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.     

Generating orthogonal matrix polynomials satisfying second order differential equations from a trio of triangular matrices

The method developed in [A.J. Durán, F.A. Grünbaum, Orthogonal matrix polynomials satisfying second order differential equations, Int. Math. Res. Not. 10 (2004) 461–484] led us to consider matrix polynomials that are orthogonal with respect to weight matrices W(t) of the form , , and (1−t)^α(1+t)^βT(t)T^*(t), with T satisfying T^′=(2Bt+A)T, T(0)=I, T^′=(A+B/t)T, T(1)=I, and T^′(t)=(−A/(1−t)+B/(1+t))T, T(0)=I, respectively. Here A and B are in general two non-commuting matrices. We are interested in sequences of orthogonal polynomials (P_n)_n which also satisfy a second order differential equation with differential coefficients that are matrix polynomials F₂, F₁ and F₀ (independent of n) of degrees not bigger than 2, 1 and 0 respectively. To proceed further and find situations where these second order differential equations hold, we only dealt with the case when one of the matrices A or B vanishes.The purpose of this paper is to show a method which allows us to deal with the case when A, B and F₀ are simultaneously triangularizable (but without making any commutativity assumption).     

Limit distributions and one-parameter groups of linear operators on Banach spaces

Let = {U_t: t > 0} be a strongly continuous one-parameter group of operators on a Banach space X and Q be any subset of a set (X) of all probability measures on X. By (Q; ) we denote the class of all limit measures of {U_tn(μ₁ * μ₂*…*μ_n)*δ_xn}, where {μ_n}Q, {x_n}X and measures U_tnμ_j (j=1, 2,…, n; N=1, 2,…) form an infinitesimal triangular array. We define classes L_m( ) as follows: L₀( )= ( (X); ), L_m( )= (L_m−1( ); ) for m=1, 2,… and L_∞( )=_m=0^∞L_m( ). These classes are analogous to those defined earlier by Urbanik on the real line. Probability distributions from L_m( ), m=0, 1, 2,…, ∞, are described in terms of their characteristic functionals and their generalized Poisson exponents and Gaussian covariance operators.     

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

     

Stability interval for explicit difference schemes for multi-dimensional second-order hyperbolic equations with significant first-order space derivative terms

In this piece of work, we introduce a new idea and obtain stability interval for explicit difference schemes of O(k²+h²) for one, two and three space dimensional second-order hyperbolic equations u_tt=a(x,t)u_xx+α(x,t)u_x-2η²(x,t)u,u_tt=a(x,y,t)u_xx+b(x,y,t)u_yy+α(x,y,t)u_x+β(x,y,t)u_y-2η²(x,y,t)u, and u_tt=a(x,y,z,t)u_xx+b(x,y,z,t)u_yy+c(x,y,z,t)u_zz+α(x,y,z,t)u_x+β(x,y,z,t)u_y+γ(x,y,z,t)u_z-2η²(x,y,z,t)u,0<x,y,z<1,t>0 subject to appropriate initial and Dirichlet boundary conditions, where h>0 and k>0 are grid sizes in space and time coordinates, respectively. A new idea is also introduced to obtain explicit difference schemes of O(k²) in order to obtain numerical solution of u at first time step in a different manner.     

Oscillations of first-order neutral delay differential equations

Consider the neutral delay differential equation (*) (d/dt)[y(t) + py(t − τ)] + qy(t − σ) = 0, t t₀, where τ, q, and σ are positive constants, while p ε (−∞, −1) (0, + ∞). (For the case p ε [−1, 0] see Ladas and Sficas, Oscillations of neutral delay differential equations (to appear)). The following results are then proved. Theorem 1. Assume p < − 1. Then every nonoscillatory solution y(t) of Eq. (*) tends to ± ∞ as t → ∞. Theorem 2. Assume p < − 1, τ > σ, and q(σ − τ)/(1 + p) > (1/e). Then every solution of Eq. (*) oscillates. Theorems 3. Assume p > 0. Then every nonoscillatory solution y(t) of Eq. (*) tends to zero as t → ∞. Theorem 4. Assume p > 0. Then a necessary condition for all solutions of Eq. (*) to oscillate is that σ > τ. Theorem 5. Assume p > 0, σ > τ, andq(σ − τ)/(1 + p) > (1/e). Then every solution of Eq. (*) oscillates. Extensions of these results to equations with variable coefficients are also obtained.     

