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Sparse interpolation refers to the exact recovery of a function as a short linear combination of basis functions from a limited number of evaluations. For multivariate functions, the case of the monomial basis is well studied, as is now the basis of exponential functions. Beyond the multivariate Chebyshev polynomial obtained as tensor products of univariate Chebyshev polynomials, the theory of root systems allows to define a variety of generalized multivariate Chebyshev polynomials that have connections to topics such as Fourier analysis and representations of Lie algebras. We present a deterministic algorithm to recover a function that is the linear combination of at most r such polynomials from the knowledge of r and an explicitly bounded number of evaluations of this function. [ABSTRACT FROM AUTHOR]

In this article we determine a generating set of rational invariants of minimal cardinality for the action of the orthogonal group O 3 on the space R [ x , y , z ] 2 d of ternary forms of even degree 2d. The construction relies on two key ingredients: on the one hand, the Slice Lemma allows us to reduce the problem to determining the invariants for the action on a subspace of the finite subgroup B 3 of signed permutations. On the other hand, our construction relies in a fundamental way on specific bases of harmonic polynomials. These bases provide maps with prescribed B 3 -equivariance properties. Our explicit construction of these bases should be relevant well beyond the scope of this paper. The expression of the B 3 -invariants can then be given in a compact form as the composition of two equivariant maps. Instead of providing (cumbersome) explicit expressions for the O 3 -invariants, we provide efficient algorithms for their evaluation and rewriting. We also use the constructed B 3 -invariants to determine the O 3 -orbit locus and provide an algorithm for the inverse problem of finding an element in R [ x , y , z ] 2 d with prescribed values for its invariants. These computational issues are relevant in brain imaging. [ABSTRACT FROM AUTHOR]

We reconstruct an n-dimensional convex polytope from the knowledge of its directional moments. The directional moments are related to the projection of the polytope vertices on a particular direction. To extract the vertex coordinates from the moment information we combine established numerical algorithms such as generalized eigenvalue computation and linear interval interpolation. Numerical illustrations are given for the reconstruction of 2-d and 3-d convex polytopes. [ABSTRACT FROM AUTHOR]

Lagrangian curves in $\mathbb {R}^{4}$ entertain intriguing relationships with second order deformation of plane curves under the special affine group and null curves in a 3-dimensional Lorentzian space form. We provide a natural affine symplectic frame for Lagrangian curves. It allows us to classify Lagrangian curves with constant symplectic curvatures, to construct a class of Lagrangian tori in $\mathbb {R}^{4}$ and determine Lagrangian geodesics. [ABSTRACT FROM AUTHOR]

Scalings form a class of group actions that have theoretical and practical importance. A scaling is accurately described by a matrix of integers. Tools from linear algebra over the integers are exploited to compute their invariants, rational sections (a.k.a. global cross-sections), and offer an algorithmic scheme for the symmetry reduction of dynamical systems. A special case of the symmetry reduction algorithm applies to reduce the number of parameters in physical, chemical or biological models. [ABSTRACT FROM AUTHOR]

We provide an algebraic formulation of the moving frame method for constructing local smooth invariants on a manifold under an action of a Lie group. This formulation gives rise to algorithms for constructing rational and replacement invariants. The latter are algebraic over the field of rational invariants and play a role analogous to Cartan's normalized invariants in the smooth theory. The algebraic algorithms can be used for computing fundamental sets of differential invariants. [ABSTRACT FROM AUTHOR]

We show that the generic zeros of a differential ideal [ A]: H ∞ A defined by a differential chain A are birationally equivalent to the general zeros of a single regular differential polynomial. This provides a generalization of both the cyclic vector construction for system of linear differential equations and the rational univariate representation of algebraic zero dimensional radical ideals. In order to achieve generality, we prove new results on differential dimension and relative orders which are of independent interest. [ABSTRACT FROM AUTHOR]

A foreword to "Foundations of Computational Mathematics" is presented.