相容核的拓扑解释

给出了相容核的拓扑解释。给定论域上的一个相容关系R，指出该相容关系诱导的拟离散闭包空间的内集拓扑是以每个元素的R_相关集构成的集族为拓扑基生成的。进而利用该内集拓扑所对应的指定预序，证明了每个元素的相容核就是内集拓扑中单点集的闭包。     

The termination of the higher-dimensional tarai functions

We prove that for every integer n≥3$n\ge 3$, the n-dimensional tarai function, proposed by D. Knuth, terminates with call-by-need. The proof is formalized by Coq.     

Decision tree approximations of Boolean functions

Decision trees are popular representations of Boolean functions. We show that, given an alternative representation of a Boolean function f, say as a read-once branching program, one can find a decision tree T which approximates f to any desired amount of accuracy. Moreover, the size of the decision tree is at most that of the smallest decision tree which can represent f and this construction can be obtained in quasi-polynomial time. We also extend this result to the case where one has access only to a source of random evaluations of the Boolean function f instead of a complete representation. In this case, we show that a similar approximation can be obtained with any specified amount of confidence (as opposed to the absolute certainty of the former case.) This latter result implies proper PAC-learnability of decision trees under the uniform distribution without using membership queries.     

Topological necessary conditions of smooth stabilization in the large

Topological necessary conditions of smooth stabilization in the large are obtained. In particular, if a smooth single-input nonlinear system is smoothly stabilizable in the large at some point of a connected component of an equilibrium set, then the connected component is an unbounded curve.     

求解非线性方程组的BFGS差分进化算法

针对差分进化算法进化后期收敛缓慢和稳定性不强的缺陷,将BFGS算法插入差分进化算法当中,提出了一种BFGS差分进化算法,用来求解非线性方程组。通过5个非线性方程组和一个工程实例的实验,说明：算法收敛精度较高、收敛速度较快、鲁棒性强、收敛成功率高,是一种较好的解决非线性方程组的方法。     

Numerical study of a novel MFIE formulation using monopolar RWG basis functions

Numerical character of a new magnetic field integral equation (MFIE) formulation solved by the method of moments (MoM) is studied. In this new formulation, monopolar Rao-Wilton-Glisson (RWG) functions are used as both the basis and the testing functions. Calculation details for the impedance matrix elements of this new formulation are presented with a form suitable for programming. Numerical results for electromagnetic scattering analysis of several small sharp-edged conducting objects show that the new formulation using the monopolar RWG basis functions is much more accurate than the traditional one using the RWG and the monopolar RWG basis functions.     

Stability analysis of nonlinear quadratic systems via polyhedral Lyapunov functions

Quadratic systems play an important role in the modeling of a wide class of nonlinear processes (electrical, robotic, biological, etc.). For such systems it is mandatory not only to determine whether the origin of the state space is locally asymptotically stable, but also to ensure that the operative range is included into the convergence region of the equilibrium. Based on this observation, this paper considers the following problem: given the zero equilibrium point of a nonlinear quadratic system, assumed to be locally asymptotically stable, and a certain polytope in the state space containing the origin, determine whether this polytope belongs to the domain of attraction of the equilibrium. The proposed approach is based on polyhedral Lyapunov functions, rather than on the classical quadratic Lyapunov functions. An example shows that our methodology may return less conservative results than those obtainable with previous approaches.     

On the formulation of cost functions for torque-optimized control of rigid bodies

In the context of controlling the attitude of a rigid body, this communique uses recent results on representations of torques (moments) to establish cost functions. The resulting cost functions are both properly invariant under whatever choice of Euler angles is used to parameterize the rotation of the rigid body and have transparent physical interpretations. The function is related to existing works in geometric control theory and applications of optimal control theory to biomechanical systems.     

求解非线性方程组的混合粒子群算法

结合Hooke-Jeeves和粒子群的优点,提出了一种混合粒子群算法,用于求解非线性方程组,以克服Hooke-Jeeves算法对初始值敏感和粒子群容易陷入局部极值而导致解的精度不够的缺陷。该算法充分发挥了粒子群强大的全局搜索能力和Hooke-Jeeves的局部精细搜索能力,数值实验结果表明：能够以满意的精度求出对未知数具有敏感性的非线性方程组的解,具有良好的鲁棒性和较快的收敛速度和较高的搜索精度。     

Topological properties of reachable sets and the control of quantum bits

We consider the problem of controlling the state of a two-level quantum system (quantum bit) via an externally applied electro-magnetic field. The describing model is a bilinear right-invariant system whose state varies on the Lie group of 2×2 special unitary matrices. We study the topological structure of the reachable sets. If two or more independent controls are used, then every state can be achieved in arbitrary time. However, this is no longer true if only one control is available and, in this case, we give an exact characterization of states reachable in arbitrary time. We prove small time local controllability for any state and the existence of a critical time which is the smallest time after which every transfer of state is possible. We provide upper and lower bounds for such a time. The mathematical development is motivated by the problem of manipulating the state of a quantum bit. Every transfer of state may be interpreted as a quantum logic operation and not every logic operation can be obtained in arbitrary time. The analysis we present provides information about the feasibility of a given operation as well as estimates for the speed of a quantum computer.     

