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- On the kernel of a singular integral operator with shiftPublication . Marreiros, Rui
Expand Some estimates for the dimension of the kernel of the singular integral operator I - cUP(+) : L-p(n)(T) -> L-p(n)(T), p is an element of (1, infinity), with a non-Carleman shift are obtained, where P+ is the Cauchy projector, U is an isometric shift operator and c(t) is a continuous matrix function on the unit circle T. It is supposed that the shift has a finite set of fixed points and all the eigenvalues of the matrix c(t) at the fixed points, simultaneously belong either to the interior of the unit circle T or to its exterior. The case of an operator with a general shift is also considered. Some relations between those estimates and the resolvent set of the operator cU are pointed out.Expand - Interactive introduction to first-order ordinary differential and its applicationsPublication . Coelho, Celestino; Marreiros, Rui; Conceição, Ana C.
Expand Differential equations constitute a large and very important branch of modern mathematics. From the early days of the calculus this subject has been an area of great theoretical research and practical applications in several branches of science. Despite this importance, the largest part of the students reveals strong difficulties to understand the theory of differential equations and its applications. As a consequence of what was mentioned above we decided to create a new educational tool that accomplishes some of those goals, describing how this dynamic device can be used in the classroom when teaching the first approach of the first- order Ordinary Differential Equations (ODEs).Expand - On the study of the dimension of the kernel of singular integral operators with non-carleman shift using mathematica softwarePublication . Marreiros, Rui
Expand We consider the singular integral operator T=I-cUP_+ : L_2 (T)→L_2 (T), with a non- Carleman shift, where I is the identity operator, c∈C(T) is a continuous matrix function on the unit circle T , U is the isometric shift operator and P_+ is the Cauchy projector. It is supposed that the shift has a finite set of fixed points and the modulus of the function c(t) at the fixed points of the shift is less than one. Under these conditions, an estimate for the dimension of the kernel of the operator T , is obtained. We consider some examples to illustrate and show that the obtained estimate, in a certain sense, is sharp.Expand - Solving second-order linear ordinary differential equations by using interactive softwarePublication . Coelho, Celestino; Marreiros, Rui
Expand Differential equations constitute an area of great theoretical research and applications in several branches of science and technology. The scope of this work is to present new software that is able to show all the steps in the process of solving a linear second-order ordinary differential equation with constant coefficients.Expand - Symbolic computation applied to the study of the kernel of a singular integral operator with non-carleman shift and conjugationPublication . Conceição, Ana C.; Marreiros, Rui; Pereira, José C.
Expand On the Hilbert space the singular integral operator with non-Carleman shift and conjugation is considered, where are the Cauchy projectors, , , , are continuous functions on the unit circle , U is the shift operator and C is the operator of complex conjugation. We show how the symbolic computation capabilities of the computer algebra system Mathematica can be used to explore the dimension of the kernel of the operator K. The analytical algorithm [ADimKer-NonCarleman] is presented; several nontrivial examples are given.Expand