Exact Decay Rate Research Articles

Existence and properties of ancient solutions of the Yamabe flow

Let $n≥ 3$ and $m=\frac{n-2}{n+2}$. We construct $5$-parameters, $4$-parameters, and $3$-parameters ancient solutions of the equation $v_t=(v^m)_{xx}+v-v^m$, $v>0$, in $\mathbb{R}× (-∞, T)$ for some $T∈\mathbb{R}$. This equation arises in the study of Yamabe flow. We obtain various properties of the ancient solutions of this equation including exact decay rate of ancient solutions as $|x|\to∞$. We also prove that both the $3$-parameters ancient solution and the $4$-parameters ancient solution are singular limit solution of the $5$-parameters ancient solutions. We also prove the uniqueness of the $4$-parameters ancient solutions. As a consequence we prove that the $4$-parameters ancient solutions that we construct coincide with the $4$-parameters ancient solutions constructed by P. Daskalopoulos, M. del Pino, J. King, and N. Sesum in [ 8 ].

Decay rates for the quadratic and super-quadratic tilt-excess of integral varifolds

This paper concerns integral varifolds of arbitrary dimension in an open subset of Euclidean space satisfying integrability conditions on their first variation. Firstly, the study of pointwise power decay rates almost everywhere of the quadratic tilt-excess is completed by establishing the precise decay rate for two-dimensional integral varifolds of locally bounded first variation. In order to obtain the exact decay rate, a coercive estimate involving a height-excess quantity measured in Orlicz spaces is established. Moreover, counter-examples to pointwise power decay rates almost everywhere of the super-quadratic tilt-excess are obtained. These examples are optimal in terms of the dimension of the varifold and the exponent of the integrability condition in most cases, for example if the varifold is not two-dimensional. These examples also demonstrate that within the scale of Lebesgue spaces no local higher integrability of the second fundamental form, of an at least two-dimensional curvature varifold, may be deduced from boundedness of its generalised mean curvature vector. Amongst the tools are Cartesian products of curvature varifolds.

Traveling wave fronts in a delayed lattice competitive system

This paper is concerned with the existence, asymptotic behavior, strict monotonicity, and uniqueness of traveling wave fronts connecting two half-positive equilibria in a delayed lattice competitive system. We first prove the existence of traveling wave fronts by constructing upper and lower solutions and Schauder’s fixed point theorem, and then, for sufficiently small intraspecific competitive delays, prove that these traveling wave fronts decay exponentially at both infinities. Furthermore, for system without intraspecific competitive delays, the strict monotonicity and uniqueness of traveling wave fronts are established by means of the sliding method. In addition, we give the exact decay rate of the stronger competitor under some technique conditions by appealing to uniqueness.

Exact Random Coding Secrecy Exponents for the Wiretap Channel

We analyze the exact exponential decay rate of the expected amount of information leaked to the wiretapper in Wyner's wiretap channel setting using wiretap channel codes constructed from both i.i.d. and constant-composition random codes. Our analysis for those sampled from i.i.d. random coding ensemble shows that the previously-known achievable secrecy exponent using this ensemble is indeed the exact exponent for an average code in the ensemble. Furthermore, our analysis on wiretap channel codes constructed from the ensemble of constant-composition random codes leads to an exponent which, in addition to being the exact exponent for an average code, is larger than the achievable secrecy exponent that has been established so far in the literature for this ensemble (which in turn was known to be smaller than that achievable by wiretap channel codes sampled from i.i.d. random coding ensemble). We show examples where the exact secrecy exponent for the wiretap channel codes constructed from random constant-composition codes is larger than that of those constructed from i.i.d. random codes and examples where the exact secrecy exponent for the wiretap channel codes constructed from i.i.d. random codes is larger than that of those constructed from constant-composition random codes. We, hence, conclude that, unlike the error correction problem, there is no general ordering between the two random coding ensembles in terms of their secrecy exponent.

Exact and memory-dependent decay rates to the non-hyperbolic equilibrium of differential equations with unbounded delay and maximum functional

In this paper, we obtain the exact rates of decay to the non--hyperbolic equilibrium of the solution of a functional differential equation with maxima and unbounded delay. We study the convergence rates for both locally and globally stable solutions. We also give examples showing how the rate of growth of decay of solutions depends on the rate of growth of the unbounded delay as well as the nonlinearity local to the equilibrium.

Asymptotic behavior and uniqueness of traveling wave fronts in a competitive recursion system

In this paper, we consider the asymptotic behavior and uniqueness of traveling wave fronts connecting two half-positive equilibria in a competitive recursion system. We first prove that these traveling wave fronts have the exponential decay rates at the minus/plus infinity, where for a given wave speed, there are three possible asymptotic behaviors for the first component of the wave profile at the plus infinity and the second component of the wave profile at the minus infinity, respectively. And then we use the sliding method to prove the uniqueness of traveling wave fronts for this system. Furthermore, by combining uniqueness and upper and lower solutions technique, we also give the exact decay rate of the weaker competitor under certain conditions.