Minimization of Haar wavelet series and Haar spectral decision diagrams for discrete functions

In this paper, a minimization of Haar wavelet series for simplification of circuits and Haar based decision diagrams representing discrete multiple-valued functions is proposed. The minimization is performed by permutation of indices of generalized Haar functions. Experimental results show that this method provides reasonable reduction in the number of non-zero coefficients. The Haar series reduced this way can be useful in the circuit synthesis for realization of multiple-valued functions. The same algorithm can be also used to reduce the number of paths in decision diagrams related to the Haar wavelet transforms. In many cases, this reduction provides smaller size of such decision diagrams.     

Granular type-2 membership functions: A new approach to formation of footprint of uncertainty in type-2 fuzzy sets

In this study, a new approach for the formation of type-2 membership functions is introduced. The footprint of uncertainty is formed by using rectangular type-2 fuzzy granules and the resulting membership function is named as granular type-2 membership function. This new approach provides more degrees of freedom and design flexibility in type-2 fuzzy logic systems. Uncertainties on the grades of membership functions can be represented independently for any region in the universe of discourse and free of any functional form. So, the designer could produce nonlinear, discontinuous or hybrid membership functions in granular formation and therefore could model any desired discontinuity and nonlinearity. The effectiveness of the proposed granular type-2 membership functions is firstly demonstrated by simulations done on noise corrupted Mackey–Glass time series prediction. Secondly, flexible design feature of granular type-2 membership functions is illustrated by modeling a nonlinear system having dead zone with uncertain system parameters. The simulation results show that type-2 fuzzy logic systems formed by granular type-2 membership functions have more modeling capabilities than the systems using conventional type-2 membership functions and they are more robust to system parameter changes and noisy inputs.     

Dynamics of the Binary Euclidean Algorithm: Functional Analysis and Operators

We provide here a complete average-case analysis of the binary continued fraction representation of a random rational whose numerator and denominator are odd and less than N . We analyze the three main parameters of the binary continued fraction expansion, namely, the height, the number of steps of the binary Euclidean algorithm, and finally the sum of the exponents of powers of 2 contained in the numerators of the binary continued fraction. The average values of these parameters are shown to be asymptotic to A _i log N , and the three constants A _i are related to the invariant measure of the Perron—Frobenius operator linked to this dynamical system. The binary Euclidean algorithm was previously studied in 1976 by Brent who provided a partial analysis of the number of steps, based on a heuristic model and some unproven conjecture. Our methods are quite different, not relying on unproven assumptions, and more general, since they allow us to study all the parameters of the binary continued fraction expansion. Received February 9, 1998; revised March 15, 1998.     

Modeling of thermodynamic properties of ABr-PrBr3 (A=Li-Cs ) systems

The thermodynamic properties of ABr-PrBr₃(A=Li-Cs) systems were assessed by the CALPHAD method. The liquid phase in the systems was described by the non-stoichiometric associate model. The entropies of mixing in the liquid were evaluated from experimental liquidus and enthalpy of mixing data. For the pseudobinary compounds A₃PrBr₆,APr₂Br₇, and A₂PrBr₅ (A=K,Rb) and Cs₃PrBr₆ and CsPr₂Br₇, the dependences of Gibbs energies of formation on temperature were calculated. The anomalies of sequences of thermodynamic properties in RbBr-PrBr₃ were observed and discussed. The nature of the liquid phase and precision of calculations of the Rb₂PrBr₅(s) compound were discussed.     

Rudiments of rough sets

Worldwide, there has been a rapid growth in interest in rough set theory and its applications in recent years. Evidence of this can be found in the increasing number of high-quality articles on rough sets and related topics that have been published in a variety of international journals, symposia, workshops, and international conferences in recent years. In addition, many international workshops and conferences have included special sessions on the theory and applications of rough sets in their programs. Rough set theory has led to many interesting applications and extensions. It seems that the rough set approach is fundamentally important in artificial intelligence and cognitive sciences, especially in research areas such as machine learning, intelligent systems, inductive reasoning, pattern recognition, mereology, knowledge discovery, decision analysis, and expert systems. In the article, we present the basic concepts of rough set theory and point out some rough set-based research directions and applications.