The concept of cyclic tridiagonal pairs is introduced, and explicit examples are given. For a fairly general class of cyclic tridiagonal pairs with cyclicity N , we associate a pair of 'divided polynomials'. The properties of this pair generalize the ones of tridiagonal pairs of Racah type. The algebra generated by the pair of divided polynomials is identified as a higher-order generalization of the Onsager algebra. It can be viewed as a subalgebra of the q−Onsager algebra for a proper specialization at q the primitive 2N th root of unity. Orthogonal polynomials beyond the Leonard duality are revisited in light of this framework. In particular, certain second-order Dunkl shift operators provide a realization of the divided polynomials at N = 2 or q = i.

Abstract. Let A, A * be the generators of the q−Onsager algebra. Analogues of Lusztig's r − th higher order relations are proposed. In a first part, based on the properties of tridiagonal pairs of q−Racah type which satisfy the defining relations of the q−Onsager algebra, higher order relations are derived for r generic. The coefficients entering in the relations are determined from a two-variable polynomial generating function. In a second part, it is conjectured that A, A * satisfy the higher order relations previously obtained. The conjecture is proven for r = 2, 3. For r generic, using an inductive argument recursive formulae for the coefficients are derived. The conjecture is checked for several values of r ≥ 4. Consequences for coideal subalgebras and integrable systems with boundaries at q a root of unity are pointed out.MSC: 81R50; 81R10; 81U15; 81T40.

The concept of cyclic tridiagonal pairs is introduced, and explicit examples are given. For a fairly general class of cyclic tridiagonal pairs with cyclicity N , we associate a pair of 'divided polynomials'. The properties of this pair generalize the ones of tridiagonal pairs of Racah type. The algebra generated by the pair of divided polynomials is identified as a higher-order generalization of the Onsager algebra. It can be viewed as a subalgebra of the q−Onsager algebra for a proper specialization at q the primitive 2N th root of unity. Orthogonal polynomials beyond the Leonard duality are revisited in light of this framework. In particular, certain second-order Dunkl shift operators provide a realization of the divided polynomials at N = 2 or q = i.

Abstract. Let {A j |j = 0, 1, ..., rank(g)} be the fundamental generators of the generalized q−Onsager algebra Oq( g) introduced in [BB], where g is a simply-laced affine Lie algebra. New relations between certain monomials of the fundamental generators -indexed by the integer r ∈ Z + -are conjectured. These relations can be seen as deformed analogues of Lusztig's r−th higher order q−Serre relations associated with Uq( g), which are recovered as special cases. The relations are proven for r ≤ 5. For r generic, several supporting evidences are presented.MSC: 81R50; 81R10; 81U15; 81T40.

Thin films based on octadecylamine (ODA) material have been successfully fabricated by different methods: Langmuir-Blodgett (LB) method, spin-coating, spray-coating and dip-coating methods. The films obtained on the glass substrate were investigated for their optical properties (Uv-vis), film morphology (digital microscope, SEM) and contact angle. The solution and thin film absorption spectra of ODA and the mixtures of ODA and polymer RTV/SR do not absorb in the wavelength of the visible light region. The SEM analysis results show that the film morphology of ODA strongly depends on the nature of the ODA solution and the method of film formation and these are important factors affecting the different hydrophobicity of ODA films. Comparison of the droplet contact angle of thin films obtained by the above methods shows that the films fabricated by LB, spin-coating and spray-coating all increase the hydrophobicity of the glass substrate with the droplet contact angle range from 100o-145o, while dip-coating it is possible to fabricate superhydrophobic films with contact angles up to 161o.

Using differential scanning calorimetry and thermogravimetric analysis showed the marking temperature of coal tar pitch is 107.8°C. The thermal decomposition of the pitch is divided into three main stages, corresponding to three phases α, β, γ, with a coke yield of 47% at 800°C. The change in the XRD diagram showed a clear transition from the amorphous state of carbon to the highly ordered crystalline structure of graphite after treatment at 2,200 and 2,700°С. The purity of the pitch sample and the structure of the CCC carbon-carbon composite material after heat treatment at 2,700°C were studied using scanning electron microscopy and energy-dispersive X-ray spectroscopy methods. The results showed that after heat treatment, the C content in the sample reached more than 99.5%, and the coke residues after graphitization were bound and connected to the carbon fabric into a mass of graphite material.