The linear preservers of real diagonalizable matrices

Using a recent result of Bogdanov and Guterman on the linear preservers of pairs of simultaneously diagonalizable matrices, we determine all the automorphisms of the vector space M_n(R) which stabilize the set of diagonalizable matrices. To do so, we investigate the structure of linear subspaces of diagonalizable matrices of M_n(R) with maximal dimension.Comment: 14 page

In the Honour of Tristram Engelhardt, Jr.: On the Sources of the Narrative Self

Modern philosophy is based on the presupposition of the certainty of the ego’s experience. Both Descartes and Kant assume this certitude as the basis for certain knowledge. Here the argument is developed that this ego has its sources not only in Scholastic philosophy, but also in the narrative of the emotional self as developed by both the troubadours and the medieval mystics. This narrative self has three moments: salvation, self-irony, and nostalgia. While salvation is rooted in the Christian tradition, self-irony and nostalgia are first addressed in twelfth-century troubadour poetry in Occitania. Their integration into a narrative self was developed in late medieval mysticism, and reached its fullest articulation in St. Teresa of Avila, whom Descartes read

Neighborliness of the symmetric moment curve

We consider the convex hull B_k of the symmetric moment curve U(t)=(cos t, sin t, cos 3t, sin 3t, ..., cos (2k-1)t, sin (2k-1)t) in R^{2k}, where t ranges over the unit circle S= R/2pi Z. The curve U(t) is locally neighborly: as long as t_1, ..., t_k lie in an open arc of S of a certain length phi_k>0, the convex hull of the points U(t_1), ..., U(t_k) is a face of B_k. We characterize the maximum possible length phi_k, proving, in particular, that phi_k > pi/2 for all k and that the limit of phi_k is pi/2 as k grows. This allows us to construct centrally symmetric polytopes with a record number of faces.Comment: 28 pages, proofs are simplified and results are strengthened somewha

Numerical radius and zero pattern of matrices

We give tight upper bounds on the numerical range of square matrices in terms of their Frobenius (Euclidian) norm and a combinatorial parameter similar to the clique number of graphs. Our results imply a concise form of the fundamental theorem of Turan in extremal graph theory

Cooperative classes of finite sets in one and more dimensions

AbstractFor x1<x2<x3 let σ=(x1−x2)/(x2−x3), ρ=max(σ, 1/σ). For α>1, the class Cα of 3-sets with ρ<α is non-cooperative: for n≥α log 4+2, no n-set has all its 3-subsets in Cα. For n≥the Ramsey number N([α log 4+3], q, q; 3), every n-set has a q-subset none of whose 3-subsets is in Cα. Likewise for q>d≥1, ∈>0, α>1 there is a bound B such that every general n-ad, n≥B, in d-space has a comonotone q-sub-ad whose 2-sub-ads form with a certain ray R an angle<∈ and whose 3-sub-ads, orthogonally projected on R, are not in Cα. Finally, for every general N(7, 7;4)-set S in 3-space there exists a knotted polygon whose vertices belong to S

Moduli spaces of framed perverse instantons on P^3

We study moduli spaces of framed perverse instantons on P^3. As an open subset it contains the (set-theoretical) moduli space of framed instantons studied by I. Frenkel and M. Jardim. We also construct a few counterexamples to earlier conjectures and results concerning these moduli spaces.Comment: 50 page

Convex Hull of Planar H-Polyhedra

Suppose are planar (convex) H-polyhedra, that is, $A_i \in \mathbb{R}^{n_i \times 2}$ and $\vec{c}_i \in \mathbb{R}^{n_i}$. Let $P_i = \{\vec{x} \in \mathbb{R}^2 \mid A_i\vec{x} \leq \vec{c}_i \}$ and $n = n_1 + n_2$. We present an $O(n \log n)$ algorithm for calculating an H-polyhedron with the smallest $P = \{\vec{x} \in \mathbb{R}^2 \mid A\vec{x} \leq \vec{c} \}$ such that $P_1 \cup P_2 \subseteq P$

A Complete Characterization of the Gap between Convexity and SOS-Convexity

Our first contribution in this paper is to prove that three natural sum of squares (sos) based sufficient conditions for convexity of polynomials, via the definition of convexity, its first order characterization, and its second order characterization, are equivalent. These three equivalent algebraic conditions, henceforth referred to as sos-convexity, can be checked by semidefinite programming whereas deciding convexity is NP-hard. If we denote the set of convex and sos-convex polynomials in $n$ variables of degree $d$ with $\tilde{C}_{n,d}$ and $\tilde{\Sigma C}_{n,d}$ respectively, then our main contribution is to prove that $\tilde{C}_{n,d}=\tilde{\Sigma C}_{n,d}$ if and only if $n=1$ or $d=2$ or $(n,d)=(2,4)$. We also present a complete characterization for forms (homogeneous polynomials) except for the case $(n,d)=(3,4)$ which is joint work with G. Blekherman and is to be published elsewhere. Our result states that the set $C_{n,d}$ of convex forms in $n$ variables of degree $d$ equals the set $\Sigma C_{n,d}$ of sos-convex forms if and only if $n=2$ or $d=2$ or $(n,d)=(3,4)$. To prove these results, we present in particular explicit examples of polynomials in $\tilde{C}_{2,6}\setminus\tilde{\Sigma C}_{2,6}$ and $\tilde{C}_{3,4}\setminus\tilde{\Sigma C}_{3,4}$ and forms in $C_{3,6}\setminus\Sigma C_{3,6}$ and $C_{4,4}\setminus\Sigma C_{4,4}$, and a general procedure for constructing forms in $C_{n,d+2}\setminus\Sigma C_{n,d+2}$ from nonnegative but not sos forms in $n$ variables and degree $d$. Although for disparate reasons, the remarkable outcome is that convex polynomials (resp. forms) are sos-convex exactly in cases where nonnegative polynomials (resp. forms) are sums of squares, as characterized by Hilbert.Comment: 25 pages; minor editorial revisions made; formal certificates for computer assisted proofs of the paper added to arXi

Converting between quadrilateral and standard solution sets in normal surface theory

The enumeration of normal surfaces is a crucial but very slow operation in algorithmic 3-manifold topology. At the heart of this operation is a polytope vertex enumeration in a high-dimensional space (standard coordinates). Tollefson's Q-theory speeds up this operation by using a much smaller space (quadrilateral coordinates), at the cost of a reduced solution set that might not always be sufficient for our needs. In this paper we present algorithms for converting between solution sets in quadrilateral and standard coordinates. As a consequence we obtain a new algorithm for enumerating all standard vertex normal surfaces, yielding both the speed of quadrilateral coordinates and the wider applicability of standard coordinates. Experimentation with the software package Regina shows this new algorithm to be extremely fast in practice, improving speed for large cases by factors from thousands up to millions.Comment: 55 pages, 10 figures; v2: minor fixes only, plus a reformat for the journal styl