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- ItemOcean rogue waves and their phase space dynamics in the limit of a linear interference model([London] : Macmillan Publishers Limited, part of Springer Nature, 2016) Birkholz, Simon; Brée, Carsten; Veselić, Ivan; Demircan, Ayhan; Steinmeyer, GünterWe reanalyse the probability for formation of extreme waves using the simple model of linear interference of a finite number of elementary waves with fixed amplitude and random phase fluctuations. Under these model assumptions no rogue waves appear when less than 10 elementary waves interfere with each other. Above this threshold rogue wave formation becomes increasingly likely, with appearance frequencies that may even exceed long-term observations by an order of magnitude. For estimation of the effective number of interfering waves, we suggest the Grassberger-Procaccia dimensional analysis of individual time series. For the ocean system, it is further shown that the resulting phase space dimension may vary, such that the threshold for rogue wave formation is not always reached. Time series analysis as well as the appearance of particular focusing wind conditions may enable an effective forecast of such rogue-wave prone situations. In particular, extracting the dimension from ocean time series allows much more specific estimation of the rogue wave probability.
- ItemOn an effective variation of Kronecker’s approximation theorem avoiding algebraic sets(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2017) Fukshansky, Lenny; German, Oleg; Moshchevitin, NikolayLet Λ⊂Rn be an algebraic lattice, coming from a projective module over the ring of integers of a number field K. Let Z⊂Rn be the zero locus of a finite collection of polynomials such that Λ⊈Z or a finite union of proper full-rank sublattices of Λ. Let K1 be the number field generated over K by coordinates of vectors in Λ, and let L1,…,Lt be linear forms in n variables with algebraic coefficients satisfying an appropriate linear independence condition over K1. For each ε>0 and a∈Rn, we prove the existence of a vector x∈Λ∖Z of explicitly bounded sup-norm such that ∥Li(x)−ai∥<ε for each 1≤i≤t, where ∥ ∥ stands for the distance to the nearest integer. The bound on sup-norm of x depends on ε, as well as on Λ, K, Z and heights of linear forms. This presents a generalization of Kronecker's approximation theorem, establishing an effective result on density of the image of Λ∖Z under the linear forms L1,…,Lt in the t-torus~Rt/Zt. In the appendix, we also discuss a construction of badly approximable matrices, a subject closely related to our proof of effective Kronecker's theorem, via Liouville-type inequalities and algebraic transference principles.