Decay Assumptions Research Articles

Approximation rates for neural networks with general activation functions

We prove some new results concerning the approximation rate of neural networks with general activation functions. Our first result concerns the rate of approximation of a two layer neural network with a polynomially-decaying non-sigmoidal activation function. We extend the dimension independent approximation rates previously obtained to this new class of activation functions. Our second result gives a weaker, but still dimension independent, approximation rate for a larger class of activation functions, removing the polynomial decay assumption. This result applies to any bounded, integrable activation function. Finally, we show that a stratified sampling approach can be used to improve the approximation rate for polynomially decaying activation functions under mild additional assumptions.

Velocity decay estimates for Boltzmann equation with hard potentials

We establish pointwise polynomial decay estimates in velocity space for the spatially inhomogeneous Boltzmann equation without cutoff, in the case of hard potentials (γ + 2s > 2), under the assumption that the mass, energy, and entropy densities are bounded above, and the mass density is bounded below. These estimates are self-generating, i.e. they do not require corresponding decay assumptions on the initial data. Our results extend the recent work of Imbert et al [2020 J. École Polytech. 7 143–83], which addressed the case of moderately soft potentials (γ + 2s ∈ [0, 2]).

Monotonicity of positive solutions for fractional [formula omitted]-systems in unbounded Lipschitz domains

In this paper, without imposing any decay assumptions on the solutions at infinity, we first develop an unbounded narrow region principle for the fractional p-Laplacian systems, an essential ingredient to carry on the direct method of moving planes; then we combine this direct method with the sliding method to obtain the monotonicity of positive bounded solutions for cooperative systems involving the fractional p-Laplacian in unbounded Lipschitz domains. We believe that the new ideas introduced here will become useful tools in studying qualitative properties of solutions for other types of nonlinear nonlocal systems in various unbounded domains.

Peeling of Dirac field on Kerr spacetime

In a recent paper with Nicolas [Ann. Henri Poincaré 20(10), 3419–3470 (2019); arXiv:1801.08996], we studied the peeling for scalar fields on Kerr metrics. The present work extends these results to Dirac fields on the same geometrical background. We follow the approach initiated by Mason and Nicolas [J. Inst. Math. Jussieu 8(1), 179–208 (2009); arXiv:gr-qc/0701049 and L. Mason and J.-P. Nicolas, J. Geom. Phys. 62(4), 867–889 (2012); arXiv:1101.4333] on the Schwarzschild spacetime and extended to Kerr metrics for scalar fields. The method combines the Penrose conformal compactification and geometric energy estimates in order to work out a definition of peeling at all orders in terms of Sobolev regularity near I, instead of Ck regularity at I, and then provides the optimal spaces of initial data such that the associated solution satisfies peeling at a given order. The results confirm that the analogous decay and regularity assumptions on initial data in the Minkowski and Kerr spacetimes produce the same regularity across null infinity. Our results are local near spacelike infinity and are valid for all values of the angular momentum of the spacetime, including for fast Kerr metrics.

Symmetry of solutions for a fractional p-Laplacian equation of Choquard type

Let [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] be a positive solution of the equation [Formula: see text] We prove that if [Formula: see text] satisfies some decay assumption at infinity, then [Formula: see text] must be radially symmetric and monotone decreasing about some point in [Formula: see text]. Instead of using equivalent fractional systems, we exploit a generalized direct method of moving planes for fractional [Formula: see text]-Laplacian equations with nonlocal nonlinearities. This new approach enables us to cover the full range [Formula: see text] in our results.

Spatial behavior of the solution to the linearized Boltzmann equation with hard potentials

The main goal is to understand the quantitative spatial decay of the solution to the linearized Boltzmann equation for hard potentials with cutoff. We obtain the quantitative space-time behavior of the linearized Boltzmann equation under some slow velocity decay assumption but without any regularity assumption on the initial data. This result extends the classical results of Liu and Yu [Commun. Pure Appl. Math. 57, 1543–1608 (2004); Bull. Inst. Math. Acad. Sin. 1, 1–78 (2006); and ibid. 6, 151–243 (2011)] from hard sphere to hard potential under the same assumption on the initial condition [compactly support in the space variable and polynomial decay in the velocity variable with power β = (3/2)+].

