The Cauchy Boundary Value Problems on Closed Piecewise Smooth Manifolds in <Emphasis Type="Italic">C</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Suppose that D is a bounded domain with a piecewise C^1 smooth boundary in C^n. Let ψ∈C^1 α(δD). By using the Hadamard principal value of the higher order singular integral and solid angle coefficient method of points on the boundary, we give the Plemelj formula of the higher order singular integral with the Boehner-Martinelli kernel, which has integral density ψ. Moreover, by means of the Plemelj formula and methods of complex partial differential equations, we discuss the corresponding Cauehy boundary value problem with the Boehner-Martinelli kernel on a closed piecewise smooth manifold and obtain its unique branch complex harmonic solution.     

Gradient estimates and Harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds

We derive the gradient estimates and Harnack inequalities for positive solutions of nonlinear parabolic and nonlinear elliptic equations (Δ − ∂/∂t) u(x, t) + h(x, t)u^α(x, t) = 0 and Δu + b · u + hu^α = 0 on Riemannian manifolds. We also obtain a theorem of Liouville type for positive solutions of the nonlinear elliptic equation.     

decay problem for the dissipative wave equation in odd dimensions

Consider the Cauchy problem in odd dimensions for the dissipative wave equation: (□+∂_t)u=0 in with (u,∂_tu)|_t=0=(u₀,u₁). Because the L² estimates and the L^∞ estimates of the solution u(t) are well known, in this paper we pay attention to the L^p estimates with 1p<2 (in particular, p=1) of the solution u(t) for t0. In order to derive L^p estimates we first give the representation formulas of the solution u(t)=∂_tS(t)u₀+S(t)(u₀+u₁) and then we directly estimate the exact solution S(t)g and its derivative ∂_tS(t)g of the dissipative wave equation with the initial data (u₀,u₁)=(0,g). In particular, when p=1 and n1, we get the L¹ estimate: u(t)_L¹Ce^−t/4(u_0W^n,1+u_1W^n−1,1)+C(u_0L¹+u_1L¹) for t0.     

An explicit extension formula of bounded holomorphic functions from analytic varieties to strictly convex domains

We consider a strictly convex domain D ⁿ and m holomorphic functions, φ₁,…, φ_m, in a domain . We set V = {z ε Ω: φ₁(z) = ··· = φ_m(z) = 0}, M = V ∩ D and ∂M = V ∩ ∂D. Under the assumptions that the variety V has no singular point on ∂M and that V meets ∂D transversally we construct an explicit kernel K(ζ, z) defined for ζ ε ∂M and z ε D so that the integral operator Ef(z) = ∝ _{ζ ε ∂M} f(ζ) K(ζ, z) (z ε D), defined for f ε H^∞(M) (using the boundary values f(ζ) for a.e. ζ ε ∂M), is an extension operator, i.e., Ef(z) = f(z) for z ε M and furthermore E is a bounded operator from H^∞ to H^∞(D).     

Norm Estimate of the Cauchy Transform on L P (Ω)

We give two sided estimates of the norm of Cauchy transform on L^p(Ω)(1 ≤ p < ∞) in the case when Ω is the unit disc or the bounded simply connected domain with piecewise C¹ boundary. As a consequence we get the better constant in Poincare inequality. Also, we conjecture the exact value norm of Cauchy transform on L^p(D), where D is the unit disc.     

On deviations between empirical and quantile processes for mixing random variables

Let {X_n} be a strictly stationary φ-mixing process with Σ_j=1^∞ φ^1/2(j) < ∞. It is shown in the paper that if X₁ is uniformly distributed on the unit interval, then, for any t [0, 1], |F_n⁻¹(t) − t + F_n(t) − t| = O(n^−3/4(log log n)^3/4) a.s. and sup_0≤t≤1 |F_n⁻¹(t) − t + F_n(t) − t| = (O(n^−3/4(log n)^1/2(log log n)^1/4) a.s., where F_n and F_n⁻¹(t) denote the sample distribution function and tth sample quantile, respectively. In case {X_n} is strong mixing with exponentially decaying mixing coefficients, it is shown that, for any t [0, 1], |F_n⁻¹(t) − t + F_n(t) − t| = O(n^−3/4(log n)^1/2(log log n)^3/4) a.s. and sup_0≤t≤1 |F_n⁻¹(t) − t + F_n(t) − t| = O(n^−3/4(log n)(log log n)^1/4) a.s. The results are further extended to general distributions, including some nonregular cases, when the underlying distribution function is not differentiable. The results for φ-mixing processes give the sharpest possible orders in view of the corresponding results of Kiefer for independent random variables.     

The nonlinear heat equation with absorption: Effects of variable coefficients

We consider the nonnegative solutions to the nonlinear degenerate parabolic equation u_t = (D(x, t)u^{m − 1}u_x)_x − b(x, t)u^p with m > 1, 0 < p < 1, and positive D(x, t), b(x, t). After obtaining the uniqueness and Hölder regularity results, we investigate the dependence of such phenomena as extinction in finite time and instantaneous shrinking of the support on the behaviour of D(x, t) and b(x, t).     

