Iterative approximation of solutions to nonlinear equations involving m-accretive operators in Banach spaces

Let E be an arbitrary real Banach space and T :E→E be a Lipschitz continuous accretive operator. Under the lack of the assumption lim_n→∞α_n=lim_n→∞β_n=0, we prove that the Ishikawa iterative sequence with errors converges strongly to the unique solution of the equation x+Tx=f. Moreover, this result provides a convergence rate estimate for some special cases of such a sequence. Utilizing this result, we imply that if T :E→E is a Lipschitz continuous strongly accretive operator then the Ishikawa iterative sequence with errors converges strongly to the unique solution of the equation Tx=f. Our results improve, generalize and unify the ones of Liu, Chidume and Osilike, and to some extent, of Reich.     

Sequences with equi-distributed extreme points in uniform polynomial approximation

Let E be a compact set in with connected complement and positive logarithmic capacity. For any f continuous on E and analytic in the interior of E, we consider the distribution of extreme points of the error of best uniform polynomial approximation on E. Let Λ=(n_j) be a subsequence of such that n_j+1/n_j→1. If, for nΛ, A_n( f)∂E denotes the set of extreme points of the error function, we prove that there is a subsequence Λ′ of Λ such that the distribution of any (n+2)th Fekete point set of A_n( f) tends weakly to the equilibrium distribution on E as n→∞ in Λ′. Furthermore, we prove a discrepancy result for the distribution of the point sets if the boundary of E is smooth enough.     

Rates of Best Rational Approximation of Analytic Functions

Let E be a compact set in the extended complex plane C and let f be holomorphic on E. Denote by ρ_n the distance from f to the class of all rational functions of order at most n, measured with respect to the uniform norm on E. We obtain results characterizing the relationship between estimates of lim inf_n→∞ ρ^1/n_n and lim sup_n→∞ ρ^1/n_n.     

An Inequality on the Size of a Set in a Cartesian Product

Let Ω be a finite subset of the Cartesian productW₁  ×   × W_nof n sets. ForA    {1, 2, , n }, denote by Ω_Athe projection ofΩ onto the Cartesian product of W_i, i   A. Generalizing an inequality given in an article by Shen, we prove that | Ω |² ≤  |Ω_A1 || Ω_Ak| provided that { A₁, , A_k} is a double cover of {1, 2, , n }. This inequality is applied to give some bounds on the numbers of special subgraphs of a graph.     

Traveling wave solutions of Lotka–Volterra type two predators‐one prey model

In this work, we consider a model with one basal resource and two species of predators feeding by the same resource. There are three non‐trivial boundary equilibria. One is the saturated state E_K of the prey without any predator. Other two equilibria, E₁ and E₂, are the coexistence states of the prey with only one species of predators. Using a high‐dimensional shooting method, the Wazewski' principle, we establish the conditions for the existence of traveling wave solutions from E_K to E₂ and from E₁ to E₂. These results show that the advantageous species v₂ always win in the competition and exclude species v₁ eventually. Finally, some numerical simulations are presented, and biological interpretations are given. Copyright © 2016 John Wiley & Sons, Ltd.     

Tails of Bipartite Distance-regular Graphs

Let Γ denote a bipartite distance-regular graph with diameterD  ≥  4 and valency k ≥  3. Let θ 0  > θ 1  >  >  θD denote the eigenvalues of Γ and let E₀, E₁, , E_Ddenote the associated primitive idempotents. Fix s(1  ≤  s ≤  D −  1 ) and abbreviate E: =  E_s. We say E is a tail whenever the entrywise product E   E is a linear combination of E₀, E and at most one other primitive idempotent of Γ. Letq_ij^{σi + 1 h} (0  ≤ h , i, j ≤  D) denote the Krein parameters of Γ and letΔ denote the undirected graph with vertices 0, 1, , D where two vertices i, j are adjacent whenever i ≠  =  j andq_ij^σi + 1s  ≠  =  0. We show E is a tail if and only if one of (i)–(iii) holds: (i) Δ is a path; (ii) Δ has two connected components, each of which is a path; (iii) D =  6 and Δ has two connected components, one of which is a path on four vertices and the other of which is a clique on three vertices.     

