Probabilities of maximal deviations for nonparametric regression function estimates

Let (X, Y) have regression function m(x) = E(Y | X = x), and let X have a marginal density f₁(x). We consider two nonparameteric estimates of m(x): the Watson estimate when f₁ is known and the Yang estimate when f₁ is known or unknown. For both estimates the asymptotic distribution of the maximal deviation from m(x) is proved, thus extending results of Bickel and Rosenblatt for the estimation of density functions.     

Error Estimates for Two Filters Based on Polynomial Interpolation for Recovering a Function from Its Fourier Coefficients

In this paper we derive error estimates for two filters based on piecewise polynomial interpolations of zeroth and first degrees. For a piecewise smooth function f(x) in [0,1], we show that, if all the discontinuity points of f(x) are nodes then, using these filters, we can reconstruct point values of f(x) accurately even near discontinuity points. If f(x) is a piecewise constant or a linear function, the reconstruction formulas are exact. We also propose reconstruction formulas such that we can compute the (approximate ) point values of f(x) using the fast Fourier transform, even when using non-uniform meshes. Several numerical experiments are also provided to illustrate the results.     

Weighted stationary phase of higher orders

The subject matter of this paper is an integral with exponential oscillation of phase f(x) weighted by g(x) on a finite interval [α β]: When the phase f(x) has a single stationary point in (α β), an nth-order asymptotic expansion of this integral is proved for n ≥ 2: This asymptotic expansion sharpens the classical result for n = 1 by M. N. Huxley. A similar asymptotic expansion was proved by V. Blomer, R. Khan and M. Young under the assumptions that f(x) and g(x) are smooth and g(x) is compactly supported on R: In the present paper, however, these functions are only assumed to be continuously differentiable on [α β] 2n + 3 and 2n + 1 times, respectively. Because there are no requirements on the vanishing of g(x) and its derivatives at the endpoints α and β, the present asymptotic expansion contains explicit boundary terms in the main and error terms. The asymptotic expansion in this paper is thus applicable to a wider class of problems in analysis, analytic number theory, and other fields.     

Asymptotic estimates of sums involving the Moebius function

Let p(n) denote the smallest prime factor of an integer n>1 and let p(1)=∞. We study the asymptotic behavior of the sum M(x,y)=Σ_{1≤n≤x,p(n)>y}μ(n) and use this to estimate the size of A(x)=max_|f|≤1|Σ_2≤n<xμ(n)f(p(n))|, where μ(n) is the Moebius function. Applications of bounds for A(x), M(x,y) and similar quantities are discussed.     

Hazard rate estimation on random fields

Consider observations (representing lifelengths) taken on a random field indexed by lattice points. Our purpose is to estimate the hazard rate r(x), which is the rate of failure at time x for the survivors up to time x. We estimate r(x) by the nonparametric estimator constructed in terms of a kernel-type estimator for f(x) and the natural estimator for . Under some general mixing assumptions, the limiting distribution of the estimator at multiple points is shown to be multivariate normal. The result is useful in establishing confidence bands for r(x) with x in an interval.     

Stability condition for functional equation through fixed point in the large

Stability and asymptotic stability conditions for the solution of functional equation of the formφ(x)=f(x)+g(x)φ[α(x)] through the method of fixed point in the large has been investigated.     

Recurrent points and non-wandering points of graph maps

Let G be a graph and f:G→G be a continuous map. Denote by P(f), R(f) and Ω(f) the sets of periodic points, recurrent points and non-wandering points of f, respectively. In this paper we show that: (1) If L=(x,y) is an open arc contained in an edge of G such that {fm(x),fk(y)}⊂(x,y) for some m,k∈N, then R(f)∩(x,y)≠∅; (2) Any isolated point of P(f) is also an isolated point of Ω(f); (3) If x∈Ω(f)−Ω(fn) for some n∈N, then x is an eventually periodic point. These generalize the corresponding results in W. Huang and X. Ye (2001) [9] and J. Xiong (1983, 1986) [17] and [19] on interval maps or tree maps.     