Pointwise Estimates for a Class of Oscillatory Integrals and Related Lp–Lq Estimates

In this article we establish pointwise estimates for a class of oscillatory integrals related to the fundamental solution of the dispersive equation ∂_t u − iP (D_x)u = 0, where P(D_x) is a differential operator whose symbol P(ξ) is a real polynomial in one variable. Then we use such pointwise estimates to establish L^p–L^q estimates for the solution of this equation.     

On positive solutions of some nonlinear fourth-order beam equations

The existence, uniqueness and multiplicity of positive solutions of the following boundary value problem is considered:

u⁽⁴⁾(t)−λf(t,u(t))=0, for 0<t<1,u(0)=u(1)=u″(0)=u″(1)=0,

where λ>0 is a constant, f :[0,1]×[0,+∞)→[0,+∞) is continuous.     

Summability of orthogonal expansions of several variables

Summability of spherical h-harmonic expansions with respect to the weight function ∏_j=1^d |x_j|^2κ_j (κ_j0) on the unit sphere S^d−1 is studied. The main result characterizes the critical index of summability of the Cesàro (C,δ) means of the h-harmonic expansion; it is proved that the (C,δ) means of any continuous function converge uniformly in the norm of C(S^d−1) if and only if δ>(d−2)/2+∑_j=1^d κ_j−min_1jd κ_j. Moreover, it is shown that for each point not on the great circles defined by the intersection of the coordinate planes and S^d−1, the (C,δ) means of the h-harmonic expansion of a continuous function f converges pointwisely to f if δ>(d−2)/2. Similar results are established for the orthogonal expansions with respect to the weight functions ∏_j=1^d |x_j|^2κ_j(1−|x|²)^μ−1/2 on the unit ball B^d and ∏_j=1^d x_j^κ_j−1/2(1−|x|₁)^μ−1/2 on the simplex T^d. As a related result, the Cesàro summability of the generalized Gegenbauer expansions associated to the weight function |t|^2μ(1−t²)^λ−1/2 on [−1,1] is studied, which is of interest in itself.     

Tests for the mean direction of the Langevin distribution with large concentration parameter

In this paper we study the asymptotic behaviors of the likelihood ratio criterion (T_L^(s)), Watson statistic (T_W^(s)) and Rao statistic (T_R^(s)) for testing H_0s: μ (a given subspace) against H_1s: μ , based on a sample of size n from a p-variate Langevin distribution M_p(μ, κ) when κ is large. For the case when κ is known, asymptotic expansions of the null and nonnull distributions of these statistics are obtained. It is shown that the powers of these statistics are coincident up to the order κ⁻¹. For the case when κ is unknown, it is shown that T_R^(s) T_L^(s) T_W^(s) in their powers up to the order κ⁻¹.     

Boundary value problems for second order nonlinear differential equations on infinite intervals

In this paper, we consider boundary value problems for nonlinear differential equations on the semi-axis (0,∞) and also on the whole axis (−∞,∞), under the assumption that the left-hand side being a second order linear differential expression belongs to the Weyl limit-circle case. The boundary value problems are considered in the Hilbert spaces L²(0,∞) and L²(−∞,∞), and include boundary conditions at infinity. The existence and uniqueness results for solutions of the considered boundary value problems are established.     

Commutative Extended Complex Numbers and Connected Trigonometry

In order to construct an extension of the complex numbers, we consider an n-dimensional commutative algebra generated by the n vectors 1, e, ..., eⁿ⁻¹ where the fundamental element satisfies the basic relation eⁿ = −1. These spaces can be classified according to the values of n: prime number, power of a prime number, general number. The question of the invertibility leads to the definition of a pseudo-norm for which the triangle inequality is not satisfied (the n = 1, 2 cases excepted). When one tries to pass from the polar form the cartesian one, one obtains functions generalizing the usual circular and hyperbolic functions and their inverse. The extended sine and hyperbolic sine functions thus constructed satisfy a determinantal-type relation and they lay the foundation of a new trigonometry for which summation and derivative formulas are given. An extended 2π quantity is defined as the periodicity of the generalized circular functions. This formalism is applied to solve the nth order differential equations (∑ⁿ⁻¹_i=1 (∂n/∂φⁿ_i) ± ω) ƒ(φ) = 0. As a further application, the solutions of the n-laplacian operator are derived.     

Lateral connections for asymptotic solutions around higher order turning points

Asymptotic expansions as ε → 0⁺ or x → ∞ for fundamental systems of solutions for ε²u″(x) − p(x) u(x) = 0 were obtained by Evgrafov and Fedoryuk on unbounded canonical domains with neighborhoods deleted around turning points. When p(x) is a polynomial, lateral connection formulas were found by Evgrafov and Fedoryuk, and Leung for two fundamental systems of solutions with known expansions in the interior of two different unbounded overlapping canonical regions with common first or second order turning point at their boundaries. In this paper, lateral connection formulas are found when the turning points are of any higher order. Recent result of uniform simplification by Sibuya around higher order turning point is used.     

