Asymptotic Properties of Orthogonal Polynomials with Respect to a Non-discrete Jacobi-Sobolev Inner Product

Let {Q _n ^(α,β) (x)} _n=0 ^∞ denote the sequence of polynomials orthogonal with respect to the non-discrete Sobolev inner product

$\langle f,g\rangle=\int_{-1}^{1}f(x)g(x)d\mu_{\alpha,\beta}(x)+\lambda\int_{-1}^{1}f'(x)g'(x)d\nu_{\alpha,\beta}(x)$

where λ>0 and d μ _α,β(x)=(x?a)(1?x)^α?1(1+x)^β?1 dx, d ν _α,β(x)=(1?x) ^α (1+x) ^β dx with aα,β>0. Their inner strong asymptotics on (?1,1), a Mehler-Heine type formula as well as some estimates of the Sobolev norms of Q _n ^(α,β) are obtained.

     

A Cohen type inequality for Laguerre-Sobolev expansions

Let introduce the Sobolev type inner product

     

Vanishing Results for Toric Varieties Associated To GL n and G2

Toric varieties associated with root systems appeared very naturally in the theory of group compactifications. Here they are considered in a very different context. We prove the vanishing of higher cohomology groups for certain line bundles on toric varieties associated to GL _n and G₂. This can be considered of general interest and it improves the previously known results for these varieties. We also show how these results give a simple proof of a converse to Mazur’s inequality for GL _n and G₂ respectively. It is known that the latter imply the nonemptiness of some affine Deligne–Lusztig varieties. Dedicated to Scarlett MccGwire and Dr. Christian Duhamel     

Generalised Risk-Sensitive Control with Full and Partial State Observation

This paper generalises the risk-sensitive cost functional by introducing noise dependent penalties on the state and control variables. The optimal control problems for the full and partial state observation are considered. Using a change of probability measure approach, explicit closed-form solutions are found in both cases. This has resulted in a new risk-sensitive regulator and filter, which are generalisations of the well-known classical results.     

Neumann series and Lommel functions of two variables

We obtain several new results for Neumann series of Bessel functions as well as for its various special cases. The generalization of some well-known results for these kind of series, such as the Graf's addition theorem, are also established.     

Tuning the C/N Ratio of C-Rich Graphitic Carbon Nitride (g-C3N4) Materials by the Melamine/Carboxylic Acid Adduct Route

C-rich graphitic carbonitride materials (CN_x) with a large range of compositions have been prepared thanks to the self-assembly, in different ratios, of melamine (M) and a panel of polycarboxylic acids (A) such as oxalic, tartaric and citric acid. The thermal conversion of the formed adducts (MA_y), led to CN_x phases, with x ranging from 0.66 to 1.4 (x=1.33 for g-C₃N₄ for comparison). The properties of these materials were examined by different techniques (XRD, Raman spectroscopy, TEM, TGA, XPS and DRIFT). It appears that the increase in the C content is associated with the disappearance of the long-range order of heptazine units and an increase in the sub-nanometer carbon-rich cluster size within the graphitic g-C₃N₄ structure. This trend is followed by a significant increase in the interlayer spacing and a lower proportion of N=C−N bonds compared to C=C bonds. The thermal stability under an inert atmosphere of these phases and their UV-Visible absorbance properties were also investigated.     

Partial fraction expansion of the hypergeometric functions

In this paper, the partial fractions expansions for the generalized hypergeometric function will be presented. Our results generalized several well-known results in the literature.     

On dominance and minuscule Weyl group elements

Fix a Dynkin graph and let λ be a coweight. When does there exist an element w of the corresponding Weyl group such that w is λ-minuscule and w(λ) is dominant? We answer this question for general Coxeter groups. We express and prove these results using a variant of Mozes’ game of numbers.     

Controllability and Controller-Observer Design for a Class of Linear Time-Varying Systems

In this paper a class of linear time-varying control systems is considered. The time variation consists of a scalar time-varying coefficient multiplying the state matrix of an otherwise time-invariant system. Under very weak assumptions of this coefficient, we show that the controllability can be assessed by an algebraic rank condition, Kalman canonical decomposition is possible, and we give a method for designing a linear state-feedback controller and Luenberger observer.