Polynomial Rate Research Articles

Explosion in a growth model with cooperative interaction on an infinite graph

In the process of adsorption, which is the process of accumulation of molecules on a surface, particles are allocated to vertices of a graph accordingly to a specific rule. Such phenomenon has great technological importance in the purification and separation of mixtures in petroleum industries, in effluent treatment, and in the manufacture of biomaterials such as prostheses and implants. The subject of this paper is to study the asymptotic behavior of a growth process, in our case it is the process of adsorption of particles in an infinite number of sites, using a probabilistic approach. For this we distinguish explosion/non-explosion of a continuos time growth process with cooperative interaction on $${\mathbb {Z}}_+$$ . We consider symmetric neighborhood and different types of rate functions and prove that explosion occurs for exponential rates, and for some cases with polynomial rates, but not for other cases. We also present some simulations to illustrate the types of explosions to check whether the explosion occurs horizontally or vertically.

Extremes of a class of non-stationary Gaussian processes and maximal deviation of projection density estimates

In this paper, we consider the distribution of the supremum of non-stationary Gaussian processes, and present a new theoretical result on the asymptotic behaviour of this distribution. We focus on the case when the processes have finite number of points attaining their maximal variance, but, unlike previously known facts in this field, our main theorem yields the asymptotic representation of the corresponding distribution function with exponentially decaying remainder term. This result can be efficiently used for studying the projection density estimates, based, for instance, on Legendre polynomials. More precisely, we construct the sequence of accompanying laws, which approximates the distribution of maximal deviation of the considered estimates with polynomial rate. Moreover, we construct the confidence bands for densities, which are honest at polynomial rate to a broad class of densities.

Structural properties of NFAs and growth rates of nondeterminism measures

Tree width (respectively, string path width) measures the number of partial (respectively, complete) computations of a nondeterministic finite automaton (NFA) on a given input. We characterize polynomial and exponential growth rates of tree width and string path width by structural properties of NFAs. Polynomial growth rates, roughly speaking, require that there exist computations going through a bounded number of cycles where the strings of characters labeling the cycles satisfy certain requirements. As the main result we show that the degrees of the polynomials bounding the tree width and string path width of an NFA differ from each other by at most one.More generally, an NFA is said to have cycle height K if any computation can visit at most K distinct cycles. We give a polynomial time algorithm to decide whether an NFA has finite cycle height and, in the positive case, to compute its optimal cycle height.

Convergence of maximum likelihood supertree reconstruction

<abstract><p>Supertree methods are tree reconstruction techniques that combine several smaller gene trees (possibly on different sets of species) to build a larger species tree. The question of interest is whether the reconstructed supertree converges to the true species tree as the number of gene trees increases (that is, the consistency of supertree methods). In this paper, we are particularly interested in the convergence rate of the maximum likelihood supertree. Previous studies on the maximum likelihood supertree approach often formulate the question of interest as a discrete problem and focus on reconstructing the correct topology of the species tree. Aiming to reconstruct both the topology and the branch lengths of the species tree, we propose an analytic approach for analyzing the convergence of the maximum likelihood supertree method. Specifically, we consider each tree as one point of a metric space and prove that the distance between the maximum likelihood supertree and the species tree converges to zero at a polynomial rate under some mild conditions. We further verify these conditions for the popular exponential error model of gene trees.</p></abstract>

On the discrepancy of random subsequences of $\{n\alpha \}$, II

Let $\alpha$ be an irrational number, let $X_1, X_2, \ldots$ be independent, identically distributed, integer-valued random variables, and put $S_k=\sum_{j=1}^k X_j$. Assuming that $X_1$ has finite variance or heavy tails $P (|X_1|>t)\sim ct^{-\beta}$, $0<\beta<2$, in Part I of this paper we proved that, up to logarithmic factors, the order of magnitude of the discrepancy $D_N (S_k \alpha)$ of the first $N$ terms of the sequence $\{S_k \alpha\}$ is $O(N^{-\tau})$, where $\tau= \min (1/(\beta \gamma), 1/2)$ (with $\beta=2$ in the case of finite variances) and $\gamma$ is the strong Diophantine type of $\alpha$. This shows a change of behavior of the discrepancy at $\beta\gamma=2$. In this paper we determine the exact order of magnitude of $D_N (S_k \alpha)$ for $\beta\gamma<1$, and determine the limit distribution of $N^{-1/2} D_N (S_k \alpha)$. We also prove a functional version of these results describing the asymptotic behavior of a wide class of functionals of the sequence $\{S_k \alpha\}$. Finally, we extend our results to the discrepancy of $\{S_k\}$ for general random walks $S_k$ without arithmetic conditions on $X_1$, assuming only a mild polynomial rate on the weak convergence of $\{S_k\}$ to the uniform distribution.

