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- 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.
- 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.
- 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.
- ItemSharp constants in the classical weak form of the John-Nirenberg inequality(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2013) Vasyunin, Vasily; Volberg, AlexanderThe sharp constants in the classical John-Nirenberg inequality are found by using Bellman function approach.
- ItemLow rank differential equations for Hamiltonian matrix nearness problems(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2013) Guglielmi, Nicola; Kressner, Daniel; Lubich, ChristianFor a Hamiltonian matrix with purely imaginary eigenvalues, we aim to determine the nearest Hamiltonian matrix such that some or all eigenvalues leave the imaginary axis. Conversely, for a Hamiltonian matrix with all eigenvalues lying off the imaginary axis, we look for a nearest Hamiltonian matrix that has a pair of imaginary eigenvalues. The Hamiltonian matrices can be allowed to be complex or restricted to be real. Such Hamiltonian matrix nearness problems are motivated by applications such as the analysis of passive control systems. They are closely related to the problem of determining extremal points of Hamiltonian pseudospectra. We obtain a characterization of optimal perturbations, which turn out to be of low rank and are attractive stationary points of low-rank differential equations that we derive. This permits us to give fast algorithms - which show quadratic convergence - for solving the considered Hamiltonian matrix nearness problems.
- ItemGrassmannian connection between three- and four-qubit observables, Mermin’s contextuality and black holes(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2013) Lévay, Péter; Planat, Michel; Saniga, MetodWe invoke some ideas from finite geometry to map bijectively 135 heptads of mutually commuting three -qubit observables into 135 symmetric four -qubit ones. After labeling the elements of the former set in terms of a seven-dimensional Clifford algebra, we present the bijective map and most pronounced actions of the associated symplectic group on both sets in explicit forms. This formalism is then employed to shed novel light on recently- discovered structural and cardinality properties of an aggregate of three-qubit Mermin’s “magic” pentagrams. Moreover, some intriguing connections with the so-called black-hole– qubit correspondence are also pointed out.
- ItemThe algebraic combinatorial approach for low-rank matrix completion(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2013) Király, Franz J.; Theran, Louis; Tomioka, Ryota; Uno, TakeakiWe propose an algebraic combinatorial framework for the problem of completing partially observed low-rank matrices. We show that the intrinsic properties of the problem, including which entries can be reconstructed, and the degrees of freedom in the reconstruction, do not depend on the values of the observed entries, but only on their position. We associate combinatorial and algebraic objects, differentials and matroids, which are descriptors of the particular reconstruction task, to the set of observed entries, and apply them to obtain reconstruction bounds. We show how similar techniques can be used to obtain reconstruction bounds on general compressed sensing problems with algebraic compression constraints. Using the new theory, we develop several algorithms for low-rank matrix completion, which allow to determine which set of entries can be potentially reconstructed and which not, and how, and we present algorithms which apply algebraic combinatorial methods in order to reconstruct the missing entries.
- ItemEnhanced spatial skin–effect for free vibrations of a thick cascade junction with “super heavy” concentrated masses(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2013) Chechkin, Gregory A.; Mel'nyk, Taras A.The asymptotic behavior (as ε→0) of eigenvalues and eigenfunctions of a boundary-value problem for the Laplace operator in a thick cascade junction with concentrated masses is studied. This cascade junction consists of the junction’s body and a great number 5N=O(ε−1) of ε-alternating thin rods belonging to two classes. One class consists of rods of finite length and the second one consists of rods of small length of order O(ε). The mass density is of order O(ε−α) on the rods from the second class and O(1) outside of them. There exist five qualitatively different cases in the asymptotic behavior of eigen-magnitudes as ε→0, namely the case of “light” concentrated masses (a∈(0,1)), “intermediate” concentrated masses (α=1) and “heavy” concentrated masses (α∈(1,+∞")) that we divide into “slightly heavy” concentrated masses (α∈(1,2)), “moderate heavy” concentrated masses (α=2), and “super heavy” concentrated masses (alpha>2). In the paper we study the influence of the concentrated masses on the asymptotic behavior of the eigen-magnitudes in the cases α=2 and α>2. The leading terms of asymptotic expansions both for the eigenvalues and eigenfunctions are constructed and the corresponding asymptotic estimates are proved. In addition, a new kind of high-frequency vibrations is found.