MathSciDoc: An Archive for Mathematician ∫

G-equations are popular front propagation models in combustion literature and describe the front motion law of normal velocity equal to a constant plus the normal projection of fluid velocity. G-equations are Hamilton-Jacobi equations with convex but non-coercive Hamiltonians. We prove homogenization of the inviscid G-equation for space periodic incompressible flows. This extends a two space dimensional result in" Periodic homogenization of G-equations and viscosity effects," Nonlinearity, to appear. We construct approximate correctors to bypass the lack of compactness due to the non-coercive Hamiltonian. The existence of approximate correctors rely on a local reachability property of the controlled flow trajectory as well as incompressibility of the flow. Homogenization then follows from the comparison principle and the perturbed test function method. The effective Hamiltonian is convex and homogeneous of

We study the minimization problem of a non-convex sparsity promoting penalty function, the transformed l_1 (TL1), and its application in compressed sensing (CS). The TL1 penalty interpolates l_1 and l_1 norms through a nonnegative parameter l_1 , similar to l_1 with l_1 , and is known to satisfy unbiasedness, sparsity and Lipschitz continuity properties. We first consider the constrained minimization problem and discuss the exact recovery of l_1 norm minimal solution based on the null space property (NSP). We then prove the stable recovery of l_1 norm minimal solution if the sensing matrix l_1 satisfies a restricted isometry property (RIP). Next, we present difference of convex algorithms for TL1 (DCATL1) in computing TL1-regularized constrained and unconstrained problems in CS. The inner loop concerns an l_1 minimization problem on which we employ the Alternating Direction Method of Multipliers (ADMM). For the unconstrained problem, we prove convergence of DCATL1 to a stationary point satisfying the first order optimality condition. In numerical experiments, we identify the optimal value l_1 , and compare DCATL1 with other CS algorithms on two classes of sensing matrices: Gaussian random matrices and over-sampled discrete cosine transform matrices (DCT). We find that for both classes of sensing matrices, the performance of DCATL1 algorithm (initiated with l_1 minimization) always ranks near the top (if not the top), and is the most robust choice insensitive to the conditioning of the sensing matrix l_1 . DCATL1 is also competitive in comparison with DCA on other non-convex penalty functions commonly used in statistics with two

Feedback modules, which appear ubiquitously in biological regulations, are often subject to disturbances from the input, leading to fluctuations in the output. Thus, the question becomes how a feedback system can produce a faithful response with a noisy input. We employed multiple time scale analysis, Fluctuation Dissipation Theorem, linear stability, and numerical simulations to investigate a module with one positive feedback loop driven by an external stimulus, and we obtained a critical quantity in noise attenuation, termed as signed activation time. We then studied the signed activation time for a system of two positive feedback loops, a system of one positive feedback loop and one negative feedback loop, and six other existing biological models consisting of multiple components along with positive and negative feedback loops. An inverse relationship is found between the noise amplification rate and the signed activation time, defined as the difference between the deactivation and activation time scales of the noise-free system, normalized by the frequency of noises presented in the input. Thus, the combination of fast activation and slow deactivation provides the best noise attenuation, and it can be attained in a single positive feedback loop system. An additional positive feedback loop often leads to a marked decrease in activation time, decrease or slight increase of deactivation time and allows larger kinetic rate variations for slow deactivation and fast activation. On the other hand, a negative feedback loop may increase the activation and deactivation times. The negative relationship between the noise amplification rate and the signed

We study the two-point correlation function of a freely decaying scalar in Kraichnan's model of advection by a Gaussian random velocity field that is stationary and white noise in time, but fractional Brownian in space with roughness exponent 0&lt;<i></i>&lt;2, appropriate to the inertial-convective range of the scalar. We find all self-similar solutions by transforming the scaling equation to Kummer's equation. It is shown that only those scaling solutions with scalar energy decay exponent <i>a</i>(<i>d</i>/<i></i>)+1 are statistically realizable, where <i>d</i> is space dimension and <i></i>=2<i></i>. An infinite sequence of invariants <i>J</i> _p, <i>p</i>=0,1,2,..., is pointed out, where <i>J</i> ₀ is Corrsin's integral invariant but the higher invariants appear to be new. We show that at least one of the invariants <i>J</i> ₀ or <i>J</i> ₁ must be nonzero (possibly infinite) for realizable initial data. Initial datum with a finite, nonzero invariantthe first being <i>J</i> _pconverges at long times to a scaling

