Super-exponential Decay Research Articles

Microlocal analysis for Gelfand–Shilov spaces

We introduce an anisotropic global wave front set of Gelfand–Shilov ultradistributions with different indices for regularity and decay at infinity. The concept is defined by the lack of super-exponential decay along power type curves in the phase space of the short-time Fourier transform. This wave front set captures the phase space behaviour of oscillations of power monomial type, a k a chirp signals. A microlocal result is proved with respect to pseudodifferential operators with symbol classes that give rise to continuous operators on Gelfand–Shilov spaces. We determine the wave front set of certain series of derivatives of the Dirac delta, and exponential functions.

Density functions for QuickQuant and QuickVal

We prove that, for every 0≤t≤1, the limiting distribution of the scale-normalized number of key comparisons used by the celebrated algorithm QuickQuant to find the tth quantile in a randomly ordered list has a Lipschitz continuous density function ft that is bounded above by 10. Furthermore, this density ft(x) is positive for every x>min{t,1−t} and, uniformly in t, enjoys superexponential decay in the right tail. We also prove that the survival function 1−Ft(x)= ∫x∞ft(y)dy and the density function ft(x) both have the right tail asymptotics exp[−xlnx−xlnlnx+O(x)]. We use the right-tail asymptotics to bound large deviations for the scale-normalized number of key comparisons used by QuickQuant.

Super-localization of elliptic multiscale problems

Numerical homogenization aims to efficiently and accurately approximate the solution space of an elliptic partial differential operator with arbitrarily rough coefficients in a d d -dimensional domain. The application of the inverse operator to some standard finite element space defines an approximation space with uniform algebraic approximation rates with respect to the mesh size parameter H H . This holds even for under-resolved rough coefficients. However, the true challenge of numerical homogenization is the localized computation of a localized basis for such an operator-dependent approximation space. This paper presents a novel localization technique that leads to a super-exponential decay of its basis relative to H H . This suggests that basis functions with supports of width O ( H | log ⁡ H | ( d − 1 ) / d ) \mathcal O(H|\log H|^{(d-1)/d}) are sufficient to preserve the optimal algebraic rates of convergence in H H without pre-asymptotic effects. A sequence of numerical experiments illustrates the significance of the new localization technique when compared to the so far best localization to supports of width O ( H | log ⁡ H | ) \mathcal O(H|\log H|) .

Mapping the myelin bilayer with short-T2 MRI: Methods validation and reference data for healthy human brain.

To explore the properties of short-T2 signals in human brain, investigate the impact of various experimental procedures on these properties and evaluate the performance of three-component analysis. Eight samples of non-pathological human brain tissue were subjected to different combinations of experimental procedures including D2 O exchange and frozen storage. Short-T2 imaging techniques were employed to acquire multi-TE (33-2067 μs) data, to which a three-component complex model was fitted in two steps to recover the properties of the underlying signal components and produce amplitude maps of each component. For validation of the component amplitude maps, the samples underwent immunohistochemical myelin staining. The signal component representing the myelin bilayer exhibited super-exponential decay with T2,min of 5.48 μs and a chemical shift of 1.07ppm, and its amplitude could be successfully mapped in both white and gray matter in all samples. These myelin maps corresponded well to myelin-stained tissue sections. Gray matter signals exhibited somewhat different components than white matter signals, but both tissue types were well represented by the signal model. Frozen tissue storage did not alter the signal components but influenced component amplitudes. D2 O exchange was necessary to characterize the non-aqueous signal components, but component amplitude mapping could be reliably performed also in the presence of H2 O signals. The myelin mapping approach explored here produced reasonable and stable results for all samples. The extensive tissue and methodological investigations performed in this work form a basis for signal interpretation in future studies both ex vivo and in vivo.

On the Crossing Estimates for Simple Conformal Loop Ensembles

Abstract We prove a super-exponential decay of probabilities that there exist $n$ crossings of a given quad for a simple $\textrm {CLE}_\kappa (\Omega )$, $\frac {8}{3}<\kappa \le 4$, as $n$ goes to infinity. Besides, being of independent interest, this also provides the missing ingredient in [2] for proving the convergence of probabilities of cylindrical events for the double-dimer loop ensembles to those for the nested $\textrm {CLE}_4(\Omega )$.

