Persistence, contractivity and global stability in logistic equations with piecewise constant delays

We establish sufficient conditions for the persistence and the contractivity of solutions and the global asymptotic stability for the positive equilibrium N^*=1/(a+∑_i=0^mb_i) of the following differential equation with piecewise constant arguments:

where r(t) is a nonnegative continuous function on [0,+∞), r(t)0, ∑_i=0^mb_i>0, b_i0, i=0,1,2,…,m, and a+∑_i=0^mb_i>0. These new conditions depend on a,b₀ and ∑_i=1^mb_i, and hence these are other type conditions than those given by So and Yu (Hokkaido Math. J. 24 (1995) 269–286) and others. In particular, in the case m=0 and r(t)≡r>0, we offer necessary and sufficient conditions for the persistence and contractivity of solutions. We also investigate the following differential equation with nonlinear delay terms:

where r(t) is a nonnegative continuous function on [0,+∞), r(t)0, 1−ax−g(x,x,…,x)=0 has a unique solution x^*>0 and g(x₀,x₁,…,x_m)C¹[(0,+∞)×(0,+∞)××(0,+∞)].     

On numerical improvement of Gauss–Legendre quadrature rules

Among all integration rules with n points, it is well-known that n-point Gauss–Legendre quadrature rule∫₋₁¹f(x) dx∑_i=1ⁿw_if(x_i)has the highest possible precision degree and is analytically exact for polynomials of degree at most 2n−1, where nodes x_i are zeros of Legendre polynomial P_n(x), and w_i's are corresponding weights.In this paper we are going to estimate numerical values of nodes x_i and weights w_i so that the absolute error of introduced quadrature rule is less than a preassigned tolerance ε₀, say ε₀=10⁻⁸, for monomial functionsf(x)=x^j, j=0,1,…,2n+1.(Two monomials more than precision degree of Gauss–Legendre quadrature rules.) We also consider some conditions under which the new rules act, numerically, more accurate than the corresponding Gauss–Legendre rules. Some examples are given to show the numerical superiority of presented rules.     

A note on asymptotic expansions for Markov chains using operator theory

We consider asymptotic expansions for sums S_n on the form S_n = ƒ₀(X₀) + ƒ(X₁, X₀) + … + ƒ(X_n, X_n−1), where X_i is a Markov chain. Under different ergodicity conditions on the Markov chain and certain conditional moment conditions on ƒ(X_i, X_i−1), a simple representation of the characteristic function of S_n is obtained. The representation is in term of the maximal eigenvalue of the linear operator sending a function g(x) into the function x → E(g(X_i)exp[itƒ(X_i, x)]|X_i−1 = x).     

On Some Problems of P. Turán Concerning Lm Extremal Polynomials and Quadrature Formulas

The L_m extremal polynomials in an explicit form with respect to the weights (1−x)^−1/2 (1+x)^(m−1)/2 and (1−x)^(m−1)/2 (1+x)^−1/2 for even m are given. Also, an explicit representation for the Cotes numbers of the corresponding Turán quadrature formulas and their asymptotic behavior is provided.     

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 .     

L∞-upper bound of L2-projections onto splines at a geometric mesh

For an integer k 1 and a geometric mesh (qⁱ)_−∞^∞ with q ε (0, ∞), let M_i,k(x): = k[q^{i + k}](· − x)₊^{k − 1}, N_i,k(x): = (q^{i + k} − qⁱM_i,k(x)/k, and let A_k(q) be the Gram matrix (∝M_i,kN_j,k)_i,jεz. It is known that A_k(q)⁻¹_∞ is bounded independently of q. In this paper it is shown that A_k(q)⁻¹_∞ is strictly decreasing for q in [1, ∞). In particular, the sharp upper bound and lower bound for A_k (q)⁻¹ are obtained: for all q ε (0, ∞).     

Rewarding Maps: On Greedy Optimization of Set Functions

Given a set function, that is, a map ƒ: (E) → {−∞} from the set (E) of subsets of a finite set E into the reals including −∞, the standard greedy algorithm (GA) for optimizing ƒ starts with the empty set and then proceeds by enlarging this set greedily, element by element. A set function ƒ is said to be tractable if in this way a sequence x₀ , x₁, . . ., x_N E (N #E) of subsets with max(ƒ) {ƒ(x₀), ƒ(x₁), . . ., ƒ(x_N)} will always be found. In this note, we will reinterpret and transcend the traditions of classical GA-theory (cf., e.g., [KLS]) by establishing necessary and sufficient conditions for a set function ƒ not just to be tractable as it stands, but to give rise to a whole family of tractable set functions ƒ(η) : (E) → : x ƒ(x) + Σ_{e x}η(e), where η runs through all real valued weighting schemes η : E → , in which case ƒ will be called rewarding. In addition, we will characterize two important subclasses of rewarding maps, viz. truncatably rewarding (or well-layered) maps, that is, set functions ƒ such that [formula] is rewarding for every i = 1, . . ., N, and matroidal maps, that is, set functions ƒ such that for every η : E → and every ƒ_eta-greedy sequence x₀, x₁, . . ., x_N as above, one has max(ƒ_η) = ƒ_η(x_i) for the unique i {0, . . ., N} with ƒ_η(x₀) < ƒ_η(x₁) < ··· < ƒ_η(x_i) ≥ ƒ_η(x_{i + 1}).     

