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.     

Monoid-labeled transition systems

Given a -complete (semi)lattice , we consider -labeled transition systems as coalgebras of a functor ⁽⁻⁾, associating with a set X the set ^X of all -fuzzy subsets. We describe simulations and bisimulations of -coalgebras to show that L⁽⁻⁾ weakly preserves nonempty kernel pairs iff it weakly preserves nonempty pullbacks iff L is join infinitely distributive (JID).Exchanging for a commutative monoid , we consider the functor ⁽⁻⁾_ω which associates with a set X all finite multisets containing elements of X with multiplicities m M. The corresponding functor weakly preserves nonempty pullbacks along injectives iff 0 is the only invertible element of , and it preserves nonempty kernel pairs iff is refinable, in the sense that two sum representations of the same value, r₁ + … + r_m = c₁ + … + c_n, have a common refinement matrix (m_{(i, j)}) whose k-th row sums to r_k and whose l-th column sums to c_l for any 1≤ k ≤ m and 1 ≤ l ≤ n.     

Complexity of Computing the Local Dimension of a Semialgebraic Set

The paper describes several algorithms related to a problem of computing the local dimension of a semialgebraic set. Let a semialgebraic set V be defined by a system of k inequalities of the formf  ≥  0 with f  R [ X₁, ,X_n ], deg(f)  < d , andx   V . An algorithm is constructed for computing the dimension of the Zariski tangent space to V at x in time (kd)^O(n). Let x belong to a stratum of codimension l_xin V with respect to a smooth stratification ofV . Another algorithm computes the local dimension dim_x(V) with the complexity (k(l_x +  1)d)^O(l_x2n). Ifl  = max_x  Vl_x, and for every connected component the local dimension is the same at each point, then the algorithm computes the dimension of every connected component with complexity (k(l +  1)d)^O(l2n). If V is a real algebraic variety defined by a system of equations, then the complexity of the algorithm is less thankd^O(l2n) , and the algorithm also finds the dimension of the tangent space to V at x in time kd^O(n). Whenl is fixed, like in the case of a smooth V , the complexity bounds for computing the local dimension are (kd)^O(n)andkd^O(n) respectively. A third algorithm finds the singular locus ofV in time (kd)^O(n2).     

An extension of Christopher's method†

A proposal is made to extend the method of Christopher (1973). which gives an accurate approximation to equations of the form

$

to equations of the form

where f(x) is cither a polynomial of the form

$

or can be approximated by such a polynomial.

The approach suggested is the approximation of f(x) by the cubic, c ₁ x + c ₃ x ³, in a Chebychev sense. Having thus obtained the coefficients c ₁, and c ₃, Christopher's method can then be applied to the resulting approximate equation.     

Stabilizability of two-dimensional linear systems via switched output feedback

The problem of stabilizing a second-order SISO LTI system of the form , y=Cx with feedback of the form u(x)=v(x)Cx is considered, where v(x) is real-valued and has domain which is all of . It is shown that, when stabilization is possible, v(x) can be chosen to take on no more than two values throughout the entire state-space (i.e., v(x){k₁,k₂} for all x and for some k₁,k₂), and an algorithm for finding a specific choice of v(x) is presented. It is also shown that the classical root locus of the corresponding transfer function C(sI-A)^-1B has a strong connection to this stabilization problem, and its utility is demonstrated through examples.     

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.     

The Solution of Linear Probabilistic Recurrence Relations

Abstract. Linear probabilistic divide-and-conquer recurrence relations arise when analyzing the running time of divide-and-conquer randomized algorithms. We consider first the problem of finding the expected value of the random process T(x) , described as the output of a randomized recursive algorithm T . On input x , T generates a sample (h ₁ ,···,h _k ) from a given probability distribution on [0,1] ^k and recurses by returning g(x) + Σ _i=1 ^k c _i T(h _i x) until a constant is returned when x becomes less than a given number. Under some minor assumptions on the problem parameters, we present a closed-form asymptotic solution of the expected value of T(x) . We show that E[T(x)] = Θ( x ^p + x ^p ∈t ₁ ^x (g(u)/ u ^p+1 ) du) , where p is the nonnegative unique solution of the equation Σ _i=1 ^k c _i E[h _i ^p ] = 1 . This generalizes the result in [1] where we considered the deterministic version of the recurrence. Then, following [2], we argue that the solution holds under a broad class of perturbations including floors and ceilings that usually accompany the recurrences that arise when analyzing randomized divide-and-conquer algorithms.     