Modelling spin evolution of magnetars

ABSTRACT The origin and fate of magnetars [young, extremely magnetized neutron stars (NSs)] remains unsolved. Probing their evolution is therefore crucial for investigating possible links to other species of isolated NSs, such as the X-ray dim NSs (XDINSs) and rotating radio transients (RRATs). Here, we investigate the spin evolution of magnetars. Two avenues of evolution are considered: one with exponentially decaying B-fields, the other with sub- and superexponential decay. Using Monte Carlo methods, we synthesize magnetar populations using different input distributions and physical parameters, such as for the initial spin period, its time derivative, and the B-field decay time-scale. Additionally, we introduce a fade-away procedure that can account for the fading of old magnetars, and we briefly discuss the effect of alignment of the B-field and spin axes. Imposing the Galactic core-collapse supernova rate of ∼20 kyr−1 as a strict upper limit on the magnetar birthrate and comparing the synthetic populations to the observed one using both manual and automatic optimization algorithms for our input parameter study, we find that the B-field must decay exponentially or superexponentially with a characteristic decay time-scale of 0.5−10 kyr (with a best value of ∼4 kyr). In addition, the initial spin period must be less than 2 s. If these constraints are kept, we conclude that there are multiple choices of input physics that can reproduce the observed magnetar population reasonably well. We also conclude that magnetars may well be evolutionary linked to the population of XDINSs, whereas they are in general unlikely to evolve into RRATs.

The lower tail of the half-space KPZ equation

We establish the first tight bound on the lower tail probability of the half-space KPZ equation with Neumann boundary parameter A=−1/2 and narrow-wedge initial data. The lower bound demonstrates a crossover between two regimes of super-exponential decay with exponents 52 and 3; the upper bound demonstrates a crossover between regimes with exponents 32 and 3. Given a crude leading-order asymptotic in the Stokes region for the Ablowitz–Segur solution to Painlevé II (Definition 1.8), we improve the upper bound to demonstrate the same crossover as the lower bound. We also establish novel bounds on the large deviations of the GOE point process.

Instantons for rare events in heavy-tailed distributions

Large deviation theory and instanton calculus for stochastic systems are widely used to gain insight into the evolution and probability of rare events. At its core lies the fact that rare events are, under the right circumstances, dominated by their least unlikely realization. Their computation through a saddle-point approximation of the path integral for the corresponding stochastic field theory then reduces an inefficient stochastic sampling problem into a deterministic optimization problem: finding the path of smallest action, the instanton. In the presence of heavy tails, though, standard algorithms to compute the instanton critically fail to converge. The reason for this failure is the divergence of the scaled cumulant generating function (CGF) due to a non-convex large deviation rate function. We propose a solution to this problem by ‘convexifying’ the rate function through a nonlinear reparametrization of the observable, which allows us to compute instantons even in the presence of super-exponential or algebraic tail decay. The approach is generalizable to other situations where the existence of the CGF is required, such as exponential tilting in importance sampling for Monte-Carlo algorithms. We demonstrate the proposed formalism by applying it to rare events in several stochastic systems with heavy tails, including extreme power spikes in fiber optics induced by soliton formation.

When is the Chernoff Exponent for Quantum Operations Finite?

We consider the problem of testing two hypotheses of quantum operations in a setting of many uses where an arbitrary prior probability distribution is given. The Chernoff exponent for quantum operations is investigated to track the minimal average error probability of discriminating two quantum operations asymptotically. We answer the question, "When is the Chernoff exponent for quantum operations finite?" We show that either two quantum operations can be perfectly distinguished with finite uses, or the minimal discrimination error decays exponentially with respect to the number of uses asymptotically. That is, the Chernoff exponent is finite if and only if the quantum operations can not be perfectly distinguished with finite uses. This rules out the possibility of super-exponential decay of error probability. Upper bounds of the Chernoff exponent for quantum operations are provided.

Stability estimates for reconstruction from the Fourier transform on the ball

Abstract We prove Hölder-logarithmic stability estimates for the problem of finding an integrable function v on ℝ d {{\mathbb{R}}^{d}} with a super-exponential decay at infinity from its Fourier transform ℱ ⁢ v {\mathcal{F}v} given on the ball B r {B_{r}} . These estimates arise from a Hölder-stable extrapolation of ℱ ⁢ v {\mathcal{F}v} from B r {B_{r}} to a larger ball. We also present instability examples showing an optimality of our results.

Asymptotics of Fundamental Solutions for Time Fractional Equations with Convolution Kernels

Abstract The paper deals with the large time asymptotic of the fundamental solution for a time fractional evolution equation with a convolution type operator. In this equation we use a Caputo time derivative of order α ∈ (0, 1), and assume that the convolution kernel of the spatial operator is symmetric, integrable and shows a super-exponential decay at infinity. Under these assumptions we describe the point-wise asymptotic behavior of the fundamental solution in all space-time regions.

