The ground plan in order to disentangle the hard problem of modelling the
motion of a bicycle is to start from a very simple model and to outline the
proper mathematical scheme: for this reason the first step we perform lies in a
planar rigid body (simulating the bicylcle frame) pivoting on a horizontal
segment whose extremities, subjected to nonslip conditions, oversimplify the
wheels. Even in this former case, which is the topic of lots of papers in
literature, we find it worthwhile to pay close attention to the formulation of
the mathematical model and to focus on writing the proper equations of motion
and on the possible existence of conserved quantities. In addition to the first
case, being essentially an inverted pendulum on a skate, we discuss a second
model, where rude handlebars are added and two rigid bodies are joined. The
geometrical method of Appell is used to formulate the dynamics and to deal with
the nonholonomic constraints in a correct way. At the same time the equations
are explained in the context of the cardinal equations, whose use is habitual
for this kind of problems. The paper aims to a threefold purpose: to formulate
the mathematical scheme in the most suitable way (by means of the
pseudovelocities), to achieve results about stability, to examine the
legitimacy of certain assumptions and the compatibility of some conserved
quantities claimed in part of the literature.

Abstract:
We show that most of the empirical or semi-empirical isotherms proposed to extend the Langmuir formula to sorption (adsorption, chimisorption and biosorption) on heterogeneous surfaces in the gaseous and liquid phase belong to the family and subfamily of the Burr_{XII} cumulative distribution functions. As a consequence they obey relatively simple differential equations which describe birth and death phenomena resulting from mesoscopic and microscopic physicochemical processes. Using the probability theory, it is thus possible to give a physical meaning to their empirical coefficients, to calculate well defined quantities and to compare the results obtained from different isotherms. Another interesting consequence of this finding is that it is possible to relate the shape of the isotherm to the distribution of sorption energies which we have calculated for each isotherm. In particular, we show that the energy distribution corresponding to the Brouers-Sotolongo (BS) isotherm [1] is the Gumbel extreme value distribution. We propose a generalized GBS isotherm, calculate its relevant statistical properties and recover all the previous results by giving well defined values to its coefficients. Finally we show that the Langmuir, the Hill-Sips, the BS and GBS isotherms satisfy the maximum Bolzmann-Shannon entropy principle and therefore should be favoured.

Abstract:
We
discuss the Brouers-Sotolongo fractal (BSf) kinetics model. This
formalism interpolates between the first and second order kinetics. But more
importantly, it introduces not only a fractional order n but also a fractal time parameter a which characterizes the time variation of the rate constant. This
exponent appears in non-exponential relaxation and complex reaction models as
demonstrated by the extended use of the Weibull and Hill kinetics which are the two
most popular approximations of the BSf (n, a) kinetic equation as well in
non-Debye relaxation formulas. We show that the use of nonlinear programs
allows an easy and precise fitting of the data yielding the BSf parameters
which have simple physical interpretations.

Abstract:
The main
purpose of the paper consists in illustrating a procedure for expressing the
equations of motion for a general time-dependent constrained system.
Constraints are both of geometrical and differential type. The use of
quasi-velocities as variables of the mathematical problem opens the possibility
of incorporating some remarkable and classic cases of equations of motion.
Afterwards, the scheme of equations is implemented for a pair of substantial
examples, which are presented in a double version, acting either as a
scleronomic system and as a rheonomic system.

Abstract:
The properties of a ball-shaped semiconductor particles and metal particles with a semiconductor thin film on the surface thereof are established. So the dimensionless thermoelectric figure of merit of a material consisting of a large number of these particles is equal to 10 - 100.