In the direct product space of a complete metric linear space X and its related space Y, a fuzzy mapping G is introduced as an operator by which we can define a projective fuzzy set G(x,y) for any xX and yY. An original system is represented by a completely continuous operator f(x)Y, e.g., in the form x=λ(f(x)), (λ is a linear operator), and a nondeterministic or fuzzy fluctuation induced into the original system is represented by a generalized form of system equation x_βG(x,f(x)). By establishing a new fixed point theorem for the fuzzy mapping G, the existence and evaluation problems of solution are discussed for this generalized equation. The analysis developed here for the fluctuation problem goes beyond the scope of the perturbation theory.

This paper discusses psychophysical aspects of human sensory system through a differential-geometrical formulation. The discussions reveal relationships among three fundamental functions--the psychophysical, the DL and the JND functions, which characterize sensory system.

This paper describes new methematical tools, taken from quantum field theory (QFT), which may make it possible to characterize localized excitations (including solitons, but also including chaotic modes) generated by PDE systems. The significance to computer hardware and neurocomputing is also discussed. This mathematics--IF further developed--may also have the potential to reorganize and simplify our understanding of QFT itself--a topic of very great intellectual and practical importance. The paper concludes by describing three new possibilities for research, which will be very important to achieving these goals.

Various models of a neuron have been proposed and many studies about them and their networks have been reported. Among these neurons, this paper describes a study about the model of a neuron providing its own feedback input and possesing a chaotic dynamics. Using a return map or a histogram of laminar length, type-I intermittency is recognized in a recurrent neuron and its network. A posibility of controlling dynamics in recurrent neural networks is also mentioned a little in this paper.

By adding a linear resistor in series with the inductor in Chua's circuit, we obtain a circuit whose state equation is topologically conjugate (i.e., equivalent) to a 21-parameter family C of continuous odd-symmetric piecewise-linear equations in R³. In particular, except for a subset of measure zero, every system or vector field belonging to the family C, can be mapped via an explicit non-singular linear transformation into this circuit, which is uniquely determined by 7 parameters. Since no circuit with less than 7 parameters has this property, this augmented circuit is called an unfolding of Chua's circuit--it is analogous to that of "unfolding a vector field" in a small neighborhood of a singular point. Our unfolding, however, is global since it applies to the entire state space R³. The significance of the unfolded Chua's Circuit is that the qualitative dynamics of every autonomous 3rd-order chaotic circuit, system, and differential equation, containing one odd-symmetric 3-segment piecewise-linear function can be mapped into this circuit, thereby making their separate analysis unnecessary. This immense power of unification reduces the investigation of the many heretofore unrelated publications on chaotic circuits and systems to the analysis of only one canonical circuit. This unified approach is illustrated by many examples selected from a zoo of more than 30 strange attractors extracted from the literature. In addition, a gallery of 18 strange attractors in full color is included to demonstrate the immensely rich and complex dynamics of this simplest among all chaotic circuits.

In this paper, a synthesizing method is proposed for the design of discrete-time cellular neural networks for binary image processing. Based on the theory of digital-logical design paradigm of threshold logic, the template parameters of the discrete-time cellular neural network for a prescribed binary image processing problem are calculated. Application examples including edge detection, connected component detection, and hole filling are given to demonstrate the merits and limitations of the proposed method. For a given realization of the parameters of the cloning template, a guideline for the selection of the offset I_c for maximum error tolerance is also considered.

The problem of local minima is inevitable when solving combinatorial optimization problems by conventional methods such as the Hopfield network, relying on the minimization of an objective function E(X). Such a problem arises from the search mechanism in which only the local information about the objective function E(X) is used. In this paper we propose a new approach called the Single Minimum Method (SMM) which uses the global information in searching for the solutions to combinatorial optimization problems. In this approach, we add a function -TS(X) to the original objective function E(X) to construct the function F(X)=E(X)-TS(X) which has only one minimum, one which can be easily found by any general gradiet method including the Hopfield network. Based on an analogy between thermodynamic systems and neural networks, it is shown that the global information about the original objective function E(X) is included in the single minimum of the function F(X) and can be used for finding the global minimum of the objective function E(X). In order to show how to apply the Single Minimum Method to a combinatorial optimization problem we give an algorithm for the TSP problem based on our method. The simulation results show that the algorithm can almost always find the shortest or near shortest paths. Finally, a modified SMM, which has some great advantages for hardware implementation, is also given.

