Scientific Reports
The preprint collection of the Department of Mathematics.
Full-text generally not available for preprints prior to may 2006.
Found 868 products
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QDD152 - 04/03/2013
Grillo, G.; Muratori, M.
Radial fast diffusion on the hyperbolic space | Abstract | | We consider positive radial solutions to the fast diffusion equation on the hyperbolic space. By radial we mean solutions depending only on the geodesic distance from a given point. We investigate the fine asymptotics of solutions near the extinction time, in terms of a separable solution, showing convergence in relative error of the former to the latter. Solutions are smooth, and bounds on derivatives are given as well. In particular, sharp convergence results are shown for spatial derivatives, again in the form of convergence in relative error. |
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QDD153 - 04/03/2013
Grillo, G.; Muratori, M.
Sharp asymptotics for the porous media equation in low dimensions via Gagliardo-Nirenberg 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 Gagliardo-Nirenberg 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 Gagliardo-Nirenberg inequalities hold true. Moreover, we show equivalence between such functional inequalities and the mentioned asymptotic bounds for solutions. |
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QDD149 - 03/25/2013
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. |
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QDD150 - 03/25/2013
Terracini, S.; Verzini, G; Zilio, A.
Uniform Hölder regularity with small exponent in competition-fractional diffusion systems | Abstract | | For a class of competition-diffusion nonlinear systems involving fractional powers of the Laplacian, including as a special the fractional Gross-Pitaevskii 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. |
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QDD151 - 03/25/2013
Soave, N.; Zilio, A.
Entire solutions with exponential growth for an elliptic system modelling phase-transition | Abstract | | We prove the existence of entire solutions with exponential growth for an semilinear elliptic system appearing in the contest of phase separation for two-mixtures of Bose-Einstein condensates. Our construction is based on an approximation procedure, whose convergence is ensured by suitable Almgren-type monotonicity formulae. The construction of some solutions is extended to systems with k components, for every k > 2. |
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QDD148 - 03/08/2013
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. |
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QDD147 - 02/21/2013
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 self-excited 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. |
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QDD146 - 02/01/2013
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. |
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