Positively invariant sets of discrete-time systems with constrained inputs

Linear discrete-time dynamical systems x_k + _I = Ax_k + _k with constrained inputs c_k ∈ ω, for which the matrix A possesses the property of leaving a proper cone AK + positively invariant, i.e. AK + ? K + . Necessary and sufficient conditions guarantee that a non-empty set 𝒟(K; a, b) ? Rⁿ, obtained from the intersection of translated proper cones, is positively invariant for motions of the system. Both the homogeneous and inhomogeous cases are considered. In the latter case, the external behaviour of motions, i.e. for trajectories originating from x₀ ? Rⁿ/𝒟(K; a, b) (respectively,x_o ? Rⁿ) is studied in terms of attractive and contractivity of the set 𝒟(K; a, b). The global attractivity conditions of 𝒟(K; a, b) are also given. It is shown how the results presented can be used to solve the saturated state feedback regulator problem.     

The Computational Complexity of Some Julia Sets

Numerous computer programs have been written to compute sets of points which approximate Julia sets [4]. Usually, no error estimations are added so that it remains unclear, how good such approximations are. Furthermore, high precision pictures are unreliable because of rounding errors, since the realizing computer programs use fixed length floating point numbers. Computable error estimation w.r.t. the Hausdorff metric d_H means that the set is recursive [10]. Many Julia sets J are recursive [11]. Recursive compact subsets of the Euclidean plane have a computable Turing machine time complexity [10]. In this paper we prove that the Julia set of a complex function f(z) = z² + c for c < 1/4 can be computed locally in time O(k²M(k)) (where M(k) is a time bound for multiplication of k-bit integers). Roughly speaking, the local time complexity is the number of Turing machine steps to decide for a single point whether it belongs to a grid K_k (2^−k · )² such that d_H(K_k,J) ≤ = 2^−k.     

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.     

Finding the Extrema of a Distributed Multiset

We consider the problem of finding the extrema of a distributed multiset in a ring, that is, of determining the minimum and the maximum values,x_minandx_max, of a multisetX= {x₀,x₂, ...,x_n−1} whose elements are drawn from a totally ordered universeUand stored at thenentities of a ring network. This problem is unsolvable if the ring size is not known to the entities, and it has complexity Θ(n²) in the case of asynchronous rings of known size. We show that, in synchronous rings of known size, this problem can always be solved inO((c+ logn) ·n) bits andO(n·c·x^1/c) time for any integerc> 0, wherex= Max{|x_min|, |x_max|}. The previous solutions requiredO(n²) bits and the same amount of time. Based on these results, we also present a bit-optimal solution to the problem of finding the multiplicity of the extrema.     

Existence of set-interpolating and energy-minimizing curves

We consider existence of curves which minimize an energy of the form ∫c^(k)p (k=1,2,… , 1<p<∞) under side-conditions of the form G_j(c(t_1,j),…,c^(k−1)(t_k,j))M_j, where G_j is a continuous function, t_i,j[0,1], M_j is some closed set, and the indices j range in some index set J. This includes the problem of finding energy minimizing interpolants restricted to surfaces, and also variational near-interpolating problems. The norm used for vectors does not have to be Euclidean.It is shown that such an energy minimizer exists if there exists a curve satisfying the side conditions at all, and if among the interpolation conditions there are at least k points to be interpolated. In the case k=1, some relations to arc length are shown.     

Lumped model of a non-linear distributed system by finite element method

The finite element method has been used to find an approximate lumped parameter model of a non-linear distributed parameter system. A one dimensional non-linear dispersion system is considered. The space domain is divided into a finite set of k elements. Each element, has n nodes. Within each element the concentration is represented by C(x,t)^(e) = [N][C] ^T where [N] = [n₁(x),n₂(x), [tdot] n_n(x)] and [C] = [C₁(t),C₂(t), [tdot] C_n(t)]. By using Galerkin's criterion a set of (k × n ? n+ 1) first order differential equations are obtained for C_i(t). These equations are solved by an iterative method. The concepts are illustrated by an example taking five three-node elements in the space domain. The results are compared with those obtained by a finite difference method. It is shown that the finite element method can be used effectively in modelling of a distributed system by a lumped system.     

