Convergence of a Halpern-type iteration algorithm for a class of pseudo-contractive mappings

Let E be a real reflexive Banach space with uniformly Gâteaux differentiable norm. Let K be a nonempty bounded closed and convex subset of E. Let T:K→K be a strictly pseudo-contractive map and let L>0 denote its Lipschitz constant. Assume F(T){xK:Tx=x}≠0/ and let zF(T). Fix δ(0,1) and let δ^* be such that δ^*δL(0,1). Define , where δ_n(0,1) and limδ_n=0. Let {α_n} be a real sequence in (0,1) which satisfies the following conditions: . For arbitrary x₀,uK, define a sequence {x_n}K by x_n+1=α_nu+(1−α_n)S_nx_n. Then, {x_n} converges strongly to a fixed point of T.     

Enveloppes convexes des processus gaussiens

Let X={X(t), t[0,1]} be a process on [0,1] and V_X=Conv{(t,x)t[0,1], x=X(t)} be the convex hull of its path.The structure of the set ext(V_X) of extreme points of V_X is studied. For a Gaussian process X with stationary increments it is proved that:

• The set ext(V_X) is negligible if X is non-differentiable.

• If X is absolutely continuous process and its derivative X′ is continuous but non-differentiable, then ext(V_X) is also negligible and moreover it is a Cantor set.

It is proved also that these properties are stable under the transformations of the type Y(t)=f(X(t)), if f is a sufficiently smooth function.     

The index of maximum ambiguous density for irreducible non-powerful sign pattern matrices

Let INS_n,p be the set of n×n irreducible non-powerful (generalized) sign pattern matrices with period p, and let AINS_n,p. In this paper, we introduce a new parameter called the index of maximum ambiguous density of A. Furthermore, the generalized index of maximum ambiguous density of A, which generalizes the concept of the index of maximum ambiguous density, is introduced. Moreover, some bounds on these indices are obtained, and we exhibit a system of gaps in the set of the index of maximum ambiguous density for AINS_n,p. Finally, the index and the generalized index of maximum ambiguous density for irreducible non-powerful zero-symmetric sign pattern matrices are discussed.     

Asymptotic distribution of the LR statistic for equality of the smallest eigenvalues in high-dimensional principal component analysis

This paper deals with the distribution of the LR statistic for testing the hypothesis that the smallest eigenvalues of a covariance matrix are equal. We derive an asymptotic null distribution of the LR statistic when the dimension p and the sample size N approach infinity, while the ratio p/N converging on a finite nonzero limit c(0,1). Numerical simulations revealed that our approximation is more accurate than the classical chi-square-type approximation as p increases in value.     

On constants in some inequalities for intermediate derivatives on a finite interval

Let L_q (1q<∞) be the space of functions f measurable on I=[−1,1] and integrable to the power q, with normL_∞ is the space of functions measurable on I with normWe denote by AC the set of all functions absolutely continuous on I. For nN, q[1,∞] we setW_n,q={f:f⁽ⁿ⁻¹⁾AC, f⁽ⁿ⁾L_q}.In this paper, we consider the problem of accuracy of constants A, B in the inequalities (1)|| f^(m)||_qA|| f||_p+B|| f^(m+k+1)||_r, mN, kW; p,q,r[1,∞], fW_m+k+1,r.     

Improved random graph isomorphism

Canonical labeling of a graph consists of assigning a unique label to each vertex such that the labels are invariant under isomorphism. Such a labeling can be used to solve the graph isomorphism problem. We give a simple, linear time, high probability algorithm for the canonical labeling of a G(n,p) random graph for p[ω(ln⁴n/nlnlnn),1−ω(ln⁴n/nlnlnn)]. Our result covers a gap in the range of p in which no algorithm was known to work with high probability. Together with a previous result by Bollobás, the random graph isomorphism problem can be solved efficiently for p[Θ(lnn/n),1−Θ(lnn/n)].     

Constrained versions of Sauer’s lemma

Let [n]={1,…,n}. For a function h:[n]→{0,1}, x[n] and y{0,1} define by the width ω_h(x,y) of h at x the largest nonnegative integer a such that h(z)=y on x−a≤z≤x+a. We consider finite VC-dimension classes of functions h constrained to have a width ω_h(x_i,y_i) which is larger than N for all points in a sample or a width no larger than N over the whole domain [n]. Extending Sauer’s lemma, a tight upper bound with closed-form estimates is obtained on the cardinality of several such classes.     

Approximation of infinitely differentiable multivariate functions is intractable

We prove that L_∞-approximation of C^∞-functions defined on [0,1]^d is intractable and suffers from the curse of dimensionality. This is done by showing that the minimal number of linear functionals needed to obtain an algorithm with worst case error at most ε(0,1) is exponential in d. This holds despite the fact that the rate of convergence is infinite.     

On Generalized Hermite–Fejér Interpolation of Lagrange Type on the Chebyshev Nodes

For fC[−1, 1], let H_m, n(f, x) denote the (0, 1, …,anbsp;m) Hermite–Fejér (HF) interpolation polynomial of f based on the Chebyshev nodes. That is, H_m, n(f, x) is the polynomial of least degree which interpolates f(x) and has its first m derivatives vanish at each of the zeros of the nth Chebyshev polynomial of the first kind. In this paper a precise pointwise estimate for the approximation error |H_2m, n(f, x)−f(x)| is developed, and an equiconvergence result for Lagrange and (0, 1, …, 2m) HF interpolation on the Chebyshev nodes is obtained. This equiconvergence result is then used to show that a rational interpolatory process, obtained by combining the divergent Lagrange and (0, 1, …, 2m) HF interpolation methods on the Chebyshev nodes, is convergent for all fC[−1, 1].     

