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.     

Exponential stability in mean square of neutral stochastic differential functional equations

The aim of this paper is to investigate the exponential stability in mean square for a neutral stochastic differential functional equation of the form d[x(t) − G(x_t)] = [f(t,x(t)) + g(t, x_t)]dt + σ(t, x_t)dw(t), where x_t = {x(t + s): − τ s 0}, with τ > 0, is the past history of the solution. Several interesting examples are a given for illustration.     

New explicit filters and smoothers for diffusions with nonlinear drift and measurements

The optimal least-squares filtering of a diffusion x(t) from its noisy measurements {y(τ); 0 τ t} is given by the conditional mean E[x(t)|y(τ); 0 τ t]. When x(t) satisfies the stochastic diffusion equation dx(t) = f(x(t)) dt + dw(t) and y(t) = ∫₀^tx(s) ds + b(t), where f(·) is a global solution of the Riccati equation /xf(x) + f(x)² = f(x)² = αx² + βx + γ, for some , and w(·), b(·) are independent Brownian motions, Benes gave an explicit formula for computing the conditional mean. This paper extends Benes results to measurements y(t) = ∫₀^tx(s) ds + ∫₀^t dx(s) + b(t) (and its multidimensional version) without imposing additional conditions on f(·). Analogous results are also derived for the optimal least-squares smoothed estimate E[x(s)|y(τ); 0 τ t], s < t. The methodology relies on Girsanov's measure transformations, gauge transformations, function space integrations, Lie algebras, and the Duncan-Mortensen-Zakai equation.     

On the non-existence of stationary diffusions which satisfy the Bene condition

The one-dimensional diffusion x_t satisfying dx_t = f(x_t)dt + dw_t, where w_t is a standard Brownian motion and f(x) satisfies the Bene condition f′(x) + f²(x) = ax² + bx + c for all real x, is considered. It is shown that this diffusion does not admit a stationary probability measure except for the linear case f(x) = αx + β, α < 0.     

Exponential Stability of Slowly Time-Varying Nonlinear Systems

Let be a time-varying vector field depending on t containing a regular and a slow time scale (α large). Assume there exist a k (τ)≥1 and a γ(τ) such that ∥x _τ(t, t ₀, x ₀)∥≤k(τ) e ^{−γ(τ)(t−t0)}∥x ₀∥, with x _τ(t, t ₀, x ₀) the solution of the parametrized system with initial state x ₀ at t ₀. We show that for α sufficiently large is exponentially stable when “on average”γ(τ) is positive. The use of this result is illustrated by means of two examples. First, we extend the circle criterion. Second, exponential stability for a pendulum with a nonlinear slowly time-varying friction attaining positive and negative values is discussed. Date received: January 22, 2000. Date revised: April 14, 2001.     

Exact finite-dimensional filters via systems realization for a class of discrete-time nonlinear systems

We obtain necessary and sufficient conditions for the existence of a finite-dimensional filter for the discrete-time nonlinear system (ε x_k+1 =φ(x_k), y_k = h(x_k)+η(x_k)ν_k, K=0, 1,…. This system is distinguished by the absence of noise in the dynamic and by the correlation between the state and the intensity of noise in the observations.The necessary and sufficient condition provides an explicit formula for the minimal filter and various system-theoretic properties of (ε) and of the minimal filter.     

Stochastic stabilisation of functional differential equations

In this paper we investigate the problem of stochastic stabilisation for a general nonlinear functional differential equation. Given an unstable functional differential equation dx(t)/dt=f(t,x_t), we stochastically perturb it into a stochastic functional differential equation , where Σ is a matrix and B(t) a Brownian motion while X_t={X(t+θ):-τθ0}. Under the condition that f satisfies the local Lipschitz condition and obeys the one-side linear bound, we show that if the time lag τ is sufficiently small, there are many matrices Σ for which the stochastic functional differential equation is almost surely exponentially stable while the corresponding functional differential equation dx(t)/dt=f(t,x_t) may be unstable.     

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 joint nonlinear filtering-smoothing of diffusion processes

The smoothing of diffusions dx_t = f(x_t) dt + σ(x_t) dw_t, measured by a noisy sensor dy_t = h(x_t) dt + dv_t, where w_t and v_t are independent Wiener processes, is considered in this paper. By focussing our attention on the joint p.d.f. of (x_τ x_t), 0 ≤ τ < t, conditioned on the observation path {y_s, 0 ≤ s ≤ t}, the smoothing problem is represented as a solution of an appropriate joint filtering problem of the process, together with its random initial conditions. The filtering problem thus obtained possesses a solution represented by a Zakai-type forward equation. This solution of the smoothing problem differs from the common approach where, by concentrating on the conditional p.d.f. of x_τ alone, a set of ‘forward and reverse’ equations needs to be solved.     

