Precise Asymptotic Formula Research Articles

An Asymptotic Integral Representation for Carleman Orthogonal Polynomials

In this paper, we investigate the asymptotic behavior of polynomials that are orthonormal over the interior domain of an analytic Jordan curve L with respect to area measure. We prove that, inside L, these polynomials behave asymptotically like a sequence of certain integrals involving the canonical conformal map of the exterior of L onto the exterior of the unit circle and certain meromorphic kernel function defined in terms of a conformal map of the interior of L onto the unit disk. We then use this result to obtain more precise asymptotic formulas for the polynomials under certain additional geometric considerations. These formulas yield, in turn, fine results on the location, limiting distribution, and accumulation points of the zeros of the polynomials.

Local structure of bifurcation curves for nonlinear Sturm–Liouville problems

We consider the nonlinear bifurcation problem arising in population dynamics and nonlinear Schrödinger equation: − u ″ ( t ) = f ( λ , u ( t ) ) , u ( t ) > 0 , t ∈ I : = ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 , where λ > 0 is a parameter. We mainly treat the case where f ( u ) = λ u ± u p ( p > 1 ) and establish the precise asymptotic expansion formulas for the bifurcation curve near the bifurcation point λ = π 2 in L q -framework. Together with the result of the global behavior of the bifurcation curve, we understand completely the structure of the bifurcation curve. We also consider the nodal solution u n , λ of the equation − u ″ ( t ) = λ ( u ( t ) + | u ( t ) | p − 1 u ( t ) ) with u ( 0 ) = u ( π ) = 0 and establish an asymptotic expansion formula for λ with respect to the gradient norm of the solution associated with λ as λ → n 2 .

On the Maximum Exceedance of a Sequence of Random Variables Over a Renewal Threshold

In this paper we study the tail behavior of the maximum exceedance of a sequence of independent and identically distributed random variables over a random walk. For both light-tailed and heavy-tailed cases, we derive a precise asymptotic formula, which extends and unifies some existing results in the recent literature of applied probability.

On the Maximum Exceedance of a Sequence of Random Variables Over a Renewal Threshold

In this paper we study the tail behavior of the maximum exceedance of a sequence of independent and identically distributed random variables over a random walk. For both light-tailed and heavy-tailed cases, we derive a precise asymptotic formula, which extends and unifies some existing results in the recent literature of applied probability.

Intrinsic ultracontractivity of a Schrödinger semigroup in [formula omitted

We give a (possibly sharp) sufficient condition on the electric potential q : R N → [ 0 , ∞ ) in the Schrödinger operator A = − Δ + q ( x ) • on L 2 ( R N ) that guarantees that the Schrödinger heat semigroup { e − A t : t ⩾ 0 } on L 2 ( R N ) generated by − A is intrinsically ultracontractive. Moreover, if q ( x ) ≡ q ( | x | ) is radially symmetric, we show that our condition on q is also necessary (i.e., truly sharp); it reads ∫ r 0 ∞ q ( r ) − 1 / 2 d r < ∞ for some r 0 ∈ ( 0 , ∞ ) . Our proofs make essential use of techniques based on a logarithmic Sobolev inequality, Rosen's inequality (proved via a new Fenchel–Young inequality), and a very precise asymptotic formula due to Hartman and Wintner.

Spectral asymptotics for inverse nonlinear Sturm-Liouville problems

We consider the nonlinear Sturm-Liouville problem −u ′′ (t) + f(u(t),u ′ (t)) = �u(t), u(t) > 0, t ∈ I := (−1/2,1/2), u(±1/2) = 0, where f(x,y) = |x| p−1 x − |y| m , p > 1,1 ≤ m 0 is an eigenvalue parameter. To understand well the global structure of the bifurcation branch of positive solutions in R+ ×L q (I) (1 ≤ q < ∞) from a viewpoint of inverse problems, we establish the precise asymptotic formulas for the eigenvalue � = �q(�) as � :=

Eigenvalue Asymptotics of the Even‐Dimensional Exterior Landau‐Neumann Hamiltonian

We study the Schrödinger operator with a constant magnetic field in the exterior of a compact domain in ℝ2d, d ≥ 1. The spectrum of this operator consists of clusters of eigenvalues around the Landau levels. We give asymptotic formulas for the rate of accumulation of eigenvalues in these clusters. When the compact is a Reinhardt domain we are able to show a more precise asymptotic formula.

On the Instability Intervals of the Mathieu–Hill Operator

Consider the Mathieu–Hill operator $${Ly=-y^{\prime\prime}+(2h {\rm cos}2x)y,\quad -\infty < x < +\infty,}$$ in \({L_{2}({\mathfrak{R}})}\) , where \({h \in {\mathfrak{R}}\setminus\{0\}}\) . We obtain the precise asymptotic formulas for the widths γk of the instability intervals of L. The formula states the isolated terms of arbitrary number in the asymptotics of the sequence γk for large k and verifies the results of Harrell (Am J Math suppl:139–150, 1981) and Avron and Simon (Ann Phys 134:76–84, 1981).

