Decay Conditions Research Articles

Uniform truncation error for Shannon sampling expansion from local averages

Let \(B_\Omega ^p \), 1 ≤ p ≤ ∞, be the set of all bounded functions in Lp(ℝ) which can be extended to entire functions of exponential type Ω. The uniform bounds for truncation error of Shannon sampling expansion from local averages are obtained for functions \(L^p (\mathbb{R})\) with the decay condition $$\left| {f(t)} \right| \leqslant \frac{A} {{\left| t \right|^\delta }},t \ne 0, $$ where A and δ are positive constants. Furthermore we also establish similar results for non-bandlimit functions in Besov classes with the same decay condition as above.

Remark on Luo-Hou’s Ansatz for a Self-similar Solution to the 3D Euler Equations

In this note we show that Luo-Hou's ansatz for the self-similar solution to the axisymmetric solution to the 3D Euler equations leads to triviality of the solution under suitable decay condition of the blow-up profile. The equations for the blow-up profile reduces to an over-determined system of partial differential equations, whose only solution with decay is the trivial solution. We also propose a generalization of Luo-Hou's ansatz. Using the vanishing of the normal velocity at the boundary, we show that this generalized self-similar ansatz also leads to a trivial solution. These results show that the self-similar ansatz may be valid either only in a time-dependent region which shrinks to the boundary circle at the self-similar rate, or under different boundary conditions at spatial infinity of the self-similar profile.

Complete non-compact gradient Ricci solitons with nonnegative Ricci curvature

In this paper, we give a rigidity theorem for a complete non-compact expanding (or steady) Ricci soliton with nonnegative Ricci curvature and certain scalar curvature decay condition. As an application, we prove that any complete non-compact expanding (or steady) Kähler-Ricci solitons with positively pinched Ricci curvature should be Ricci flat. The result answers a question proposed by Chow, Lu and Ni in case of Kähler-Ricci solitons.

A nodal solution of the scalar field equation at the second minimax level

We prove the existence of a sign-changing eigenfunction at the second minimax level of the eigenvalue problem for the scalar field equation under a slow decay condition on the potential near infinity. The proof involves constructing a set consisting of sign-changing functions that is dual to the second minimax class. We also obtain a nonradial sign-changing eigenfunction at this level when the potential is radial.

A multiplicity result for the scalar field equation

Abstract We prove the existence of N - 1 distinct pairs of nontrivial solutions of the scalar field equation in ℝ N under a slow decay condition on the potential near infinity, without any symmetry assumptions. Our result gives more solutions than the existing results in the literature when N ≥ 6. When the ground state is the only positive solution, we also obtain the stronger result that at least N - 1 of the first N minimax levels are critical, i.e., we locate our solutions on particular energy levels with variational characterizations. Finally we prove a symmetry breaking result when the potential is radial. To overcome the difficulties arising from the lack of compactness we use the concentration compactness principle of Lions, expressed as a suitable profile decomposition for critical sequences.

Weighted estimates for multilinear pseudodifferential operators

In this paper, we study the weighted estimates for multilinear pseudodifferential operators. We show that a multilinear pseudodifferential operator is bounded with respect to multiple weights whenever its symbol satisfies some smoothness and decay conditions. Our result generalizes similar ones from the classical Ap weights to multiple weights.

Spectral problems and scattering on noncompact star-shaped graphs containing finite rays

Abstract We consider Schrödinger operators on noncompact star-shaped graphs. The following topics will be treated under suitable decay conditions on the potential: characterization of the set of eigenvalues and expression of each eigen-projection; low-energy behavior of the resolvent, spectral representations of the operator restricted on infinite rays; determination of the Marchenko equations and the inverse scattering problems.

Euler’s equations and the maximum principle

In this paper we use maximum principle in the far field region for the time dependent self-similar Euler equations to exclude discretely self-similar blow-up for the Euler equations of the incompressible fluid flows. Our decay conditions near spatial infinity of the blow-up profile are given explicitly in terms the coefficient in the equations. We also deduce triviality of the discretely self-similar solution to the magnetohydrodynamic system, under suitable decay conditions near spatial infinity than the previous one. Applying similar argument directly to the Euler equations, we obtain a priori estimate of the vorticity in the far field region.

