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We give a necessary and sufficient smoothness condition for the scheme parameterizing the n-dimensional representations of a finitely generated associative algebra over an algebraically closed field of characteristic zero. In particular,…
The purpose of this paper is to prove that the symbolic Rees rings of ideals defining certain finite sets of points in the projective plane over an algebraically closed field are finitely generated using a ring theoretical criterion which…
This paper studies finite projective dimension of finitely generated modules over a Noetherian local ring, by means of spectral sequence methods related to generalized local cohomology. Our main goal is to address a question raised by D.…
In this paper, we are concerned with the problem of counting the multiplicities of a zero-dimensional regular set's zeros. We generalize the squarefree decomposition of univariate polynomials to the so-called pseudo squarefree decomposition…
We propose a version of the classical shape lemma for zero-dimensional ideals of a commutative multivariate polynomial ring to the noncommutative setting of zero-dimensional ideals in an algebra of differential operators.
Spherical Designs are finite sets of points on the sphere $\mathbb{S}^{d}$ with the property that the average of certain (low-degree) polynomials in these points coincides with the global average of the polynomial on $\mathbb{S}^{d}$. They…
Measures generated by Iterated Function Systems composed of uncountably many one--dimensional affine maps are studied. We present numerical techniques as well as rigorous results that establish whether these measures are absolutely or…
We construct an example of an $A_{\infty}$ algebra structure defined over a finite dimensional graded vector space.
A symmetric ideal is an ideal in a polynomial ring which is stable under all permutations of the variables. In this paper we initiate a global study of zero-dimensional symmetric ideals. By this we mean a geometric study of the invariant…
In this article ideas from Kit Fine's theory of arbitrary objects are applied to questions regarding mathematical structuralism. I discuss how sui generic mathematical structures can be viewed as generic systems of mathematical objects,…
The numerical range of a matrix is studied geometrically via the cone of positive semidefinite matrices (or semidefinite cone for short). In particular it is shown that the feasible set of a two-dimensional linear matrix inequality (LMI),…
The numerical range of a matrix is studied geometrically via the cone of positive semidefinite matrices (or semidefinite cone for short). In particular it is shown that the feasible set of a two-dimensional linear matrix inequality (LMI),…
A border basis scheme is an affine scheme that can be viewed as an open subscheme of the Hilbert scheme of \mu points of affine n-space. We study syzygies of the generators of a border basis scheme's defining ideal. These generators arise…
We define standardized constructions of finite fields, and standardized generators of (multiplicative) cyclic subgroups in these fields. The motivation is to provide a substitute for Conway polynomials which can be used by various software…
Our main aim in this paper is to introduce a general concept of multidimensional fixed point of a mapping in spaces with distance and establish various multidimensional fixed point results. This new concept simplifies the similar notion…
Definition. A symmetric with respect to 0 bounded closed convex set A in a finite dimensional normed space X is called a sufficient enlargement for X (or of B(X)) if for arbitrary isometric embedding of X into a Banach space Y there exists…
We describe all Mathieu-Zhao spaces of $k[x_1,\cdots,x_n]$ ($k$ is an algebraically closed field of characteristic zero) which contains an ideal of finite codimension. Furthermore we give an algorithm to decide if a subspace of the form…
We compare three notions of genericity of separable metric structures. Our analysis provides a general model theoretic technique of showing that structures are generic in descriptive set theoretic (topological) sense and in measure…
We characterize countable dimensionality and strong countable dimensionality by means of an infinite game.
In this paper the notion of Dirac structure in finite dimension is extended to the convenient setting. In particular, we introduce the notion of partial Dirac structure on convenient Lie algebroids and manifolds. We then look for those…