New asymptotic formula for eigenvalues of nonlinear Sturm-Liouville problems

We study the nonlinear Sturm-Liouville problem $$ - u''(t) + f(u(t)) = \lambda u(t), u(t) > 0, t \in {\rm I}: = (0,1), u(0) = u(1) = 0,$$ where λ > 0 is an eigenvalue parameter and f(u) is a rapidly increasing function. For better understanding of the global behavior of the bifurcation branch in R+ × L 2(I), we establish precise asymptotic formulas up to the third term for the eigenvalue λ(α) associated with the eigenfunction u α with ‖u α‖2 = α, as α → ∞. We show that there exists a new type of asymptotic formula for λ (α) as α → ∞.

Precise Asymptotic Shapes of Solutions to Nonlinear Two-Parameter Problems

We consider the nonlinear two-parameter problem, which comes from a perturbed simple pendulum problem $$-u^{\prime\prime}(t)+\mu f(u(t)) = \lambda g(u(t)), t \in I: = (-T, T), u(t) > 0,\quad t \in I,\quad u(\pm T) = 0,$$ where μ, λ > 0 are parameters and T > 0 is a constant. For a given μ > 0, there exists a solution triple (\(\mu, \lambda(\mu), u_{\mu}) \in \rm{R}^{2}_{+}\times C^{2}(\bar{I}),\) which is obtained by a variational method, such that uμ is almost flat inside I and develops boundary layers as \(\mu \rightarrow \infty\). We establish the precise asymptotic formulas for \(\|u_{\mu}\|_{q}(1 \leq q \leq \infty), u^{\prime}_{\mu} (\pm T)\) and the variational eigencurve λ(μ) as \(\mu \rightarrow \infty\). By these formulas, we understand well not only the local behavior of uμ as \(\mu \rightarrow \infty\), but also the total shape of uμ. Furthermore, we find the precise asymptotics of \(\|u_{\mu}\|_{q}\) as \(q \rightarrow \infty\). By this, we understand well how \(\|u_{\mu}\|_{q}\) tends to \(\|u_{\mu}\|_{\infty}\) as \(q \rightarrow \infty\).

Global behavior of the branch of positive solutions for nonlinear Sturm–Liouville problems

We consider the nonlinear Sturm–Liouville problem $$ -u''(t) + f(u(t)) = \lambda u(t),\quad u(t) > 0, \quad t \in I := (0, 1),\quad u(0) = u(1) = 0, $$ (1) where λ > 0 is an eigenvalue parameter. To understand well the global behavior of the bifurcation branch in R+ × L2(I), we establish the precise asymptotic formula for λ(α), which is associated with eigenfunction uα with ‖ uα ‖2 = α, as α → ∞. It is shown that if for some constant p > 1 the function h(u) ≔ f(u)/up satisfies adequate assumptions, including a slow growth at ∞, then λ(α) ∼ αp−1h(α) as α → ∞ and the second term of λ(α) as α → ∞ is determined by limu → ∞uh′(u).

Counting $d$-Polytopes with $d+3$ Vertices

We completely solve the problem of enumerating combinatorially inequivalent $d$-dimensional polytopes with $d+3$ vertices. A first solution of this problem, by Lloyd, was published in 1970. But the obtained counting formula was not correct, as pointed out in the new edition of Grünbaum's book. We both correct the mistake of Lloyd and propose a more detailed and self-contained solution, relying on similar preliminaries but using then a different enumeration method involving automata. In addition, we introduce and solve the problem of counting oriented and achiral (i.e., stable under reflection) $d$-polytopes with $d+3$ vertices. The complexity of computing tables of coefficients of a given size is then analyzed. Finally, we derive precise asymptotic formulas for the numbers of $d$-polytopes, oriented $d$-polytopes and achiral $d$-polytopes with $d+3$ vertices. This refines a first asymptotic estimate given by Perles.

$L^q$ spectral asymptotics for nonlinear Sturm-Liouville problems

We consider the nonlinear Sturm-Liouville problem $$ -u''(t) + f(u(t)) = \lambda u(t), \ \ u(t) > 0, \quad t \in I := (0, 1), \ \ u(0) = u(1) = 0, $$ where $\lambda > 0$ is an eigenvalue parameter. For better understanding of the global behavior of the branch of positive solutions in $\mbox{\bf R}_+ \times L^q(I)$ ($1 \le q \le \infty$), we establish precise asymptotic formulas for the eigenvalue $\lambda$ with respect to $\Vert u_\lambda\Vert_q$, where $u_\lambda$ is the unique solution associated with given $\lambda > \pi^2$.