A note on positive evanescent solutions for a certain class of elliptic problems

This paper is devoted to the existence and properties of solutions of the following class of nonlinear elliptic differential equations Δu(x)+f(x,u(x))+g(‖x‖)x⋅∇u(x)=0, x∈Rn, ‖x‖>R. We prove existence of positive solutions vanishing at positive infinity. Our approach is based on the subsolution and supersolution method. The nonlinearity f covers both sublinear and superlinear cases and does not necessarily satisfy f(x,0)≡0. The asymptotic behavior of solutions is also described.     

On an interpolatory product rule for evaluating Cauchy principal value integrals

Convergence results for interpolatory product rules for evaluating Cauchy principal value integrals of the form f _?1 ¹ v(x)f(x)/x ? λ dx wherev is an admissible weight function have been extended to integrals of the form f _?1 ¹ k(x)f(x)/x ? λ dx wherek is an arbitrary integrable function subject to certain conditions. Further, whereas the above convergence results were shown when the interpolation points were the Gauss points with respect to some admissible weight functionw, they are now shown to hold when the interpolation points are Radau or Lobatto points with respect tow.     

The least common multiple of a sequence of products of linear polynomials

Let f(x) be the product of several linear polynomials with integer coefficients. In this paper, we obtain the estimate log?lcm?(f(1),…,f(n))～An as n→∞, where A is a constant depending on?f.     

On the precise growth, covering, and distortion theorems for normalized biholomorphic mappings

In this paper, we consider a normalized biholomorphic mapping f(x) defined on the unit ball in a complex Banach space, where the origin 0 is a zero of order k+1 of f(x)−x. The precise growth and covering theorem for f(x) is obtained when f(x) is a starlike mapping or a starlike mapping of order α. Especially, the precise growth and covering theorem for f(x) is also established when f(x) is a quasi-convex mapping. Moreover, the precise distortion theorem for f(x) is given when f(x) is a convex mapping. Our result includes many known results.     

A second-order estimate for blow-up solutions of elliptic equations

We investigate second-term asymptotic behavior of boundary blow-up solutions to the problems Δu=b(x)f(u), x∈Ω, subject to the singular boundary condition u(x)=∞, in a bounded smooth domain Ω⊂R^N. b(x) is a non-negative weight function. The nonlinearly f is regularly varying at infinity with index ρ>1 (that is lim_u→∞f(ξu)/f(u)=ξ^ρ for every ξ>0) and the mapping f(u)/u is increasing on (0,+∞). The main results show how the mean curvature of the boundary ∂Ω appears in the asymptotic expansion of the solution u(x). Our analysis relies on suitable upper and lower solutions and the Karamata regular variation theory.     

A three-point Taylor algorithm for three-point boundary value problems

We consider second-order linear differential equations φ(x)y^″+f(x)y^′+g(x)y=h(x) in the interval (−1,1) with Dirichlet, Neumann or mixed Dirichlet-Neumann boundary conditions given at three points of the interval: the two extreme points x=±1 and an interior point x=s∈(−1,1). We consider φ(x), f(x), g(x) and h(x) analytic in a Cassini disk with foci at x=±1 and x=s containing the interval [−1,1]. The three-point Taylor expansion of the solution y(x) at the extreme points ±1 and at x=s is used to give a criterion for the existence and uniqueness of the solution of the boundary value problem. This method is constructive and provides the three-point Taylor approximation of the solution when it exists. We give several examples to illustrate the application of this technique.     

Uniqueness results and asymptotic behavior of solutions with boundary blow-up for logistic-type porous media equations

Under the proper structure conditions on the nonlinear term f(u) and weight function b(x), the paper shows the uniqueness and asymptotic behavior near the boundary of boundary blow-up solutions to the porous media equations of logistic type ?Δu = a(x)u ^1/m ? b(x)f(u) with m > 1.     

