Quaderni di Dipartimento
Collezione dei preprint del Dipartimento di Matematica. La presenza del fulltext è lacunosa per i prodotti antecedenti maggio 2006.
Trovati 867 prodotti

QDD153  03/04/2013
Grillo, G.; Muratori, M.
Sharp asymptotics for the porous media equation in low dimensions via GagliardoNirenberg inequalities  Abstract   We prove sharp asymptotic bounds for solutions to the porous media equation with homogeneous Dirichlet or Neumann boundary conditions on a bounded Euclidean domain, in dimension one and two. This is achieved by making use of appropriate GagliardoNirenberg inequalities only. The generality of the discussion allows to prove similar bounds for weighted porous media equations, provided one deals with weights for which suitable GagliardoNirenberg inequalities hold true. Moreover, we show equivalence between such functional inequalities and the mentioned asymptotic bounds for solutions. 

QDD149  25/03/2013
Arioli, G.; Gazzola, F.
A new mathematical explanation of the Tacoma Narrows Bridge collapse  Abstract   The spectacular collapse of the Tacoma Narrows Bridge, which occurred in 1940, has attracted the attention of engineers, physicists, and mathematicians in the last 70 years. There have been many attempts to explain this amazing event.
Nevertheless, none of these attempts gives a satisfactory and universally accepted explanation of the phenomena visible the day of the collapse.
The purpose of the present paper is to suggest a new mathematical model for the study of the dynamical behavior of suspension bridges which provides a realistic explanation of the Tacoma collapse. 

QDD150  25/03/2013
Terracini, S.; Verzini, G; Zilio, A.
Uniform Hölder regularity with small exponent in competitionfractional diffusion systems  Abstract   For a class of competitiondiffusion nonlinear systems involving fractional powers of the Laplacian, including as a special the fractional GrossPitaevskii system, we prove that uniform boundedness implies Hölder boundedness for sufficiently small positive exponents, uniformly as the interspecific competition parameter diverges. This implies strong convergence for the family of solutions as the segregation of their supports occurs. 

QDD151  25/03/2013
Soave, N.; Zilio, A.
Entire solutions with exponential growth for an elliptic system modelling phasetransition  Abstract   We prove the existence of entire solutions with exponential growth for an semilinear elliptic system appearing in the contest of phase separation for twomixtures of BoseEinstein condensates. Our construction is based on an approximation procedure, whose convergence is ensured by suitable Almgrentype monotonicity formulae. The construction of some solutions is extended to systems with k components, for every k > 2. 

QDD148  08/03/2013
Cerqueti, R.; Marazzina, D.; Ventura, M.
Optimal investments in a patent race  Abstract   This paper explores the optimal expenditure rate that a firm should employ to develop a new technology and pursue the registration of the related patent. Our model takes into account an economic environment with industrial competition among firms operating in the same sector and in presence of
sources of uncertainty in knowledge accumulation. We develop a stochastic optimal control problem with random horizon, and solve it theoretically by adopting a dynamic programming approach. An extensive numerical analysis confirms, as suggested by previous research, that the optimal expenditure
rate is an increasing function of the knowledge accumulated by the firm, but also brings out that its pattern is steeper in the early stage of the race. 

QDD147  21/02/2013
Gazzola, F.
Nonlinearity in oscillating bridges  Abstract   We first recall several historical oscillating bridges that, in some cases, led to collapses. Some of them are quite recent and show that, nowadays, oscillations in suspension bridges are not yet well understood. Next, we survey some attempts to model bridges with differential equations. Although these equations arise from quite different scientific communities, they display some common features. One of them, which we believe to be incorrect, is the acceptance of the linear Hooke law in elasticity. This law should be used only in presence of small deviations from equilibrium, a situation which does not occur in strongly oscillating bridges. Then we discuss a couple of recent models whose solutions exhibit selfexcited oscillations,
the phenomenon visible in real bridges. This suggests a different point of view in modeling equations and gives a strong hint how to modify the existing models in order to obtain a reliable theory. The purpose of this paper is precisely to highlight the necessity of revisiting classical models, to introduce reliable models, and to indicate the steps we believe necessary to reach this target. 

QDD146  01/02/2013
Marchionna, C.
Free vibrations in space of the single mode for the Kirchhoff string  Abstract   We study a single mode for the Kirchhoff string vibrating in space. In 3D a single mode is generally almost periodic in contrast to the 2D periodic case. In order to show a complete geometrical description of a single mode we prove some monotonicity properties of the almost periods of the solution, with respect to the mechanical energy and the momentum. As a consequence of these properties, we observe that a planar single mode in 3D is always unstable, while it is known that a single mode in 2D is stable (under a suitable definition of stability), if the energy is small. 

QDD145  17/01/2013
Barchielli, A.; Gregoratti, M.
Quantum continuous measurements: The stochastic Schroedinger equations and the spectrum of the output  Abstract   The stochastic Schroedinger equation, of classical or quantum type, allows to describe open quantum systems under measurement in continuous time. In this paper we review the link between these two descriptions and we study the properties of the output of the measurement. For simplicity we deal only with the diusive case. Firstly, we discuss the quantum stochastic Schroedinger equation, which is based on quantum stochastic calculus, and we show how to transform it into the classical stochastic Schroedinger equation by diagonalization of suitable quantum observables, based on the isomorphism between Fock space and Wiener space. Then, we give the a posteriori state, the conditional system state at time t given the output up to that time and we link its evolution to the classical stochastic Schroedinger equation. Finally, we study the output of the continuous measurement, which is a stochastic process with probability distribution given by the rules of quantum mechanics. When the output process is stationary, at least in the long run, the spectrum of the process can be introduced and its properties studied. In particular we show how the Heisenberg uncertainty relations give rise to characteristic bounds on the possible spectra and we discuss how this is related to the typical quantum phenomenon of squeezing. We use a simple quantum system, a twolevel atom stimulated by a laser, to discuss the dierences between homodyne and heterodyne detection and to explicitly show squeezing and antisqueezing and the Mollow triplet in the fluorescence spectrum. 