Gehring’s Lemma and Reverse Hölder Classes on Metric Measure Spaces

This work extends some Euclidean results on the structure of the reverse Holder classes to metric measure spaces with a doubling measure satisfying an annular decay condition. We also give an alternative proof of the celebrated Gehring’s lemma. Muckenhoupt weights are used extensively throughout the work.

Optimal convergence rates of high order Parzen windows with unbounded sampling

Abstract High order Parzen windows are considered with data drawn from unbounded sampling processes. Convergence analysis is established by imposing a moment hypothesis on the unbounded sampling outputs and some decay conditions on the marginal distributions. Our estimate of convergence rate is consistent with the previous result with bounded sampling.

삶은 달걀의 부패에 따른 부위별 냄새물질의 발생특성 연구

In this study, the concentration of odorants released from albumin (EA) and yolk (EY) portions of boiled egg samples were determined as a function of storage time. The concentrations were measured at storage days of 0, 1, 3, 6, and 9 under room temperature. As such, odorants produced during both fresh and decay conditions were measured through time. A total of 19 compounds were selected as the main target odorants along with 12 reference compounds. GC-MS (for VOC) and GC-PFPD system (for sulfur gases) equipped with thermal desorption (TD) system were employed for odorant analysis in this work. The initial concentrations measured from the chamber system were converted into flux terms (ng∙g∙min). The EA showed the highest concentration of H2S (234 ng∙g ∙min) at EA-0, and the concentrations of AT (Acetone) was also seen clearly in the range of 11.7 (EA-0) to 58.6 ng∙g∙ min (EA-9). The EY showed similar patterns. EtAl (Ethyl alcohol) increased 9.47 (EA-1) to 96.7 ng∙g∙min (EA-9) in EA samples. Ketone, alcohol, sulfur groups generally exhibited high concentrations compared to other odorants. These data were also compared in relation to olfactometry related dilution-to-threshold (D/T) ratio by air dilution sensory (ADS) test and sum of odor intensity (SOI).

Stability of entire solutions to supercritical elliptic problems involving advection

We examine the equation given by (1)−Δu+a(x)⋅∇u=upin RN, where p>1 and a(x) is a smooth vector field satisfying some decay conditions. We show that for p<pc, the Joseph–Lundgren exponent, there is no positive stable solution of (1) provided one imposes a smallness condition on a along with a divergence free condition. In the other direction we show that for N≥4 and p>N−1N−3 there exists a positive solution of (1) provided a satisfies a smallness condition. For p>pc we show the existence of a positive stable solution of (1) provided a is divergence free and satisfies a smallness condition.

A conservative spectral method for the Boltzmann equation with anisotropic scattering and the grazing collisions limit

We present the formulation of a conservative spectral method for the Boltzmann collision operator with anisotropic scattering cross-sections. The method is an extension of the conservative spectral method of Gamba and Tharkabhushanam [17,18], which uses the weak form of the collision operator to represent the collisional term as a weighted convolution in Fourier space. The method is tested by computing the collision operator with a suitably cut-off angular cross section and comparing the results with the solution of the Landau equation. We analytically study the convergence rate of the Fourier transformed Boltzmann collision operator in the grazing collisions limit to the Fourier transformed Landau collision operator under the assumption of some regularity and decay conditions of the solution to the Boltzmann equation. Our results show that the angular singularity which corresponds to the Rutherford scattering cross section is the critical singularity for which a grazing collision limit exists for the Boltzmann operator. Additionally, we numerically study the differences between homogeneous solutions of the Boltzmann equation with the Rutherford scattering cross section and an artificial cross section, which give convergence to solutions of the Landau equation at different asymptotic rates. We numerically show the rate of the approximation as well as the consequences for the rate of entropy decay for homogeneous solutions of the Boltzmann equation and Landau equation.