The structured controllability radius of linear delay systems

In this article, we shall deal with the problem of calculation of the controllability radius of a delay dynamical systems of the form x′(t)?=?A ₀ x(t)?+?A ₁ x(t???h ₁)?+?···?+?A _k x(t???h _k )?+?Bu(t). By using multi-valued linear operators, we are able to derive computable formulas for the controllability radius of a controllable delay system in the case where the system's coefficient matrices are subjected to structured perturbations. Some examples are provided to illustrate the obtained results.     

Stability of Some Difference Equations with Two Delays

In the framework of the general problem for clarifying the stability of the zero solution of the equation x(n) = a ₁ x(n – m) – a ₂ x(n – k) with delays k and m, some partial problems are solved. An appreciable dependence of the stability on the divisibility of one delay by the other is revealed.     

On selecting thek largest with median tests

LetW _itk(n) be the minimax complexity of selecting thek largest elements ofn numbersx ₁,x ₂,...,x _n by pairwise comparisonsx _i :x _j . It is well known thatW ₂(n) =n–2+ [lgn], andW _k (n) = n + (k–1)lg n +O(1) for all fixed k 3. In this paper we studyW _k (n), the minimax complexity of selecting thek largest, when tests of the form Isx _i the median of {x _i ,x _j ,x _t }? are also allowed. It is proved thatW₂(n) =n–2+ [lgn], andW _k (n) =n + (k–1)lg₂ n +O(1) for all fixedk3.This research was supported in part by the National Science Foundation under Grant No. DCR-8308109.     

The μ-basis of a planar rational curve—properties and computation

A moving line L(x,y;t)=0 is a family of lines with one parameter t in a plane. A moving line L(x,y;t)=0 is said to follow a rational curve P(t) if the point P(t₀) is on the line L(x,y;t₀)=0 for any parameter value t₀. A μ-basis of a rational curve P(t) is a pair of lowest degree moving lines that constitute a basis of the module formed by all the moving lines following P(t), which is the syzygy module of P(t). The study of moving lines, especially the μ-basis, has recently led to an efficient method, called the moving line method, for computing the implicit equation of a rational curve [3 and 6]. In this paper, we present properties and equivalent definitions of a μ-basis of a planar rational curve. Several of these properties and definitions are new, and they help to clarify an earlier definition of the μ-basis [3]. Furthermore, based on some of these newly established properties, an efficient algorithm is presented to compute a μ-basis of a planar rational curve. This algorithm applies vector elimination to the moving line module of P(t), and has O(n²) time complexity, where n is the degree of P(t). We show that the new algorithm is more efficient than the fastest previous algorithm [7].     

Stochastic stabilization and destabilization

It is shown in this paper that any nonlinear systems in ^d can be stabilized by Brownian motion provided |ƒ(x,t)| ≤ K|x| for some K > 0. On the other hand, this system can also be destabilized by Brownian motion if the dimension d ≥ 2. Similar results are also obtained for any given stochastic differential equation dx(t) = ƒ(x(t), t) + g(x(t), t) dW(t).     

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.     

A note on two-sided stochastic control problems

We consider a class of two-sided stochastic control problems. For each continuous process π_t = π_t⁺ − π_t⁻ with bounded variation, the state process (x_t) is defined by x_t = B_t + f₀^t I(x_s - a)^d_πs+ − f₀^t I(x_s a)^dπ_s−, where a is a positive constant and (B_t) is a standard Brownian motion. We show the existence of an optimal policy so as to minimize the cost function J(π) = E [f₀^∞ e^−αsX_s² ds], with discount rate α > 0, associated with π.     

Decomposition of Polynomials with Respect to the Cyclic Group of Orderm

Given a polynomial solution of a differential equation, its m -ary decomposition, i.e. its decomposition as a sum of m polynomials P^{[ j ]}(x)  = ∑_kα_j,kx^{λ_{j, k}}containing only exponentsλ_{j, k} with λ_{j,k  + 1} − λ_j,k = m, is considered. A general algorithm is proposed in order to build holonomic equations for the m -ary parts P^{[ j ]}(x) starting from the initial one, which, in addition, provides a factorized form of them. Moreover, these differential equations are used to compute expansions of the m -ary parts of a given polynomial in terms of classical orthogonal polynomials. As illustration, binary and ternary decomposition of these classical families are worked out in detail.     

Distributions having bases which generate finite dimensional Lie algebras

Let X¹,…, X^k be real analytic vector fields on an n-dimensional manifold M, k < n, which are linearly independent at a point p ε M and which, together with their Lie products at p, span the tangent space TM_p. Then X¹,…, X^k form a local basis for a real analytic k-dimensional distribution x→D^k(x)=span{X¹(x),…,X^k(x)}. We study the question of when D^k admits a basis which generates a nilpotent, or solvable (or finite dimensional) Lie algebra. If this is the case the study of affine control systems, or partial differential operators, described via X¹,…, X^k can often be greatly simplified.