Let $ (G_n)_{n=0}^{\infty } $ be a nondegenerate linear recurrence sequence whose power sum representation is given by $ G_n = a_1(n) \alpha _1^n + \cdots + a_t(n) \alpha _t^n $. We prove a function field analogue of the well-known result in the number field case that, under some nonrestrictive conditions, $ |{G_n}| \geq ( \max _{j=1,\ldots ,t} |{\alpha _j}| )^{n(1-\varepsilon )} $ for $ n $ large enough.

Polymers with stable radical groups are promising materials for organic electronic devices due to their unique redox activity. Block copolymers with one redox active block could be used in nanostructured devices for electronic applications. We report on the synthesis and characterization of such multifunctional block copolymers in which phase separation on the 10 nm (half pitch) scale is achieved by using fluorinated blocks. Fluorination of one block increases the degree of phase separation and leads to smaller accessible domain sizes. Block copolymers with 60%, 80% and 90% of a stable radical containing block and either fluorinated or non-fluorinated second blocks were made by atom transfer radical polymerization, and their microstructure formation as a function of fluorine content is described after solvent vapor or thermal annealing. Electrical characterization of such a partly fluorinated block copolymer shows their potential for electronic devices.

In this paper we completely solve a family of Thue inequalities defined over the field of functions , namely deg (X4−4cX3Y+(6c+2)X2Y2+4cXY3+Y4) ≤ deg c, where the solutions x,y come from the ring and the parameter is some non-constant polynomial.

It is proved that there does not exist a set of four positive integers with the property that the product of any two of its distinct elements plus their sum is a perfect square. This settles an old problem investigated by Diophantus and Euler.