Supersymmetric rigidity of asymptotically locally hyperbolic manifolds

Let (M, g) be an Asymptotically Locally Hyperbolic (ALH) manifold which is the interior of a conformally compact manifold and (∂M, [γ]) its conformal infinity. Suppose that the Ricci tensor of (M, g) dominates that of the hyperbolic space and that its scalar curvature satisfies a certain decay condition at infinity. If the Yamabe invariant of (∂M, [γ]) is non-negative, we prove that there exists a conformal metric on M with non-negative scalar curvature and whose boundary ∂M has either positive or zero constant inner mean curvature. In the spin case, we make use of a previous estimate obtained by X. Zhang and the authors for the Dirac operator of the induced metric on ∂M. As a consequence, we generalize and simplify the proof of the result by Andersson and Dahl in [Scalar curvature rigidity for asymptotically locally hyperbolic manifolds, Ann. Global Anal. Geom.16 (1998) 1–27] about the rigidity of the hyperbolic space when the prescribed conformal infinity ∂M is a round sphere. We also provide non-existence results for conformally compact ALH spin metrics when ∂M is conformal to a Riemannian manifold with special holonomy.

Existence and nonexistence of local/global solutions for a nonhomogeneous heat equation

In this paper, we study the existence of local/global solutions to the Cauchy problem \begin{eqnarray}\rho(x)u_t=\Delta u+q(x)u^p, (x,t)\in R^N \times (0,T),\\u(x,0)=u_{0}(x)\ge 0, x \in R^N\end{eqnarray}with $p > 0$ and $N\ge 3$. We describe the sharp decay conditions on $\rho, q$ and the data $u_0$ at infinity that guarantee the local/global existence of nonnegative solutions.

On large deviations for some sequences of weighted means of Gaussian processes

In this paper we study some sequences of weighted means of continuous real valued Gaussian processes. More precisely we consider suitable generalizations of both arithmetic and logarithmic means of a Gaussian process with covariance function which satisfies either an exponential decay condition or a power decay condition. Our aim is to provide limits of variances of functionals of such weighted means which allow the application of some large deviation results in the literature.

Bounding hyperbolic and spherical coefficients of Maass forms

On développe une nouvelle méthode pour majorer les coefficients de Fourier hyperboliques et sphériques des formes de Maass définies par rapport à des réseaux uniformes généraux.

Anisotropic Two-Microlocal Spaces and Regularity

We defineD-u-anisotropic two-microlocal spaces by decay conditions on anisotropic wavelet coefficients on anyD-u-anisotropic wavelet basis ofL2(Rd). We prove that these spaces allow the characterizing of pointwise anisotropic Hölder regularity. We also prove an anisotropic wavelet criterion for anisotropic uniform regularity. We finally prove that both this criterion and anisotropicD-u-two-microlocal spaces are independent of the chosen anisotropicD-u-orthonormal wavelet basis.

Weighted Radon Transforms and First Order Differential Systems on the Plane

We consider weighted Radon transforms on the plane, where weights are given as finite Fourier series in angle variable. By means of additive Riemann-Hilbert problem techniques, we reduce inversion of these transforms to solving first order differential systems on $\R^2=\C$ with a decay condition at infinity. As a corollary, we obtain new injectivity and inversion results for weighted Radon transforms on the plane.

Infinitely many positive solutions for nonlinear equations with non-symmetric potentials

We consider the following nonlinear Schrodinger equation $$\begin{aligned} \left\{ \begin{array}{l} \Delta u-(1+\delta V)u+f(u)=0 \ \ \hbox { in }\mathbb {R}^N,\\ u>0 \ \hbox {in} \ \mathbb {R}^N, u\in H^1(\mathbb {R}^N) \end{array} \right. \end{aligned}$$ where $$V$$ is a continuous potential and $$ f(u)$$ is a nonlinearity satisfying some decay condition and some non-degeneracy condition, respectively. Using localized energy method, we prove that there exists a $$\delta _0$$ such that for $$0<\delta <\delta _0$$ , the above problem has infinitely many positive solutions. This generalizes and gives a new proof of the results by Cerami et al. (Comm. Pure Appl. Math. 66, 372–413, 2013). The new techniques allow us to establish the existence of infinitely many positive bound states for elliptic systems.