Markov-Type Inequalities for Products of Müntz Polynomials

Let Λ(λ_j)^∞_j=0 be a sequence of distinct real numbers. The span of {x^λ₀, x^λ₁, …, x^λ_n} over is denoted by M_n(Λ)span{x^λ₀, x^λ₁, …, x^λ_n}. Elements of M_n(Λ) are called Müntz polynomials. The principal result of this paper is the following Markov-type inequality for products of Müntz polynomials. T 2.1. LetΛ(λ_j)^∞_j=0andΓ(γ_j)^∞_j=0be increasing sequences of nonnegative real numbers. Let

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Then

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18(n+m+1)(λ_n+γ_m).In particular ,

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Under some necessary extra assumptions, an analog of the above Markov-type inequality is extended to the cases when the factor x is dropped, and when the interval [0, 1] is replaced by [a, b](0, ∞).     

The Existence of Positive Solutions to Neutral Differential Equations

In this paper, we shall consider a class of neutral differential equations of the form

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where τ (0, ∞), σ [0, ∞), Q(t) C([t₀, ∞), R ⁺ ), r(t) C([t₀, ∞), (0, ∞)) with r(t) nondecreasing on [t₀ − τ, ∞). We shall show that all positive solutions of ( * ) can be classified into four types, A, B, C, and D, and we shall obtain sufficient and necessary conditions for the existence of A-type, B-type, and D-type positive solutions of ( * ), respectively. A sufficient condition for the existence of C-type positive solutions of ( * ) is also given. Finally, we shall offer a sharp oscillation result for all solutions of ( * ). Our results generalize and improve those established in B. Yang and B. G. Zhang (Funkcial. Ekvac.39 (1996), 347–362).     

Positive solution to a special singular second-order boundary value problem

Let λ be a nonnegative parameter. The existence of a positive solution is studied for a semipositone second-order boundary value problem

where d>0,α≥0,β≥0,α+β>0, q(t)f(t,u,v)≥0 on a suitable subset of [0,1]×[0,+∞)×(−∞,+∞) and f(t,u,v) is allowed to be singular at t=0,t=1 and u=0. The proofs are based on the Leray–Schauder fixed point theorem and the localization method.     

Zolotarevω-Polynomials inWH[0, 1]

The main result of this paper characterizes generalizationsof Zolotarev polynomials as extremal functions in the Kolmogorov–Landauproblem

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whereω(t) is a concave modulus of continuity,r, m: 1mr,are integers, andBB₀(r, m, ω). We show that theextremal functionsZ_Bhaver+1 points of alternance andthe full modulus of continuity ofZ^(r)_B: ω(Z^(r)_B; t)=ω(t) for allt[0, 1]. This generalizesthe Karlin's result on the extremality of classical Zolotarevpolynomials in the problem () forω(t)=tand allBB_r.     

A connection between a generalized Pascal matrix and the hypergeometric function

The n × n generalized Pascal matrix P(t) whose elements are related to the hypergeometric function ₂F₁(a, b; c; x) is presented and the Cholesky decomposition of P(t) is obtained. As a result, it is shown that

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is the solution of the Gauss's hypergeometric differential equation,

x(1 − x)y″ + [1 + (a + b − 1)x]y′ − ABY = 0

. where a and b are any nonnegative integers. Moreover, a recurrence relation for generating the elements of P(t) is given.     

Unbounded divergence of simple quadrature formulas

Given a sequence of real or complex coefficients c_i and a sequence of distinct nodes t_i in a compact interval T, we prove the divergence and the unbounded divergence on superdense sets in the space C(T) of the simple quadrature formulas ∝_Tx(t)du(t) = Q_n(x) + R_n(x) and ∝_Tw(t)x(t)dt = Q_n(x) + R_n(x), where Q_n(x)=∑_i=1^m_n c_ix(t_i), ε C(T).The divergence (not certainly unbounded) for at most one continuous function of the first simple quadrature formula, with m_n = n and u(t) = t, was established by P. J. Davis in 1953.     

