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We consider the problem of interpolating a function given on scattered points using Hermite-Birkhoff formulas on the sphere and other manifolds. We express each proposed interpolant as a linear combination of basis functions, the combination coefficients being incomplete Taylor expansions of the interpolated function at the interpolation points. The basis functions have the following features: (i) depend on the geodesic distance; (ii) are orthonormal with respect to the point-evaluation functionals; and (iii) have all derivatives equal zero up to a certain order at the interpolation points. Moreover, the construction of such interpolants, which belong to the class of partition of unity methods, takes advantage of not requiring any solution of linear systems.

The purpose of this paper is twofold. Firstly, we provide explicit and compact formulas for computing both Caputo and (modified) Riemann-Liouville (RL) fractional pseudospectral differentiation matrices (F-PSDMs) of any order at general Jacobi-Gauss-Lobatto (JGL) points. We show that in the Caputo case, it suffices to compute F-PSDM of order $\mu\in (0,1)$ to compute that of any order $k+\mu$ with integer $k\ge 0,$ while in the modified RL case, it is only necessary to evaluate a fractional integral matrix of order $\mu\in (0,1).$ Secondly, we introduce suitable fractional JGL Birkhoff interpolation problems leading to new interpolation polynomial basis functions with remarkable properties: (i) the matrix generated from the new basis yields the exact inverse of F-PSDM at "interior" JGL points; (ii) the matrix of the highest fractional derivative in a collocation scheme under the new basis is diagonal; and (iii) the resulted linear system is well-conditioned in the Caputo case, while in the modified RL case, the eigenvalues of the coefficient matrix are highly concentrated. In both cases, the linear systems of the collocation schemes using the new basis can solved by an iterative solver within a few iterations. Notably, the inverse can be computed in a very stable manner, so this offers optimal preconditioners for usual fractional collocation methods for fractional differential equations (FDEs). It is also noteworthy that the choice of certain special JGL points with parameters related to the order of the equations can ease the implementation. We highlight that the use of the Bateman's fractional integral formulas and fast transforms between Jacobi polynomials with different parameters, are essential for our algorithm development.

In this paper we give lower bounds for the representation of real univariate polynomials as sums of powers of degree 1 polynomials. We present two families of polynomials of degree d such that the number of powers that are required in such a representation must be at least of order d. This is clearly optimal up to a constant factor. Previous lower bounds for this problem were only of order $\Omega$($\sqrt$ d), and were obtained from arguments based on Wronskian determinants and "shifted derivatives." We obtain this improvement thanks to a new lower bound method based on Birkhoff interpolation (also known as "lacunary polynomial interpolation").

Although it is important both in theory as well as in applications, a theory of Birkhoff interpolation with main emphasis on the shape of the set of nodes is still missing. Although we will consider various shapes (e.g. we find all the shapes for which the associated Lagrange problem has unique solution), we concentrate on one of the simplest shapes:``rectangular'' (also called "cartesian grids"). The ultimate goal is to obtain a geometrical understanding of the solvability. We partially achieve this by describing several regularity criteria, which we illustrate by many examples. At the end we discuss several conjectures which, we think, are important in understanding the behaviour of Birkhoff interpolation schemes in higer dimensions. Although we prove these conjectures in many unrelated cases, we believe that a ``complete proof'' requires new ideas which go beyond the usual methods in interpolation theory (and may reach areas such as algebraic geometry or algebraic topology).

Although it is important both in theory as well as in applications, a theory of Birkhoff interpolation with main emphasis on the shape of the set of nodes is still missing. Although we will consider various shapes (e.g. we find all the shapes for which the associated Lagrange problem has unique solution), we concentrate on one of the simplest shapes:``rectangular'' (also called "cartesian grids"). The ultimate goal is to obtain a geometrical understanding of the solvability. We partially achieve this by describing several regularity criteria, which we illustrate by many examples. At the end we discuss several conjectures which, we think, are important in understanding the behaviour of Birkhoff interpolation schemes in higer dimensions. Although we prove these conjectures in many unrelated cases, we believe that a ``complete proof'' requires new ideas which go beyond the usual methods in interpolation theory (and may reach areas such as algebraic geometry or algebraic topology).

We consider a specific scheme of multivariate Birkhoff polynomial interpolation. Our samples are derivatives of various orders $k_j$ at fixed points $v_j$ along fixed straight lines through $v_j$ in directions $u_j$, under the following assumption: the total number of sampled derivatives of order $k, \ k=0,1,\ldots$ is equal to the dimension of the space homogeneous polynomials of degree $k$. We show that this scheme is regular for general directions. Specifically this scheme is regular independent of the position of the interpolation nodes. In the planar case, we show that this scheme is regular for distinct directions. Next we prove a "Birkhoff-Remez" inequality for our sampling scheme extended to larger sampling sets. It bounds the norm of the interpolation polynomial through the norm of the samples, in terms of the geometry of the sampling set.

We introduce a new family of multivalue and multistage methods based on Hermite–Birkhoff interpolation for solving nonlinear Volterra integro differential equations. The proposed methods that have high order and ex tensive stability region, use the approximated values of the first derivative of the solution in the m collocation points and the approximated values of the solution as well as its first derivative in the r previous steps. Convergence order of the new methods is determined and their linear stability is analyzed. Efficiency of the methods is shown by some numerical experiments.

The infinite (upper triangular) Pascal matrix is $T = [\binom{j}{i}]$ for $0\leq i,j$. It is easy to see that that submatrix $T(0:n,0:n)$ is triangular with determinant 1, hence in particular, it is invertible. But what about other submatrices $T(r,x)$ for selections $r=[r_0,...,r_d]$ and $x=[x_0,...,x_d]$ of the rows and columns of $T$? The goal of this paper is provide a necessary and sufficient condition for invertibility based on a connection to polynomial interpolation. In particular, we generalize the theory of Birkhoff interpolation and P\"olya systems, and then adapt it to this problem. The result is simple: $T(r,x)$ is invertible iff $r \leq x$, or equivalently, iff all diagonal entries are nonzero.

The infinite (upper triangular) Pascal matrix is $T = [\binom{j}{i}]$ for $0\leq i,j$. It is easy to see that that submatrix $T(0:n,0:n)$ is triangular with determinant 1, hence in particular, it is invertible. But what about other submatrices $T(r,x)$ for selections $r=[r_0,...,r_d]$ and $x=[x_0,...,x_d]$ of the rows and columns of $T$? The goal of this paper is provide a necessary and sufficient condition for invertibility based on a connection to polynomial interpolation. In particular, we generalize the theory of Birkhoff interpolation and P\"olya systems, and then adapt it to this problem. The result is simple: $T(r,x)$ is invertible iff $r \leq x$, or equivalently, iff all diagonal entries are nonzero.

When Albert Einstein published his first paper on relativity theory, it caused a stir in the physicists' community. When more and more evidence was gathered to prove the theory correct, even laymen became interested in it. Since the theory of relativity uses involved higher mathematics, it is considered notoriously difficult to grasp, and at the time it was published, it was claimed that only 12 people in the world were able to fully understand it. One of these was the Dutch physicist Hendrik Lorentz, who wrote the articles collected in this book for a lay audience. He explains the basics of the theory in clear and concise terms without needing any mathematics. All that is needed to fo follow his arguments is a bit of patience and time. (Summary by Availle)