In this lecture we learn to solve nonhomogeneous equations of the form ay′′ + by′ + cy = g(t) using the “method of undetermined coefficients”. In previous lectures we learned to how to solve any equation of the form ay′′ + by′ + cy = 0. We will apply these techniques to help us solve the nonhomogeneous equation ay′′ + by′ + cy = g(t) by understanding the connection between solutions to homogene...

Let {Rn} and {Pn} be two polynomial systems which induce signed polynomial hypergroup structures on N0. We investigate when the Banach algebra l(N0, h) can be continuously embedded into or is isomorphic to l(N0, h ). We find sufficient conditions on the connection coefficients cnk given by Rn = ∑n k=0 cnkPk, for the existence of such an embedding or isomorphism. Finally we apply these results t...

In this paper a semilinear elliptic PDE with rapidly oscillating coefficients is homogenized. The novetly of our result lies in the fact that we allow the second order part of the differential operator to be degenerate in some portion of Rd. Our fully probabilistic method is based on the connection between PDEs and BSDEs with random terminal time and the weak convergence of a class of diffusion...

It being required that the determinant g of gij be different from zero, and in fact negative, a contravariant symmetric tensor giJ is defined by g'Jg,, = Vk, where 4Vk is the Kronecker delta, and where a repeated superscript and subscript, as i in this case, means the sum as the index takes the values 1 to 4. This notation holds throughout the paper. For any coefficients of connection rijh, sym...

For a given base scheme, a correspondence is established between a class of sheaves on curves over this base scheme and certain points of infinite Grassmannians. This equivalence extends to a characterization of commutative algebras of ordinary differential operators with coefficients in the ring of formal power series over a given kalgebra. Our construction generalizes the approach of M.Mulase...

The network of 5823 cities of Mexico with a population more than 5000 inhabitants is studied. Our analysis is focused to the spectral properties of the adjacency matrix, the small-world properties of the network, the distribution of the clustering coefficients and the degree distribution of the vertices. The connection of these features with the spread of epidemics on this network is also discu...

In this paper, an application of the Picard's iterative method for finding the solution of two phase Stefan problem is presented. In the proposed method an iterative connection is formulated, which allows to determine the temperature distribution in considered domain. Another unknown function, describing position of the moving interface, is approximated with the aid of linear combination of som...

A tile composed of two pieces, which we refer to as a fence tile, is introduced to give the Tribonacci numbers a new combinatorial interpretation. The interpretation is used to construct a Pascal-like triangle and various identities concerning the triangle are proven. An intuitive proof of a general identity for the Tribonacci numbers in terms of sums of products of the binomial coefficients is...

By starting from known graded Lie algebras, including Virasoro algebras, new kinds of time-dependent evolution equations are found possessing graded symmetry algebras. The modified KP equations are taken as an illustrative example: new modified KP equations with m arbitrary time-dependent coefficients are obtained possessing symmetries involving m arbitrary functions of time. A particular grade...