Some oscillation criteria for second order nonlinear functional ordinary differential equations

The main objective of this article is to study the oscillatory behavior of the solutions of the following nonlinear functional differential equations (a(t)x'(t))' δ1p(t)x'(t) δ2q(t)f(x(g(t))) = 0,for 0 ≤ t0 ≤ t, where δ1 = ±1 and δ2 = ±1. The functions p,q,g : [t0, ∞) → R, f :R → R are continuous, a(t) ＞ 0, p(t) ≥ 0,q(t) ≥ 0 for t ≥ t0, limt→∞ g(t) = ∞, and q is not identically zero on any subinterval of [t0, ∞). Moreover, the functions q(t),g(t), and a(t) are continuously differentiable.     

Approximation of elements of a generalized Orlicz sequence space by nonlinear, singular kernels

Let l be a generalized Orlicz sequence space generated by a modular (x) = ∑_{i − 0}^∞ _i(¦t_i¦), X = (t_i), with s-convex functions _i, 0 < s 1, and let K_w,j: R₊ → R₊ for j=0,1,2,…, w ε Wwhere is an abstract set of indices. Assuming certain singularity assumptions on the nonlinear kernel K_w,j and setting T_wx = ((T_wx)_i)_{i = 0}^∞, with (T_wx)_i = ∑_{j = 0}ⁱ K_{w,i − j}(¦t_j¦) for x = (t_j), convergence results: T_wx → x in l are obtained (both modular convergence and norm convergence), with respect to a filter of subsets of the set .     

Nonradial large solutions of sublinear elliptic problems

Let p be a nonnegative locally bounded function on , N3, and 0<γ<1. Assuming that the oscillation sup_|x|=rp(x)−inf_|x|=rp(x) tends to zero as r→∞ at a specified rate, it is shown that the equation Δu=p(x)u^γ admits a positive solution in satisfying lim_|x|→∞u(x)=∞ if and only if

     

A recursive algorithm for the infinity-norm fixed point problem

We present the PFix algorithm for the fixed point problem f(x)=x on a nonempty domain [a,b], where d1, , and f is a Lipschitz continuous function with respect to the infinity norm, with constant q1. The computed approximation satisfies the residual criterion , where >0. In general, the algorithm requires no more than ∑_i=1^dsⁱ function component evaluations, where s≡max(1,log₂(||b−a||_∞/))+1. This upper bound has order as →0. For the domain [0,1]^d with <0.5 we prove a stronger result, i.e., an upper bound on the number of function component evaluations is , where r≡log₂(1/). This bound approaches as r→∞ (→0) and as d→∞. We show that when q<1 the algorithm can also compute an approximation satisfying the absolute criterion , where x^* is the unique fixed point of f. The complexity in this case resembles the complexity of the residual criterion problem, but with tolerance (1−q) instead of . We show that when q>1 the absolute criterion problem has infinite worst-case complexity when information consists of function evaluations. Finally, we report several numerical tests in which the actual number of evaluations is usually much smaller than the upper complexity bound.     

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).     

RELATIVE WIDTH OF SMOOTH CLASSES OF MULTIVARIATE PERIODIC FUNCTIONS WITH RESTRICTIONS ON ITERATED LAPLACE DERIVATIVES IN THE L2-METRIC

For two subsets W and V of a Banach space X, let Kn(W, V, X) denote the relative Kolmogorov n-width of W relative to V defined by Kn (W, V, X) := inf sup Ln f∈W g∈V∩Ln inf ‖f-g‖x,where the infimum is taken over all n-dimensional linear subspaces Ln of X. Let W2(△r) denote the class of 2w-periodic functions f with d-variables satisfying ∫[-π,π]d |△rf(x)|2dx ≤ 1,while △r is the r-iterate of Laplace operator △. This article discusses the relative Kolmogorov n-width of W2(△r) relative to W2(△r) in Lq([-r, πr]d) (1 ≤ q ≤∞), and obtain its weak asymptotic result.     

