The -basis and implicitization of a rational parametric surface

The concept of a μ-basis was introduced in the case of parametrized curves in 1998 and generalized to the case of rational ruled surfaces in 2001. The μ-basis can be used to recover the parametric equation as well as to derive the implicit equation of a rational curve or surface. Furthermore, it can be used for surface reparametrization and computation of singular points. In this paper, we generalize the notion of a μ-basis to an arbitrary rational parametric surface. We show that: (1) the μ-basis of a rational surface always exists, the geometric significance of which is that any rational surface can be expressed as the intersection of three moving planes without extraneous factors; (2) the μ-basis is in fact a basis of the moving plane module of the rational surface; and (3) the μ-basis is a basis of the corresponding moving surface ideal of the rational surface when the base points are local complete intersections. As a by-product, a new algorithm is presented for computing the implicit equation of a rational surface from the μ-basis. Examples provide evidence that the new algorithm is superior than the traditional algorithm based on direct computation of a Gröbner basis. Problems for further research are also discussed.     

μ-Bases and singularities of rational planar curves

We provide a technique to detect the singularities of rational planar curves and to compute the correct order of each singularity including the infinitely near singularities without resorting to blow ups. Our approach employs the given parametrization of the curve and uses a μ-basis for the parametrization to construct two planar algebraic curves whose intersection points correspond to the parameters of the singularities including infinitely near singularities with proper multiplicity. This approach extends Abhyankar's method of t-resultants from planar polynomial curves to rational planar curves. We also derive the classical result that for a rational planar curve of degree n the sum of all the singularities with proper multiplicity is (n−1)(n−2)/2. Examples are provided to flesh out our results.     

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.     

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 Attainable Order of Collocation Methods for the Neutral Functional-Differential Equations with Proportional Delays

In this paper, we extend the recent results of H. Brunner in BIT (1997) for the DDE y′(t)= by(qt), y(0)=1 and the DVIE y(t)=1+∫₀ ^t by(qs)ds with proportional delay qt, 0<q≤1, to the neutral functional-differential equation (NFDE): and the delay Volterra integro-differential equation (DVIDE) : with proportional delays p _i t and q _i t, 0<p _i ,q _i ≤1 and complex numbers a,b _i and c _i . We analyze the attainable order of m-stage implicit (collocation-based) Runge-Kutta methods at the first mesh point t=h for the collocation solution v(t) of the NFDE and the `iterated collocation solution u _it (t)' of the DVIDE to the solution y(t), and investigate the existence of the collocation polynomials M _m (t) of v(th) or M^ _m (t) of u _it (th), t∈[0,1] such that the rational approximant v(h) or u _it (h) is the (m,m)-Padé approximant to y(h) and satisfies |v(h)−y(h)|=O(h ^{2 m +1}). If they exist, then we actually give the conditions of M _m (t) and M^ _m (t), respectively. Received September 17, 1998; revised September 30, 1999     

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.     

Stabilization and decay estimate for distributed bilinear systems

In this paper, we consider the problem of feedback stabilization for the distributed bilinear system y′(t)=Ay(t)+u(t)By(t). Here A is the infinitesimal generator of a linear C⁰ semigroup of contractions on a Hilbert space H and B:H→H is a linear bounded operator. A sufficient condition for feedback stabilization is given and explicit decay estimate is established. Applications to vibrating systems are presented.     

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.     

Errors in reduction methods

A mathematical basis is given for comparing the relative merits of various techniques used to reduce the order of large linear and non-linear dynamics problems during their numerical integration. In such techniques as Guyan-Irons, path derivatives, selected eigenvectors, Ritz vectors, etc., the nth order initial value problem of $\text{[}\text{/.y}\text{=}\text{f(y)}$ for t > 0, y(0) given] is typically reduced to the mth order (m ? n) problem of $\text{z}\text{?}\text{= g(z)}$ for t > 0, z(0) given] by the transformation $\text{y}\text{=}\text{Pz}$ where P changes from technique to technique. This paper gives an explicit approximate expression for the reduction error e_i in terms of P and the Jacobian of f. It is shown that: (a) reduction techniques are more accurate when the time rate of change of the response y is relatively small; (b) the change in response between two successive stations contributes to the errors at future stations after the change in response is transformed by a filtering matrix H, defined in terms of P; (c) the error committed at a station propagates to future stations by a mixing and scaling matrix G, defined in terms of P, Jacobian of f, and time increment h. The paper discusses the conditions under which the reduction errors may be minimized and gives guidelines for selecting the reduction basis vector, i.e. the columns of P.     

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]:

Full-size image

, 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.     

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.     

Control of singular problem via differentiation of a Min-Max

For Ω a smooth domain in Rⁿ with boundary Λ = Λ₀Λ₁, we are concerned with the wave equation y″ − Δy = S in Q_T =]0, T[ × Ω with = ∂/∂t, at source term satisfying S, S′ ε L¹(0, T L² (Ω)). A Dirichlet condition is imposed on Λ₀ and we consider an absorbing condition ∂y/∂n + uy′ = 0 in [0, T] × gL₁ where u is the control.parameter. We introduce the cost function. and using the Min-Max formulation of J we by-pasas the sensitivity analysis of u → y and obtain the gradient of J with a usual adjoint problem. We first present an abstract frame for this kind of problems. using the differentiability results of a Min-Max [1,2], which we very shortly deduce here, we show that the well posedness of the adjoint equation implies differentiability of the cost function governed by a linear well posed problem.     

