A further result on the oscillation of delay difference equations

In this paper, we are concerned with the delay difference equations of the form

(*)

y_n+1 − y_n + p_ny_n−k = 0, N = 0, 1, 2, …,

(*)where p_n ≥ 0 and k is a positive integer. We prove by using a new technique that

guarantees that all solutions of equation (*) oscillate, which improves many previous well-known results. In particular, our theorems also fit the case where Σⁿ⁻¹_i=n−kp_i ≤ k^k+1/(k + 1)^k+1. In addition, we present a nonoscillation sufficient condition for equation (*).     

Asymptotic and oscillatory behavior of solutions of general nonlinear difference equations of second order

The asymptotic and oscillatory behavior of solutions of some general second-order nonlinear difference equations of the form

δ(a_nh(y_n+1)δy_n)+p_nδy_n+q_n+1f(y_σ(n+1))=0 nZ,

is studied. Oscillation criteria for their solutions, when “p_n” is of constant sign, are established. Results are also presented for the damped-forced equation

δ(a_nh(y_n+1)δy_n)+p_nδy_n+q_n+1f(y_σ(n+1))=e_n nZ.

Examples are inserted in the text for illustrative purposes.     

Generalized dilations and the stabilization of uncertain systems via measurement feedback

We study the problem of semiglobally stabilizing uncertain nonlinear system

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, with (A,B) in Brunowski form. We prove that if p₁(z,u,t)u and p₂(z,u,t)u are of order greater than 1 and 0, respectively, with “generalized” dilation δ_l(z,u)=(l¹⁻ⁿz₁,…,l⁻¹z_n−1,z_n,lu) and uniformly with respect to t, where z_i is the ith component of z, then we can achieve semiglobal stabilization via arbitrarily bounded linear measurement feedback.     

Oscillation criteria of certain nonlinear partial difference equations

This paper is concerned with the nonlinear partial difference equation with continuous variables

,where a, σ_i, τ_i are positive numbers, h_i(x, y, u) ε C(R⁺ × R⁺ × R, R), uh_i(x, y, u) > 0 for u ≠ 0, h_i is nondecreasing in u, i = 1, …, m. Some oscillation criteria of this equation are obtained.     

Filtering of some nonlinear diffusions satisfying the general Bene condition

Under some regularity assumptions and the following generalization of the well-known Bene condition [1]:

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, where F(t,z) = g⁻²(t)∫f(t,z)dz, F_t, F_z, F_zz, are partial derivatives of F, we obtain explicit formulas for the unnormalized conditional density q_t(z, x) α Px_t ε dz| y_s, 0 st, where diffusion x_t on R¹ solves x₀ = x, dx_t = [β(t) + α(t)x_t + f(t, x_t] dt + g(t) dw₁, and observation y_t = ∫_o^th(s)x_s ds + ∫_o^t(s) dw_2t, with w = (w₁, w₂) a two-dimensional Wiener process.     

On the qualitative behaviors of solutions of third order nonlinear differential equations

We present some new results about oscillation and asymptotic behavior of solutions of third order nonlinear differential equations of the form

(r₂(t)(r₁(t)y^′)^′)^′+p(t)y^′+q(t)f(y(g(t)))=0.     

A unified presentation of some classes of meromorphically multivalent functions

The authors introduce and investigate various properties of a general class

U_k[p,a,β,A,B]

p, k ε N {1, 2, 3,…,}; 0 α < p; β 0

;

−1 A < B 1; 0 < B 1)

,which unifies and extends several (known or new) subclasses of meromorphically multivalent functions. The properties and characteristics of this general class, which are presented here, include growth and distortion theorems; they also involve Hadamard products (or convolution) of functions belonging to the class U_k[p,α,β,A,B].     

Existence of triple positive solutions of two-point right focal boundary value problems on time scales

We consider the following boundary value problem, (−1)ⁿ⁻¹y^Δn(t)=(−1)^p+1F(t,y(σⁿ⁻¹(t))),t[a,b]∩T, y^Δn(a)=0,0≤i≤p−1, y^Δn(σ(b))=0,p≤i≤n−1,where n ≥ 2, 1 ≤ p ≤ n - 1 is fixed and T is a time scale. By applying fixed-point theorems for operators on a cone, existence criteria are developed for triple positive solutions of the boundary value problem. We also include examples to illustrate the usefulness of the results obtained.     

