a semi-batch crystallization system is used to determine growth kinetics and morphology of barium carbonate crystals, under a constant ph value and constant temperature in the various relative supersaturations. results show that crystal growth rate satisfies a first-order kinetic expression at high relative supersaturation, while at relative supersaturation values lower than 3,  the kinetic exp...

Liouvillian functions are functions that are built up from rational functions using exponentiation, integration, and algebraic functions. We show that if a system of differential equations has a generic solution that satisfies a liouvillian relation, that is, there is a liouvillian function of several variables vanishing on the curve defined by this solution, then the system has a liouvillian f...

Searching for geometric loci belongs to the traditional part of mathematics school curricula all over the world. This topic is generally considered to be quite difficult for students, despite many well–known loci are around us, such as lines, circles or conics. Nowadays new computational technologies substantially facilitate investigation of loci, especially in a plane. Dynamic geometry softwar...

A characteristic pair is a pair (G, C) of polynomial sets in which G is a reduced lexicographic Gröbner basis, C is the minimal triangular set contained in G, and C is normal. In this paper, we show that any finite polynomial set P can be decomposed algorithmically into finitely many characteristic pairs with associated zero relations, which provide representations for the zero set of P in term...

The Ritt problem asks if there is an algorithm that tells whether one prime differential ideal is contained in another one if both are given by their characteristic sets. We give several equivalent formulations of this problem. In particular, we show that it is equivalent to testing if a differential polynomial is a zero divisor modulo a radical differential ideal. The technique used in the pro...

The classical Ritt’s Theorems state several properties of univariate polynomial decomposition. In this paper we present new counterexamples to the first Ritt theorem, which states the equality of length of decomposition chains of a polynomial, in the case of rational functions. Namely, we provide an explicit example of a rational function with coefficients in Q and two decompositions of differe...