow-up of a Sobolev norm in finite existence times. In particular, combining this with an upcoming work of local well-posedness of the Cauchy problem, it follows that for smooth initial data with compact support, here exists a (smooth) unique global solution in dimension n=1,2n = 1, 2. We also give a proof of the uniqueness of solutions that are bounded in these Sobolev norms

    Shortest Distances as Enumeration Problem

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    We investigate the single source shortest distance (SSSD) and all pairs shortest distance (APSD) problems as enumeration problems (on unweighted and integer weighted graphs), meaning that the elements (u,v,d(u,v))(u, v, d(u, v)) -- where uu and vv are vertices with shortest distance d(u,v)d(u, v) -- are produced and listed one by one without repetition. The performance is measured in the RAM model of computation with respect to preprocessing time and delay, i.e., the maximum time that elapses between two consecutive outputs. This point of view reveals that specific types of output (e.g., excluding the non-reachable pairs (u,v,)(u, v, \infty), or excluding the self-distances (u,u,0)(u, u, 0)) and the order of enumeration (e.g., sorted by distance, sorted row-wise with respect to the distance matrix) have a huge impact on the complexity of APSD while they appear to have no effect on SSSD. In particular, we show for APSD that enumeration without output restrictions is possible with delay in the order of the average degree. Excluding non-reachable pairs, or requesting the output to be sorted by distance, increases this delay to the order of the maximum degree. Further, for weighted graphs, a delay in the order of the average degree is also not possible without preprocessing or considering self-distances as output. In contrast, for SSSD we find that a delay in the order of the maximum degree without preprocessing is attainable and unavoidable for any of these requirements.Comment: Updated version adds the study of space complexit

    An update on statistical boosting in biomedicine

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    Statistical boosting algorithms have triggered a lot of research during the last decade. They combine a powerful machine-learning approach with classical statistical modelling, offering various practical advantages like automated variable selection and implicit regularization of effect estimates. They are extremely flexible, as the underlying base-learners (regression functions defining the type of effect for the explanatory variables) can be combined with any kind of loss function (target function to be optimized, defining the type of regression setting). In this review article, we highlight the most recent methodological developments on statistical boosting regarding variable selection, functional regression and advanced time-to-event modelling. Additionally, we provide a short overview on relevant applications of statistical boosting in biomedicine

    Dissipative Dynamics with Trapping in Dimers

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    The trapping of excitations in systems coupled to an environment allows to study the quantum to classical crossover by different means. We show how to combine the phenomenological description by a non-hermitian Liouville-von Neumann Equation (LvNE) approach with the numerically exact path integral Monte-Carlo (PIMC) method, and exemplify our results for a system of two coupled two-level systems. By varying the strength of the coupling to the environment we are able to estimate the parameter range in which the LvNE approach yields satisfactory results. Moreover, by matching the PIMC results with the LvNE calculations we have a powerful tool to extrapolate the numerically exact PIMC method to long times.Comment: 5 pages, 2 figure
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