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The origins of bisimulation and bisimilarity are examined, in the three fields where they have been independently discovered: Computer Science, Philosophical Logic (precisely, Modal Logic), Set Theory. Bisimulation and bisimilarity are coinductive notions, and as such are intimately related to fixed points, in particular greatest fixed points. Therefore also the appearance of coinduction and fixed points is discussed, though in this case only within Computer Science. The paper ends with some historical remarks on the main fixed-point theorems (such as Knaster-Tarski) that underpin the fixed-point theory presented. Categories and Subject Descriptors: F.4.1 [Mathematical Logic and Formal Languages]: Mathematical Logic--Computational logic; modal logic; set theory; F.4.0 [Mathematical Logic and Formal Languages]: General Genera] Terms: Theory, Verification Additional Key Words and Phrases: Bisimulation, coinduction, fixed points, greatest fixed points, history

A method is presented for the optimal choice of the fixed-point reluctivity in nonlinear eddy current problems solved for steady-state in the frequency and the time domain. The optimal selection is shown to ensure a minimal contraction number for the fixed-point method. The proposed technique is compared to earlier approaches using a 2-D example. Index Terms--Convergence of numerical methods, eddy currents, Fourier transforms, numerical analysis.

The fixed-point method together with the Fourier decomposition of time dependent quantities can be used to determine the steady state solution of sinusoidaily driven electrodynamic problems in the presence of nonlinear materials. The advantage of this method is to obtain the steady state immediately without having to calculate transient processes. However, the convergence rate of the method is heavily influenced by the so called fixed-point permeability. An appropriate choice of this parameter can speed up the convergence whereas an unsuitable value leads the method to divergence. The aim of this paper is to present a strategy to choose the value for the fixed-point permeability in order to enhance the convergence rate. The result is illustrated by two 2-D examples. Index Terms--Eddy currents, frequency domain analysis, nonlinearities, numerical analysis.

Partial blind source separation (BSS) methods are proposed that separate P supposedly nonstationary sources of interest. The results have shown that the differential algorithms are more robust to noise sources than the standard fast independent component analysis (FastICA) and C-FICA algorithms.

In this letter, we propose a coordinate rotation digital computer (CORDIC)-like algorithm for computing the feed-forward phase of a support vector machine (SVM) in fixed-point arithmetic, using only shift and add operations and avoiding resource-consuming multiplications. This result is obtained thanks to a hardware-friendly kernel, which greatly simplifies the SVM feed-forward phase computation and, at the same time, maintains good classification performance respect to the conventional Gaussian kernel. Index Terms--Coordinate rotation digital computer (CORDIC), embedded systems, fixed-point arithmetic, support vector machine (SVM).

Massively parallel processor-per-pixel single-instruction multiple data arrays are being successfully used for early vision applications in smart sensor systems; however, they are inherently inefficient when executing algorithms involving propagation of binary signals, such as the geodesic reconstruction. Yet, these algorithms, at the interface between pixel-level and object-level image processing, should be implemented on the vision chip to facilitate data reduction at the sensor level. A cellular asynchronous network is presented in this paper, which can be used to execute binary propagation operations. The proposed circuit is optimized in terms of speed and power consumption. In 0.35-[micro]m technology, the simulated propagation speed is 0.18 ns per pixel and the total energy expended per propagation is 0.37 pJ per cell. In this brief, implementation issues are discussed and simulation results including image processing examples are presented. Index Terms--Cellular arrays, image processing, parallel architectures, VLSI.

Neural models of associative memories are usually concerned with the storage and the retrieval of binary or bipolar patterns. Thus far, the emphasis in research on morphological associative memory systems has been on binary models, although a number of notable features of autoassociative morphological memories (AMMs) such as optimal absolute storage capacity and one-step convergence have been shown to hold in the general, gray-scale setting. In this paper, we make extensive use of minimax algebra to analyze gray-scale autoassociative morphological memories. Specifically, we provide a complete characterization of the fixed points and basins of attractions which allows us to describe the storage and recall mechanisms of gray-scale AMMs. Computer simulations using gray-scale images illustrate our rigorous mathematical results on the storage capacity and the noise tolerance of gray-scale morphological associative memories (MAMs). Finally, we introduce a modified gray-scale AMM model that yields a fixed point which is closest to the input pattern with respect to the Chebyshev distance and show how gray-scale AMMs can be used as classifiers. Index Terms--Basin of attraction, fixed point, gray-scale morphological associative memory, minimax algebra, morphological neural network.