Lower tail of the KPZ equation

We provide the first tight bounds on the lower tail probability of the one point distribution of the KPZ equation with narrow wedge initial data. Our bounds hold for all sufficiently large times $T$ and demonstrates a crossover between super-exponential decay with exponent $5/2$ (and leading pre-factor $\frac{4}{15\pi} T^{1/3}$) for tail depth greater than $T^{2/3}$, and exponent $3$ (with leading pre-factor $\frac{1}{12}$) for tail depth less than $T^{2/3}$.

KPZ equation tails for general initial data

We consider the upper and lower tail probabilities for the centered (by time$/24$) and scaled (according to KPZ time$^{1/3}$ scaling) one-point distribution of the Cole-Hopf solution of the KPZ equation when started with initial data drawn from a very general class. For the lower tail, we prove an upper bound which demonstrates a crossover from super-exponential decay with exponent $3$ in the shallow tail to an exponent $5/2$ in the deep tail. For the upper tail, we prove super-exponential decay bounds with exponent $3/2$ at all depths in the tail.

Dynamics of a class-A nonlinear mirror mode-locked laser.

Using a delay differential equation model we study theoretically the dynamics of a unidirectional class-A ring laser with a nonlinear amplifying loop mirror. We perform linear stability analysis of the continuous-wave regimes in the large delay limit and demonstrate that these regimes can be destabilized via modulational and Turing-type instabilities, as well as by an instability leading to the appearance of square-waves. We investigate the formation of square waves and mode-locked pulses in the system. We show that mode-locked pulses are asymmetric with exponential decay of the trailing edge in positive time and faster-than-exponential (superexponential) decay of the leading edge in negative time. We discuss asymmetric interaction of these pulses leading to a formation of harmonic mode-locked regimes.

Super-exponential decay rates for eigenvalues and singular values of integral operators on the sphere

This paper brings results about the behavior of sequences of eigenvalues or singular values of integral operators generated by square-integrable kernels on the real m-dimensional unit sphere, m≥2. Under smoothness assumptions on the generating kernels, given via Laplace–Beltrami differentiability, we obtain super-exponential decay rates for the eigenvalues of the generated positive integral operators and for singular values of those integral operators which are non-positive. We show an optimal-type result and provide a list of parametric families of kernels which are of interest for numerical analysis and geostatistical communities and satisfy the smoothness assumptions for the positive case.

Intrinsic Ultracontractivity and Ground State Estimates of Non-local Dirichlet forms on Unbounded Open Sets

In this paper we consider a large class of symmetric Markov processes $${X=(X_t)_{t\ge0}}$$ on $${\mathbb{R}^d}$$ generated by non-local Dirichlet forms, which include jump processes with small jumps of $${\alpha}$$ -stable-like type and with large jumps of super-exponential decay. Let $${D\subset \mathbb{R}^d}$$ be an open (not necessarily bounded and connected) set, and $${X^D=(X_t^D)_{t \ge 0}}$$ be the killed process of X on exiting D. We obtain explicit criterion for the compactness and the intrinsic ultracontractivity of the Dirichlet Markov semigroup $${(P^{D}_t)_{t\ge0}}$$ of XD. When D is a horn-shaped region, we further obtain two-sided estimates of ground state in terms of jumping kernel of X and the reference function of the horn-shaped region D.

Contact-Patch-Size Distribution and Limits of Self-Affinity in Contacts between Randomly Rough Surfaces

True contact between solids with randomly rough surfaces tends to occur at a large number of microscopic contact patches. Thus far, two scaling regimes have been identified for the number density n ( A ) of contact-patch sizes A in elastic, non-adhesive, self-affine contacts. At small A, n ( A ) is approximately constant, while n ( A ) decreases as a power law at large A. Using Green’s function molecular dynamics, we identify a characteristic (maximum) contact area A c above which a superexponential decay of n ( A ) becomes apparent if the contact pressure is below the pressure p cp at which contact percolates. We also find that A c increases with load relatively slowly far away from contact percolation. Results for A c can be estimated from the stress autocorrelation function G σ σ ( r ) with the following argument: the radius of characteristic contact patches, r c , cannot be so large that G σ σ ( r c ) is much less than p cp 2 . Our findings provide a possible mechanism for the breakdown of the proportionality between friction and wear with load at large contact pressures and/or for surfaces with a large roll-off wavelength.

