We analyze a class of piecewise linear parabolic maps on the torus, namely those obtained by considering a linear map with double eigenvalue one and taking modulo one in each component. We show that within this two parameter family of maps, the set of noninvertible maps is open and dense. For cases where the entries in the matrix are rational we show that the maximal invariant set has positive ...

We study the normal forms for incompressible flows and maps in the neighborhood of an equilibrium or fixed point with a triple eigenvalue. We prove that when a divergence free vector field in R has nilpotent linearization with maximal Jordan block then, to arbitrary degree, coordinates can be chosen so that the nonlinear terms occur as a single function of two variables in the third component. ...

Abstract: The exact stability bound of the parasitic parameter for a discrete-time singularly perturbed system is determined by the linear fractional transformation (LFT) framework. Two systematic approaches including time-domain and frequency-domain methods are proposed to solve this problem based on the unified LFT framework. One employs the Kronecker product of LFTs and the guardian map theo...

Teo Banica Block-modified Wishart matrices and free Poisson laws We study the random matrices of type W̃ = (id ⊗ φ)W , where W is a complex Wishart matrix of parameters (dn, dm), and φ : Mn(C) → Mn(C) is a self-adjoint linear map. We prove that, under suitable assumptions, we have the d→∞ eigenvalue distribution formula δmW̃ ∼ πmnρ ν, where ρ is the law of φ, viewed as a square matrix, π is the f...

Let φ : X → X be a map on an projective variety. It is known that whenever φ∗ : Pic(X) ⊗ R → Pic(X) ⊗ R has an eigenvalue α > 1, we can build a canonical measure, a canonical height and a canonical metric associated to φ. In the present work, we establish the following fact: if two commuting maps φ, ψ : X → X satisfy these conditions, for eigenvalues α and β and the same eigenvector L, then the...

The A-polynomial was introduced in [3] (see also [5]), and we present an alternative definition here. Let M be a compact 3-manifold with boundary a torus T. Pick a basis λ,μ of π " T, which we shall refer to as the longitude and meridian. Consider the subset R U of the affine algebraic variety R ̄Hom(π " M, SL # #) having the property that ρ(λ) and ρ(μ) are upper triangular. This is an algebraic...

We analyze a class of piecewise linear parabolic maps on the torus, namely those obtained by considering a linear map with double eigenvalue one and taking modulo one in each component. We show that within this two parameter family of maps, the set of noninvertible maps is open and dense. For cases where the entries in the matrix are rational we show that the maximal invariant set has positive ...

Linear Discriminative Analysis techniques have been used in pattern recognition to map feature vectors to achieve optimal classification. Kernel Discriminative Analysis(KDA) seeks to introduce non-linearity in this approach by mapping the features to a non-linear space before applying LDA analysis. The formulation is expressed as an eigenvalue problem resolution. Using a different kernel, one c...

Estimating the effect of design changes on the stability of robots designed to run over rough terrain is a difficult task. No current predictor or measure is currently universally accepted. The most common techniques, including the ‘stability margins’ commonly used in walking machines and the return-map eigenvalue analysis used on simple models, are not fully applicable to these fast and comple...

Some properties that nominally involve the eigenvalues of Gaussian Unitary Ensemble (GUE) can instead be phrased in terms of singular values. By discarding the signs of the eigenvalues, we gain access to a surprising decomposition: the singular values of the GUE are distributed as the union of the singular values of two independent ensembles of Laguerre type. This independence is remarkable giv...