Factorization of disconjugate higher-order Sturm-Liouville difference operators

Using a recently proved equivalence between disconjugacy of the 2n^th-order difference equation

and solvability of the corresponding Riccati matrix difference equation, it is shown that the equation L(y) = 0 is disconjugate on a given interval if and only if the operator L admits the factorization of the form

L(y)_k+n=M^*(c_kM(y)_k)_k+n,

where M and its adjoint M* are certain n^th-order difference operators and c_k is a sequence of positive numbers.     

Global stability for separable nonlinear delay differential equations

Consider the following separable nonlinear delay differential equation

, where we assume that, there is a strictly monotone increasing function f(x) on (−∞, +∞) such that

In this paper, to the above separable nonlinear delay differential equation, we establish conditions of global asymptotic stability for the zero solution. In particular, for a special wide class of f(x) which contains a case of f(x) = e^x−1, we give more explicit conditions. Applying these, we offer conditions of global asymptotic stability for solutions of nonautonomous logistic equations with delays.     

Existence and uniqueness theorems for fourth-order singular boundary value problems

By constructing a special cone and using cone compression and expansion fixed point theorem, the existence and uniqueness are established for the following singular fourth-order boundary value problems:

where f(t,x,y) may be singular at t=0,1; x=0 and y=0.     

Nonlinear eigenvalue problems for quasilinear systems

The paper deals with the existence of positive solutions for the quasilinear system (Φ(u'))' + λh(t)f(u) = 0,0 < t < 1 with the boundary condition u(0) = u(1) = 0. The vector-valued function Φ is defined by Φ(u) = (q(t)(p(t)u₁), …, q(t)(p(t)u_n)), where u = (u₁, …, u_n), andcovers the two important cases (u) = u and (u) = up > 1, h(t) = diag[h₁(t), …, h_n(t)] and f(u) = (f¹(u), …, fⁿ (u)). Assume that fⁱ and h_i are nonnegative continuous. For u = (u₁, …, u_n), let

, f₀ = maxf¹₀, …, fⁿ₀ and f_∞ = maxf¹_∞, …, fⁿ_∞. We prove that the boundary value problem has a positive solution, for certain finite intervals of λ, if one of f₀ and f_∞ is large enough and the other one is small enough. Our methods employ fixed-point theorem in a cone.     

Dynamics of the difference equation

We study the invariant interval, the character of semicycles, the global stability, and the boundedness of the difference equation

where the initial conditions x_−k,…,x₋₁,x₀ are nonnegative, and p,q>0.     

The query complexity of order-finding

We consider the problem where π is an unknown permutation on {0,1,…,2ⁿ−1}, y₀{0,1,…,2ⁿ−1}, and the goal is to determine the minimum r>0 such that π^r(y₀)=1. Information about π is available only via queries that yield π^x(y) from any x{0,1,…,2^m−1} and y{0,1,…,2ⁿ−1} (where m is polynomial in n). The main resource under consideration is the number of these queries. We show that the number of queries necessary to solve the problem in the classical probabilistic bounded-error model is exponential in n. This contrasts sharply with the quantum bounded-error model, where a constant number of queries suffices.     

Elements of small norm in Shanks’ cubic extensions of imaginary quadratic fields

Let be an imaginary quadratic number field with ring of integers Z_k and let k(α) be the cubic extension of k generated by the polynomial f_t(x)=x³−(t−1)x²−(t+2)x−1 with tZ_k. In the present paper we characterize all elements γZ_k[α] with norms satisfying |N_k(α)/k|≤|2t+1| for |t|≥14. This generalizes a corresponding result by Lemmermeyer and Pethő for Shanks’ cubic fields over the rationals.     

QFT template generation for time-delay plants based on zero-inclusion test

Plant template generation is the key step in applying quantitative feedback theory (QFT) to design robust control for uncertain systems. In this paper we propose a technique for generating plant templates for a class of linear systems with an uncertain time delay and affine parameter perturbations in coefficients. The main contribution lies in presenting a necessary and sufficient condition for the zero inclusion of the value set f(T,Q)={f(τ,q): τT[τ⁻,τ⁺], qQ∏_k=0^m−1[q_k⁻,q_k⁺]}, where f(τ,q)=g(q)+h(q)e^−jτω*, g(q) and h(q) are both complex-valued affine functions of the m-dimensional real vector q, and ω^* is a fixed frequency. Based on this condition, an efficient algorithm which involves, in the worst case, evaluation of m algebraic inequalities and solution of m2^m−1 one-variable quadratic equations, is developed for testing the zero inclusion of the value set f(T,Q). This zero-inclusion test algorithm allows one to utilize a pivoting procedure to generate the outer boundary of a plant template with a prescribed accuracy or resolution. The proposed template generation technique has a linear computational complexity in resolution and is, therefore, more efficient than the parameter gridding and interval methods. A numerical example illustrating the proposed technique and its computational superiority over the interval method is included.     

