Open Source Development in a Differentiated Duopoly

Open source software is released under an open source license giving individuals the right to use, modify, and redistribute freely the programs. This paper proposes a model of differentiated duopoly in which firms invest in the development of proprietary or open source software. The main findings are: (i) firms invest more when the products are substitutes; (ii) for substitute products, firms’ investment in software development is greatest when the software is open source; (iii) for close to perfect complements, firms’ investment in software development is greatest when the software is proprietary; and (iv) for substitute products, investment in open source software yields higher profits than investment in proprietary software.

Contraction and optimality properties of an adaptive Legendre-Galerkin method: the multi-dimensional case

We analyze the theoretical properties of an adaptive Legendre-Galerkin method in the multidimensional case. After the recent investigations for Fourier-Galerkin methods in a periodic box and for Legendre-Galerkin methods in the one dimensional setting, the present study represents a further step towards a mathematically rigorous understanding of adaptive spectral/$hp$ discretizations of elliptic boundary-value problems. The main contribution of the paper is a careful construction of a multidimensional Riesz basis in $H^1$, based on a quasi-orthonormalization procedure. This allows us to design an adaptive algorithm, to prove its convergence by a contraction argument, and to discuss its optimality properties (in the sense of non-linear approximation theory) in certain sparsity classes of Gevrey type

On the decay of the inverse of matrices that are sum of Kronecker products

Decay patterns of matrix inverses have recently attracted considerable interest, due to their relevance in numerical analysis, and in applications requiring matrix function approximations. In this paper we analyze the decay pattern of the inverse of banded matrices in the form $S=M \otimes I_n + I_n \otimes M$ where $M$ is tridiagonal, symmetric and positive definite, $I_n$ is the identity matrix, and $\otimes$ stands for the Kronecker product. It is well known that the inverses of banded matrices exhibit an exponential decay pattern away from the main diagonal. However, the entries in $S^{-1}$ show a non-monotonic decay, which is not caught by classical bounds. By using an alternative expression for $S^{-1}$, we derive computable upper bounds that closely capture the actual behavior of its entries. We also show that similar estimates can be obtained when $M$ has a larger bandwidth, or when the sum of Kronecker products involves two different matrices. Numerical experiments illustrating the new bounds are also reported

Optimal control in ink-jet printing via instantaneous control

This paper concerns the optimal control of a free surface flow with moving contact line, inspired by an application in ink-jet printing. Surface tension, contact angle and wall friction are taken into account by means of the generalized Navier boundary condition. The time-dependent differential system is discretized by an arbitrary Lagrangian-Eulerian finite element method, and a control problem is addressed by an instantaneous control approach, based on the time discretization of the flow equations. The resulting control procedure is computationally highly efficient and its assessment by numerical tests show its effectiveness in deadening the natural oscillations that occur inside the nozzle and reducing significantly the duration of the transient preceding the attainment of the equilibrium configuration

Adaptive Fourier-Galerkin Methods

We study the performance of adaptive Fourier-Galerkin methods in a periodic box in $\mathbb{R}^d$ with dimension $d\ge 1$. These methods offer unlimited approximation power only restricted by solution and data regularity. They are of intrinsic interest but are also a first step towards understanding adaptivity for the $hp$-FEM. We examine two nonlinear approximation classes, one classical corresponding to algebraic decay of Fourier coefficients and another associated with exponential decay. We study the sparsity classes of the residual and show that they are the same as the solution for the algebraic class but not for the exponential one. This possible sparsity degradation for the exponential class can be compensated with coarsening, which we discuss in detail. We present several adaptive Fourier algorithms, and prove their contraction and optimal cardinality properties.Comment: 48 page

An optimal adaptive Fictitious Domain Method

We consider a Fictitious Domain formulation of an elliptic partial differential equation and approximate the resulting saddle-point system using an inexact preconditioned Uzawa iterative algorithm. Each iteration entails the approximation of an elliptic problems performed using adaptive finite element methods. We prove that the overall method converges with the best possible rate and illustrate numerically our theoretical findings

Analisi dinamica comparativa di due motori motociclistici ad alte prestazioni

Le moderne tecnologie informatiche consentono di svolgere studi sulla dinamica motore, replicando con modelli numerici il comportamento al banco prova dei propulsori. Dopo una modellazione di base dei due propulsori bicilindrici ad “L”, tramite software Multibody, sono stati evoluti tali modelli con l’uso integrato FEM-Multibody, concentrando l’attenzione sul componente albero motore. Gli obiettivi principali sono stati: validazione dei due modelli mediante confronto con i dati sperimentali dei valori ottenuti dalle simulazioni; realizzazione di uno studio per la definizione di un “modus operandi” adeguato per l’azienda DUCATI MOTOR HOLDING S.p.a. per l’uso del software Multibody, infine, il confronto del comportamento dinamico e delle prestazioni dei due propulsori oggetto di studio