Nonnegative and strictly positive linearization of Jacobi and generalized Chebyshev polynomials

In the theory of orthogonal polynomials, as well as in its intersection with harmonic analysis, it is an important problem to decide whether a given orthogonal polynomial sequence $(P_n(x))_{n\in\mathbb{N}_0}$ satisfies nonnegative linearization of products, i.e., the product of any two $P_m(x),P_n(x)$ is a conical combination of the polynomials $P_{|m-n|}(x),\ldots,P_{m+n}(x)$. Since the coefficients in the arising expansions are often of cumbersome structure or not explicitly available, such considerations are generally very nontrivial. In 1970, G. Gasper was able to determine the set $V$ of all pairs $(\alpha,\beta)\in(-1,\infty)^2$ for which the corresponding Jacobi polynomials $(R_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$, normalized by $R_n^{(\alpha,\beta)}(1)\equiv1$, satisfy nonnegative linearization of products. In 2005, R. Szwarc asked to solve the analogous problem for the generalized Chebyshev polynomials $(T_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$, which are the quadratic transformations of the Jacobi polynomials and orthogonal w.r.t. the measure $(1-x^2)^{\alpha}|x|^{2\beta+1}\chi_{(-1,1)}(x)\,\mathrm{d}x$. In this paper, we give the solution and show that $(T_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$ satisfies nonnegative linearization of products if and only if $(\alpha,\beta)\in V$, so the generalized Chebyshev polynomials share this property with the Jacobi polynomials. Moreover, we reconsider the Jacobi polynomials themselves, simplify Gasper's original proof and characterize strict positivity of the linearization coefficients. Our results can also be regarded as sharpenings of Gasper's one.Comment: The second version puts more emphasis on strictly positive linearization of Jacobi polynomials. We reorganized the structure, added several references and corrected a few typos. We added a geometric interpretation of the set $V^{\prime}$ and some comments on its interior. We added a detailed comparison to Gasper's classical result. Title and abstract were changed. These are the main change

Tur\'{a}n's inequality, nonnegative linearization and amenability properties for associated symmetric Pollaczek polynomials

An elegant and fruitful way to bring harmonic analysis into the theory of orthogonal polynomials and special functions, or to associate certain Banach algebras with orthogonal polynomials satisfying a specific but frequently satisfied nonnegative linearization property, is the concept of a polynomial hypergroup. Polynomial hypergroups (or the underlying polynomials, respectively) are accompanied by $L^1$-algebras and a rich, well-developed and unified harmonic analysis. However, the individual behavior strongly depends on the underlying polynomials. We study the associated symmetric Pollaczek polynomials, which are a two-parameter generalization of the ultraspherical polynomials. Considering the associated $L^1$-algebras, we will provide complete characterizations of weak amenability and point amenability by specifying the corresponding parameter regions. In particular, we shall see that there is a large parameter region for which none of these amenability properties holds (which is very different to $L^1$-algebras of locally compact groups). Moreover, we will rule out right character amenability. The crucial underlying nonnegative linearization property will be established, too, which particularly establishes a conjecture of R. Lasser (1994). Furthermore, we shall prove Tur\'{a}n's inequality for associated symmetric Pollaczek polynomials. Our strategy relies on chain sequences, asymptotic behavior, further Tur\'{a}n type inequalities and transformations into more convenient orthogonal polynomial systems.Comment: Main changes towards first version: The part on associated symmetric Pollaczek polynomials was extended (with more emphasis on Tur\'{a}n's inequality and including a larger parameter region), and the part on little $q$-Legendre polynomials became a separate paper. We added several references and corrected a few typos. Title, abstract and MSC class were change

Leadership Selection in the Major Multilaterals

Leadership selection in the major global economic organizations produced unprecedented levels of public conflict during the 1990s. The convention that awards the IMF managing directorship to a European and the World Bank presidency to an American sparked conflict between the United States and Europe as well as growing discontent on the part of Japan and the developing countries. At the WTO, successive conflicts demonstrate deeper shortcomings in governance as membership expands rapidly and consensus decision making fails. Protracted efforts to choose new heads of these increasingly important organizations have undermined their legitimacy and distracted members from their core agendas. This selection process and its flaws provide a central theme for the analysis and prescriptions presented in this study, which focuses on the major international financial institutions (IFIs) and other global and regional multilaterals. Author Miles Kahler looks at the sources of conflict and presents recommendations for reform: in the short run, changes in the process, such as the use of search committees; in the long run, the dismantling of the US-European convention at the IFIs and changes in representation at the WTO. The author's diagnosis and policy recommendations have important implications for leadership selection in other regional and global organizations.

Solar origins of coronal mass ejections

The large scale properties of coronal mass ejections (CMEs), such as morphology, leading edge speed, and angular width and position, have been cataloged for many events observed with coronagraphs on the Skylab, P-78, and SMM spacecraft. While considerable study has been devoted to the characteristics of the SMEs, their solar origins are still only poorly understood. Recent observational work has involved statistical associations of CMEs with flares and filament eruptions, and some evidence exists that the flare and eruptive-filament associated CMEs define two classes of events, with the former being generally more energetic. Nevertheless, it is found that eruptive-filament CMEs can at times be very energetic, giving rise to interplanetary shocks and energetic particle events. The size of the impulsive phase in a flare-associated CME seems to play no significant role in the size or speed of the CME, but the angular sizes of CMEs may correlate with the scale sizes of the 1-8 angstrom x-ray flares. At the present time, He 10830 angstrom observations should be useful in studying the late development of double-ribbon flares and transient coronal holes to yield insights into the CME aftermath. The recently available white-light synoptic maps may also prove fruitful in defining the coronal conditions giving rise to CMEs

Discrete hardy spaces

We study the boundary behavior of discrete monogenic functions, i.e. null-solutions of a discrete Dirac operator, in the upper and lower half space. Calculating the Fourier symbol of the boundary operator we construct the corresponding discrete Hilbert transforms, the projection operators arising from them, and discuss the notion of discrete Hardy spaces. Hereby, we focus on the 3D-case with the generalization to the n-dimensional case being straightforward

Multi-core computation of transfer matrices for strip lattices in the Potts model

The transfer-matrix technique is a convenient way for studying strip lattices in the Potts model since the compu- tational costs depend just on the periodic part of the lattice and not on the whole. However, even when the cost is reduced, the transfer-matrix technique is still an NP-hard problem since the time T(|V|, |E|) needed to compute the matrix grows ex- ponentially as a function of the graph width. In this work, we present a parallel transfer-matrix implementation that scales performance under multi-core architectures. The construction of the matrix is based on several repetitions of the deletion- contraction technique, allowing parallelism suitable to multi-core machines. Our experimental results show that the multi-core implementation achieves speedups of 3.7X with p = 4 processors and 5.7X with p = 8. The efficiency of the implementation lies between 60% and 95%, achieving the best balance of speedup and efficiency at p = 4 processors for actual multi-core architectures. The algorithm also takes advantage of the lattice symmetry, making the transfer matrix computation to run up to 2X faster than its non-symmetric counterpart and use up to a quarter of the original space

Observational data formats and identification of supplementary documentation for the National Geodetic Satellite Program

Observational formats for geodetic tracking data and supporting documentation for data reductio