Boundary layer and variational eigencurve in two-parameter single pendulum type equations

We consider the nonlinear two-parameter single pendulum type equation $-u''(t) + \mu f(u(t)) = \lambda\sin u(t), t \in I$ :$= (-T, T), u(t) > 0, t \in I, u(\pm T) = 0$, where $T > 0$ is a constant and $\mu, \lambda > 0$ are parameters. For a given $\mu > 0$, there exists a solution triple $(\mu, \lambda(\mu), u_\mu) \in \mbox{\bf R}_+^2 \times C^2(\bar{I})$, which is obtained by a variational method, such that $u_\mu$ develops a boundary layer as $\mu \to \infty$. We establish the precise asymptotic formulas for $||u_\mu||_\infty, u_\mu'(\pm T)$ and the variational eigencurve $\lambda(\mu)$ as $\mu \to \infty$.

Asymptotic Distribution of Eigenvalues for Damped String Equation: Numerical Approach

In the present paper, we consider a one-parameter family of the nonself-adjoint operators, which are the dynamics generators for systems governed by the wave equations containing dissipative terms. The equations contain viscous damping terms and are equipped with the boundary conditions involving an arbitrary complex parameter. In the current engineering literature, this type of boundary condition is used to model the action of smart materials (self-sensing/self-straining actuators). In the previous research of the first writer, the aforementioned dynamics generators have been studied analytically and precise asymptotic formulas for the eigenvalues have been derived (the asymptotic when the number of the eigenvalues tends to infinity). The goal of the present paper is to demonstrate that the analytic formulas are not only important theoretically, but also extremely efficient practically. Namely, we show that the leading terms in the asymptotic formulas approximate the actual eigenvalues with excellent acc...

Variational method for precise asymptotic formulas for nonlinear eigenvalue problems

We consider the nonlinear eigenvalue problem motivated by the perturbed elliptic sine-Gordon equation where p >1 is a constant and λ ∈ R is an eigenvalue parameter. Our aim is to clarify the asymptotic relationship between L p+1-norm, L ∞-norm of the solutions and λ when these norms are large. To this end, we consider an associated variational problem with this equation on a manifold new parameter), and obtain a solution pair . We establish the precise asymptotic formulas for .

Asymptotic expansion of the variational eigencurve for two-parameter simple pendulum-type equations

We consider the nonlinear two-parameter simple pendulum-type equation −u″(t)+μu p=λ sin u(t), u(t)>0, t∈I=(−T,T), u(±T)=0, where p>1 is a constant and μ, λ>0 are parameters. For a given μ>0, there exists a solution triple (μ,λ(μ),u μ)∈ R + 2×C 2( I ̄ ) , which is obtained by a variational method, such that || u μ || ∞→0 as μ→∞. We establish the precise asymptotic formula for the variational eigencurve λ( μ) as μ→∞, whose growth order is sublinear.

PRECISE SPECTRAL ASYMPTOTICS FOR LOGISTIC EQUATIONS OF POPULATION DYNAMICS

AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Abstract The semilinear elliptic eigenvalue problem with superlinear pure power nonlinearity is considered. This problem is treated from the standpoint of L2-theory and the precise asymptotic formula for the eigenvalue parameter λ = λ(α) as α → ∞ is established, where α is the L2-norm of the solution u associated with λ. 2000 Mathematics Subject Classification 35P30 (primary), 35J60 (secondary). Citing Literature Volume36, Issue1January 2004Pages 59-64 RelatedInformation

Asymptotic Formulas for Boundary Layers and Eigencurves for Nonlinear Elliptic Eigenvalue Problems

Nonlinear elliptic eigenvalue problem −Δu + up = λuin Ω u > 0in Ω, u = 0on ∂Ωis studied, where (N ≥ 2) is a bounded appropriately smooth domain, p > 1is a constant and λ > 0is an eigenvalue parameter. We establish the precise asymptotic formulas for the boundary layers of uλ as λ→∞. Moreover, we establish the precise asymptotic formulas for eigencurve λ(α)(associated with eigenfunction uα with ) as α→ ∞. In particular, the exact second term of “the width of the boundary layers” () (λ →∞) and λ(α)(α →∞) are obtained when Ωare a ball and an annulus.

Ornstein-Zernike theory for the Bernoulli bond percolation on $\mathbb{Z}^d$

We derive a precise Ornstein–Zernike asymptotic formula for the decay of the two-point function $\mathbb{P}_p (0 \leftrightarrow x)$ of the Bernoulli bond percolation on the integer lattice $\mathbb{Z}^d$ in any dimension $d \geq 2$, in any direction $x$ and for any subcritical value of $p < p_c (d)$.