Singular semilinear elliptic problems with asymptotically linear reaction terms

We consider the problem{−Δu=λK(x)f(u)in B1c,u=0on ∂B1,u(x)→0as |x|→∞, where B1c={x∈Rn||x|>1},n>2, λ is a positive parameter, K belongs to a class of functions which satisfy certain decay assumptions and f belongs to a class of functions which are asymptotically linear and may be singular at the origin. We prove the existence of positive solutions to such problems for certain values of parameter λ. Existence results to similar problems in Rn are also obtained. Our existence results are proved using the Schauder fixed point theorem and the method of sub and super solutions.

On the well-posedness and decay rates of strong solutions to a multi-dimensional non-conservative viscous compressible two-fluid system

The present paper deals with the Cauchy problem of a multi-dimensional non-conservative viscous compressible two-fluid system. We first study the well-posedness of the model in spaces with critical regularity indices with respect to the scaling of the associated equations. In the functional setting as close as possible to the physical energy spaces, we prove the unique global solvability of strong solutions close to a stable equilibrium state. Furthermore, under a mild additional decay assumption involving only the low frequencies of the data, we establish the time decay rates for the constructed global solutions. The proof relies on an application of Fourier analysis to a complicated parabolic-hyperbolic system, and on a refined time-weighted inequality.

Infinitely many bound states for Choquard equations with local nonlinearities

In this paper, we consider the following Choquard equation (CH)−Δu+u=(Iα∗|u|p)|u|p−2u+V(x)|u|q−2uinRN,u∈H1(RN),where N≥3, α∈((N−4)+,N), p∈[2,N+αN−2), q∈(2,2NN−2)∩(1+(p−1)(N−2)αN2,1+2N(N−α)+N2(p−1)(N−2)α) and Iα is the Riesz potential. Under some suitable decay assumptions but without any symmetry property on V(x), we prove that the problem has infinitely many solutions, whose energy can be arbitrarily large.

The large-time behavior of the multi-dimensional hyperbolic-parabolic model arising from chemotaxis

The present paper is dedicated to the study of large-time behavior of global strong solutions to the initial value problem for the hyperbolic-parabolic system derived from chemotaxis models in any dimension d ≥ 2. Under a suitable additional decay assumption involving only the low frequencies of the data and in L2-critical regularity framework, we exhibit the decay rates of strong solutions to the system for initial data close to a stable equilibrium state. The proof relies on a refined time-weighted energy functional in the Fourier space and the Littlewood-Paley decomposition technology.The present paper is dedicated to the study of large-time behavior of global strong solutions to the initial value problem for the hyperbolic-parabolic system derived from chemotaxis models in any dimension d ≥ 2. Under a suitable additional decay assumption involving only the low frequencies of the data and in L2-critical regularity framework, we exhibit the decay rates of strong solutions to the system for initial data close to a stable equilibrium state. The proof relies on a refined time-weighted energy functional in the Fourier space and the Littlewood-Paley decomposition technology.

Mathematical modeling of light curves of RHESSI and AGILE terrestrial gamma-ray flashes

We consider different distribution functions as possible fits for the light curves (time profiles) of Terrestrial Gamma-ray Flashes (TGFs): the piecewise Gaussian and the piecewise exponential, which correspond to the assumption of exponential growth and decay of the electrons that emit the radiation, and the inverse Gaussian and Ornstein-Uhlenbeck functions, which correspond to the assumption of an exit-time process. We compute maximum likelihood estimations (MLEs) for each of these four functions for a 100 TGFs recorded by the RHESSI and AGILE satellites, and compared those values with MLEs of the lognormal distribution function, which has been very widely used. The analysis of time profiles of these TGFs shows that all five distributions fit the data equally well. The results obtained in this work, combined with the results obtained for the analysis of BATSE and FERMI TGFs (Abukhaled et al. in J. Geophys. Res. Space Phys. 119:5918–5930, 2014) show that the TGF data as recorded thus far by various satellite gamma-ray detectors, are not powerful enough to discriminate between those various mathematical functions, which, if one function were preferred, would identify some physical features of the phenomenon.

Finite dimensionality of the global attractor for a fractional Schrödinger equation on [formula omitted

The finite dimensionality of the global attractor is proved for a damped fractional Schrödinger equation on the real line. The force merely belongs to L2(R), in particular needs no decay assumptions at the infinity. This solves a problem posed by Goubet and Zahrouni (2017).

