Extensions of the Cauchy-Goursat Integral Theorem

It is natural to conjecture that if a function f is continuous on the closed region determined by a rectifiable 1-cycle Γ and complex-differentiable on the open region then Γ_∫f=0. The main result is an extension of the classical Cauchy-Goursat Theorem: the equality conjectured holds (with no boundary condition on f^′) under the additional hypothesis that the winding numbers of Γ define an Lp function and f satisfies a matching Hölder continuity condition near the image of Γ. (In particular, continuity suffices if p=∞.) The proof uses approximations of a rectifiable path by piecewise linear paths.     

On the solutions of the linear integral equations of Volterra type

Some boundaries about the solution of the linear Volterra integral equations of the form f(t)=1?K*f were obtained as |f(t)|?1, |f(t)|?2 and |f(t)|?4 in (J. Math. Anal. Appl. 1978; 64 :381–397; Int. J. Math. Math. Sci. 1982; 5 (1):123–131). The boundary of the solution function of an equation in this type was found as |f(t)|?2ⁿ in (Integr. Equ. Oper. Theory 2002; 43 :466–479), where t∈[0, ∞) and n is a natural number such that n?2. In (Math. Comp. 2006; 75 :1175–1199), it is shown that the boundary of the solution function of an equation in the same form can also be derived as that of (Integr. Equ. Oper. Theory 2002; 43 :466–479) under different conditions than those of (Integr. Equ. Oper. Theory 2002; 43 :466–479). In the present paper, the sufficient conditions for the boundedness of functions f, f′, f′′, …, f⁽ⁿ⁺³⁾, (n∈?) defined on the infinite interval [0, ∞) are given by our method, where f is the solution of the equation f(t)=1?K*f. Copyright © 2007 John Wiley & Sons, Ltd.     

Infinitely many mountain pass solutions on a kind of fourth-order Neumann boundary value problem

In this paper, the existence of infinitely many mountain pass solutions are obtained for the fourth-order boundary value problem (BVP) u⁽⁴⁾(t)-2u^″(t)+u(t)=f(u(t)),0<t<1, u^′(0)=u^′(1)=u^?(0)=u^?(1)=0, where f:R→R is continuous. The study of the problem is based on the variational methods and critical point theory. We prove the conclusion by using sub-sup solution method, Mountain Pass Theorem in Order Intervals, Leray-Schauder degree theory and Morse theory.     

A Komlós Theorem for abstract Banach lattices of measurable functions

Consider a Banach function space X(μ) of (classes of) locally integrable functions over a σ-finite measure space (Ω,Σ,μ) with the weak σ-Fatou property. Day and Lennard (2010) [9] proved that the theorem of Komlós on convergence of Cesàro sums in L¹[0,1] holds also in these spaces; i.e. for every bounded sequence n(fn) in X(μ), there exists a subsequence k(fnk) and a function f∈X(μ) such that for any further subsequence j(hj) of k(fnk), the series converges μ-a.e. to f. In this paper we generalize this result to a more general class of Banach spaces of classes of measurable functions — spaces L¹(ν) of integrable functions with respect to a vector measure ν on a δ-ring — and explore to which point the Fatou property and the Komlós property are equivalent. In particular we prove that this always holds for ideals of spaces L¹(ν) with the weak σ-Fatou property, and provide an example of a Banach lattice of measurable functions that is Fatou but do not satisfy the Komlós Theorem.     

On spanning connected graphs

A k-containerC(u,v) of G between u and v is a set of k internally disjoint paths between u and v. A k-container C(u,v) of G is a k^*-container if the set of the vertices of all the paths in C(u,v) contains all the vertices of G. A graph G is k^*-connected if there exists a k^*-container between any two distinct vertices. Therefore, a graph is 1^*-connected (respectively, 2^*-connected) if and only if it is hamiltonian connected (respectively, hamiltonian). In this paper, a classical theorem of Ore, providing sufficient conditional for a graph to be hamiltonian (respectively, hamiltonian connected), is generalized to k^*-connected graphs.     

