Gauss-Bonnet theorems in the affine group and the group of rigid motions of the Minkowski plane

In this paper, we compute sub-Riemannian limits of Gaussian curvature for a Euclidean C~2-smooth surface in the affine group and the group of rigid motions of the Minkowski plane away from characteristic points and signed geodesic curvature for Euclidean C~2-smooth curves on surfaces. We get Gauss-Bonnet theorems in the affine group and the group of rigid motions of the Minkowski plane.     

Polynomial affine algorithms for linear programming

The method of steepest descent with scaling (affine scaling) applied to the potential functionq logcx — _i=1 ⁿ logx _i solves the linear programming problem in polynomial time forq n. Ifq = n, then the algorithm terminates in no more than O(n ² L) iterations; if q n + withq = O(n) then it takes no more than O(nL) iterations. A modified algorithm using rank-1 updates for matrix inversions achieves respectively O(n ⁴ L) and O(n ^3.5 L) arithmetic computions.     

On the properly discontinuous subgroups of affine motions

The fundamental groupΓ of a compact complete affine manifold is represented as an affine crystallographic subgroup of Aff(n). L.S.Auslander conjectured thatΓ is virtually solvable. Our purpose is to find the algebraic condition onΓ which leads affirmative answer to the conjecture.     

Racah operators for the group of motions

The author [5] introduced Racah operators for unitary representations of topological groups of type I. In the present paper, we indicate the explicit form of these operators for representations of the group of motions of three-dimensional Euclidean space.     

Polynomial root radius optimization with affine constraints

The root radius of a polynomial is the maximum of the moduli of its roots (zeros). We consider the following optimization problem: minimize the root radius over monic polynomials of degree n, with either real or complex coefficients, subject to k linearly independent affine constraints on the coefficients. We show that there always exists an optimal polynomial with at most \(k-1\) inactive roots, that is, roots whose moduli are strictly less than the optimal root radius. We illustrate our results using some examples arising in feedback control.     

Roots of the affine Cremona group

Let k ^[n] = k[x ₁,…, x _n ] be the polynomial algebra in n variables and let \mathbbAⁿ = \textSpec  \boldk^{[ n ]} {\mathbb{A}^n} = {\text{Spec}}\;{{\bold{k}}^{\left[ n \right]}} . In this note we show that the root vectors of \textAu\textt^*( \mathbbAⁿ ) {\text{Au}}{{\text{t}}^*}\left( {{\mathbb{A}^n}} \right) , the subgroup of volume preserving automorphisms in the affine Cremona group \textAut( \mathbbAⁿ ) {\text{Aut}}\left( {{\mathbb{A}^n}} \right) , with respect to the diagonal torus are exactly the locally nilpotent derivations x ^α (∂/∂x _i ), where x ^α is any monomial not depending on x _i . This answers a question posed by Popov.     

Algebraic monoids with affine unit group are affine

In this short paper we prove that any irreducible algebraic monoid whose unit group is an affine algebraic group is affine.     

Mendelsohn designs admitting the affine group

All Mendelsohn designs containing a Frobenius group with cyclic complement of orderv – 1 as a subgroup of the automorphism are found. Furthermore, the automorphism group of each of the designs is constructed. These designs generalize Mendelsohn's construction of Mendelsohn designs containing a certain doubly transitive automorphism group.The research on this paper was partially supported by North Texas State Faculty Research Grant #35524.     

On affine motions and universal rigidity of tensegrity frameworks

Recently, Alfakih and Ye (2013) [4] proved that if an r  -dimensional bar framework (G,p)$(G,p)$ on n?r+2$n?r+2$ nodes in general position in R^r${\mathbb{R}}^r$ admits a positive semidefinite stress matrix with rank n−r−1$n-r-1$, then (G,p)$(G,p)$ is universally rigid. In this paper, we generalize this result in two directions. First, we extend this result to tensegrity frameworks. Second, we replace the general position assumption by the weaker assumption that in configuration p, each point and its neighbors in G   affinely span R^r${\mathbb{R}}^r$.     

Alternating group covers of the affine line

For an odd prime p t= 2 mod 3, we prove Abhyankar’s Inertia Conjecture for the alternating group A _p+2, by showing that every possible inertia group occurs for a (wildly ramified) A _p+2-Galois cover of the projective k-line branched only at infinity where k is an algebraically closed field of characteristic p > 0. More generally, when 2 ≤ s < p and gcd(p−1, s+1) = 1, we prove that all but finitely many rational numbers which satisfy the obvious necessary conditions occur as the upper jump in the filtration of higher ramification groups of an A _p+s -Galois cover of the projective line branched only at infinity.     

An estimate for a first-order Riesz operator on the affine group

On the affine group of the line, which is a solvable Lie group of exponential growth, we consider a right-invariant Laplacian . For a certain right-invariant vector field , we prove that the first-order Riesz operator is of weak type (1, 1) with respect to the left Haar measure of the group. This operator is therefore also bounded on . Locally, the operator is a standard singular integral. The main part of the proof therefore concerns the behaviour of the kernel of the operator at infinity and involves cancellation.

     

A family of affine Weyl group representations

In this paper we explicitly determine the virtual representations of the finite Weyl subgroups of the affine Weyl group on the cohomology of the space of affine flags containing a family of elementsn _t in an affine Lie algebra. We also compute the Euler characteristic of the space of partial flags containingn _t and give a connection with hyperplane arrangements.This paper forms part of my Ph.D. thesis at M.I.T. I was supported by an NSF Graduate Fellowship and NSF grant DMS 9304580.