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- ItemSpherical arc-length as a global conformal parameter for analytic curves in the Riemann sphere(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2016) Gauthier, Paul; Nestoridis, Vassili; Papadopoulos, AthanaseWe prove that for every analytic curve in the complex plane C, Euclidean and spherical arc-lengths are global conformal parameters. We also prove that for any analytic curve in the hyperbolic plane, hyperbolic arc-length is also a global parameter. We generalize some of these results to the case of analytic curves in Rn and Cn and we discuss the situation of curves in the Riemann sphere C {∞}.
- ItemEquidistribution of elements of norm 1 in cyclic extensions(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2014) Petersen, Kathleen L.; Sinclair, Christopher D.Upon quotienting by units, the elements of norm 1 in a number field K form a countable subset of a torus of dimension r1 + r2 - 1 where r1 and r2 are the numbers of real and pairs of complex embeddings. When K is Galois with cyclic Galois group we demonstrate that this countable set is equidistributed in this torus with respect to a natural partial ordering.
- ItemAnalytic structure in fibers(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2016) Aron, Richard M.; Falcó, Javier; García, Domingo; Maestre, ManuelLet BX be the open unit ball of a complex Banach space X, and let H∞(BX) and Au(BX) be, respectively, the algebra of bounded holomorphic functions on BX and the subalgebra of uniformly continuous holomorphic functions on BX. In this paper we study the analytic structure of fibers in the spectrum of these two algebras. For the case of H∞(BX), we prove that the fiber in M(H∞(Bc0)) over any point of the distinguished boundary of the closed unit ball B¯ℓ∞ of ℓ∞ contains an analytic copy of Bℓ∞. In the case of Au(BX) we prove that if there exists a polynomial whose restriction to the open unit ball of X is not weakly continuous at some point, then the fiber over every point of the open unit ball of the bidual contains an analytic copy of D.
- ItemNon-extendability of holomorphic functions with bounded or continuously extendable derivatives(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2017) Moschonas, Dionysios; Nestoridis, VassiliWe consider the spaces H∞F(Ω) and AF(Ω) containing all holomorphic functions f on an open set Ω⊆C, such that all derivatives f(l), l∈F⊆N0={0,1,...}, are bounded on Ω, or continuously extendable on Ω¯¯¯¯, respectively. We endow these spaces with their natural topologies and they become Fr\'echet spaces. We prove that the set S of non-extendable functions in each of these spaces is either void, or dense and Gδ. We give examples where S=∅ or not. Furthermore, we examine cases where F can be replaced by F˜={l∈N0:minF⩽l⩽supF}, or F˜0={l∈N0:0⩽l⩽supF} and the corresponding spaces stay unchanged.
- ItemYet another algorithm for the symmetric eigenvalue problem(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2016) Aurentz, Jared L.; Mach, Thomas; Vandebril, Raf; Watkins, David S.In this paper we present a new algorithm for solving the symmetric matrix eigenvalue problem that works by first using a Cayley transformation to convert the symmetric matrix into a unitary one and then uses Gragg’s implicitly shifted unitary QR algorithm to solve the resulting unitary eigenvalue problem. We prove that under reasonable assumptions on the symmetric matrix this algorithm is backward stable and also demonstrate that this algorithm is comparable with other well known implementations in terms of both speed and accuracy.
- ItemOn the prediction of stationary functional time series(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2014) Aue, Alexander; Norinho, Diogo Dubart; Hörmann, SiegfriedThis paper addresses the prediction of stationary functional time series. Existing contributions to this problem have largely focused on the special case of first-order functional autoregressive processes because of their technical tractability and the current lack of advanced functional time series methodology. It is shown here how standard multivariate prediction techniques can be utilized in this context. The connection between functional and multivariate predictions is made precise for the important case of vector and functional autoregressions. The proposed method is easy to implement, making use of existing statistical software packages, and may therefore be attractive to a broader, possibly non-academic, audience. Its practical applicability is enhanced through the introduction of a novel functional final prediction error model selection criterion that allows for an automatic determination of the lag structure and the dimensionality of the model. The usefulness of the proposed methodology is demonstrated in a simulation study and an application to environmental data, namely the prediction of daily pollution curves describing the concentration of particulate matter in ambient air. It is found that the proposed prediction method often significantly outperforms existing methods.
- ItemAn explicit formula for the Dirac multiplicities on lens spaces(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2014) Boldt, Sebastian; Lauret, Emilio A.We present a new description of the spectrum of the (spin-) Dirac operator D on lens spaces. Viewing a spin lens space L as a locally symmetric space n Spin(2m)= Spin(2m1) and exploiting the representation theory of the Spin groups, we obtain explicit formulas for the multiplicities of the eigenvalues of D in terms of finitely many integer operations. As a consequence, we present conditions for lens spaces to be Dirac isospectral. Tackling classic questions of spectral geometry, we prove with the tools developed that neither spin structures nor isometry classes of lens spaces are spectrally determined by giving infinite families of Dirac isospectral lens spaces. These results are complemented by examples found with the help of a computer.
- ItemThe colored Jones polynomial and Kontsevich-Zagier series for double twist knots(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2017) Lovejoy, Jeremy; Osburn, RobertUsing a result of Takata, we prove a formula for the colored Jones polynomial of the double twist knots K(−m,−p) and K(−m,p) where m and p are positive integers. In the (−m,−p) case, this leads to new families of q-hypergeometric series generalizing the Kontsevich-Zagier series. Comparing with the cyclotomic expansion of the colored Jones polynomials of K(m,p) gives a generalization of a duality at roots of unity between the Kontsevich-Zagier function and the generating function for strongly unimodal sequences.
- ItemLegendrian lens space surgeries(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2016) Geiges, Hansjörg; Onaran, SinemWe show that every tight contact structure on any of the lens spaces L(ns2 - s + 1; s2) with n ≥ 2, s ≥ 1, can be obtained by a single Legendrian surgery along a suitable Legendrian realisation of the negative torus knot T(s - (sn - 1)) in the tight or an overtwisted contact structure on the 3-sphere.
- ItemPrediction and quantification of individual athletic performance(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2015) Blythe, Duncan A.J.; Király, Franz J.We present a novel, quantitative view on the human athletic performance of individuals. We obtain a predictor for athletic running performances, a parsimonious model, and a training state summary consisting of three numbers, by application of modern validation techniques and recent advances in machine learning to the thepowerof10 database of British athletes’ performances (164,746 individuals, 1,417,432 performances). Our predictor achieves a low average prediction error (out-of-sample), e.g., 3.6 min on elite Marathon performances, and a lower error than the state-of-the-art in performance prediction (30% improvement, RMSE). We are also the first to report on a systematic comparison of predictors for athletic running performance. Our model has three parameters per athlete, and three components which are the same for all athletes. The first component of the model corresponds to a power law with exponent dependent on the athlete which achieves a better goodness-of-fit than known power laws in athletics. Many documented phenomena in quantitative sports science, such as the form of scoring tables, the success of existing prediction methods including Riegel’s formula, the Purdy points scheme, the power law for world records performances and the broken power law for world record speeds may be explained on the basis of our findings in a unified way. We provide strong evidence that the three parameters per athlete are related to physiological and/or behavioural parameters, such as training state, event specialization and age, which allows us to derive novel physiological hypotheses relating to athletic performance. We conjecture on this basis that our findings will be vital in exercise physiology, race planning, the study of aging and training regime design.