Empirical Bayes estimation of the scale parameter in a Pareto distribution

Let f(xθ) = αθ^αx^−(α+1)I(x>θ) be the pdf of a Pareto distribution with known shape parameter α>0, and unknown scale parameter θ. Let {(X_i, θ_i)} be a sequence of independent random pairs, where X_i's are independent with pdf f(xα_i), and θ_i are iid according to an unknown distribution G in a class of distributions whose supports are included in an interval (0, m), where m is a positive finite number. Under some assumption on the class and squared error loss, at (n + 1)th stage we construct a sequence of empirical Bayes estimators of θ_n+1 based on the past n independent observations X₁,…, X_n and the present observation X_n+1. This empirical Bayes estimator is shown to be asymptotically optimal with rate of convergence O(n^−1/2). It is also exhibited that this convergence rate cannot be improved beyond n^−1/2 for the priors in class .     

Universal stabilization of a class of nonlinear systems with homogeneous vector fields

Under relative-degree-one and minimum-phase assumptions, it is well known that the class of finite-dimensional, linear, single-input (u), single-output (y) systems (A,b,c) is universally stabilized by the feedback strategy u = Λ(λ)y, λ = y², where Λ is a function of Nussbaum type (the terminology “universal stabilization” being used in the sense of rendering /s(0/s) a global attractor for each member of the underlying class whilst assuring boundedness of the function λ(·)). A natural generalization of this result to a class ^k of nonlinear control systems (a,b,c), with positively homogeneous (of degree k 1) drift vector field a, is described. Specifically, under the relative-degree-one (cb ≠ 0) and minimum-phase hypotheses (the latter being interpreted as that of asymptotic stability of the equilibrium of the “zero dynamics”), it is shown that the strategy u = Λ(λ)/vby/vb^k−1y, assures ^k-universal stabilization. More generally, the strategy u = Λ(λ)exp(/vby/vb)y, assures -universal stabilization, where = _{k 1} ^k.     

Maximum flow and critical cutset as descriptors of multi-state systems with randomly capacitated components

Let G = (V, E, s, t) denote a directed network with node set V, arc set E = {1,…, n}, source node s and sink node t. Let Γ denote the set of all minimal s−t cutsets and b₁(τ), …, B_n(τ), the random arc capacities at time τ with known joint probability distribution function. Let Λ(τ) denote the maximum s−t flow at time τ and D(τ), the corresponding critical minimal s−t cutset. Let Ω denote a set of minimal s−t cutsets. This paper describes a comprehensive Monte Carlo sampling plan for efficiently estimating the probability that D(τ)εΩ-Γ and x<λ(τ)y at time τ and the probability that D(τ) Ω given that x < Λ(τ) y at time τ. The proposed method makes use of a readily obtainable upper bound on the probability that Λ(τ) > x to gain its computational advantage. Techniques are described for computing confidence intervals and credibility measures for assessing that specified accuracies have been achieved. The paper includes an algorithm for performing the Monte Carlo sampling experiment, an example to illustrate the technique and a listing of all steps needed for implementation.     

Robust stabilization of infinite dimensional systems by finite dimensional controllers

The problem of robustly stabilizing an infinite dimensional system with transfer function G, subject to an additive perturbation Δ is considered. It is assumed that: G ε ₀(σ) of systems introduced by Callier and Desoer [3]; the perturbation satisfies |W₁ΔW₂| < ε, where W₁ and W₂ are stable and minimum phase; and G and G + Δ have the same number of poles in ⁺. Now write W₁GW₂=G₁ + G₁, where G₁ is rational and totally unstable and G₂ is stable. Generalizing the finite dimensional results of Glover [12] this family of perturbed systems is shown to be stabilizable if and only if ε σ_min (G^*₁)( = the smallest Hankel singular value of G^*₁). A finite dimensional stabilizing controller is then given by where 2 is a rational approximation of G₂ such that

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) and K₁ robustly stabilizes G₁ to margin ε. The feedback system (G, K) will then be stable if |W₁ΔW₂| _∞< ε − Δ.