Weighted finite Laplace transform operator: spectral analysis and quality of approximation by its eigenfunctions

ABSTRACTFor two real numbers we study some spectral properties of the weighted finite bilateral Laplace transform operator, defined over the space by In particular, we use a technique based on the Min–Max theorem to prove that the sequence of the eigenvalues of this operator has a super-exponential decay rate to zero. Moreover, we give a lower bound with a magnitude of order for the largest eigenvalue of the operator Also, we give some local estimates and bounds of the eigenfunctions of Moreover, we show that these eigenfunctions are good candidates for the spectral approximation of a function that can be written as a weighted finite Laplace transform of an other function. Finally, we give some numerical examples that illustrate the different results of this work. In particular, we provide an example that illustrate the Laplace-based spectral method, for the inversion of the finite Laplace transform.

Convexity at infinity in Cartan–Hadamard manifolds and applications to the asymptotic Dirichlet and Plateau problems

We study the asymptotic Dirichlet and Plateau problems on Cartan-Hadamard manifolds satisfying the so-called Strict Convexity (abbr. SC) condition. The main part of the paper consists in studying the SC condition on a manifold whose sectional curvatures are bounded from above and below by certain functions depending on the distance to a fixed point. In particular, we are able to verify the SC condition on manifolds whose curvature lower bound can go to -infinity and upper bound to 0 simultaneously at certain rates, or on some manifolds whose sectional curvatures go to -infinity faster than any prescribed rate. These improve previous results of Anderson, Borb\'ely, and Ripoll and Telichevsky. We then solve the asymptotic Plateau problem for locally rectifiable currents with Z_2-multiplicity in a Cartan-Hadamard manifold satisfying the SC condition given any compact topologically embedded (k-1)-dimensional submanifold of \partial_{\infty}M, 2\leq k\leq n-1, as the boundary data. We also solve the asymptotic Plateau problem for locally rectifiable currents with Z-multiplicity on any rotationally symmetric manifold satisfying the SC condition given a smoothly embedded submanifold as the boundary data. These generalize previous results of Anderson, Bangert, and Lang. Moreover, we obtain new results on the asymptotic Dirichlet problem for a large class of PDEs. In particular, we are able to prove the solvability of this problem on manifolds with super-exponential decay (to -infinity) of the curvature.

The Finite Hankel Transform Operator: Some Explicit and Local Estimates of the Eigenfunctions and Eigenvalues Decay Rates

For fixed real numbers $$c>0,$$ $$\alpha >-\frac{1}{2},$$ the finite Hankel transform operator, denoted by $$\mathcal {H}_c^{\alpha }$$ is given by the integral operator defined on $$L^2(0,1)$$ with kernel $$K_{\alpha }(x,y)= \sqrt{c xy} J_{\alpha }(cxy).$$ To the operator $$\mathcal {H}_c^{\alpha },$$ we associate a positive, self-adjoint compact integral operator $$\mathcal Q_c^{\alpha }=c\, \mathcal {H}_c^{\alpha }\, \mathcal {H}_c^{\alpha }.$$ Note that the integral operators $$\mathcal {H}_c^{\alpha }$$ and $$\mathcal Q_c^{\alpha }$$ commute with a Sturm-Liouville differential operator $$\mathcal D_c^{\alpha }.$$ In this paper, we first give some useful estimates and bounds of the eigenfunctions $$\varphi ^{(\alpha )}_{n,c}$$ of $$\mathcal H_c^{\alpha }$$ or $$\mathcal Q_c^{\alpha }.$$ These estimates and bounds are obtained by using some special techniques from the theory of Sturm-Liouville operators, that we apply to the differential operator $$\mathcal D_c^{\alpha }.$$ If $$(\mu _{n,\alpha }(c))_n$$ and $$\lambda _{n,\alpha }(c)=c\, |\mu _{n,\alpha }(c)|^2$$ denote the infinite and countable sequence of the eigenvalues of the operators $$\mathcal {H}_c^{(\alpha )}$$ and $$\mathcal Q_c^{\alpha },$$ arranged in the decreasing order of their magnitude, then we show an unexpected result that for a given integer $$n\ge 0,$$ $$\lambda _{n,\alpha }(c)$$ is decreasing with respect to the parameter $$\alpha .$$ As a consequence, we show that for $$\alpha \ge \frac{1}{2},$$ the $$\lambda _{n,\alpha }(c)$$ and the $$\mu _{n,\alpha }(c)$$ have a super-exponential decay rate. Also, we give a lower decay rate of these eigenvalues. As it will be seen, the previous results are essential tools for the analysis of a spectral approximation scheme based on the eigenfunctions of the finite Hankel transform operator. Some numerical examples will be provided to illustrate the results of this work.