Abstract. Continuous spline functions are defined as piecewise polynomials on the faces of a polyhedral complex that agree on the intersections of two faces. Splines are used in approximation theory and numerical analysis, with applications in data interpolation, to create smooth curves in computer graphics and to find numerical solutions to partial differential equations. Gilbert, Tymoczko, and Viel [ Pacific J. Math., 281 (2016), pp. 333–364] generalized the classical splines combinatorially and algebraically: a generalized spline is a vertex labeling of a graph [Formula: see text] by elements of the ring so that the difference between the labels of any two adjacent vertices lies in the ideal generated by the corresponding edge label. We study the generalized splines on the planar graphs whose edges are labeled by two-variable polynomials of the form [Formula: see text] and whose vertices are labeled by polynomials of degree at most two. In this paper we, address the upper-bound conjecture for the dimension of degree-2 splines of smoothness 1. The dimension is expressed in terms of the rank of the extended cycle basis matrix. We also provide a combinatorial algorithm on graphs to compute the rank by contracting certain subgraphs.
Abstract. We consider the representations of the group [Formula: see text] that contain the standard representation [Formula: see text] as an irreducible component. This is a large class of representations containing the vector spaces of ternary forms of any odd degree, the space of piezoelectricity tensors, or the classical product space [Formula: see text] which appears in multiple physical and engineering situations. For any such representation, we construct a set of algebraically independent rational invariants that generates the invariant field. Our key ingredient is a constructive use of Seshadri slice lemma. This lemma establishes an isomorphism between the [Formula: see text]-invariant field on the full representation [Formula: see text] with the invariant field of a representation of [Formula: see text] on a subspace [Formula: see text]. We exhibit a set of [Formula: see text]-invariant polynomials generating this latter invariant field. The explicit expressions for the generating rational [Formula: see text]-invariants can then be obtained with symbolic manipulations. The results are then extended to actions of [Formula: see text] and several cases of interest to applications are detailed as examples.
Abstract. We consider the problem of computing matrix polynomials [Formula: see text], where [Formula: see text] is a large dense matrix, with as few matrix-matrix multiplications as possible. More precisely, let [Formula: see text] represent the set of polynomials computable with [Formula: see text] matrix-matrix multiplications, but with an arbitrary number of matrix additions and scaling operations. We characterize this set through a tabular parameterization. By deriving equivalence transformations of the tabular representation, we establish new methods that can be used to construct elements of [Formula: see text] and determine general properties of the set. The transformations allow us to eliminate variables and prove that the dimension is bounded by [Formula: see text], which is subsequently proven to be sharp, i.e., [Formula: see text]. Consequently, we have identified a parameterization that, to the best of our knowledge, is the first minimal parameterization. We also conduct a study using computational tools from algebraic geometry to determine the largest degree [Formula: see text] such that all polynomials of that degree belong to [Formula: see text] or its closure. In many cases, the computational setup is constructive in the sense that it can also be used to determine a specific evaluation scheme for a given polynomial.
We consider the representations of the group SO3(R) that contain the standard representation R3 as an irreducible component. This is a large class of representations containing the vector spaces of ternary forms of any odd degree, the space of piezoelectricity tensors, or the classical product space R3 imes \cdot \cdot \cdot imes R3 which appears in multiple physical and engineering situations. For any such representation, we construct a set of algebraically independent rational invariants that generates the invariant field. Our key ingredient is a constructive use of Seshadri slice lemma. This lemma establishes an isomorphism between the SO3(R)-invariant field on the full representation R3\oplus\scrH with the invariant field of a representation of O2(R) on a subspace R \oplus \scrH. We exhibit a set of O2(R)-invariant polynomials generating this latter invariant field. The explicit expressions for the generating rational SO3(R)-invariants can then be obtained with symbolic manipulations. The results are then extended to actions of O3(R) and several cases of interest to applications are detailed as examples.
We develop a discrete differential geometry for surfaces of non-constant negative curvature, which can be used to model various phenomena from the growth of flower petals to marine invertebrate swimming. Specifically, we derive and numerically integrate a version of the classical Lelieuvre formulas that apply to immersions of C1,1 hyperbolic surfaces of non-constant curvature. In contrast to the constant curvature case, these formulas do not provide an explicit method for constructing an immersion but rather describe an immersion via an implicit set of equations. We propose an iterative method for resolving these equations. Because we are interested in scenarios where the curvature is a function of the intrinsic material coordinates, in particular, on the geodesic distance from an origin or from an edge, we suggest a fast marching method for computing geodesic distance on manifolds. We apply our methods to generate surfaces of non-constant curvature and demonstrate how one can introduce branch points to account for the multi-generational buckling and subwrinkling observed in many applications.
