Cyclic decomposition of unitary spaces

If π is a unitary transformation that is similar to its inverse, then the vector space V is an orthogonal sum of regular π-cyclic and π-bicyclic subspaces.     

Canonical connections with an algebraic curvature tensor field on naturally reductive spaces

We study Ambrose-Singer connections with an algebraic curvature tensor on simply connected manifolds carrying a homogeneous Riemannian structure of class ₃ in the classification given by F. Tricerri and L. Vanhecke.This work was partially supported by MURST.     

Orthogonal bases of symmetrized tensor spaces

It is shown that a symmetrized tensor space does not have an orthogonal basis consisting of standard symmetrized tensors if the associated permutation group is 2-transitive. In particular, no such basis exists if the group is the symmetric group or the algernating group as conjectured by T.-Y. Tam and the author.     

Some algebraic aspects of mesoprimary decomposition

Recent results of Kahle and Miller give a method of constructing primary decompositions of binomial ideals by first constructing “mesoprimary decompositions” determined by their underlying monoid congruences. Monoid congruences (and therefore, binomial ideals) can present many subtle behaviors that must be carefully accounted for in order to produce general results, and this makes the theory complicated. In this paper, we examine their results in the presence of a positive A-grading, where certain pathologies are avoided and the theory becomes more accessible. Our approach is algebraic: while key notions for mesoprimary decomposition are developed first from a combinatorial point of view, here we state definitions and results in algebraic terms, which are moreover significantly simplified due to our (slightly) restricted setting. In the case of toral components (which are well-behaved with respect to the A-grading), we are able to obtain further simplifications under additional assumptions. We also provide counterexamples to two open questions, identifying (i) a binomial ideal whose hull is not binomial, answering a question of Eisenbud and Sturmfels, and (ii) a binomial ideal I for which $I_{\mathrm{toral}}$ is not binomial, answering a question of Dickenstein, Miller and the first author.     

酉延拓矩阵的奇异值分解及其广义逆

从普通奇异值分解出发,导出了酉延拓矩阵的奇异值和奇异向量与母矩阵的奇异值和奇异向量间的定量关系,同时对酉延拓矩阵的满秩分解及g逆,反射g逆,最小二乘g逆,最小范数g逆作了定量分析,得到了酉延拓矩阵的满秩分解矩阵F*和G*与母矩阵A的分解矩阵F和G之间的关系.最后给出了相应的快速求解算法,并举例说明该算法大大降低了分解的计算量和存储量,提高了计算效率.     

Dimension of the space of unitary equivariant translation invariant tensor valuations

Following the work of Semyon Alesker in the scalar valued case and of Thomas Wannerer in the vector valued case, the dimensions of the spaces of continuous translation invariant and unitary equivariant tensor valuations are computed. In addition, a basis in the vector valued case is presented.     

Relation between algebraic and geometric view on nurbs tensor product surfaces

NURBS (Non-Uniform Rational B-Splines) belong to special approximation curves and surfaces which are described by control points with weights and B-spline basis functions. They are often used in modern areas of computer graphics as free-form modelling, modelling of processes. In literature, NURBS surfaces are often called tensor product surfaces. In this article we try to explain the relationship between the classic algebraic point of view and the practical geometrical application on NURBS.     

Cell decomposition of almost smooth real algebraic surfaces

Let Z be a two dimensional irreducible complex component of the solution set of a system of polynomial equations with real coefficients in N complex variables. This work presents a new numerical algorithm, based on homotopy continuation methods, that begins with a numerical witness set for Z and produces a decomposition into 2-cells of any almost smooth real algebraic surface contained in Z. Each 2-cell (a face) has a generic interior point and a boundary consisting of 1-cells (edges). Similarly, the 1-cells have a generic interior point and a vertex at each end. Each 1-cell and each 2-cell has an associated homotopy for moving the generic interior point to any other point in the interior of the cell, defining an invertible map from the parameter space of the homotopy to the cell. This work draws on previous results for the curve case. Once the cell decomposition is in hand, one can sample the 2-cells and 1-cells to any resolution, limited only by the computational resources available.     

Computing the equidimensional decomposition of an algebraic closed set by means of lifting fibers

We present a new probabilistic method for solving systems of polynomial equations and inequations. Our algorithm computes the equidimensional decomposition of the Zariski closure of the solution set of such systems. Each equidimensional component is encoded by a generic fiber, that is a finite set of points obtained from the intersection of the component with a generic transverse affine subspace. Our algorithm is incremental in the number of equations to be solved. Its complexity is mainly cubic in the maximum of the degrees of the solution sets of the intermediate systems counting multiplicities.Our method is designed for coefficient fields having characteristic zero or big enough with respect to the number of solutions. If the base field is the field of the rational numbers then the resolution is first performed modulo a random prime number after we have applied a random change of coordinates. Then we search for coordinates with small integers and lift the solutions up to the rational numbers. Our implementation is available within our package Kronecker from version 0.166, which is written in the Magma computer algebra system.     

Degenerate orbits of adjoint representation of orthogonal and unitary groups regarded as algebraic submanifolds

We suggest a method for describing some types of degenerate orbits of orthogonal and unitary groups in the corresponding Lie algebras as level surfaces of a special collection of polynomial functions. This method allows one to describe orbits of the types SO(2n)/SO(2k)×SO(2) ^n?k , SO(2n+1)/SO(2k+1)×SO(2) ^n?k , and (S)U(n)/(S)(U(2k)×U(2) ^n?k ) in so(2n), so(2n+1), and (s)u(n), respectively. In addition, we show that the orbits of minimal dimensions of the groups under consideration can be described in the corresponding algebras as intersections of quadries. In particular, this approach is used for describing the orbit CP ^n?1?u(n).     

DC decomposition of nonconvex polynomials with algebraic techniques

We consider the problem of decomposing a multivariate polynomial as the difference of two convex polynomials. We introduce algebraic techniques which reduce this task to linear, second order cone, and semidefinite programming. This allows us to optimize over subsets of valid difference of convex decompositions (dcds) and find ones that speed up the convex–concave procedure. We prove, however, that optimizing over the entire set of dcds is NP-hard.     

On decomposition theory for unitary representations of locally compact groups

The main purpose of this note is to give a simple example of a unitary representation of a (discrete) group, which cannot be decomposed as a topological direct integral of factor representations. But it is shown that the regular representation of any discrete group is a direct integral (on a compact space) of factor representations.