The self-organization rule of planar neural networks has been proposed for learning of 2D distributions. However, it cannot be applied to 3D objects. In this paper, we propose a new model for global representation of the 3D objects. Based on this model, global topology reserving self-organization is achieved using parallel local competitive learning rule such as Kohonen's maps. The proposed model is able to represent the objects distributively and easily accommodate local features.

The method of estimating multiple sound source locations based on a neural network algorithm and its performance are described in this paper. An evaluation function is first defined to reflect both properties of sound propagation of spherical wave front and the uniqueness of solution. A neural network is then composed to satisfy the conditions for the above evaluation function. Locations of multiple sources are given as exciting neurons. The proposed method is evaluated and compared with the deterministic method based on the Hyperbolic Method for the case of 8 sources on a square plane of 200m200m. It is found that the solutions are obtained correctly without any pseudo or dropped-out solutions. The proposed method is also applied to another case in which 54 sound sources are composed of 9 sound groups, each of which contains 6 sound sources. The proposed method is found to be effective and sufficient for practical application.

Asynchronous machines are a topic of great interest in the research area of actuators. Due to the complexity of these systems and to the required performance, the modelling and control of asynchronous machines are complex questions. Problems arise when the control goals require accurate descriptions of the electric machine or when we need to identify some electrical parameters; in the models employed it becomes very hard to take into account all the phenomena involved and therefore to make the error amplitude adequately small. Moreover, it is well known that, though an efficient control strategy requires knowledge of the flux vector, direct measurement of this quantity, using ad hoc transducers, does not represent a suitable approach, because it results in expensive machines. It is therefore necessary to perform an estimation of this vector, based on adequate dynamic non-linear models. Several different strategies have been proposed in literature to solve the items in a suitable manner. In this paper the authors propose a neural approach both to derive NARMAX models for asynchronous machines and to design non-linear observers: the need to use complex models that may be inefficient for control aims is therefore avoided. The results obtained with the strategy proposed were compared with simulations obtained by considering a classical fifth-order non-linear model.

A new optimal nonlinear regulator design method is developed by applying a multi-layered neural network and a fixed point theorem for a nonlinear controlled system. Based on the calculus of variations and the fixed point theorem, an optimal control law containing a nonlinear mapping of the state can be derived. Because the neural network has not only a good learning ability but also an excellent nonlinear mapping ability, the neural network is used to represent the state nonlinear mapping after some learning operations and an optimal nonlinear regulator may be formed. Simulation demonstrates that the new nonlinear regulator is quite efficient and has a good robust performance as well.

Abrupt variations of attractors caused by argumental discreteness in non-Hermitian complex-valued neural networks are reported. When we apply the complex-valued associative memories to dynamical processing, the weighting matrices are constructed as non-Hermitian in general so that they have motive force to the signal vectors. It is observed that competitions between argumental rotation force and noise-suppression ability of associative memories lead to trajectory distortions and abrupt variations of the attractors.

The mechanism of environment-dependent self-organization of "positional information" in a coupled nonlinear oscillator system is proposed as a new principle of realtime coordinative control in biological distributed system. By modeling the pattern formation in tactic response of Physarum plasmodium, it is shown that a global phase gradient pattern self-organized by mutual entrainment encodes not only the positional relationship between subsystems and the total system but also the relative relationship between internal state of the system and the environment.