Random Quadratic Forms and the Bootstrap for U-Statistics

We study the bootstrap distribution for U-statistics with special emphasis on the degenerate case. For the Efron bootstrap we give a short proof of the consistency using Mallows′ metrics. We also study the i.i.d. weighted bootstrap [formula] where (X_i) and (ξ_i) are two i.i.d. sequences, independent of each other and where Eξ_i = 0, Var(ξ_i) = 1. It turns out that, conditionally given (X_i), this random quadratic form converges weakly to a Wiener-Ito double stochastic integral ∫¹₀ ∫¹₀h(F⁻¹(x), F⁻¹(y)) dW(x) dW(y). As a by-product we get an a.s. limit theorem for the eigenvalues of the matrix H_n=((1/n)h(X_i, X_j))_{1 ≤ i, j ≤ n}.     

On the evaluation of the Eigenvalues of a banded toeplitz block matrix

Let A = (a_ij) be an n × n Toeplitz matrix with bandwidth k + 1, K = r + s, that is, a_ij = a_j−i, i, J = 1,… ,n, a_i = 0 if i > s and if i < -r. We compute p(λ)= det(A - λI), as well as p(λ)/p′(λ), where p′(λ) is the first derivative of p(λ), by using O(k log k log n) arithmetic operations. Moreover, if a_i are m × m matrices, so that A is a banded Toeplitz block matrix, then we compute p(λ), as well as p(λ)/p′(λ), by using O(m³k(log² k + log n) + m²k log k log n) arithmetic operations. The algorithms can be extended to the computation of det(A − λB) and of its first derivative, where both A and B are banded Toeplitz matrices. The algorithms may be used as a basis for iterative solution of the eigenvalue problem for the matrix A and of the generalized eigenvalue problem for A and B.     

On piecewise-polynomial approximation of functions with a bounded fractional derivative in an Lp-norm

We study the error in approximating functions with a bounded (r + α)th derivative in an L_p-norm. Here r is a nonnegative integer, α ε [0, 1), and ƒ^{(r + α)} is the classical fractional derivative, i.e., ƒ^{(r + α)}(y) = ∝₀¹, ^α d(ƒ^(r)(t)). We prove that, for any such function ƒ, there exists a piecewise-polynomial of degree s that interpolates ƒ at n equally spaced points and that approximates ƒ with an error (in sup-norm) ƒ^{(r + α)}_p O(n^{−(r+α−1/p}). We also prove that no algorithm based on n function and/or derivative values of ƒ has the error equal ƒ^{(r + α)}_p O(n^{−(r+α−1/p}) for any ƒ. This implies the optimality of piecewise-polynomial interpolation. These two results generalize well-known results on approximating functions with bounded rth derivative (α = 0). We stress that the piecewise-polynomial approximation does not depend on α nor on p. It does not depend on the exact value of r as well; what matters is an upper bound s on r, s r. Hence, even without knowing the actual regularity (r, α, and p) of ƒ, we can approximate the function ƒ with an error equal (modulo a constant) to the minimal worst case error when the regularity were known.     

On the degree of weak convergence of a sequence of finite measures to the unit measure under convexity

This is a study of the degree of weak convergence under convexity of a sequence of finite measures μ_j on ^k, k 1, to the unit measure δ_x0. LetQ denote a convex and compact subset of ^k, let ƒ ε C^m(Q), m 0, satisfy a convexity condition and let μ be a finite measure on Q. Using standard moment methods, upper bounds and best upper bounds are obtained for ¦∝_Qƒdμ − ƒ(x₀)¦. They sometimes lead to sharp inequalities which are attained for particular μ and ƒ. These estimates are better than the corresponding ones found in the literature.     

Asymptotic expansions in the conditional central limit theorem

Let X_n, n , be i.i.d. with mean 0, variance 1, and E(¦X_n¦^r) < ∞ for some r 3. Assume that Cramér's condition is fulfilled. We prove that the conditional probabilities P(1/√n Σ_{i = 1}ⁿ X_i t¦B) can be approximated by a modified Edgeworth expansion up to order o(1/n^{(r − 2)/2)}), if the distances of the set B from the σ-fields σ(X₁, …, X_n) are of order O(1/n^{(r − 2)/2)}(lg n)^β), where β < −(r − 2)/2 for r and β < −r/2 for r . An example shows that if we replace β < −(r − 2)/2 by β = −(r − 2)/2 for r (β < −r/2 by β = −r/2 for r ) we can only obtain the approximation order O(1/n^{(r − 2)/2)}) for r (O(lg lgn/n^{(r − 2)/2)}) for r ).     