Inequalities for nonuniform wavelet frames

Abstract Gabardo and Nashed studied nonuniform wavelets by using the theory of spectral pairs for which the translation set Λ = { 0 , r / N } + 2 ⁢ ℤ {\Lambda=\{0,r/N\}+2\mathbb{Z}} is no longer a discrete subgroup of ℝ {\mathbb{R}} but a spectrum associated with a certain one-dimensional spectral pair. In this paper, we establish three sufficient conditions for the nonuniform wavelet system { ψ j , λ ⁢ ( x ) = ( 2 ⁢ N ) j / 2 ⁢ ψ ⁢ ( ( 2 ⁢ N ) j ⁢ x - λ ) , j ∈ ℤ , λ ∈ Λ } {\{\psi_{j,\lambda}(x)=(2N)^{j/2}\psi((2N)^{j}x-\lambda),\,j\in\mathbb{Z},\,% \lambda\in\Lambda\}} to be a frame for L 2 ⁢ ( ℝ ) {L^{2}(\mathbb{R})} . The proposed inequalities are stated in terms of Fourier transforms and hold without any decay assumptions on the generator of such a system.

Existence, Nonexistence, and Asymptotics of Deep Water Solitary Waves with Localized Vorticity

In this paper, we study solitary waves propagating along the surface of an infinitely deep body of water in two or three dimensions. The waves are acted upon by gravity and capillary effects are allowed --- but not required --- on the interface. We assume that the vorticity is localized in the sense that it satisfies certain moment conditions, and we permit there to be finitely many point vortices in the bulk of the fluid in two dimensions. We also consider a two-fluid model with a vortex sheet. Under mild decay assumptions, we obtain precise asymptotics for the velocity field and free surface, and relate this to global properties of the wave. For instance, we rule out the existence of waves whose free surface elevations have a single sign and of vortex sheets with finite angular momentum. Building on the work of Shatah, Walsh, and Zeng, we also prove the existence of families of two-dimensional capillary-gravity waves with compactly supported vorticity satisfying the above assumptions. For these waves, we further show that the free surface is positive in a neighborhood of infinity, and that the asymptotics at infinity are linked to the net vorticity.

Examining the Limits of Predictability of Human Mobility.

We challenge the upper bound of human-mobility predictability that is widely used to corroborate the accuracy of mobility prediction models. We observe that extensions of recurrent-neural network architectures achieve significantly higher prediction accuracy, surpassing this upper bound. Given this discrepancy, the central objective of our work is to show that the methodology behind the estimation of the predictability upper bound is erroneous and identify the reasons behind this discrepancy. In order to explain this anomaly, we shed light on several underlying assumptions that have contributed to this bias. In particular, we highlight the consequences of the assumed Markovian nature of human-mobility on deriving this upper bound on maximum mobility predictability. By using several statistical tests on three real-world mobility datasets, we show that human mobility exhibits scale-invariant long-distance dependencies, contrasting with the initial Markovian assumption. We show that this assumption of exponential decay of information in mobility trajectories, coupled with the inadequate usage of encoding techniques results in entropy inflation, consequently lowering the upper bound on predictability. We highlight that the current upper bound computation methodology based on Fano’s inequality tends to overlook the presence of long-range structural correlations inherent to mobility behaviors and we demonstrate its significance using an alternate encoding scheme. We further show the manifestation of not accounting for these dependencies by probing the mutual information decay in mobility trajectories. We expose the systematic bias that culminates into an inaccurate upper bound and further explain as to why the recurrent-neural architectures, designed to handle long-range structural correlations, surpass this upper limit on human mobility predictability.

The inverse backscattering for Schrödinger operators for potentials with noncompact support

We consider the inverse backscattering problem for the Schrödinger operator H = −Δ + V on , n ≥ 3, as well as the higher‐order Schrödinger operator ( − Δ)m + V, m = 2,3,…. We show that in some suitable Banach spaces, the map from the potential to the backscattering amplitude is a local diffeomorphism. This kind of problem (for m = 1) was studied by Eskin and Ralston [Comm. Math. Phys., 124(2), 169‐215 (1989)], where they assumed that . In this paper, we replace the assumption on V with certain decay assumption at infinity.