Function Values Are Enough for L_2-Approximation

We study the L_2-approximation of functions from a Hilbert space and compare the sampling numbers with the approximation numbers. The sampling number e_n is the minimal worst-case error that can be achieved with n function values, whereas the approximation number a_n is the minimal worst-case error that can be achieved with n pieces of arbitrary linear information (like derivatives or Fourier coefficients). We show that en≲1kn∑j≥knaj2,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} e_n \\,\\lesssim \\, \\sqrt{\\frac{1}{k_n} \\sum _{j\\ge k_n} a_j^2}, \\end{aligned}$$\\end{document}where k_n asymp n/log (n). This proves that the sampling numbers decay with the same polynomial rate as the approximation numbers and therefore that function values are basically as powerful as arbitrary linear information if the approximation numbers are square-summable. Our result applies, in particular, to Sobolev spaces H^s_mathrm{mix}(mathbb {T}^d) with dominating mixed smoothness s>1/2 and dimension din mathbb {N}, and we obtain en≲n-slogsd(n).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} e_n \\,\\lesssim \\, n^{-s} \\log ^{sd}(n). \\end{aligned}$$\\end{document}For d>2s+1, this improves upon all previous bounds and disproves the prevalent conjecture that Smolyak’s (sparse grid) algorithm is optimal.

Polynomial Decay Rate for a Coupled Lamé System with Viscoelastic Damping and Distributed Delay Terms

In this paper, we prove a general energy decay results of a coupled Lamé system with distributed time delay. By assuming a more general of relaxation functions and using some properties of convex functions, we establish the general energy decay results to the system by using an appropriate Lyapunov functional.

New general decay result for a system of two singular nonlocal viscoelastic equations with general source terms and a wide class of relaxation functions

This work is concerned with a system of two singular viscoelastic equations with general source terms and nonlocal boundary conditions. We discuss the stabilization of this system under a very general assumption on the behavior of the relaxation function k_{i}, namely, ki′(t)≤−ξi(t)Ψi(ki(t)),i=1,2.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} k_{i}^{\\prime }(t)\\le -\\xi _{i}(t) \\Psi _{i} \\bigl(k_{i}(t)\\bigr),\\quad i=1,2. \\end{aligned}$$ \\end{document} We establish a new general decay result that improves most of the existing results in the literature related to this system. Our result allows for a wider class of relaxation functions, from which we can recover the exponential and polynomial rates when k_{i}(s) = s^{p} and p covers the full admissible range [1, 2).

Decay Results for a Viscoelastic Problem with Nonlinear Boundary Feedback and Logarithmic Source Term

The main goal of this work is to investigate the long-time behavior of a viscoelastic equation with a logarithmic source term and a nonlinear feedback localized on a part of the boundary. In the framework of potential well, we first show the global existence. Then, we discuss the asymptotic behavior of the problem with a very general assumption on the behavior of the relaxation function g, namely, $g^{\prime }(t)\le -\xi (t) G(g(t))$ . We establish explicit and general decay results from which we can recover the well-known exponential and polynomial rates when G(s) = sp and p covers the full admissible range [1,2). Our results are obtained without imposing any restrictive growth assumption on the boundary damping term. This work generalizes and improves many earlier results in the literature.

On classical solutions of the two-species Vlasov–Poisson system in R3 with infinite charge

We are concerned with the initial value problem for the two-species Vlasov–Poisson system in the plasma physics case in R3. It is assumed that initially the phase-space densities of ions and electrons decay only polynomially with respect to the velocities, but the net charge density vanishes at infinity with a polynomial rate, which allows for defining the electrostatic field. This situation falls into the so-called infinite charge problem, and there is no rigorous result as far as we know. Local existence, uniqueness, and continuation criteria of the classical solution are established in this paper. Moreover, this article also shows the existence of a global solution in the case of radial symmetry.