Parameter estimation in linear filtering

Suppose on a probability space (Ω, F, P), a partially observable random process (x_t, y_t), t ≥ 0; is given where only the second component (y_t) is observed. Furthermore assume that (x_t, y_t) satisfy the following system of stochastic differential equations driven by independent Wiener processes (W₁(t)) and (W₂(t)): dx_t=−βx_tdt+dW₁(t), x₀=0, dy_t=αx_tdt+dW₂(t), y₀=0; α, β∞(a,b), a>0. We prove the local asymptotic normality of the model and obtain a large deviation inequality for the maximum likelihood estimator (m.l.e.) of the parameter θ = (α, β). This also implies the strong consistency, efficiency, asymptotic normality and the convergence of moments for the m.l.e. The method of proof can be easily extended to obtain similar results when vector valued instead of one-dimensional processes are considered and θ is a k-dimensional vector.     

Capture in linear functional differential games of pursuit

The problem of capture in a pursuit game which is described by a linear retarded functional differential equation is considered. The initial function belongs to the Sobolev space W₂⁽¹⁾. The target is either a subset of W₂⁽¹⁾ a point in W₂⁽¹⁾, a subset of the Euclidean space Eⁿ or a point of Eⁿ. There is capture if the initial function can be forced to the target by the pursuer no matter what the quarry does. The concept of capture therefore formalizes the concepts of controllability under unpredictable disturbances. This is proved to be equivalent to the controllability of an associated linear retarded functional differential equation. There is nothing in (2) (6) or (7) below which restricts the control sets to be of the same dimension as the phase space. Our results can be applied in (2) for example, if the constraint sets Q′, P′ are subsets of E^m and Eⁱ respectively with q(t) = C(t) q′(t), − p(t) = B(t) p′(t), q′(t) ε E^mp′(t) ε E^r and B(t) is an n × r′-matrices and C(t) an n × m-matrix.     

Multifunctions of faces for conditional expectations of selectors and Jensen's inequality

Let (T, , P) be a probability space, a P-complete sub-δ-algebra of and X a Banach space. Let multifunction t → Γ(t), t T, have a (X)-measurable graph and closed convex subsets of X for values. If x(t) ε Γ(t) P-a.e. and y(·) ε E_p x(·), then y(t) ε Γ(t) P-a.e. Conversely, x(t) ε F(Γ(t), y(t)) P-a.e., where F(Γ(t), y(t)) is the face of point y(t) in Γ(t). If X = , then the same holds true if Γ(t) is Borel and convex, only. These results imply, in particular, extensions of Jensen's inequality for conditional expectations of random convex functions and provide a complete characterization of the cases when the equality holds in the extended Jensen inequality.     

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 → ∞.     

Compact causal data interpolation

Consider a Hilbert space equipped with a time-structure, i.e., a resolution E of the identity on defined on subsets of some linearly ordered set Λ. For which x and y in is it possible to find a causal (time respecting) compact operator T, so that Tx = y? When T is required to be a Hilbert-Schmidt operator and (Λ, E) is sufficiently regular, this question is answered in terms of the “time-densities” of x and y. The condition is that the integral ∝_gLμ_x({s t})⁻¹ dμ_y(t) should be finite, where μ_x and μ_y are the measures on Λ given by μ_x(Ω) = ¦|E(Ω)x¦|² and μ_y(Ω) = ¦|E(Ω)y¦|². Further a solution is given for the related problem of minimizing the sum of ¦|Tx − y¦|² and the squared Hilbert-Schmidt norm ¦|R¦|₂² of T.     

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.     

Asymptotic distributions of regression and autoregression coefficients with martingale difference disturbances

In this paper a form of the Lindeberg condition appropriate for martingale differences is used to obtain asymptotic normality of statistics for regression and autoregression. The regression model is y_t = Bz_t + v_t. The unobserved error sequence {v_t} is a sequence of martingale differences with conditional covariance matrices {Σ_t} and satisfying sup_{t=1,…, n} {v′_tv_tI(v′_tv_t>a) |z_t, v_t−1, z_t−1, …} 0 as a → ∞. The sample covariance of the independent variables z₁, …, z_n, is assumed to have a probability limit M, constant and nonsingular; max_t=1,…,nz′_tz_t/n 0. If (1/n)Σ_t=1ⁿΣ_t Σ, constant, then √nvec( _n−B) N(0,M⁻¹Σ) and _n Σ. The autoregression model is x_t = Bx_{t − 1} + v_t with the maximum absolute value of the characteristic roots of B less than one, the above conditions on {v_t}, and (1/n)Σ_t=max(r,s)+1(Σ_tv_t−1−rv′_t−1−s) δ_rs(ΣΣ), where δ_rs is the Kronecker delta. Then √nvec( _n−B) N(0,Γ⁻¹Σ), where Γ = Σ_{s = 0}^∞B^sΣ(B′)^s.     