The heteroclinic cycle in the model of competition betweenn species and its stability

This paper gives a sufficient condition for the existence of heteroclinic cycle in the model of competition betweenn species and a criterion for determining the stability of the heteroclinic cycle. The results given in this paper extend the results obtained by May and Leonard in and by Hofbauer and Sigmund in. A conjecture on the permanence of the model and a open problem on the stability of the heteroclinic cycle for the critical case are given at the end of this paper. This research is supported by the National Natural Science Foundation of China     

Numerical properties of Chern classes of certain covering manifolds

In this paper, we study numerical properties of Chern classes of certain covering manifolds. One of the main results is the following: Let ψ : X → Pⁿ be a finite covering of the n-dimensional complex projective space branched along a hypersurface with only simple normal crossings and suppose X is nonsingular. Let c_i(X) be the i-th Chern class of X. Then (i) if the canonical divisor K_X is numerically effective, then (−1)^kc_k(X) (k ≥ 2) is numerically positive, and (ii) if X is of general type, then (−1)ⁿcⁱ_l (X) cⁱ_r, (X) > 0, where i_l + … + i_r = n. Furthermore we show that the same properties hold for certain Kummer coverings.     

On a generalization of a theorem of Erdős and Fuchs

Let A={a₁,a₂,…}(a₁<a₂<) be an infinite sequence of nonnegative integers, let k≥2 be a fixed integer and denote by r_k(A,n) the number of solutions of a_i1+a_i2++a_ik≤n. Montgomery and Vaughan proved that r₂(A,n)=cn+o(n^1/4) cannot hold for any constant c>0. In this paper, we extend this result to k>2.     

Energy dependence of the scattering operator

We prove that the scattering operator S(E) depends continuously on the energy E for a certain class of Schrodinger operators, by an abstract method using trace conditions and the dilation group. We also obtain pointwise bounds on S(E) −1₁as E → ∞, and even as E → 0 in the case of repulsive potentials.     

Zeros of orthogonal polynomials on the real line

Let p_n(x) be the orthonormal polynomials associated to a measure dμ of compact support in . If Esupp(dμ), we show there is a δ>0 so that for all n, either p_n or p_n+1 has no zeros in (E−δ,E+δ). If E is an isolated point of supp(μ), we show there is a δ so that for all n, either p_n or p_n+1 has at most one zero in (E−δ,E+δ). We provide an example where the zeros of p_n are dense in a gap of supp(dμ).     

A note on the largest eigenvalue of a large dimensional sample covariance matrix

Let {v_ij; i, J = 1, 2, …} be a family of i.i.d. random variables with E(v₁₁⁴) = ∞. For positive integers p, n with p = p(n) and p/n → y > 0 as n → ∞, let M_n = (1/n) V_n V_n^T , where V_n = (v_ij)_{1 ≤ i ≤ p, 1 ≤ j ≤ n}, and let λ_max(n) denote the largest eigenvalue of M_n. It is shown that a.s. This result verifies the boundedness of E(v₁₁⁴) to be the weakest condition known to assure the almost sure convergence of λ_max(n) for a class of sample covariance matrices.     

Stability and other limit laws for exit times of random walks from a strip or a halfplane

We show that the passage time, T^*(r), of a random walk S_n above a horizontal boundary at r (r≥0) is stable (in probability) in the sense that as r→∞ for a deterministic function C(r)>0, if and only if the random walk is relatively stable in the sense that as n→∞ for a deterministic sequence B_n>0. The stability of a passage time is an important ingredient in some proofs in sequential analysis, where it arises during applications of Anscombe's Theorem. We also prove a counterpart for the almost sure stability of T^*(r), which we show is equivalent to E|X|<∞, EX>0. Similarly, counterparts for the exit of the random walk from the strip {|y|≤r} are proved. The conditions arefurther related to the relative stability of the maximal sum and the maximum modulus of the sums. Another result shows that the exit position of the random walk outside the boundaries at ±r drifts to ∞ as r→∞ if and only if the random walk drifts to ∞.     

Estimates for the first and second derivatives of the Stieltjes polynomials

Let w_λ(x)(1−x²)^λ−1/2 and P_n^(λ) be the ultraspherical polynomials with respect to w_λ(x). Then we denote E_n+1^(λ) the Stieltjes polynomials with respect to w_λ(x) satisfyingIn this paper, we give estimates for the first and second derivatives of the Stieltjes polynomials E_n+1^(λ) and the product E_n+1^(λ)P_n^(λ) by obtaining the asymptotic differential relations. Moreover, using these differential relations we estimate the second derivatives of E_n+1^(λ)(x) and E_n+1^(λ)(x)P_n^(λ)(x) at the zeros of E_n+1^(λ)(x) and the product E_n+1^(λ)(x)P_n^(λ)(x), respectively.     

