Some results on uniform arithmetic circuit complexity

We introduce a natural set of arithmetic expressions and define the complexity class AE to consist of all those arithmetic functions (over the fieldsF _2n) that are described by these expressions. We show that AE coincides with the class of functions that are computable with constant depth and polynomial-size unbounded fan-in arithmetic circuits satisfying a natural uniformity constraint (DLOGTIME-uniformity). A 1-input and 1-output arithmetic function over the fieldsF2n may be identified with ann-input andn-output Boolean function when field elements are represented as bit strings. We prove that if some such representation is X-uniform (where X is P or DLOGTIME), then the arithmetic complexity of a function (measured with X-uniform unbounded fan-in arithmetic circuits) is identical to the Boolean complexity of this function (measured with X-uniform threshold circuits). We show the existence of a P-uniform representation and we give partial results concerning the existence of representations with more restrictive uniformity properties.The research of G. S. Frandsen was partially carried out while visiting Dartmouth College, New Hampshire. He was partially supported by the Danish Natural Science Research Council (Grant No. 11-7991) and by the ESPRIT II Basic Research Actions Program of the EC under Contract No. 3075 (project ALCOM). D. A. M. Barrington's research was supported by NSF Computer and Computation Theory Grant CCR-8714714.     

NOTE ON THE ARITHMETIC RULE BY ZADEH FOR FUZZY CONDITIONAL INFERENCE

Abstract

This paper shows that Zadeh's arithmetic rule for fuzzy conditional propositions “If x is A then y is B” and “If x is A then y is B else y is C” can infer quite reasonable consequences in a fuzzy conditional inference if new compositions of “max-[Odot] composition” and “max- composition” are used in the compositional rule of inference, though, as was pointed out before, this arithmetic rule cannot get suitable consequences in the compositional rule of inference which uses max-min composition. Moreover, it is shown that the arithmetic rule satisfies a syllogism under these two compositions.     

一类可对称化矩阵反问题的最小二乘解

1.引言 用Rn×m,ORn×n,SRn×n及ASRn×n分别表示n×m实矩阵,n阶实正交矩阵,n阶实对称矩阵和n阶实反对称矩阵的全体组成的集合.用S⊥表示集合S的正交补,A(?)B表示A和B的正交直和.设A,B∈Rn×m,定义A与B的内积为     

Orthogonal least-squares parameter estimation algorithms for non-linear stochastic systems

The derivations of orthogonal least-squares algorithms based on the principle of Hsia's method and generalized least-squares are presented. Extensions to the case of non-linear stochastic systems are discussed and the performance of the algorithms is illustrated with the identification of both simulated systems and linear models of an electric arc furnace and a gas furnace.     

Calculus Students' Use and Interpretation of Variables: Algebraic vs. Arithmetic Thinking

Abstract

The ability to use and interpret algebraic variables as generalized numbers and changing quantities is fundamental to the learning of calculus. This study considers the use of variables in these advanced ways as a component of algebraic thinking. College introductory calculus students' (n = 174) written responses to algebra problems requiring the use and interpretation of variables as changing quantities were examined for evidence of algebraic and arithmetic thinking. A framework was developed to describe and categorize examples of algebraic, transitional, and arithmetic thinking reflected in these students' uses of variables. The extent to which students' responses showed evidence of algebraic or arithmetic thinking was quantified and related to their course grades. Only one third of the responses of these entering calculus students were identified as representative of algebraic thinking. This study extends previous research by showing that evidence of algebraic thinking in students' work was positively related to successful performance in calculus.     

多种燃料锅炉效率曲线的辨识方法

多种燃料锅炉的运行优化是钢铁联合企业节约能源的一个主要方法,但是这种锅炉的效率曲线的辨识却是一个复杂而困难的非线性最小二乘问题。借鉴大系统递阶算法的思想,本文提出了一种新的两级递阶辨识的算法。这种算法通过预估关联量,将非线性最小二乘问题转化为两级线性最小二乘问题。实验的结果证明:这种算法是一种有效的算法。     

A DESCRIPTION OF A SET OF HYPOTHESES FOR SYSTEM IDENTIFICATION PROBLEM*

This paper concerns the system identification process which is a specific form of the hypothetico-deductive process. More specifically, this paper deals with the inductive inference, i.e., with the process of generating a set of hypotheses σ(E) that explain a given finite set of input-output experiments performed on a finite sequential system being identified. It is shown that Sεσ(E) if and only if there exists a homomorphism of a basic hypothesis explaining E, into S. Next, a set of hypotheses σ¹(E). defined as follows: Sεσ¹(E) if E is structure-complete w.r.t. S, is considered. Then it is proved that Sεσ¹(E) if and only if there exists a full homomorphism of a basic hypothesis onto S. Some important methodological consequences of obtained results are derived. Finally, relationship linking the properties of a system identification algorithm is investigated.     

