We review the existence results of traveling wave solutions to the reaction-diffusion equations with periodic diffusion (convection) coefficients and combustion (bistable) nonlinearities. We prove that whenever traveling waves exist, the solutions of the initial value problem with either frontlike or pulselike data propagate with the constant effective speeds of traveling waves in all suitable directions. In the case of bistable nonlinearity and one space dimension, we give an example of nonexistence of traveling waves which causes quenching (localization) of wavefront propagation. Quenching (localization) only occurs when the variations of the media from their constant mean values are large enough. Our related numerical results also provide evidence for this phenomenon in the parameter regimes not covered by the analytical example. Finally, we comment on the role of the effective wave speeds in

A collaborative convex framework for factoring a data matrix <i>X</i> into a nonnegative product <i>AS</i> , with a sparse coefficient matrix <i>S</i> , is proposed. We restrict the columns of the dictionary matrix <i>A</i> to coincide with certain columns of the data matrix <i>X</i> , thereby guaranteeing a physically meaningful dictionary and dimensionality reduction. We use <i>l</i> _1, regularization to select the dictionary from the data and show that this leads to an exact convex relaxation of <i>l</i> ₀ in the case of distinct noise-free data. We also show how to relax the restriction-to- <i>X</i> constraint by initializing an alternating minimization approach with the solution of the convex model, obtaining a dictionary close to but not necessarily in <i>X</i> . We focus on applications of the proposed framework to hyperspectral endmember and abundance identification and also show an application to blind source separation of nuclear magnetic resonance data.

A review is presented of recent results on front propagation in reaction-diffusion-advection equations in homogeneous and heterogeneous media. Formal asymptotic expansions and heuristic ideas are used to motivate the results wherever possible. The fronts include constant-speed monotone traveling fronts in homogeneous media, periodically varying traveling fronts in periodic media, and fluctuating and fractal fronts in random media. These fronts arise in a wide range of applications such as chemical kinetics, combustion, biology, transport in porous media, and industrial deposition processes. Open problems are briefly discussed along the way.

Among all the interesting spaces in topology, the spheres are no doubt beautiful objects and of most consideration since antiquity. Any nontrivial observations of them are of course very important. From the categorical point of view, we should not just focus on the objects themselves, but the morphisms between them as well. For this purpose, in algebraic topology, we do want to classify the set of continuous morphisms between spheres under the equivalent relation named homotopy, which describes a continuous deformation between two continuous maps. Let Sn be the n-sphere and k (X) be the set of homotopic equivalent based maps from Sk to X. For the reason that Sk is a double suspension when k 2, the set is actually an abelian group. A natural question is, what are these abelian groups?

We prove that the 61-sphere has a unique smooth structure. Following results of Moise [35], Kervaire-Milnor [25], Browder [10] and Hill-Hopkins-Ravenel [19], we show that the only odd dimensional spheres with a unique smooth structure are S1, S3, S5 and S61.