On the structure of binary feedforward inverses with delay 2

Let M‘=S(Mα,f)be a semi-input-memory finite automaton with input alphabet Y and output alphabet X.If X=Y={0,1},then M‘ is a feedforware inverse with delay 2 if and only if there exists a cycle c of state diagram of Mαsuch that f(y0,…,yc,λα（t)0 can be expressed in the form of f ^(1)(y0,…,yc-1,λα（t)) yc for any state t in C and y0,y1,…,yc in Y;or of f^(2)(y0,…,yc-2,λα（t)) yc-c for any state t in Cand y0,y1,…,yc in Y;or for any state t in Cand y0,y1,…yc,in Y,y0,y1…yc satisfies the D[t] condition.The socalled y0,y1…yc satisfying the D[t] condition is that:for some i,j,(i,j)∈{(1,2),(1,3),(2,1),(2,2),(3,1),(3,2)},there exists a (c 2-k)-ary function f^(k),k=1,2,3,such that the Equation(1)and Equation (2)hokl simultaneously for all y‘c-2,…,y‘c 1∈Y. Equation (1);f(y0,…,yc-i,y‘c-i 1,…y‘c,λα（t))=f^(j)(y0,…yc-i,λα（t）） y‘c-i 1 Equation (2):f(y1,…,yc-j 1,y‘c-j 2,…,y‘c 1,λα（t））=f^(j)(y1,…,yc-j 1,λα（t）） y‘c-j s where t=δα（t）and if (i,j)=(1,2)then one and only one of the following conditions C1 and C2 holds for all y‘c-1,y‘c,y‘c 1∈Y.Condition C1:there exists a c-ary function g^(1),such that f(y0,…,yc-2,y‘c-1,y‘c,λα（t））=g^(1),(y0,…,yc-2,λα（t）） y‘c-1( )y‘c;Condition C2:there exists a (c-1)-ary functiong g^(2)such that f(y1,…,yc-2,y‘c-1,y‘c,y‘c 1,λα（t））=g^(2)(y1,…,yc-2,λα（t）） y‘c-1 y‘c,where t=δα(t).     

Discrete-time linear periodic systems: A note on the reachability and controllability interval length

Discrete-time linear systems with periodic coefficients of period T are considered in this paper. The reachability and controllability indices at time t for periodic systems, μ_rt and μ_ct, are defined. It is shown that the reachability [controllability] subspace at time t does coincide with the reachability [controllability] subspace over the interval (t − μ_rt,T, t) [(t, t + μ_ct.,T)] and properly includes the reachability [controllability] subspace over the interval (t −(μ_rt−1) T, t) [(t, t +(μ_cr−1)T)].     

The persistence of nonoscillatory solutions of difference equations under impulsive perturbations

We obtain a sufficient condition for the persistence of nonoscillatory solutions of the difference equation with continuous variable, ,under the impulsive perturbations, x(t_k+τ)−x(t_k)=I_k(x(t_k)),kN(1),     

Two-sided bounds for the asymptotic behaviour of free nonlinear vibration systems with application of the differential calculus of norms

This paper deals with the free nonlinear dynamical system [xdot](t)=A x(t)+h(t, x(t)), t≥t ₀, x(t ₀)=x ₀, where A is an n×n-matrix and h a nonlinear vector function with h(t, u)=o(∥u∥). As the first novel point, a lower bound for the asymptotic behaviour on the solution x(t) is derived. Two methods are applied to determine the optimal two-sided bounds, where one of the methods is the differential calculus of norms. In this context, the second novel point enters; it consists of a new strategy to significantly reduce the computation time for the determination of the optimal constants in the two-sided bounds. The obtained results are especially of interest in engineering and cannot be obtained by the methods used so far.     

Panconnectivity and edge-pancyclicity of 3-ary <Emphasis Type="Italic">N</Emphasis>-cubes

We study two topological properties of the 3-ary n-cube Q _n ³. Given two arbitrary distinct nodes x and y in Q _n ³, we prove that there exists an x–y path of every length ranging from d(x,y) to 3 ⁿ −1, where d(x,y) is the length of a shortest path between x and y. Based on this result, we prove that Q _n ³ is edge-pancyclic by showing that every edge in Q _n ³ lies on a cycle of every length ranging from 3 to 3 ⁿ .

Canonical completeness of infinitary μ

This paper presents a new model construction for a natural cut-free infinitary version of the propositional modal μ-calculus. Based on that the completeness of and the related system K_ω(μ) can be established directly – no detour, for example through automata theory, is needed. As a side result we also obtain a finite, cut-free sound and complete system for the propositional modal μ-calculus.