Shellsort with a constant number of increments

We consider the worst-case running time of Shellsort when only a constant number,c, of increments are allowed. Forc=3, we show that Shellsort can be implemented to run inO(N ^5/3) time, which is optimal. Forc=4, we further improve the running time toO(N ^11/7), and, forc=5, we obtain a bound ofO(N ^23/15). We also show anO(N ^1+1/k ) bound for generalc, wherek=[(1+8c+1)/4]. Forc=6, this isO(N ^3/2).     

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.     

Water waves: eigenvalue placement by linear feedback

This paper considers the effect of linear feedback control on the eigenvalues of a system of small-amplitude waves on a fluid surface. When such waves are controlled by a vibrating wall, the system has the form w = A ₀ w + 〈f, w〉b, where the evolution operator A ₀ is unbounded and skew-adjoint with compact resolvent on a Hilbert space H, the fixed vector b?H reflects the design of the controlling wall (and is subject to a mild smoothness condition) and f?H is to be chosen as a control function. The kth eigenvalue of A ₀ is λ _k = ik ^1/2 + O(k?3/2), k = ±1, ±2,[tdot] with corresponding eigenvector φ _k . If 〈 _f , φ _k 〉 is denoted by b_k , then the main result of this paper is: let {η _k } be any sequence of complex numbers for which (η _k 7minus; λ _k )/b_k is square summable. Then there exists a unique control f?H for which {η _k } is precisely the set of eigenvalues of the closed-loop operator A ₀( ·) + 〈f, · 〉b, and the corresponding eigenfunctions form a basis for H. For skew-adjoint systems of the same form where λ _k = ik_r + O(k ^2?r ) for 1/2 < r < 1, the above result remains true. A weaker result is given for the case where 0 < r < 1/2.     

Interpolation of Shifted-Lacunary Polynomials

Given a “black box” function to evaluate an unknown rational polynomial f ? \mathbbQ[x]f \in {\mathbb{Q}}[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity $t \in {\mathbb{Z}}_{>0}$t \in {\mathbb{Z}}_{>0}, the shift a ? \mathbbQ\alpha \in {\mathbb{Q}}, the exponents 0 ￡ e₁ < e₂ < ? < e_t{0 \leq e_{1} < e_{2} < \cdots < e_{t}}, and the coefficients c₁, ?, c_t ? \mathbbQ \{0}c_{1}, \ldots , c_{t} \in {\mathbb{Q}} \setminus \{0\} such that

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 thatW′₂(n) =n?2+ [lgn], andW′ _k (n) =n + (k?1)lg₂ n +O(1) for all fixedk≥3.     

Tensor and border rank of certain classes of matrices and the fast evaluation of determinant inverse matrix and eigenvalues

The tensor rankA of the linear spaceA generated by the set of linearly independent matricesA ₁, A₂, …, A_p, is the least integert for wich there existt diadsu ^(r) v ^(r)τ, τ=1,2,...,t, such that . IfA=n+k,k≪n then some computational problems concerning matricesA∈A can be solyed fast. For example the parallel inversion of almost any nonsingular matrixA∈A costs 3 logn+0(log² k) steps with max(n ²+p (n+k), k² n+nk) processors, the evaluation of the determinant ofA can be performed by a parallel algorithm in logp+logn+0 (log² k) parallel steps and by a sequential algorithm inn(1+k ²)+p (n+k)+0 (k ³) multiplications. Analogous results hold to accomplish one step of bisection method, Newton's iterations method and shifted inverse power method applied toA−λB in order to compute the (generalized) eigenvalues provided thatA, B∈A. The same results hold if tensor rank is replaced by border rank. Applications to the case of banded Toeplitz matrices are shown. Dedicated to Professor S. Faedo on his 70th birthday Part of the results of this paper has been presented at the Oberwolfach Conference on Komplexitatstheorie, November 1983