Global stability for a model of competition in the chemostat with microbial inputs

We propose a model of competition of n species in a chemostat, with constant input of some species. We mainly emphasize the case that can lead to coexistence in the chemostat in a non-trivial way, i.e., where the n−1 less competitive species are in the input. We prove that if the inputs satisfy a constraint, the coexistence between the species is obtained in the form of a globally asymptotically stable (GAS) positive equilibrium, while a GAS equilibrium without the dominant species is achieved if the constraint is not satisfied. This work is round up with a thorough study of all the situations that can arise when having an arbitrary number of species in the chemostat inputs; this always results in a GAS equilibrium that either does or does not encompass one of the species that is not present in the input.     

On the approximate behavior of the posterior distribution for an extreme multivariate observation

The behavior of the posterior for a large observation is considered. Two basic situations are discussed; location vectors and natural parameters.Let X = (X₁, X₂, …, X_n) be an observation from a multivariate exponential distribution with that natural parameter Θ = (Θ₁, Θ₂, …, Θ_n). Let θ_x^* be the posterior mode. Sufficient conditions are presented for the distribution of Θ − θ_x^* given X = x to converge to a multivariate normal with mean vector 0 as |x| tends to infinity. These same conditions imply that E(Θ | X = x) − θ_x^* converges to the zero vector as |x| tends to infinity.The posterior for an observation X = (X₁, X₂, …, X_n is considered for a location vector Θ = (Θ₁, Θ₂, …, Θ_n) as x gets large along a path, γ, in Rⁿ. Sufficient conditions are given for the distribution of γ(t) − Θ given X = γ(t) to converge in law as t → ∞. Slightly stronger conditions ensure that γ(t) − E(Θ | X = γ(t)) converges to the mean of the limiting distribution.These basic results about the posterior mean are extended to cover other estimators. Loss functions which are convex functions of absolute error are considered. Let δ be a Bayes estimator for a loss function of this type. Generally, if the distribution of Θ − E(Θ | X = γ(t)) given X = γ(t) converges in law to a symmetric distribution as t → ∞, it is shown that δ(γ(t)) − E(Θ | X = γ(t)) → 0 as t → ∞.     

Signed permutations and the four color theorem

Let σ₁,σ₂ be two permutations in the symmetric group S_n. Among the many sequences of elementary transpositions τ₁,…,τ_r transforming σ₁ into σ₂=τ_rτ₁σ₁, some of them may be signable, a property introduced in this paper. We show that the four color theorem in graph theory is equivalent to the statement that, for any n≥2 and any σ₁,σ₂S_n, there exists at least one signable sequence of elementary transpositions from σ₁ to σ₂. This algebraic reformulation rests on a former geometric one in terms of signed diagonal flips, together with a codification of the triangulations of a convex polygon on n+2 vertices by permutations in S_n.     

Some remarks on the strong law of large numbers for Banach-space valued weakly integrable random variables

The purpose of this paper is to show the equivalence of almost sure convergence of S_n/n, n ≥ 1 and lim sup_n→∞S_n/n < ∞ a.e., where S_n = X₁ + X₂ + … + X_n and X₁, X₂,… are independent identically distributed random elements in a separable Banach space with EX₁ < ∞. This result disproves a result of Pop-Stojanovic [8].     

On the Multiplicities of the Primitive Idempotents of a Q-Polynomial Distance-regular Graph

Ito, Tanabe and Terwilliger recently introduced the notion of a tridiagonal pair. We apply their results to distance-regular graphs and obtain the following theorem.Theorem Let Γ denote a distance-regular graph with diameter D ≥  3. Suppose Γ is Q -polynomial with respect to the orderingE₀ , E₁, , E_Dof the primitive idempotents. For 0  ≤  i ≤  D, let m_idenote the multiplicity ofE_i . Then (i)m_i − 1 ≤ m_i (1  ≤  i ≤  D / 2),(ii)m_i  ≤ m_D − i (0  ≤  i ≤ D  / 2).By proving the above theorem we resolve a conjecture of Dennis Stanton.     

Some extensions of a property of linear representation functions

Let A={a₁,a₂,…}(a₁<a₂<) be an infinite sequence of nonnegative integers. Let k≥2 be a fixed integer and for , let R_k(A,n) be the number of solutions of a_i1++a_ik=n,a_i1,…,a_ikA, and let and denote the number of solutions with the additional restrictions a_i1<<a_ik, and a_i1≤≤a_ik respectively. Recently, Horváth proved that if d>0 is an integer, then there does not exist n₀ such that for n>n₀. In this paper, we obtain the analogous results for R_k(A,n), and .