Linear stability of slowly rotating Kerr black holes

We prove the linear stability of slowly rotating Kerr black holes as solutions of the Einstein vacuum equations: linearized perturbations of a Kerr metric decay at an inverse polynomial rate to a linearized Kerr metric plus a pure gauge term. We work in a natural wave map/DeTurck gauge and show that the pure gauge term can be taken to lie in a fixed 7-dimensional space with a simple geometric interpretation. Our proof rests on a robust general framework, based on recent advances in microlocal analysis and non-elliptic Fredholm theory, for the analysis of resolvents of operators on asymptotically flat spaces. With the mode stability of the Schwarzschild metric as well as of certain scalar and 1-form wave operators on the Schwarzschild spacetime as an input, we establish the linear stability of slowly rotating Kerr black holes using perturbative arguments; in particular, our proof does not make any use of special algebraic properties of the Kerr metric. The heart of the paper is a detailed description of the resolvent of the linearization of a suitable hyperbolic gauge-fixed Einstein operator at low energies. As in previous work by the second and third authors on the nonlinear stability of cosmological black holes, constraint damping plays an important role. Here, it eliminates certain pathological generalized zero energy states; it also ensures that solutions of our hyperbolic formulation of the linearized Einstein equations have the stated asymptotics and decay for general initial data and forcing terms, which is a useful feature in nonlinear and numerical applications.

TEST OF SIGNIFICANCE FOR HIGH-DIMENSIONAL LONGITUDINAL DATA.

This paper concerns statistical inference for longitudinal data with ultrahigh dimensional covariates. We first study the problem of constructing confidence intervals and hypothesis tests for a low dimensional parameter of interest. The major challenge is how to construct a powerful test statistic in the presence of high-dimensional nuisance parameters and sophisticated within-subject correlation of longitudinal data. To deal with the challenge, we propose a new quadratic decorrelated inference function approach, which simultaneously removes the impact of nuisance parameters and incorporates the correlation to enhance the efficiency of the estimation procedure. When the parameter of interest is of fixed dimension, we prove that the proposed estimator is asymptotically normal and attains the semiparametric information bound, based on which we can construct an optimal Wald test statistic. We further extend this result and establish the limiting distribution of the estimator under the setting with the dimension of the parameter of interest growing with the sample size at a polynomial rate. Finally, we study how to control the false discovery rate (FDR) when a vector of high-dimensional regression parameters is of interest. We prove that applying the Storey (2002)'s procedure to the proposed test statistics for each regression parameter controls FDR asymptotically in longitudinal data. We conduct simulation studies to assess the finite sample performance of the proposed procedures. Our simulation results imply that the newly proposed procedure can control both Type I error for testing a low dimensional parameter of interest and the FDR in the multiple testing problem. We also apply the proposed procedure to a real data example.

Polynomial decay rate of C 0 semigroup in Hilbert spaces

In the paper, we characterize the polynomial stability of semigroup with generator A on Hilbert space . Let A have compact resolvent and there be a sequence of the eigenvectors of A that forms a Riesz basis for . By the asymptotic relation of the real part and imaginary part of eigenvalues of A, we give the optimal decay rate of polynomial stability of . Moreover, we give the zeros distribution of certain equations. As an application, we illustrate our general results by an acoustical system.

Functional regression on the manifold with contamination

Summary We propose a new method for functional nonparametric regression with a predictor that resides on a finite-dimensional manifold, but is observable only in an infinite-dimensional space. Contamination of the predictor due to discrete or noisy measurements is also accounted for. By using functional local linear manifold smoothing, the proposed estimator enjoys a polynomial rate of convergence that adapts to the intrinsic manifold dimension and the contamination level. This is in contrast to the logarithmic convergence rate in the literature of functional nonparametric regression. We also observe a phase transition phenomenon related to the interplay between the manifold dimension and the contamination level. We demonstrate via simulated and real data examples that the proposed method has favourable numerical performance relative to existing commonly used methods.

Convergence rate for eigenvalues of the elastic Neumann–Poincaré operator in two dimensions

In this paper, we consider the Neumann–Poincaré type operator associated with the Lamé system of linear elasticity. It is known that if the boundary of a planar domain is smooth enough, it has eigenvalues converging to two different points determined by Lamé parameters. We show that eigenvalues converge at a polynomial rate on smooth boundaries and the convergence rate is determined by smoothness of the boundary. We also show that they converge at an exponential rate if the boundary of the domain is real analytic.