Asymptotic behavior and uniqueness of traveling wave fronts in a delayed nonlocal dispersal competitive system

This paper is concerned with the asymptotic behavior and uniqueness of traveling wave fronts connecting two half-positive equilibria in a delayed nonlocal dispersal competitive system. We first prove the existence results by applying abstract theories. And then, we show that the traveling wave fronts decay exponentially at both infinities. At last, the strict monotonicity and uniqueness of traveling wave fronts are obtained by using the sliding method in the absent of intraspecific competitive delays. Based on the uniqueness, the exact decay rate of the stronger competitor is established under certain conditions.

Finding the exact decay rate of all solutions to some second order evolution equations with dissipation

We consider an abstract second order evolution equation with damping. The “elastic” term is represented by a self-adjoint nonnegative operator A with discrete spectrum, and the nonlinear term has order greater than one at the origin. We investigate the asymptotic behavior of solutions.We prove the coexistence of slow solutions and fast solutions. Slow solutions live close to the kernel of A, and decay as negative powers of t as solutions of the first order equation obtained by neglecting the operator A and the second order time-derivatives in the original equation. Fast solutions live close to the range of A and decay exponentially as solutions of the linear homogeneous equation obtained by neglecting the nonlinear terms in the original equation.The abstract results apply to semilinear dissipative hyperbolic equations.

Decay rates for approximation numbers of composition operators

A general method for estimating the approximation numbers of composition operators on the Hardy space H 2, using finite-dimensional model subspaces, is studied and applied in the case when the symbol of the operator maps the unit disc to a domain whose boundary meets the unit circle at just one point. The exact rate of decay of the approximation numbers is identified when this map is sufficiently smooth at the point of tangency; it follows that a composition operator with any prescribed slow decay of its approximation numbers can be explicitly constructed. Similarly, an asymptotic expression for the approximation numbers is found when the mapping has a sharp cusp at the distinguished boundary point. Precise asymptotic estimates in the intermediate cases, including that of maps with a corner at the distinguished boundary point, are also established.

Exact decay rate of a nonlinear elliptic equation related to the Yamabe flow

Let 0 2, $\alpha=(2\beta +\rho)/(1-m)$ and $\beta>m\rho/(n-2-mn)$ for some constant $\rho>0$. Suppose v is a radially symmetric symmetric solution of $\frac{n-1}{m}\Delta v^m+\alpha v+\beta x\cdot\nabla v=0$, v>0, in $R^n$. When m=(n-2)/(n+2), the metric $g=v^{4/(n+2)}dx^2$ corresponds to a locally conformally flat Yamabe shrinking gradient soliton with positive sectional curvature. We prove that the solution $v$ of the above nonlinear elliptic equation has the exact decay rate $\lim_{r\to\infty}r^2v(r)^{1-m}=\frac{2(n-1)(n(1-m)-2)}{(1-m)(\alpha (1-m)-2\beta)}$.

L 2-Decay rate for non-ergodic Jackson network

We establish the additive theorem of L2-decay rate for multidimensional Markov process with independent marginal processes. Using this and the decomposition method, we obtain explicit upper and lower bounds for decay rate of non-ergodic Jackson network. In some cases, we get the exact decay rate.

Approximation numbers of Sobolev embeddings—Sharp constants and tractability

We investigate optimal linear approximations (approximation numbers) in the context of periodic Sobolev spaces Hs(Td) of fractional smoothness s>0 for various equivalent norms including the classical one. The error is always measured in L2(Td). Particular emphasis is given to the dependence of all constants on the dimension d. We capture the exact decay rate in n and the exact decay order of the constants with respect to d, which is in fact polynomial. As a consequence we observe that none of our considered approximation problems suffers from the curse of dimensionality. Surprisingly, the square integrability of all weak derivatives up to order three (classical Sobolev norm) guarantees weak tractability of the associated multivariate approximation problem.

Sharp decay estimates of [formula omitted]-norms for nonnegative Schrödinger heat semigroups

Let H=−Δ+V be a nonnegative Schrödinger operator on L2(RN), where N⩾3 and V is a radially symmetric nonpositive function in RN decaying quadratically at the space infinity. For any 1⩽p⩽q⩽∞, we denote by ‖e−tH‖q,p the operator norm of the Schrödinger heat semigroup e−tH from Lp(RN) to Lq(RN). In this paper, under suitable conditions on V, we give the exact and optimal decay rates of ‖e−tH‖q,p as t→∞ for all 1⩽p⩽q⩽∞. The decay rates of ‖e−tH‖q,p depend on whether the operator H is subcritical or critical and on the behavior of the positive harmonic function for the operator H.