A new orthogonal series approach to state-space analysis and identification

The contribution of this paper is two-fold. First, it introduces a new orthogonal series approach to the state-space analysis of linear time-invariant systems. This approach yields explicit expressions for the state and output vector coefficient matrices. These expressions only involve the multiplication of matrices of small dimensions. No algebraic system of equations needs to be solved, and therefore no inversion of large matrices is required here, as compared to other known techniques. The second contribution consists of using this new orthogonal series technique to solve the state-space identification problem. It is shown that by appropriately manipulating the aforementioned state-space analysis results, an algorithm is derived which yields the state-space system matrix A. This algorithm gives a new outlook and a better insight into the state-space identification problem.     

System identification with generalized orthonormal basis functions

A least-squares identification method is studied that estimates a finite number of expansion coefficients in the series expansion of a transfer function, where the expansion is in terms of recently introduced generalized basis functions. The basis functions are orthogonal in ₂, and generalize the pulse, Laguerre and Kautz bases. One of their important properties is that, when chosen properly, they can substantially increase the speed of convergence of the series expansion. This leads to accurate approximate models with only a few coefficients to be estimated. Explicit bounds are derived for the bias and variance errors that occur in parameter estimates as well as in the resulting transfer function estimates.     

Hybrid curve fitting

We consider a parameterized family of closed planar curves and introduce an evolution process for identifying a member of the family that approximates a given unorganized point cloud {p _i } _{i =1,..., N} . The evolution is driven by the normal velocities at the closest (or foot) points (f _i ) to the data points, which are found by approximating the corresponding difference vectors p _i -f _i in the least-squares sense. In the particular case of parametrically defined curves, this process is shown to be equivalent to normal (or tangent) distance minimization, see [3], [19]. Moreover, it can be generalized to very general representations of curves. These include hybrid curves, which are a collection of parametrically and implicitly defined curve segments, pieced together with certain degrees of geometric continuity.     

AN ITERATIVE INFERENCE MECHANISM FOR THE PROBABILISTIC EXPERT SYSTEM PES

Abstract

The paper suggests a new inference mechanism based on iterative use of the Bayesian inference scheme. The procedure iteratively computes optimal component weights of a distribution mixture from a class called generalized knowledge base. It is proved that the iterative process converges to a unique limit whereby the resulting probability distribution can be defined as the information-divergence projection of input distribution on the generalized knowledge base. The iterative inference mechanism resembles natural process of cognition as iteratively improving understanding of the input information.     

A PARALLEL ALGORITHM FOR HOUSEHOLDER'S TRID1AGONALIZATION OF A SYMMETRIC MATRIX

Abstract

The tridiagonalization of symmetric matrices appears frequently in the solution of mathematical problems. In this paper we present the parallelization of Householder's method in hypercube computers with distributed memory and message-passing communications, using a single program-multiple data mode of parallelism. Results for the NCUBE/10 and for a SUPERNODE system, composed of eighi transputers T800 interconnected as a hypercube, are included. We have designed a parallel algorithm getting minimum data redundancy (the matrices are symmetric) and a very high computational load balance among the processing elements.     

A split least-squares characteristic mixed element method for nonlinear nonstationary convection–diffusion problem

In this paper, a split least-squares characteristic mixed finite element method is proposed for solving nonlinear nonstationary convection–diffusion problem. By selecting the least-squares functional property, the resulting least-squares procedure can be split into two independent symmetric positive definite sub-schemes. The first sub-scheme is for the unknown variable u, which is the same as the standard characteristic Galerkin finite element approximation. The second sub-scheme is for the unknown flux σ. Theoretical analysis shows that the method yields the approximate solutions with optimal accuracy in L ²(Ω) norm for the primal unknown and in H(div; Ω) norm for the unknown flux, respectively. Some numerical examples are given to confirm our theory results.     

A programmable ternary CPU using hybrid CMOS/memristor circuits

Abstract

The carry propagation of arithmetic operations is one of the major shortcomings of common binary number encodings as the two’s complement. Signed-digit arithmetic allows the addition of two numbers without carry propagation and in asymptotically constant time in dependence of the word length, while at the same time requiring a digit representation with more than two states. With the advent of memristors, it has become possible to store multiple states within a single memory cell. This paper proposes an implementation of a general purpose CPU using signed-digit arithmetic by exploiting memristors in order to implement multi-value registers. The proposed model of the CPU is evaluated by the execution of various image processing algorithms. It is shown that a break-even point exists at which signed-digit algorithms outperform conventional binary arithmetic operations. Furthermore, simulation results prove that the memristor device lends itself to store signed-digit data efficiently.     