AN ADAPTIVE TEST OF STOCHASTIC MONOTONICITY

We propose a new nonparametric test of stochastic monotonicity which adapts to the unknown smoothness of the conditional distribution of interest, possesses desirable asymptotic properties, is conceptually easy to implement, and computationally attractive. In particular, we show that the test asymptotically controls size at a polynomial rate, is nonconservative, and detects certain smooth local alternatives that converge to the null with the fastest possible rate. Our test is based on a data-driven bandwidth value and the critical value for the test takes this randomness into account. Monte Carlo simulations indicate that the test performs well in finite samples. In particular, the simulations show that the test controls size and, under some alternatives, is significantly more powerful than existing procedures.

THE GROWTH RATE OF THE DIGITS IN THE LÜROTH EXPANSIONS

For any [Formula: see text], let [Formula: see text] be its Lüroth expansion with digits [Formula: see text]. This paper is concerned with the growth rate of the digits in the Lüroth expansions. Let [Formula: see text] be a function satisfying [Formula: see text] as [Formula: see text] and [Formula: see text]. In this paper, we consider the set [Formula: see text] and we quantify the size of [Formula: see text] in the sense of Hausdorff dimension. As applications, we get the Hausdorff dimensions of the sets of points for which [Formula: see text] grows with polynomial and exponential rate.

Asymptotic Behavior of a Tumor Angiogenesis Model with Haptotaxis

This paper considers the existence and asymptotic behavior of solutions to the angiogenesis system p t = Δ p − ρ ∇ · ( p ∇ w ) + λ p ( 1 − p ) , w t = − γ p w β in a bounded smooth domain Ω ⊂ R N ( N = 1 , 2 ) , where ρ , λ , γ &gt; 0 and β ≥ 1 . More precisely, it is shown that the corresponding solution ( p , w ) converges to ( 1 , 0 ) with an explicit exponential rate if β = 1 , and polynomial rate if β &gt; 1 as t → ∞ , respectively, in L ∞ -norm.

Tunability of auto resonance network

This paper proposes a new type of Artificial Neural Network called Auto-Resonance Network (ARN) derived from synergistic control of biological joints. The network can be tuned to any real valued input without any degradation of learning rate. Neuronal density of the network is low and grows at a linear or low order polynomial rate with input classification. Input coverage of the neuron can be tuned dynamically to match properties of input data. ARN can be used as a part of hierarchical structures to support deep learning applications.

Level shift estimation in the presence of non-stationary volatility with an application to the unit root testing problem

This paper focuses on the estimation of the location of level breaks in time series whose shocks display non-stationary volatility (permanent changes in unconditional volatility). We propose a new feasible weighted least squares (WLS) estimator, based on an adaptive estimate of the volatility path of the shocks. We show that this estimator belongs to a generic class of weighted residual sum of squares which also contains the ordinary least squares (OLS) and WLS estimators, the latter based on the true volatility process. For fixed magnitude breaks we show that the consistency rate of the generic estimator is unaffected by non-stationary volatility. We also provide local limiting distribution theory for cases where the break magnitude is either local-to-zero at some polynomial rate in the sample size or is exactly zero. The former includes the Pitman drift rate which is shown via Monte Carlo experiments to predict well the key features of the finite sample behaviour of both the OLS and our feasible WLS estimators. The simulations highlight the importance of the break location, break magnitude, and the form of non-stationary volatility for the finite sample performance of these estimators, and show that our proposed feasible WLS estimator can deliver significant improvements over the OLS estimator under heteroskedasticity. We discuss how these results can be applied, by using level break fraction estimators on the first differences of the data, when testing for a unit root in the presence of trend breaks and/or non-stationary volatility. Methods to select between the break and no break cases, using standard information criteria and feasible weighted information criteria based on our adaptive volatility estimator, are also discussed. Simulation evidence suggests that unit root tests based on these weighted quantities can display significantly improved finite sample behaviour under heteroskedasticity relative to their unweighted counterparts. An empirical illustration to U.S. and U.K. real GDP is also considered.