Some New Applications of the Subspace Theorem

We present some applications of the Subspace Theorem to the investigation of the arithmetic of the values of Laurent series f(z) at S-unit points. For instance we prove that if f(q ⁿ ) is an algebraic integer for infinitely many n, then h(f(q ⁿ )) must grow faster than n. By similar principles, we also prove diophantine results about power sums and transcendency results for lacunary series; these include as very special cases classical theorems of Mahler. Our arguments often appear to be independent of previous techniques in the context.     

On the spanning connectivity of graphs

A k-containerC(u,v) of G between u and v is a set of k internally disjoint paths between u and v. A k-container C(u,v) of G is a k^*-container if it contains all vertices of G. A graph G is k^*-connected if there exists a k^*-container between any two distinct vertices. The spanning connectivity of G, κ^*(G), is defined to be the largest integer k such that G is w^*-connected for all 1?w?k if G is a 1^*-connected graph. In this paper, we prove that κ^*(G)?2δ(G)-n(G)+2 if (n(G)/2)+1?δ(G)?n(G)-2. Furthermore, we prove that κ^*(G-T)?2δ(G)-n(G)+2-|T| if T is a vertex subset with |T|?2δ(G)-n(G)-1.     

An Application of a Mountain Pass Theorem

We are concerned with the following Dirichlet problem: −Δu(x) = f(x, u), x∈Ω, u∈H ¹ ₀(Ω), (P) where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L ^∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0, 0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR) is no longer true, where F(x, s) = ∫ ^s ₀ f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming (AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞. Received June 24, 1998, Accepted January 14, 2000.     

On necessary and sufficient conditions for uniform strong consistency of estimators of a density and its derivatives

Based on a random sample from a population with (unknown) probability density f, this note exhibits a class of statistics f^(p) for each fixed integer $\text{p ≧ 0}$. It is shown that f^(p) are uniformly strongly consistent estimators of f^(p), the pth order derivative of f, if and only iff^(p)is bounded and uniformly continuous.     

Chaotic behaviour of the map x ↦ ω(x, f)

Let K(2^?) be the class of compact subsets of the Cantor space 2^?, furnished with the Hausdorff metric. Let f ∈ C(2^?). We study the map ω _f : 2 ^? → K(2^?) defined as ω _f (x) = ω(x, f), the ω-limit set of x under f. Unlike the case of n-dimensional manifolds, n ≥ 1, we show that ω _f is continuous for the generic self-map f of the Cantor space, even though the set of functions for which ω _f is everywhere discontinuous on a subsystem is dense in C(2^?). The relationships between the continuity of ω _f and some forms of chaos are investigated.     

Newton-type methods of high order and domains of semilocal and global convergence

We present the geometric construction of some classical iterative methods that have global convergence and “infinite” speed of convergence when they are applied to solve certain nonlinear equations f(t)=0. In particular, for nonlinear equations with the degree of logarithmic convexity of f^′, L_f^′(t)=f^′(t)f^?(t)/f^″(t)², is constant, a family of Newton-type iterative methods of high orders of convergence is constructed. We see that this family of iterations includes the classical iterative methods. The convergence of the family is studied in the real line and the complex plane, and domains of semilocal and global convergence are located.     

Mappings of Finite Distortion of Polynomial Type

Suppose f:? ⁿ →? ⁿ is a mapping of K-bounded p-mean distortion for some p>n?1. We prove the equivalence of the following properties of f: the doubling condition for J(x,f) over big balls centered at the origin, the boundedness of the multiplicity function N(f,? ⁿ ), the polynomial type of f, and the polynomial growth condition for f.     