A priori estimation is presented for a computational complexity of the homotopy method applying to a certain class of strongly monotone nonlinear equations. In the present papers, a condition is presented for a certain class of uniquely solvable equations, under which an upper bound of a computational complexity of the Newton type homotopy method can be a priori estimated. In this paper, a condition is considered in a case of linear homotopy equations including the Newton type homotopy equations. In the first place, the homotopy algorithm based on the simplified Newton method is introduced. Then by using Urabe type theorem, which gives a sufficient condition guaranteeing the convergence of the simplified Newton method, a condition is presented under which an upper bound of a computational complexity of the algorithm can be a priori estimated, when it is applied to a certain class of strongly monotone nonlinear equations. The presented condition is demonstrated by numerical experiments.

In this paper, we are concerned with a problem of obtaining an approximate solution of a finite-dimensional nonlinear equation with guaranteed accuracy. Assuming that an approximate solution of a nonlinear equation is already calculated by a certain numerical method, we present computable conditions to validate whether there exists an exact solution in a neighborhood of this approximate solution or not. In order to check such conditions by computers, we present a method using rational arithmetic. In this method, both the effects of the truncation errors and the rounding errors of numerical computation are taken into consideration. Moreover, based on rational arithmetic we propose a new modified Newton interation to obtain an improved approximate solution with desired accuracy.

In this paper, a new time series analysis method is proposed. The proposed method uses the exponential (EXP) model. The residual signal is assumed to be identically and independently distributed (IID). To achieve accurate and efficient estimates, the parameter of the system model is calculated by maximizing the logarithm of the likelihood of the residual signal which is assumed to be IID t-distribution. The EXP model theoretically assures the stability of the system. This model is appropriate for analyzing signals which have not only poles, but also poles and zeroes. The asymptotic efficiency of the EXP model is addressed. The optimal solution is calculated by the Newton-Raphson iteration method. Simulation results show that only a small number of iterations are necessary to reach stationary points which are always local minimum points. When the method is used to estimate the spectrum of synthetic signals, by using small α we can achieve a more accurate and efficient estimate than that with large α. To reduce the calculation burden an alternative algorithm is also proposed. In this algorithm, the estimated parameter is updated in every sampling instant using an imperfect, short-term, gradient method which is similar to the LMS algorithm.

Erasure-and-error decoding is a general form of channel decoding and is a basis of important coding schemes, such as the concatenated coding scheme and coded ARQ. However, there do not exist enough discussions on the interrelationship between erasure-and-error decoding schemes. In this paper, threshold decoding schemes are discussed in a systematic manner and compared with Forney's optimal scheme. Some confusions in known results are pointed out and new results on threshold decoding are shown.

The paper deals with the level-crossing intervals of a stationary random process. Probability densities of the level-crossing intervals of a process consisting of a sine wave plus Gaussian random noise are experimentally investigated by digital simulation. The Gaussian random noise is selected to be of 7-th order Butterworth power spectrum density. The obtained probability densities appear with a number of isolated peaks much larger than the number observed on the corresponding probability densities of the Gaussian noise alone. When the sine wave frequency is greater than or approximately equal to the half of the cutoff frequency of the noise spectrum, experimental data suggest that each isolated peak has a Gaussian-like density. The mean time interval associated with each Gaussian-like density is equal to a multiple number of the period of the sine wave, while the variance is found to be fairly the same for all peaks of a same probability density. The assumption of "quasi-independence" is not valid for the level-crossing intervals of the present process.

In this article a Neural Network learning scheme is described, which is a generalization of VQ (Vector Quantization) and ART2a (a simplified version of Adaptive Resonance Theory 2). The basic differences between VQ and ART2a will be exhibited and it will be shown how these differences are covered by the generalized scheme. The generalized scheme enables a rich set of variations on VQ and ART2a. One such variation uses the expression ||I||²+||z_j||²/||z_j||sin(I,z_j), as the distance measure between input vector I and weight vector z_j. This variation tends to be more robust to noise than ART2a, as is shown by experiments we performed. These experiments use the same data-set as the ART2a experiments in Ref.(3).