IASBS Mathematics Colloquium

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Spring 1398

1398-01-20
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1398-01-27
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1398-02-03
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1398-02-10
Speaker: Arash Ghorbanalizadeh (IASBS)
Title: Function spaces with variable exponent and some approximation problems
Abstract: In recent years, variable exponent function spaces and approximation problems in variable exponent Lebesgue spaces $L_p(x)$ have attracted more attention. Many authors have obtained analogs of classical results in function space with variable exponents because of their applications in elasticity theory, fluid mechanics, differential operators, nonlinear Dirichlet boundary value problems, nonstandard growth, and variational calculus. In this talk, first we introduce variable exponent Lebesgue spaces $L_p(x)$ and some properties of this function space and then we discuss the approximation by integral functions of finite degree in the variable exponent Lebesgue spaces defined on the real axis.

1398-02-17
Speaker: Abbas Nasrollah Nejad (IASBS)
Title: The Gauss Algebra of a Toric Variety
Abstract: Let $S=K[x_0,\ldots,x_d]$ be a polynomial ring over an algebraically closed field $K$ of characteristic zero and $A=K[g_0,\ldots,g_n]\subseteq S$ a $k$ -subalgebra of dimension $d+1$ generated by homogeneous polynomials of the same degree. We define the Gauss algebra $\mathbb{G}(A)$ of $A$ as $K$ -subalgebra of $S$ generated by $d+1$ -minors of the Jacobian matrix of $g_0,\ldots,g_n$ . The Gauss algebra $\mathbb{G}(A)$ is isomorphic to the coordinate ring of the Gauss image of the projective variety defined parametrically by $g_0,\ldots,g_n$ in the Pluecker embedding of the Grassmannian $\mathbb{G}(d,n)$ of $d$ -planes. In this talk, we describe the generators and the structure of $\mathbb{G}(A)$ , when A is a Borel fixed algebra or a squarefree Veronese algebra generated in degree $2$ .

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1398-02-31
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1398-03-07
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1398-03-21
Speaker: Saad Varsaie (IASBS)
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1398-03-28
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1398-04-12
Speaker: Amirhossein Amiraslani (STEM Department, University of Hawaii Maui College, USA and Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran)
Title: Explicit integration matrices and their applications to numerical solution of integral equations
Abstract: The purpose of this talk is twofold. We first explain the process of finding the differentiation as well as integration operational matrices for some degree-graded and nondegree-graded polynomial bases. One of the main advantages of having explicit formulas for operational matrices in a basis is that we do not need to change the basis given that conversion between bases can be typically unstable from a numerical point of view. Of particular interest is to obtain the integration operational matrices in terms of new orthogonal polynomials bases derived from the spectral methods for solving operator equations. This leads to well-conditioned simple structure matrices and reduce the matrix operations. We then focus on some applications of the proposed integration matrices in a matrix quadrature rule for numerical integration in various bases. Such a quadrature enables us to choose arbitrary degrees for the interpolating polynomials in each subinterval. We also present a numerical algorithm for the approximate solution of multi-dimensional nonlinear integral equations in terms of bivariate degree-graded orthogonal polynomial bases that arise in the theory of nonlinear parabolic boundary-value problems as well as the mathematical modeling of the spatio-temporal development of an epidemic. Finally, some numerical experiments are reported which show high reduction of the computational complexity with reasonable higher order accuracy.