Summability of Product Jacobi Expansions

Orthogonal expansions in product Jacobi polynomials with respect to the weight function W_α, β(x)=∏^d_j=1 (1−x_j)^α_j (1+x_j)^β_j on [−1, 1]^d are studied. For α_j, β_j>−1 and α_j+β_j−1, the Cesàro (C, δ) means of the product Jacobi expansion converge in the norm of L^p(W_{α, β}, [−1, 1]^d), 1p<∞, and C([−1, 1]^d) if

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Moreover, for α_j, β_j−1/2, the (C, δ) means define a positive linear operator if and only if δ∑^d_i=1 (α_i+β_i)+3d−1.     

Boundary effects on convergence rates for Tikhonov regularization

We consider the Tikhonov regularizer f_λ of a smooth function f ε H^2m[0, 1], defined as the solution (see [1]) to We prove that if f^(j)(0) = f^(j)(1) = 0, J = m, …, k < 2m − 1, then ¦f − f_λ¦_j² Rλ^{(2k − 2j + 3)/2m}, J = 0, …, m. A detailed analysis is given of the effect of the boundary on convergence rates.     

A Chaotic Continuous Map Generates All Probability Distributions

Let ƒ be a continuous map of the compact unit interval I = [0, 1], such that ƒ², the second iterate of ƒ, is topologically transitive in I. If for some x and y in I and any t in I there exists lim(1/n) # {i ≤ n; |ƒⁱ(x) − ƒⁱ(y)| < t} for n → ∞, denote it by φ_xy(t). In the paper we consider the class (ƒ) if all φ_xy. The main results are that (ƒ) is convex and pointwise closed. Using this we show that (ƒ) is always bigger than the class (ƒ) of probability distributions generated analogously by single trajectories (and corresponding to the class of probability invariant measures of ƒ), and prove that there are universal generators of probability distributions, i.e., maps ƒ such that (ƒ) is the class of all non-decreasing functions I I (contrary to this, (ƒ) for no ƒ). These results can be extended to more general continuous maps. One of the possible applications is to use the size of (ƒ) as a measure of the degree of chaos of ƒ.     

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.     

Estimates of Freud-Christoffel functions for some weights with the whole real line as support

Upper and lower bounds for generalized Christoffel functions, called Freud-Christoffel functions, are obtained. These have the form λ_n,p(W,j,x) = infPW_Lp(R)/|P^(j)(X)| where the infimum is taken over all polynomials P(x) of degree at most n − 1. The upper and lower bounds for λ_n,p(W,j,x) are obtained for all 0 < p ∞ and J = 0, 1, 2, 3,… for weights W(x) = exp(−Q(x)), where, among other things, Q(x) is bounded in [− A, A], and Q″ is continuous in β(−A, A) for some A > 0. For p = ∞, the lower bounds give a simple proof of local and global Markov-Bernstein inequalities. For p = 2, the results remove some restrictions on Q in Freud's work. The weights considered include W(x) = exp(− ¦x¦^α/2), α > 0, and W(x) = exp(− exp(¦x¦)), > 0.     

Schur convexity and Schur multiplicative convexity for a class of symmetric functions with applications

For x = (x ₁, x ₂, …, x _n ) ∈ (0, 1 ] ⁿ and r ∈ { 1, 2, … , n}, a symmetric function F _n (x, r) is defined by the relation

On theL 1 exact penalty function with locally lipschitz functions

In this paper, we discuss the following inequality constrained optimization problem (P) minf(x) subject tog(x)0,g(x)=(g ₁(x), ...,g _r (x)) , wheref(x),g _j (x)(j=1, ...,r) are locally Lipschitz functions. TheL ₁ exact penalty function of the problem (P) is (PC) minf(x)+cp(x) subject tox R ⁿ , wherep(x)=max {0,g ₁(x), ...,g _r (x)},c>0. We will discuss the relationships between (P) and (PC). In particular, we will prove that under some (mild) conditions a local minimum of (PC) is also a local minimum of (P).     

Construction of multiwavelets with high approximation order and symmetry

In this paper, based on existing symmetric multiwavelets, we give an explicit algorithm for constructing multiwavelets with high approximation order and symmetry. Concretely, suppose Φ(x) := (φ1(x), . . . , φr(x))T is a symmetric refinable function vectors with approximation order m. For an arbitrary nonnegative integer n, a new symmetric refinable function vector Φnew(x) := (φn1 ew(x), . . . , φrn ew(x))T with approximation order m + n can be constructed through the algorithm mentioned above. Additionally,...