On the Dynamics of Zero-Speed Solutions for Camassa–Holm-Type Equations

Abstract In this paper, we consider globally defined solutions of Camassa–Holm (CH)-type equations outside the well-known nonzero-speed, peakon region. These equations include the standard CH and Degasperis–Procesi (DP) equations, as well as nonintegrable generalizations such as the $b$-family, elastic rod, and Benjamin-Bona-Mahony (BBM) equations. Having globally defined solutions for these models, we introduce the notion of zero-speed and breather solutions, i.e., solutions that do not decay to zero as $t\to +\infty $ on compact intervals of space. We prove that, under suitable decay assumptions, such solutions do not exist because the identically zero solution is the global attractor of the dynamics, at least in a spatial interval of size $|x|\lesssim t^{1/2-}$ as $t\to +\infty $. As a consequence, we also show scattering and decay in CH-type equations with long-range nonlinearities. Our proof relies in the introduction of suitable virial functionals à la Martel–Merle in the spirit of the works of [74, 75] and [50] adapted to CH-, DP-, and BBM-type dynamics, one of them placed in $L^1_x$ and the 2nd one in the energy space $H^1_x$. Both functionals combined lead to local-in-space decay to zero in $|x|\lesssim t^{1/2-}$ as $t\to +\infty $. Our methods do not rely on the integrable character of the equation, applying to other nonintegrable families of CH-type equations as well.

Moments and regularity for a Boltzmann equation via Wigner transform

In this paper, we continue our study of the Boltzmann equation by use of tools originating from the analysis of dispersive equations in quantum dynamics. Specifically, we focus on properties of solutions to the Boltzmann equation with collision kernel equal to a constant in the spatial domain $ \mathbb{R}^d $, $ d\geq 2 $, which we use as a model in this paper. Local well-posedness for this equation has been proven using the Wigner transform when $ \left< v \right>^\beta f_0 \in L^2_v H^\alpha_x $ for $ \min (\alpha,\beta) > \frac{d-1}{2} $. We prove that if $ \alpha,\beta $ are large enough, then it is possible to propagate moments in $ x $ and derivatives in $ v $ (for instance, $ \left< x \right>^k \left< \nabla_v \right>^\ell f \in L^\infty_T L^2_{x,v} $ if $ f_0 $ is nice enough). The mechanism is an exchange of regularity in return for moments of the (inverse) Wigner transform of $ f $. We also prove a persistence of regularity result for the scale of Sobolev spaces $ H^{\alpha,\beta} $; and, continuity of the solution map in $ H^{\alpha,\beta} $. Altogether, these results allow us to conclude non-negativity of solutions, conservation of energy, and the $ H $-theorem for sufficiently regular solutions constructed via the Wigner transform. Non-negativity in particular is proven to hold in $ H^{\alpha,\beta} $ for any $ \alpha,\beta > \frac{d-1}{2} $, without any additional regularity or decay assumptions.

On asymptotic dynamics for L2 critical generalized KdV equations with a saturated perturbation

In this paper, we consider the $L^2$ critical gKdV equation with a saturated perturbation: $\partial_t u+(u_{xx}+u^5-\gamma u|u|^{q-1})_x=0$, where $q>5$ and $0<\gamma\ll1$. For any initial data $u_0\in H^1$, the corresponding solution is always global and bounded in $H^1$. This equation has a family of solitons, and our goal is to classify the dynamics near soliton. Together with a suitable decay assumption, there are only three possibilities: (i) the solution converges asymptotically to a solitary wave, whose $H^1$ norm is of size $\gamma^{-2/(q-1)}$, as $\gamma\rightarrow0$; (ii) the solution is always in a small neighborhood of the modulated family of solitary waves, but blows down at $+\infty$; (iii) the solution leaves any small neighborhood of the modulated family of the solitary waves. This extends the classification of the rigidity dynamics near the ground state for the unperturbed $L^2$ critical gKdV (corresponding to $\gamma=0$) by Martel, Merle and Rapha\"el. However, the blow-down behavior (ii) is completely new, and the dynamics of the saturated equation cannot be viewed as a perturbation of the $L^2$ critical dynamics of the unperturbed equation. This is the first example of classification of the dynamics near ground state for a saturated equation in this context. The cases of $L^2$ critical NLS and $L^2$ supercritical gKdV, where similar classification results are expected, are completely open.

Liouville's theorem for a fractional elliptic system

In this paper, we investigate the following fractional elliptic system \begin{document}$\left\{ \begin{array}{*{35}{l}} {{(-\Delta )}^{\alpha /2}}u(x) = f(x){{v}^{q}}(x),&amp;x\in {{R}^{n}}, \\ {{(-\Delta )}^{\beta /2}}v(x) = h(x){{u}^{p}}(x),&amp;x\in {{R}^{n}}, \\\end{array} \right.$ \end{document} where $1≤p, q &lt; ∞$, $0 &lt; α, β &lt; 2$, $f(x)$ and $h(x)$ satisfy suitable conditions. Applying the method of moving planes, we prove monotonicity without any decay assumption at infinity. Furthermore, if $ α = β$, a Liouville theorem is established.