Existence and multiplicity of positive solutions of a nonlinear eigenvalue problem with indefinite weight function

This paper is concerned with the existence, multiplicity and stability of positive solutions of an indefinite weight boundary value problem

where aC[0,1] changes sign. The proof of our main result is based upon bifurcation techniques.     

Regularity results for stable-like operators

For α[1,2) we consider operators of the form

and for α(0,1) we consider the same operator but where the f term is omitted. We prove, under appropriate conditions on A(x,h), that any solution u to will be in C^α+β if fC^β.     

Existence results for -point boundary value problem of second order ordinary differential equations

This study concerns the existence of positive solutions to the boundary value problemwhere ξ_i(0,1) with 0<ξ₁<ξ₂<<ξ_n-2<1, a_i, b_i[0,∞) with and . By applying the Krasnoselskii's fixed-point theorem in Banach spaces, some sufficient conditions guaranteeing the existence of at least one positive solution or at least two positive solutions are established for the above general n-point boundary value problem.     

Combination of partially non-distinct beliefs: The cautious-adaptive rule

The combination rule is critical in an evidence based fusion process. The conjunctive rule is most common eventhough when the cognitive independence – distinctness – assumption is often questionable. A new combination rule is tested here in both discrete and continuous cases, accounting for a partial non-distinctness between evidences. It is based on ‘generalized discounting’, that we define for separable basic belief assignments (bbas) or basic belief densities (bbds), to be applied to the source correlation derived from the cautious rule. This correlation can be specified in both considered cases of consonant bbas/bbds (as proposed by Dubois et al.) and separable bbas/bbds (as proposed by Denœux). Then, the so-called ‘cautious-adaptive’ rule varies between the conjunctive rule and the cautious one, depending on the discounting level. In the Gaussian case with standard deviation σ, the evidence non-distinctness will be parameterized by a factor [0,1] dividing σ. It leads to the generalized discounting needed in the cautious-adaptive formulation.     

On universal cycles for multisets

A Universal Cycle for t-multisets of [n]={1,…,n} is a cyclic sequence of integers from [n] with the property that each t-multiset of [n] appears exactly once consecutively in the sequence. For such a sequence to exist it is necessary that n divides , and it is reasonable to conjecture that this condition is sufficient for large enough n in terms of t. We prove the conjecture completely for t{2,3} and partially for t{4,6}. These results also support a positive answer to a question of Knuth.     

Constant-sign solutions for systems of singular integral equations of Hammerstein type

We consider the system of Hammerstein integral equations

where T>0 is fixed, ρ_i’s are given functions and the nonlinearities f_i(t,x₁,x₂,…,x_n) can be singular at t=0 and x_j=0 where j{1,2,,n}. Criteria are offered for the existence of constant-sign solutions, i.e., θ_iu_i(t)≥0 for t[0,T] and 1≤i≤n, where θ_i{1,−1} is fixed. The tools used are a nonlinear alternative of Leray–Schauder type, Krasnosel’skii’s fixed point theorem in a cone and Schauder’s fixed point theorem. We also include examples and applications to illustrate the usefulness of the results obtained.     

Optimal Recovery of the Derivative of Periodic Analytic Functions from Hardy Classes

LetS_β{z : |Im z|<β}. For 2π-periodic functions which are analytic inS_βwithp-integrable boundary values, we construct an optimal method of recovery off′(ξ), ξS_β, using information about the valuesf(x₁), mldr;, f(x_n), x_j[0, 2π).     

High-dimensional asymptotic expansions for the distributions of canonical correlations

This paper examines asymptotic distributions of the canonical correlations between and with q≤p, based on a sample of size of N=n+1. The asymptotic distributions of the canonical correlations have been studied extensively when the dimensions q and p are fixed and the sample size N tends toward infinity. However, these approximations worsen when q or p is large in comparison to N. To overcome this weakness, this paper first derives asymptotic distributions of the canonical correlations under a high-dimensional framework such that q is fixed, m=n−p→∞ and c=p/n→c₀∈[0,1), assuming that and have a joint (q+p)-variate normal distribution. An extended Fisher’s z-transformation is proposed. Then, the asymptotic distributions are improved further by deriving their asymptotic expansions. Numerical simulations revealed that our approximations are more accurate than the classical approximations for a large range of p,q, and n and the population canonical correlations.     

Uniqueness and nonuniqueness of nodal radial solutions of sublinear elliptic equations in a ball

The following Dirichlet problem

(1.1)

is considered, where , N≥2, KC²[0,1] and K(r)>0 for 0≤r≤1, , sf(s)>0 for s≠0. Assume moreover that f satisfies the following sublinear condition: f(s)/s>f^′(s) for s≠0. A sufficient condition is derived for the uniqueness of radial solutions of (1.1) possessing exactly k−1 nodes, where . It is also shown that there exists KC^∞[0,1] such that (1.1) has three radial solutions having exactly one node in the case N=3.     

Some exact rates in the functional law of the iterated logarithm

We find exact convergence rate in the Strassen's functional law of the iterated logarithm for a class of elements on the boundary of the limit set. Our result applies, in particular, to the power functions c_αx^α with α ]1/2,1[, thus solving a small ball estimate problem which was open for ten years.     

A new regularized method for two dimensional nonhomogeneous backward heat problem

We consider the problem of finding, from the final data u(x,y,T)=g(x,y), the initial data u(x,y,0) of the temperature function u(x,y,t),(x,y)I=(0,π)×(0,π),t[0,T] satisfying the following system

The problem is severely ill-posed. In this paper a simple and convenient new regularization method for solving this problem is considered. Meanwhile, some quite sharp error estimates between the approximate solution and exact solution are provided. A numerical example also shows that the method works effectively.