Heat flow and Hardy inequality in complete Riemannian manifolds with singular initial conditions

Upper bounds are obtained for the heat content of an open set D with singular initial condition f on a complete Riemannian manifold, provided (i) the Dirichlet-Laplace-Beltrami operator satisfies a strong Hardy inequality, and (ii) f satisfies an integrability condition. Precise asymptotic results for the heat content are obtained for an open bounded and connected set D in Euclidean space with C² boundary, and with initial condition f(x)=δ(x)^−α,0<α<2, where δ(x) is the distance from x to the boundary of D.     

The stability of a cubic type functional equation with the fixed point alternative

In this note we investigate the generalized Hyers-Ulam-Rassias stability for the new cubic type functional equation f(x+y+2z)+f(x+y−2z)+f(2x)+f(2y)=2[f(x+y)+2f(x+z)+2f(x−z)+2f(y+z)+2f(y−z)] by using the fixed point alternative. The first systematic study of fixed point theorems in nonlinear analysis is due to G. Isac and Th.M. Rassias [Internat. J. Math. Math. Sci. 19 (1996) 219-228].     

Multi-layered stationary solutions for a spatially inhomogeneous Allen-Cahn equation

We consider stationary solutions of a spatially inhomogeneous Allen-Cahn-type nonlinear diffusion equation in one space dimension. The equation involves a small parameter ε, and its nonlinearity has the form h(x)²f(u), where h(x) represents the spatial inhomogeneity and f(u) is derived from a double-well potential with equal well-depth. When ε is very small, stationary solutions develop transition layers. We first show that those transition layers can appear only near the local minimum and local maximum points of the coefficient h(x) and that at most a single layer can appear near each local minimum point of h(x). We then discuss the stability of layered stationary solutions and prove that the Morse index of a solution coincides with the total number of its layers that appear near the local maximum points of h(x). We also show the existence of stationary solutions having clustering layers at the local maximum points of h(x).     

Transformation methods for finding multiple roots of nonlinear equations

In this paper, to estimate a multiple root p of an equation f(x) = 0, we transform the function f(x) to a hyper tangent function combined with a simple difference formula whose value changes from −1 to 1 as x passes through the root p. Then we apply the so-called numerical integration method to the transformed equation, which may result in a specious approximate root. Furthermore, in order to enhance the accuracy of the approximation we propose a Steffensen-type iterative method, which does not require any derivatives of f(x) nor is quite affected by an initial approximation. It is shown that the convergence order of the proposed method becomes cubic by simultaneous approximation to the root and its multiplicity. Results for some numerical examples show the efficiency of the new method.     

Remarks on large solutions of a class of semilinear elliptic equations

In this paper, we show existence, uniqueness and exact asymptotic behavior of solutions near the boundary to a class of semilinear elliptic equations −Δu=λg(u)−b(x)f(u) in Ω, where λ is a real number, b(x)>0 in Ω and vanishes on ∂Ω. The special feature is to consider g(u) and f(u) to be regularly varying at infinity and b(x) is vanishing on the boundary with a more general rate function. The vanishing rate of b(x) determines the exact blow-up rate of the large solutions. And the exact blow-up rate allows us to obtain the uniqueness result.     

A singularly perturbed semilinear reaction-diffusion problem in a polygonal domain

The semilinear reaction-diffusion equation −ε²Δu+b(x,u)=0 with Dirichlet boundary conditions is considered in a convex polygonal domain. The singular perturbation parameter ε is arbitrarily small, and the “reduced equation” b(x,u₀(x))=0 may have multiple solutions. An asymptotic expansion for u is constructed that involves boundary and corner layer functions. By perturbing this asymptotic expansion, we obtain certain sub- and super-solutions and thus show the existence of a solution u that is close to the constructed asymptotic expansion. The polygonal boundary forces the study of the nonlinear autonomous elliptic equation −Δz+f(z)=0 posed in an infinite sector, and then well-posedness of the corresponding linearized problem.