Stability conditions for systems of linear nonautonomous delay differential equations

Systems of linear nonautonomous delay differential equations are considered which are of the form y′_i(t) = ∑_{k = 1}ⁿ ∝₀^T b_ik(t, s) y_k(t − s) dη_ik(s) − c_i(t) y_i(t), where I = 1,…, n. Sufficient conditions are derived for both the asymptotic stability and the instability of the zero solution. The main result is found by a monotone technique using elementary methods only. Moreover, additional criteria are obtained by using the method of Lyapunov functionals.     

Bounds on margin distributions in learning problems

Let be a probability space and let P_n be the empirical measure based on i.i.d. sample (X₁,…,X_n) from P. Let be a class of measurable real valued functions on For define F_f(t):=P{ft} and F_n,f(t):=P_n{ft}. Given γ(0,1], define _n,γ(δ):=1/(n^1−γ/2δ^γ). We show that if the L₂(P_n)-entropy of the class grows as ^−α for some α(0,2), then, for all and all δ(0,Δ_n), Δ_n=O(n^1/2),

and

where and c(σ)↓1 as σ↓0 (the above inequalities hold for any fixed σ(0,1] with a high probability). Also, define

Then for all

uniformly in and with probability 1 (for the above ratio is bounded away from 0 and from ∞). The results are motivated by recent developments in machine learning, where they are used to bound the generalization error of learning algorithms. We also prove some more general results of similar nature, show the sharpness of the conditions and discuss the applications in learning theory.     

On existence of singular solutions

In the paper sufficient conditions are given under which the differential equation y⁽ⁿ⁾=f(t,y,…,y⁽ⁿ⁻²⁾)g(y⁽ⁿ⁻¹⁾) has a singular solution y :[T,τ)→R, τ<∞ fulfilling

     

Asymptotic Behavior of Sobolev-Type Orthogonal Polynomials on the Unit Circle

We study the asymptotic behavior of the sequence of polynomials orthogonal with respect to the discrete Sobolev inner product on the unit circle

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where f(Z)=(f(z₁), …, f^(l₁)(z₁), …, f(z_m), …, f^(l_m)(z_m)), A is a M×M positive definite matrix or a positive semidefinite diagonal block matrix, M=l₁+…+l_m+m, dμ belongs to a certain class of measures, and |z_i|>1, i=1, 2, …, m.     

Uniform persistence for Lotka–Volterra-type delay differential systems

Consider the uniform persistence (permanence) of models governed by the following Lotka–Volterra-type delay differential system:

where each r_i(t) is a nonnegative continuous function on [0,+∞), r_i(t)0, each a_i0 and τ_ij^k(t)t, 1i,jn, 0km.In this paper, we establish sufficient conditions of the uniform persistence and contractivity for solutions (and global asymptotic stability). In particular, we extend the results in Wang and Ma (J. Math. Anal. Appl. 158 (1991) 256) for a predator–prey system and Lu and Takeuchi (Nonlinear Anal. TMA 22 (1994) 847) for a competitive system in the case n=2, to the above system with n2.     

Permanence and global stability in a Lotka-Volterra predator-prey system with delays

Consider the permanence and global asymptotic stability of models governed by the following Lotka-Volterra-type system:

, with initial conditions

x_i(t) = φ_i(t) ≥ o, t ≤ t₀, and φ_i(t₀) > 0. 1 ≤ i ≤n

. We define x₀(t) = x_n+1(t)≡0 and suppose that φ_i(t), 1 ≤ i ≤ n, are bounded continuous functions on [t₀, + ∞) and γ_i, α_i, c_i > 0,γ_i,j ≥ 0, for all relevant i,j.Extending a technique of Saito, Hara and Ma[1] for n = 2 to the above system for n ≥ 2, we offer sufficient conditions for permanence and global asymptotic stability of the solutions which improve the well-known result of Gopalsamy.     

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.     

On Rado numbers for

For all integers m3 and all natural numbers a₁,a₂,…,a_m−1, let n=R(a₁,a₂,…,a_m−1) represent the least integer such that for every 2-coloring of the set {1,2,…,n} there exists a monochromatic solution to

a₁x₁+a₂x₂++a_m−1x_m−1=x_m.