Computations that require higher than double precision for robust and exact decision making

Consider the computation of deciding relative orientations of objects undergoing multiple translations and rotations. Such an orientation test involves the computation of expressions based on arithmetic operations, square roots and trigonometric functions. The computation of signs of such expressions using double precision floating-point arithmetic in modern computers may result in errors. In this article we demonstrate the existence of examples where double precision is not sufficient to compute the correct sign of an expression. We consider (i) simple expressions involving only the four basic arithmetic operations, (ii) expressions involving the square-root function and (iii) expressions representing orientation tests in two- and three-dimensions involving objects undergoing arbitrary rotations by angles given in radians, thereby requiring the computation of trigonometric functions. We develop a system that uses requisite high precision for computing the correct sign of such expressions. The system uses our floating-point filter called L-filter and the bigfloat extended precision package in LEDA (Library of Efficient Data Types and Algorithms).     

On canonical transformations between equivalent hamiltonian formulations of general relativity

Two Hamiltonian formulations of general relativity, due to Pirani, Schild and Skinner (Phys. Rev. 87, 452, 1952) and Dirac (Proc. Roy. Soc. A 246, 333, 1958), are considered. Both formulations, despite having different expressions for the constraints, allow one to derive four-dimensional diffeomorphism invariance. The relation between these two formulations at all stages of the Dirac approach to constrained Hamiltonian systems is analyzed. It is shown that the complete sets of their phase-space variables are related by a transformation which satisfies the ordinary condition of canonicity known for unconstrained Hamiltonians and, in addition, converts one total Hamiltonian into another, thus preserving form-invariance of generalized Hamiltonian equations for the constrained systems.     

Worst-case analysis of the least-squares method and related identification methods

We consider worst-case analysis of system identification under less restrictive assumptions on the noise than the l_∞ bounded error condition. It is shown that the least-squares method has a robust convergence property in l₂ identification, but lacks a corresponding property in l₁ identification (as well as in all other non-Hilbert space settings). The latter result is in stark contrast with typical results in asymptotic stochastic analysis of the least-squares method. Furthermore, it is shown that the Khintchine inequality is useful in the analysis of least l^p identification methods.     

A New Technique in Systems Analysis Under Interval Uncertainty and Ambiguity

The main subject of this work is mathematical and computational aspects of modeling of static systems under interval uncertainty and/or ambiguity. A cornerstone of the new approach we are advancing in the present paper is, first, the rigorous and consistent use of the logical quantifiers to characterize and distinguish different kinds of interval uncertainty that occur in the course of modeling, and, second, the systematic use of Kaucher complete interval arithmetic for the solution of problems that are minimax by their nature. As a formalization of the mathematical problem statement, concepts of generalized solution sets and AE-solution sets to an interval system of equations, inequalities, etc., are introduced. The major practical result of our paper is the development of a number of techniques for inner and outer estimation of the so-called AE-solution sets to interval systems of equations. We work out, among others, formal approach, generalized interval Gauss-Seidel iteration, generalized preconditioning and PPS-methods. Along with the general nonlinear case, the linear systems are treated more thoroughly.     

Analysis of the identification of closed-loop systems using least-squares methods

This paper presents theory and results concerning the analysis of the identification of closed-loop systems using least-squares methods. The least-squares technique is applied in its normal form and in a modified version developed to cope with the bias problem. The analysis has been established following a mathematical investigation of the problem, and by simulation of different identification experiments applied to different structures of closed-loop systems.

The results derived from this analysis show the conditions under which the identifiability of the open-loop process can be ensured, considering different situations such as whether or not there is noise present at the system output and whether or not external signals are used to perform the identification experiments.

Practical experiments of closed-loop identification on a micromachine system in use in the Department of Electrical Engineering of the University of Manchester are also described. Results for different experimental conditions are presented through graphs showing both the plant and the identified model outputs for the same sequence of sampled input signals.     

A self-tuning regulator for multivariable systems

The controller described in this paper is designed for multivariable plants with constant, unknown parameters. The algorithm operates on-line with the a priori information about the time delay. The order of the system may be given a priori. In the case where the order of the system is unknown it can be determined by a generalized likelihood-ratio statistical test which is described in this paper. The multivariable self tuning regulator consists of the two tasks of estimation and regulation. Estimation of the input-output system model parameters is based on the least-squares principle. The control is computed to minimize the combined cost of output deviation and control energy. Asymptotic properties of the estimation are discussed. Usefulness and simplicity of this approach are illustrated by examples.