Diffusion-generated motion by mean curvature is a simple algorithm for producing motion by mean curvature of a surface, in which the motion is generated by alternately diffusing and renormalizing a characteristic function. In this paper, we generalize diffusion-generated motion to a procedure that can be applied to the curvature motion of filaments, i.e., curves in <i>R</i> ^3, that may initially consist of a complex configuration of links. The method consists of applying diffusion to a complex-valued function whose values wind around the filament, followed by normalization. We motivate this approach by considering the essential features of the complex Ginzburg-Landau equation, which is a reaction-diffusion PDE that describes the formation and propagation of filamentary structures. The new algorithm naturally captures topological merging and breaking of filaments without fattening curves. We justify the new

We study the front dynamics of the bistable reaction-diffusion equations with periodic diffusion and/or convection coefficients in several space dimensions. When traveling wave solutions exist, the solutions of the initial value problem behave as wave fronts propagating with the effective speeds of traveling waves under various initial conditions. Yet due to the bistable nature of the nonlinearity, traveling waves may not always exist when the medium variations from the mean states are large enough. Their existence is closely related to the detailed forms of diffusion and convection coefficients, more so in multidimension than in one. We present a simple sufficient condition for the nonexistence of traveling waves (quenching) using perturbation method. Our two dimensional finite difference numerical computations show a variety of front behaviors, such as: the propagation, quenching and retreat of fronts. We found

Abstract We study the GinzburgLandau equation on the plane with initial data being the product of n wellseparated+ 1 vortices and spatially decaying perturbations. If the separation distances are O ( 1), l, we prove that the n vortices do not move on the time scale O(^2),=o(1\over); instead, they move on the time scale O(^-21\over) according to the law j= xj W, W= l j log| xl xj|, xj=(j, j) 2, the location of the jth vortex. The main ingredients of our proof consist of estimating the large space behavior of solutions, a monotonicity inequality for the energy density of solutions, and energy comparisons. Combining these, we overcome the infinite energy difficulty of the planar vortices to establish the dynamical law. John &amp; Wiley Sons, Inc.

We study sparse blind source separation (BSS) for a class of positive and partially overlapped signals. The signals are only allowed to have nonoverlapping at certain locations, while they could overlap with each other elsewhere. For nonnegative data, a novel approach has been proposed by Naanaa and Nuzillard (NN) assuming that nonoverlapping exists for each source signal at some location of acquisition variable. However, the NN method introduces errors (spurious peaks) in the output when their nonoverlapping condition is not satisfied. To resolve this problem and improve robustness of separation, postprocessing techniques are developed in two aspects. One is to detect coherent and uncertain components from NN outputs by using multiple mixture data, then removing the uncertain portion to enhance signals. The other is to find better estimation of mixing matrix by leveraging reliable source peak

We prove the existence of global solutions to a coupled system of NavierStokes, and reaction-diffusion equations (for temperature and mass fraction) with prescribed front data on an infinite vertical strip or tube. This system models a one-step exothermic chemical reaction. The heat release induced volume expansion is accounted for via the Boussinesq approximation. The solutions are time dependent moving fronts in the presence of fluid convection. In the general setting, the fronts are subject to intensive Rayleigh-Taylor and thermal-diffusive instabilities. Various physical quantities, such as fluid velocity, temperature, and front speed, can grow in time. We show that the growth is at most for large time <i>t</i> by constructing a nonlinear functional on the temperature and mass fraction components. These results hold for arbitrary order reactions in two space dimensions and for quadratic and cubic reactions in

We establish the variational principle of Kolmogorov-Petrovsky-Piskunov (KPP) front speeds in temporally random shear flows with sufficiently decaying correlations. A key quantity in the variational principle is the almost sure Lyapunov exponent of a heat operator with random potential. To prove the variational principle, we use the comparison principle of solutions, the path integral representation of solutions, and large deviation estimates of the associated stochastic flows. The variational principle then allows us to analytically bound the front speeds. The speed bounds imply the linear growth law in the regime of large root mean square shear amplitude at any fixed temporal correlation length, and the sublinear growth law if the temporal decorrelation is also large enough, the so-called bending phenomenon.