Kalman Filtering With Intermittent Observations: Tail Distribution and Critical Value

In this paper, we analyze the performance of Kalman filtering for discrete-time linear Gaussian systems, where packets containing observations are dropped according to a Markov process modeling a Gilbert-Elliot channel. To address the challenges incurred by the loss of packets, we give a new definition of non-degeneracy, which is essentially stronger than the classical definition of observability, but much weaker than one-step observability, which is usually used in the study of Kalman filtering with intermittent observations. We show that the trace of the Kalman estimation error covariance under intermittent observations follows a power decay law. Moreover, we are able to compute the exact decay rate for non-degenerate systems. Finally, we derive the critical value for non-degenerate systems based on the decay rate, improving upon the state of the art.

Singular limit and exact decay rate of a nonlinear elliptic equation

For any n≥3, 0<m≤(n−2)/n, and constants η>0, β>0, α≤β(n−2)/m, we prove the existence of radially symmetric solution of n−1mΔvm+αv+βx⋅∇v=0, v>0, in Rn, v(0)=η, without using the phase plane method. When 0<m<(n−2)/n, n≥3, we prove that v satisfies lim|x|→∞|x|2v(x)1−mlog|x|=2(n−1)(n−2−nm)β(1−m) if α=2β/(1−m)>0 and lim|x|→∞|x|α/βv(x)=A for some constant A>0 if 2β/(1−m)>max(α,0). For β>0 or α=0, we prove that the radially symmetric solution v(m) of the above elliptic equation converges uniformly on every compact subset of Rn to the solution of an elliptic equation as m→0.

Formula omitted] norms of nonnegative Schrödinger heat semigroup and the large time behavior of hot spots

This paper is concerned with the Cauchy problem for the heat equation with a potential(P){∂tu=Δu−V(|x|)uin RN×(0,∞),u(x,0)=ϕ(x)in RN, where ∂t=∂/∂t, N⩾3, ϕ∈L2(RN), and V=V(|x|) is a smooth, nonpositive, and radially symmetric function having quadratic decay at the space infinity. In this paper we assume that the Schrödinger operator H=−Δ+V is nonnegative on L2(RN), and give the exact power decay rates of Lq norm (q⩾2) of the solution e−tHϕ of (P) as t→∞. Furthermore we study the large time behavior of the solution of (P) and its hot spots.

Long memory in a linear stochastic Volterra differential equation

In this paper we consider a linear stochastic Volterra equation which has a stationary solution. We show that when the kernel of the fundamental solution is regularly varying at infinity with a log-convex tail integral, then the autocovariance function of the stationary solution is also regularly varying at infinity and its exact pointwise rate of decay can be determined. Moreover, it can be shown that this stationary process has either long memory in the sense that the autocovariance function is not integrable over the reals or is subexponential. Under certain conditions upon the kernel, even arbitrarily slow decay rates of the autocovariance function can be achieved. Analogous results are obtained for the corresponding discrete equation.

The power law for the Buffon needle probability of the four-corner Cantor set

Let $\mathcal {C}_n$ be the $n$th generation in the construction of the middle-half Cantor set. The Cartesian square $\mathcal {K}_n$ of $\mathcal {C}_n$ consists of $4^n$ squares of side length $4^{-n}$. The chance that a long needle thrown at random in the unit square will meet $\mathcal {K}_n$ is essentially the average length of the projections of $\mathcal {K}_n$, also known as the Favard length of $\mathcal {K}_n$. A classical theorem of Besicovitch implies that the Favard length of $\mathcal {K}_n$ tends to zero. It is still an open problem to determine its exact rate of decay. Until recently, the only explicit upper bound was $\exp (- c\log _* n)$, due to Peres and Solomyak ($\log _* n$ is the number of times one needs to take the log to obtain a number less than $1$, starting from $n$). In the paper, a power law bound is obtained by combining analytic and combinatorial ideas.

L p -weak regularity and asymptotic behavior of solutions for critical equations with singular potentials on Carnot groups

In this paper, we consider a large class of critical problems with singular potentials on Carnot groups. In particular, we prove sharp regularity results in Marcinkiewicz spaces and the exact rate of decay of the solutions. Our general results apply, for instance, to the equation satisfied by the extremals of some relevant Hardy–Sobolev type inequalities on Stratified groups.

Outlets of 2D invasion percolation and multiple-armed incipient infinite clusters

We study invasion percolation in two dimensions, focusing on properties of the outlets of the invasion and their relation to critical percolation and to incipient infinite clusters (IIC's). First we compute the exact decay rate of the distribution of both the weight of the kth outlet and the volume of the kth pond. Next we prove bounds for all moments of the distribution of the number of outlets in an annulus. This result leads to almost sure bounds for the number of outlets in a box B(2^n) and for the decay rate of the weight of the kth outlet to p_c. We then prove existence of multiple-armed IIC measures for any number of arms and for any color sequence which is alternating or monochromatic. We use these measures to study the invaded region near outlets and near edges in the invasion backbone far from the origin.