Oscillation of second-order linear delay differential equations

The aim of this paper is to derive sufficient conditions for the linear delay differential equation (r(t)y′(t))′ + p(t)y(τ(t)) = 0 to be oscillatory by using a generalization of the Lagrange mean-value theorem, the Riccati differential inequality and the Sturm comparison theorem.      

On the L convergence of Lagrange interpolating entire functions of exponential type

Let f: be a continuous, 2π-periodic function and for each n ε let t_n(f; ·) denote the trigonometric polynomial of degree n interpolating f in the points 2kπ/(2n + 1) (k = 0, ±1, …, ±n). It was shown by J. Marcinkiewicz that lim_{n → ∞} ∝₀^2π¦f(θ) − t_n(f θ)¦^p dθ = 0 for every p > 0. We consider Lagrange interpolation of non-periodic functions by entire functions of exponential type τ > 0 in the points kπ/τ (k = 0, ± 1, ± 2, …) and obtain a result analogous to that of Marcinkiewicz.     

Orthogonal Polynomial Solutions of Spectral Type Differential Equations: Magnus' Conjecture

Let τ=σ+ν be a point mass perturbation of a classical moment functional σ by a distribution ν with finite support. We find necessary conditions for the polynomials {Q_n(x)}^∞_n=0, orthogonal relative to τ, to be a Bochner–Krall orthogonal polynomial system (BKOPS); that is, {Q_n(x)}^∞_n=0 are eigenfunctions of a finite order linear differential operator of spectral type with polynomial coefficients: L_N[y](x)=∑^N_i=1 ℓ_i(x) y⁽ⁱ⁾(x)=λ_ny(x). In particular, when ν is of order 0 as a distribution, we find necessary and sufficient conditions for {Q_n(x)}^∞_n=0 to be a BKOPS, which strongly support and clarify Magnus' conjecture which states that any BKOPS must be orthogonal relative to a classical moment functional plus one or two point masses at the end point(s) of the interval of orthogonality. This result explains not only why the Bessel-type orthogonal polynomials (found by Hendriksen) cannot be a BKOPS but also explains the phenomena for infinite-order differential equations (found by J. Koekoek and R. Koekoek), which have the generalized Jacobi polynomials and the generalized Laguerre polynomials as eigenfunctions.     

Oscillation theorem for superlinear second order damped differential equations

In this paper, we are concerned with the oscillation of second order superlinear differential equations of the form

(a(t)y^′(t))^′+p(t)y^′(t)+q(t)f(y(t))=0.     

Numerical Solution of the Bagley-Torvik Equation

We consider the numerical solution of the Bagley-Torvik equation Ay(t) + BD _* ^3/2 y(t) + Cy(t) = f(t), as a prototype fractional differential equation with two derivatives. Approximate solutions have recently been proposed in the book and papers of Podlubny in which the solution obtained with approximate methods is compared to the exact solution. In this paper we consider the reformulation of the Bagley-Torvik equation as a system of fractional differential equations of order 1/2. This allows us to propose numerical methods for its solution which are consistent and stable and have arbitrarily high order. In this context we specifically look at fractional linear multistep methods and a predictor-corrector method of Adams type.     

A new class of symmetric polynomials defined in terms of tableaux

A new class of symmetric polynomials in n variables z = (z₁,…, z_n), denoted t_λ(z), and labelled by partitions λ = [λ₁ … λ_n] is defined in terms of standard tableaux (equivalently, in terms of Gel'fand-Weyl patterns of the general linear group GL(n,C)). The t_λ(z) are shown to be a -basis of the ring of all symmetric polynomials in n variables. In contrast to the usual basis sets such as the Schur functions e_λ(z), which are homogeneous polynomials in the z_i, the t_λ(z) are inhomogeneous. This property is reflected in the fact that the t_λ(z) are a natural basis for the expansion of certain (inhomogeneous) symmetric polynomials constructed from rising factorials. This and several other properties of the t_λ(z) are proved. Two generalizations of the t_λ(z) are also given. The first generalizes the t_λ(z) to a 1-parameter family of symmetric polynomials, T_λ(α; z), where α is an arbitrary parameter. The T_λ(α; z) are shown to possess properties similar to those of the t_λ(z). The second generalizes the t_λ(z) to a class of skew-tableau symmetric polynomials, t_λ/μ(z), for which only a few preliminary results are given.     

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.