Related papers: Singular Vectors in Real Affine Subspaces

Suppose that $f$ defines a singular, complex affine hypersurface. If the critical locus of $f$ is one-dimensional at the origin, we obtain new general bounds on the ranks of the homology groups of the Milnor fiber, $F_{f, \mathbf 0}$, of…

We give general expressions for singular vectors of the N=2 superconformal algebra in the form of {\it monomials} in the continued operators by which the universal enveloping algebra of N=2 is extended. We then show how the algebraic…

An affine iterated function system is a finite collection of affine invertible contractions and the invariant set associated to the mappings is called self-affine. In 1988, Falconer proved that, for given matrices, the Hausdorff dimension…

In this paper we consider convex improper affine maps of the 3-dimensional affine space and classify their singularities. The main tool developed is a generating family with properties that closely resembles the area function for non-convex…

We show that a uniformly Euclidean metric with isolated singularity on $M^n = T^n \# M_0$, where $4\leq n\leq 7$ or $n\geq 4$, $M_0$ spin, and nonnegative scalar curvature on the smooth part is Ricci flat and extends smoothly over the…

We prove that every closed set which is not sigma-finite with respect to the Hausdorff measure H^{N-1} carries singularities of continuous vector fields in the Euclidean space R^N for the divergence operator. We also show that finite…

The set of f-vectors of pure simplicial complexes is an important but little understood object in combinatorics and combinatorial commutative algebra. Unfortunately, its explicit characterization appears to be a virtually intractable…

Positive semidefinite Hermitian matrices that are not fully specified can be completed provided their underlying graph is chordal. If the matrix is positive definite the completion can be uniquely characterized as the matrix that maximizes…

Let $r$ and $n$ be positive integers such that $r<n$, and $\mathbb{K}$ be an arbitrary field. We determine the maximal dimension for an affine subspace of $n$ by $n$ symmetric (or alternating) matrices with entries in $\mathbb{K}$ and with…

Let K be a (commutative) field with characteristic not 2, and V be a linear subspace of n by n matrices that have at most two eigenvalues in K (respectively, at most one non-zero eigenvalue in K). We prove that the dimension of V is less…

The behavior of the leading singular values and vectors of noisy low-rank matrices is fundamental to many statistical and scientific problems. Theoretical understanding currently derives from asymptotic analysis under one of two regimes:…

For every $n \in \mathbb{N}$ and every field $K$, let $N(n,K)$ be the set of the nilpotent $n \times n$ matrices over $K$ and let $D(n,K) $ be the set of the $n \times n$ matrices over $K$ which are diagonalizable over $K$. Moreover, let…

We prove that the singular set of a multiplicity $2$ integral hypercurrent that is stationary in the sense of varifolds has a singular set of measure zero.

Two-particle scattering amplitudes for integrable relativistic quantum field theory in 1+1 dimensions can normally have at most singularities of poles and zeros along the imaginary axis in the complex rapidity plane. It has been supposed…

We explore how the combinatorial arrangement of prescribed zeros in a matrix affects the possible eigenvalues that the matrix can obtain. We demonstrate that there are inertially arbitrary patterns having a digraph with no 2-cycle, unlike…

We prove that singular vectors have measure zero with respect to any friendly measure on $\Bbb R^n$ (e.g. the volume measure on a nondegenerate submanifold). This generalizes special cases considered by Davenport-Schmidt, Baker and Bugeaud.…

An $n\times n$ zero pattern $S$, which is a matrix with entries $*$ and $0$, is called spectrally arbitrary with respect to a field $F$ if any monic polynomial $f$ of degree $n$ can be realized as the characteristic polynomial of a matrix…

We give an explicit criterion for the irreducibility of some induction products of evaluation modules of affine Hecke algebras of type A. This allows to describe the form of the singularities of the trigonometric R-matrix associated to any…

Let K be an arbitrary (commutative) field and L be an algebraic closure of it. Let V be a linear subspace of M_n(K), with n>2. We show that if every matrix of V has at most one eigenvalue in K, then dim V<=1+n(n-1)/2. If every matrix of V…

Let $\mathcal{V} \subset M$ denote any of the varieties of singular $m \times m$ complex matrices which may be general, symmetric, or skew-symmetric ($m$ even), or $m \times p$ matrices, in the corresponding space $M$ of such matrices. A…