A 3-dimensional singular kernel problem in viscoelasticity: an existence result

Materials with memory, namely those materials whose mechanical and/or thermodynamical behaviour depends on time not only via the present time, but also through its past history, are considered. Specifically, a three dimensional viscoelastic body is studied. Its mechanical behaviour is described via an integro-differential equation, whose kernel represents the relaxation modulus, characteristic of the viscoelastic material under investigation. According to the classical model, to guarantee the thermodynamical compatibility of the model itself, such a kernel satisfies regularity conditions which include the integrability of its time derivative. To adapt the model to a wider class of materials, this condition is relaxed; that is, conversely to what is generally assumed, no integrability condition is imposed on the time derivative of the relaxation modulus. Hence, the case of a relaxation modulus which is unbounded at the initial time t = 0, is considered, so that a singular kernel integro-differential equation, is studied. In this framework, the existence of a weak solution is proved in the case of a three dimensional singular kernel initial boundary value problem.Comment: 15 page

KdV-type equations linked via Baecklund transformations: remarks and perspectives

Third order nonlinear evolution equations, that is the Korteweg-deVries (KdV), modified Korteweg-deVries (mKdV) equation and other ones are considered: they all are connected via Baecklund transformations. These links can be depicted in a wide Baecklund Chart} which further extends the previous one constructed in [22]. In particular, the Baecklund transformation which links the mKdV equation to the KdV singularity manifold equation is reconsidered and the nonlinear equation for the KdV eigenfunction is shown to be linked to all the equations in the previously constructed Baecklund Chart. That is, such a Baecklund Chart is expanded to encompass the nonlinear equation for the KdV eigenfunctions [30], which finds its origin in the early days of the study of Inverse scattering Transform method, when the Lax pair for the KdV equation was constructed. The nonlinear equation for the KdV eigenfunctions is proved to enjoy a nontrivial invariance property. Furthermore, the hereditary recursion operator it admits [30 is recovered via a different method. Then, the results are extended to the whole hierarchy of nonlinear evolution equations it generates. Notably, the established links allow to show that also the nonlinear equation for the KdV eigenfunction is connected to the Dym equation since both such equations appear in the same Baecklund chart.Comment: 18 page

Some remarks on the model of rigid heat conductor with memory: unbounded heat relaxation function

The model of rigid linear heat conductor with memory is reconsidered focussing the interest on the heat relaxation function. Thus, the definitions of heat flux and thermal work are revised to understand where changes are required when the heat flux relaxation function $k$ is assumed to be unbounded at the initial time $t=0$. That is, it is represented by a regular integrable function, namely $k\in L^1(\R^+)$, but its time derivative is not integrable, that is $\dot k\notin L^1(\R^+)$. Notably, also under these relaxed assumptions on $k$, whenever the heat flux is the same also the related thermal work is the same. Thus, also in the case under investigation, the notion of equivalence is introduced and its physical relevance is pointed out

SCELTA EDUCATIVA, STATUS SOCIALE E CRESCITA

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Some remarks on the model of rigid heat conductor with memory: unbounded heat relaxation function

The model of rigid linear heat conductor with memory is reconsidered focussing the interest on the heat relaxation function. Thus, the definitions of heat flux and thermal work are revised to understand where changes are required when the heat flux relaxation function $k$ is assumed to be unbounded at the initial time $t=0$. That is, it is represented by a regular integrable function, namely $k\in L^1(\R^+)$, but its time derivative is not integrable, that is $\dot k\notin L^1(\R^+)$. Notably, also under these relaxed assumptions on $k$, whenever the heat flux is the same also the related thermal work is the same. Thus, also in the case under investigation, the notion of equivalence is introduced and its physical relevance is pointed out

Social Rewards in Science and Economic Growth

In this paper we put forward a model of basic research and long-run economic growth in which the incentives of social reward to scientific work may produce increasing returns and multiple equilibria. The state organizes production of new knowledge - a public good that improves firms technology - with taxes on the private sector. Scientists compete with one another to attain priority over a discovery and be awarded both a real prize and prestige in the scientific community. Also, scientists derive job motivation from dedication to science which provides social status. Analysis of the model shows, on the one hand, a low equilibrium where the economy is endowed with a small science sector, researchers have high relative income but low prestige, and competition for discoveries is weak. On the other hand, there is a high equilibrium where the economy has a large science sector, scientists obtain for new findings high prestige but lower relative salaries and, as the e¤ect of creative destruction is strong, there is fierce competition among researchers. Comparative statics shows that if the scientific infrastructure is poor, policies that increase the marginal benefits from a discovery have perverse e¤ects, while policies aimed at improving the selection mechanism of researchers work well. The same policies have opposite effects at the high steady state.

Some remarks on materials with memory: heat conduction and viscoelasticity

Materials with memory are here considered. The introduction of the dependence on time not only via the present, but also, via the past time represents a way, alternative to the introduction of possible non linearities, when the physical problem under investigation cannot be suitably described by any linear model. Specifically, the two different models of a rigid heat conductor, on one side, and of a viscoelastic body, on the other one, are analyzed. In them both, to evaluate the quantities of physical interest a key role is played by the past history of the material and, accordingly, the behaviour of such materials is characterized by suitable constitutive equations where Volterra type kernels appear. Specifically, in the heat conduction problem, the heat flux is related to the history of the temperature-gradient while, in isothermal viscoelasticity, the stress tensor is related to the strain history. Then, the notion of equivalence is considered to single out and associate together all those different thermal histories, or, in turn, strain histories, which produce the same work. The corresponding explicit expressions of the minimum free energy are compared

Scientific research, externalities and Economic growth

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