Some determinants of path generating functions

We evaluate four families of determinants of matrices, where the entries are sums or differences of generating functions for paths consisting of up-steps, down-steps and level steps. By specialisation, these determinant evaluations have numerous corollaries. In particular, they cover numerous determinant evaluations of combinatorial numbers - most notably of Catalan, ballot, and of Motzkin numbers - that appeared previously in the literature.Comment: 35 pages, AmS-TeX; minor corrections; final version to appear in Adv. Appl. Mat

The Reversed q-Exponential Functional Relation

After obtaining some useful identities, we prove an additional functional relation for $q$ exponentials with reversed order of multiplication, as well as the well known direct one in a completely rigorous manner.Comment: 6 pages, LaTeX, no figure

q-Analogue of Shock Soliton Solution

By using Jackson's q-exponential function we introduce the generating function, the recursive formulas and the second order q-differential equation for the q-Hermite polynomials. This allows us to solve the q-heat equation in terms of q-Kampe de Feriet polynomials with arbitrary N moving zeroes, and to find operator solution for the Initial Value Problem for the q-heat equation. By the q-analog of the Cole-Hopf transformation we construct the q-Burgers type nonlinear heat equation with quadratic dispersion and the cubic nonlinearity. In q -> 1 limit it reduces to the standard Burgers equation. Exact solutions for the q-Burgers equation in the form of moving poles, singular and regular q-shock soliton solutions are found.Comment: 13 pages, 5 figure

Carborane–β-cyclodextrin complexes as a supramolecular connector for bioactive surfaces

Supramolecular chemistry provides an attractive entry to generate dynamic and well-controlled bioactive surfaces. Novel host–guest systems are urgently needed to provide a broader affinity and applicability portfolio. A synthetic strategy to carborane–peptide bioconjugates was therefore developed to provide an entry to monovalent supramolecular functionalization of β-cyclodextrin coated surfaces. The β-cyclodextrin·carborane–cRGD surfaces are formed efficiently and with high affinity as demonstrated by IR-RAS, WCA, and QCM-D, compare favourable to existing bio-active host–guest surface assemblies, and display an efficient bioactivity, as illustrated by a strong functional effect of the supramolecular system on the cell adhesion and spreading properties. Cells seeded on the supramolecular surface displaying bioactive peptide epitopes exhibited a more elongated morphology, focal adhesions, and stronger cell adhesion compared to control surfaces. This highlights the macroscopic functionality of the novel supramolecular immobilization strategy

h analogue of Newton's binomial formula

In this letter, the $h$--analogue of Newton's binomial formula is obtained in the $h$--deformed quantum plane which does not have any $q$--analogue. For $h=0$, this is just the usual one as it should be. Furthermore, the binomial coefficients reduce to $\frac{n!}{(n-k)!}$ for $h=1$. \\ Some properties of the $h$--binomial coefficients are also given. \\ Finally, I hope that such results will contribute to an introduction of the $h$--analogue of the well--known functions, $h$--special functions and $h$--deformed analysis.Comment: 6 pages, latex Jounal-ref: J. Phys. A: Math. Gen. 31 (1998) L75