Continuous Dependence on a Parameter of Exponential Attractors for Nonclassical Diffusion Equations

In this paper, a new abstract result is given to verify the continuity of exponential attractors with respect to a parameter for the underlying semigroup. We do not impose any compact embedding on the main assumptions in the abstract result which is different from the corresponding result established by Efendiev et al. in 2004. Consequently, it can be used for equations whose solutions have no higher regularity. As an application, we prove the continuity of exponential attractors in H01 for a class of nonclassical diffusion equations with initial datum in H01.

On the instability of the Millionshchikov linear systems depending on a real parameter

The present article considers one-parameter families of second-order linear differential systems with a coefficient matrix depending on the real parameter, which is a diagonal matrix at each odd time interval of unit length. The Cauchy matrix is the rotation matrix at each odd time interval, whereas the angle is the sum of a parameter value and some real number. Earlier, it has been has proved that the upper Lyapunov exponent of each such a system, which is considered to be the function of parameter, is positive on the set of the positive Lebesque measure if the diagonal part of the coefficient matrix is independent on a parameter and separated from zero. The proof of this result essentially uses a complex matrix of special type. In recent article, the author has given another way to prove this theorem based on implementing the Parseval equality for trygonometric sums. Besides, the author considers the special case of the above systems. Now the diagonal part of the coefficient matrix is time-independent and is sufficiently big, whereas the rotation angle is defined by a maximum degree of two that divides the number of the corresponding time interval. For such a system, in the case of a continious coefficient dependence on a parameter it is proved that such a value exists, at which the corresponding system is unstable.

Visualização de integrais impróprias em um parâmetro com o auxílio do Geogebra

Resumo: A noção de integral generalizada constitui elemento obrigatório de estudo no contexto de Análise. Neste trabalho, discutiremos uma classe particular de integrais generalizadas de funções , que se caracterizam pela dependência de um parâmetro . Dentre alguns dos critérios que empregaremos com o objetivo precípuo de descrever o comportamento de algumas dessas integrais, destacamos a passagem de derivação sob o sinal da integral. Enfatizaremos, também, a visualização de propriedades possibilitadas pelo uso do software Geogebra. Com ele, mostraremos, do ponto de vista gráfico-geométrico, as relações de dependência conceitual e do comportamento (convergência/divergência) de integrais. Por fim, pela exploração de certas características qualitativas deste objeto, indicaremos vias de transmissão deste saber, com vistas ao entendimento no contexto de ensino. Palavras-chave: Integral generalizada, Visualização, Software Geogebra. VISUALIZATION OF IMPROPER INTEGRALS ON A PARAMETER BY THE USE OF GEOGEBRA SOFTWARE Abstract: The notion of generalized integral is an element of the required study in the Analysis’s context. In this paper, we discuss a particular class of generalized integrals of the functions , which are characterized by dependence on a parameter. Among some of the criteria that we will use with the primary objective to describe the behavior of some of these integrals, we highlight the passage of derivation under the integral sign. Also will emphasize the visualization of properties made ​​possible by the use of Geogebra software. With it, we will show the graphical point of view, geometric, conceptual dependency relationships and behavior (convergence / divergence) of integrals. Finally, the use of certain qualitative properties of this object, we will indicate routes of transmission of this knowledge, with a view to understanding the concept in the teaching context. Keywords: Generalized integrals, Visualization, Software Geogebra.