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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.
- ItemCalculating conjugacy classes in Sylow p-subgroups of finite Chevalley groups of rank six and seven(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2013) Goodwi, Simon M.; Mosch, Peter; Röhrle, GerhardLet G(q) be a finite Chevalley group, where q is a power of a good prime p, and let U(q) be a Sylow p-subgroup of G(q). Then a generalized version of a conjecture of Higman asserts that the number k(U(q)) of conjugacy classes in U(q) is given by a polynomial in q with integer coefficients. In [12], the first and the third authors developed an algorithm to calculate the values of k(U(q)). By implementing it into a computer program using GAP, they were able to calculate k(U(q)) for G of rank at most 5, thereby proving that for these cases k(U(q)) is given by a polynomial in q. In this paper we present some refinements and improvements of the algorithm that allow us to calculate the values of k(U(q)) for finite Chevalley groups of rank six and seven, except E7. We observe that k(U(q)) is a polynomial, so that the generalized Higman conjecture holds for these groups. Moreover, if we write k(U(q)) as a polynomial in q−1, then the coefficients are non-negative. Under the assumption that k(U(q)) is a polynomial in q−1, we also give an explicit formula for the coefficients of k(U(q)) of degrees zero, one and two.
- 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.
- ItemOn conjugacy of MASAs and the outer automorphism aroup of the Cuntz algebra(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2013) Conti, Roberto; Hong, Jeong Hee; Szyma´nski, WojciechWe investigate the structure of the outer automorphism group of the Cuntz algebra and the closely related problem of conjugacy of MASAa in On. In particular, we exhibit an uncountable family of MASAs, conjugate to the standard MASA Dn via Bogolubov automorphisms, that are not inner conjugate to Dn.