We use adiabatic limits to study foliated manifolds. The Bott connection naturally shows up as the adiabatic limit of Levi-Civita connections. As an application, we then construct certain natural elliptic operators associated to the foliation and present a direct geometric proof of a vanshing theorem of Connes [Co], which extends the Lichnerowicz vanishing theorem [L] to foliated manifolds with spin leaves, for what we call almost Riemannian foliations. Several new vanishing theorems are also proved by using our method.

We propose an approach to find constant curvature metrics on triangulated closed 3-manifolds using a finite dimensional variational method whose energy function is the volume. The concept of an angle structure on a tetrahedron and on a triangulated closed 3-manifold is introduced following the work of Casson, Murakami and Rivin. It is proved by A. Kitaev and the author that any closed 3-manifold has a triangulation supporting an angle structure. The moduli space of all angle structures on a triangulated 3-manifold is a bounded open convex polytope in a Euclidean space. The volume of an angle structure is defined. Both the angle structure and the volume are natural generalizations of tetrahedra in the constant sectional curvature spaces and their volume. It is shown that the volume functional can be extended continuously to the compact closure of the moduli space. In particular, the maximum point of the

We study the stability and oscillation of traveling fronts in a three-component, advection-reaction biodegradation model. The three components are pollutant, nutrient, and bacteria concentrations. Under an explicit condition on the biomass growth and decay coefficients, we derive reduced, two-component, semilinear hyperbolic models through a relaxation procedure, during which biomass is slaved to pollutant and nutrient concentration variables. The reduced two-component models resemble the Broadwell model of the discrete velocity gas. The traveling fronts of the reduced system are explicit and are expressed in terms of hyperbolic tangent function in the nutrient-deficient regime. We perform energy estimates to prove the asymptotic stability of these fronts under explicit conditions on the coefficients in the system. In the small damping limit, we carry out Wentzel--Kramers--Brillouin (WKB) analysis on front

A recently proposed class of multivariate Public-Key Cryptosystems, the Rainbow-Like Digital Signature Schemes, in which successive sets of central variables are obtained from previous ones by solving linear equations, seem to lead to efficient schemes (TTS, TRMS, and Rainbow) that perform well on systems of low computational resources. Recently SFLASH (C^{∗−}) was broken by Dubois, Fouque, Shamir, and Stern via a differential attack. In this paper, we exhibit similar algebraic and diffential attacks, that will reduce published Rainbow-like schemes below their security levels. We will also discuss how parameters for Rainbow and TTS schemes should be chosen for practical applications.

In this paper we study the support recovery problem for single index models Y=f(X⊺β,ε), where f is an unknown link function, X∼Np(0,𝕀p) and β is an s-sparse unit vector such that βi∈{±1s√,0}. In particular, we look into the performance of two computationally inexpensive algorithms: (a) the diagonal thresholding sliced inverse regression (DT-SIR) introduced by Lin et al. (2015); and (b) a semi-definite programming (SDP) approach inspired by Amini & Wainwright (2008). When s=O(p1−δ) for some δ>0, we demonstrate that both procedures can succeed in recovering the support of β as long as the rescaled sample size κ=nslog(p−s) is larger than a certain critical threshold. On the other hand, when κ is smaller than a critical value, any algorithm fails to recover the support with probability at least 12 asymptotically. In other words, we demonstrate that both DT-SIR and the SDP approach are optimal (up to a scalar) for recovering the support of β in terms of sample size. We provide extensive simulations, as well as a real dataset application to help verify our theoretical observations.