Weakly nonlinear discrete multipoint boundary value problems

In this paper we study nonlinear, discrete, multipoint boundary value problems of the form

x(t+1)=A(t)x(t)+?f(t,x(t))     

Maximizing the L∞ norm of the gradient of solutions to the Poisson equation

We find the maximum of ¦Du _f ¦ _L∞ when u_f is the solution, which vanishes at infinity, of the Poisson equation Δu =f on ? ⁿ in terms of the decreasing rearrangement off. Hence, we derive sharp estimates for ¦Du _f ¦ _L∞ in terms of suitable Lorentz orL ^p norms off. We also solve the problem of maximizing ¦Du _f ^B (0)¦ whenu _f ^B is the solution, vanishing on?B, to the Poisson equation in a ballB centered at 0 and the decreasing rearrangement off is assigned.     

Single Basepoint Subdivision Schemes for Manifold-valued Data: Time-Symmetry Without Space-Symmetry

This paper establishes smoothness results for a class of nonlinear subdivision schemes, known as the single basepoint manifold-valued subdivision schemes, which shows up in the construction of wavelet-like transform for manifold-valued data. This class includes the (single basepoint) Log–Exp subdivision scheme as a special case. In these schemes, the exponential map is replaced by a so-called retraction map f from the tangent bundle of a manifold to the manifold. It is known that any choice of retraction map yields a C ² scheme, provided the underlying linear scheme is C ² (this is called “C ² equivalence”). But when the underlying linear scheme is C ³, Navayazdani and Yu have shown that to guarantee C ³ equivalence, a certain tensor P _f associated to f must vanish. They also show that P _f vanishes when the underlying manifold is a symmetric space and f is the exponential map. Their analysis is based on certain “C ^k  proximity conditions” which are known to be sufficient for C ^k  equivalence. In the present paper, a geometric interpretation of the tensor P _f is given. Associated to the retraction map f is a torsion-free affine connection, which in turn defines an exponential map. The condition P _f =0 is shown to be equivalent to the condition that f agrees with the exponential map of the connection up to the third order. In particular, when f is the exponential map of a connection, one recovers the original connection and P _f vanishes. It then follows that the condition P _f =0 is satisfied by a wider class of manifolds than was previously known. Under the additional assumption that the subdivision rule satisfies a time-symmetry, it is shown that the vanishing of P _f implies that the C ⁴ proximity conditions hold, thus guaranteeing C ⁴ equivalence. Finally, the analysis in the paper shows that for k≥5, the C ^k  proximity conditions imply vanishing curvature. This suggests that vanishing curvature of the connection associated to f is likely to be a necessary condition for C ^k equivalence for k≥5.     

Multiple cross-intersecting families of signed sets

A k-signed r-set on[n]={1,…,n} is an ordered pair (A,f), where A is an r-subset of [n] and f is a function from A to [k]. Families A₁,…,Ap are said to be cross-intersecting if any set in any family Ai intersects any set in any other family Aj. Hilton proved a sharp bound for the sum of sizes of cross-intersecting families of r-subsets of [n]. Our aim is to generalise Hilton's bound to one for families of k-signed r-sets on [n]. The main tool developed is an extension of Katona's cyclic permutation argument.     

A Class of Nonlinear Averaging Integral Operators

LetHbe the class of analytic functions defined in the unit discU, and let coEdenote the convex hull of a setEinC. IfK⊂H, then an operatorI:K→His an averaging operator ifI[f](0) =f(0) andI[f](U) ⊂ cof(U), for allf∈K. The authors show that the operatorI_β,γ[f](z) ≡ [γz^−γ∫^z₀f^β(t)t^γ−1dt]^1/βis an averaging operator on certain subsets ofH.     

On the successive coefficients of close-to-convex functions

Let f∈S, f be a close-to-convex function, f_k(z)=[f(z^k)]^1/k. The relative growth of successive coefficients of f_k(z) is investigated. The sharp estimate of ||c_n+1|−|c_n|| is obtained by using the method of the subordination function.