A novel resampling mechanism for parallel processing of fixed-point particle filtering where the mechanism utilizes a particle-tagging scheme during quantization to compensate possible loss of replicated particles due to the finite precision effect is discussed. The particle filters have three important operations, generation of new particles, computation of the particle weights, and resampling, which is very important for accurate tracking.

Abstract: In this article we present two new results concerning the Smarandache Ceil function. The first result proposes an equation for the number of fixed-point number of the Smarandache ceil [...]

The authors consider the modal-calculus in terms of interpolation, the Lyndon Theorem, and the Los-Tarski Theorem. All 3 classical theorems are proven for the mu-calculus.

The change of variables for functions of any number of variables is proven by the fundamental theorem of calculus. A proof of the Brouwer fixed point theorem is derived for this purpose. The proof is then compared with other existing analytic proofs of the fixed point theorem. It is shown that the change of variables formula alone indicates Brouwer's theorem, with the integration of exterior forms over chains presupposing the change of variable formula for multiple integrals.

The monotone fixed point principle is explored in relation to Feferman's explicit mathematics. Set-theoretic models are constructed for systems of explicit mathematics in extensions of Kripke-Platek set theory, and the set theories are related to subsystems of second arithmetic.

The total variational regularization method is identified as one of the most successful regularization approaches which effectively recovers edges of an image and some blurring functions including out-of-focus blur and motion blur. An alternating minimization implicit iterative scheme is developed to identify the point spread function and recover the image. The alternating minimization/fixed point algorithm is identified as an effective and robust solution to the minimization problem.

The homotopy invariance of the index was first argued by Poincare in 1886. The theorem obtained by Poincare, same with the Brouwer fixed point theorem was the result of Miranda's theorem. It was further proven in 1909 having the same result established by Bohl in 1904. Another proof was given in the Bohl-Brouwer theorem based on the lemma of Sperner. Simple combinatorial methods of proof taken from Sperner lemma were applied and discussed.

Cellular Neural Networks (CNN's) with space-varying interconnections are considered here to implement associative memories. A fast learning method is presented to compute the interconnection weights. The algorithm was carefully tested and compared to other methods. Storage capacity, noise immunity, and spurious state avoidance capability of the proposed system are discussed. Index Terms - Associative memory, fixed point.

An analog system for fixed point computation is described. The system is derived from a continuous time analog of the classical over-relaxed fixed point iteration. The dynamical system is proved to converge for nonexpansive mappings under all p norms, p [element of] (1, [infinity]]. This extends previously established results to not necessarily differentiable maps which are nonexpansive under the [infinity]-norm. The system will always converge to a single fixed point in a connected set of fixed points. This allows the system to function as a complementary paradigm to energy minimization techniques for optimization in the analog domain. It is shown that the proposed technique is applicable to a large class of dynamic programming computations.

Byline: Piotr Piecuch (1), Josef Paldus (1) Abstract: The radii of convergence of power series expansions describingenergy of a molecule in external electrostatic field are investigated usingD'Alembert ratio test, standard and generalized Cauchy--Hadamardcriteria, and Pade approximants. The corresponding coefficients atvarious field and fieldagradient components, representing multipolemoments and (hyper)polarizabilities and including terms of tenth oreven twentieth order, are determined using an ab initio linear responsecoupledacluster theory. Most calculations are performed for the HFmolecule described by the basis set of double zeta quality, while the role ofbasis set is discussed by comparing the results with estimates of the radii ofconvergence obtained with the basis set of [5s3p2d/3s2p] quality.Emphasis is placed on the dependence of the interval of convergence of powerseries expansion describing energy of a molecule in applied electrostatic fieldon the nuclear geometry. The results might have important implications forvarious numerical methods used to calculate electrostatic molecular propertiesas functions of the internuclear geometry, including the finiteafield andfixedapointacharge approaches. Author Affiliation: (1) Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1 Article History: Registration Date: 11/09/2004

Fixed point theorems for correspondences with values in partially ordered sets are proposed. The theorems generalize the fixed point theorem introduced by Knaster (1928) and Tarski (1955). They prove the existence of a Nash equilibrium for extended supermodular games. The theorems also help show that any non-decreasing correspondence has a fixed point if the partially ordered set has a bottom element.

Brouwer Fixed Point Theorem states that stationary points exist in liquids subjected to circular motion. These stationary points are difficult to detect with the help of the theorem, which does not state the location of the stationary points. The motion and the approximate location of the stationary points are determined by the use of mathematical mapping and continuous functions. This shows that the moving surface has the property of a fixed point.