Global bifurcation of a class of periodic boundary-value problems

We prove that λ=0 is a global bifurcation point of the second-order periodic boundary-value problem (p(t)x^′(t))^′−λx(t)−λ²x^′(t)−f(t,x(t),x^′(t),x^″(t));x(0)=x(1),x^′(0)=x^′(1). We study this equation under hypotheses for which it may be solved explicitly for x^″(t). However, it is shown that the explicitly solved equation does not satisfy the usual conditions that are sufficient to conclude global bifurcation. Thus, we need to study the implicit equation with regard to global bifurcation.     

Approaching an extinction point in one-dimensional semilinear heat equations with strong absorption

This paper deals with the Cauchy problem u_t − u_xx + u^p = 0; − ∞ < x < + ∞, t>0, u(x, 0) = u₀(x); − ∞ < x < + ∞, where 0 < p < 1 and u₀(x) is continuous, nonnegative, and bounded. In this case, solutions are known to vanish in a finite time T, and interfaces separating the regions where u(x, t) > 0 and u(x, t) = 0 appear when t is close to T. We describe here all possible asymptotic behaviours of solutions and interfaces near an extinction point as the extinction time is approached. We also give conditions under which some of these behaviours actually occur.     

Characterizations of degree one bivariate measures of concordance

We call a measure of concordance κ of an ordered pair (X,Y) of two continuous random variables a bivariate measure of concordance. This κ may be considered to be a function κ(C) of the copula C associated with (X,Y). κ is considered to be of degree n if, given any two copulas A and B, the value of their convex sum, κ(tA+(1−t)B), is a polynomial in t of degree n. Examples of bivariate measures of concordance are Spearman’s rho, Blomqvist’s beta, Gini’s measure of association, and Kendall’s tau. The first three of these are of degree one, but Kendall’s tau is of degree two. We exhibit three characterizations of bivariate measures of concordance of degree one.     

The approximate identity kernels of product type for the Walsh system

At present there are only a few approximate identity kernels for the Walsh system, for example, the p^N-truncated Dirichlet kernel D_{p^N − 1}(t) = ∑_{j = 0}^{pN − 1} w_j(t) [6]; the Abel-Poisson kernel λ_γ(t) = ∑_{k = 0}^∞ γ^kw_k(t) [3], and so on. In [6], Zheng has introduced a new kind of approximate identity kernels for the Walsh system—the kernels of product type. In the present paper we discuss the approximation properties of such product type kernels. Estimates of their moments as well as a direct approximation theorem are obtained. Then, to establish an inverse approximation theorem, we need the p-adic derivative of product type kernels and we estimate this derivative in L¹-norm.     

Quadrature formula for approximating the singular integral of Cauchy type with unbounded weight function on the edges

New quadrature formulas (QFs) for evaluating the singular integral (SI) of Cauchy type with unbounded weight function on the edges is constructed. The construction of the QFs is based on the modification of discrete vortices method (MMDV) and linear spline interpolation over the finite interval [−1,1]. It is proved that the constructed QFs converge for any singular point x not coinciding with the end points of the interval [−1,1]. Numerical results are given to validate the accuracy of the QFs. The error bounds are found to be of order O(h^α|lnh|) and O(h|lnh|) in the classes of functions H^α([−1,1]) and C¹([−1,1]), respectively.     

Asymptotic distributions of regression and autoregression coefficients with martingale difference disturbances

In this paper a form of the Lindeberg condition appropriate for martingale differences is used to obtain asymptotic normality of statistics for regression and autoregression. The regression model is y_t = Bz_t + v_t. The unobserved error sequence {v_t} is a sequence of martingale differences with conditional covariance matrices {Σ_t} and satisfying sup_{t=1,…, n} {v′_tv_tI(v′_tv_t>a) |z_t, v_t−1, z_t−1, …} 0 as a → ∞. The sample covariance of the independent variables z₁, …, z_n, is assumed to have a probability limit M, constant and nonsingular; max_t=1,…,nz′_tz_t/n 0. If (1/n)Σ_t=1ⁿΣ_t Σ, constant, then √nvec( _n−B) N(0,M⁻¹Σ) and _n Σ. The autoregression model is x_t = Bx_{t − 1} + v_t with the maximum absolute value of the characteristic roots of B less than one, the above conditions on {v_t}, and (1/n)Σ_t=max(r,s)+1(Σ_tv_t−1−rv′_t−1−s) δ_rs(ΣΣ), where δ_rs is the Kronecker delta. Then √nvec( _n−B) N(0,Γ⁻¹Σ), where Γ = Σ_{s = 0}^∞B^sΣ(B′)^s.