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A geometric interpretation of the first-order Marcum Q-function, Q(a, b), is introduced as the probability that a complex, Gaussian random variable with real mean a, takes on values outside of a disk [C.sub.O,b] of radius b centered at the origin O. This interpretation engenders a fruitful approach for deriving new representations and tight, upper and lower bounds on Q(a,b). The new representations obtained involve finite-range integrals with pure exponential integrands. They are shown to be simpler and more robust than their counterparts in the literature. The new bounds obtained include the generic exponential bounds which involve an arbitrarily large number of exponential functions, and the simple erfc bounds which involve just a few erfc functions, together with exponential functions in some cases. The new generic exponential bounds approach the exact value of Q(a, b) as the number of exponential terms involved increases. These generic exponential bounds evaluated with only two terms and the new simple erfc bounds are much tighter than the existing exponential bounds in most cases, especially when the arguments a and b are large. Thus, in many applications requiring further analytical manipulations of Q(a, b), these new bounds can lead to some closed-form results which are better than the results available so far. Index Terms--Complex Gaussian distribution, geometric interpretation, Marcum Q-function, probability bounds, Rician distribution.

This paper presents a new blind carrier frequency offset (CFO) estimation scheme for orthogonal frequency division multiplexing (OFDM) systems with constant modulus (CM) signaling. Both single-input single-output (SISO) systems and multiple-input multiple-output (MIMO) systems with orthogonal space-time block coding are considered. The proposed scheme is based on the reasonable assumption that the channel frequency response changes slowly in the frequency domain, which implies that the channel frequency response on two consecutive subcarriers is approximately the same. Based on this assumption, cost functions are derived in closed-form, which minimize the difference between the signal power of two neighboring subcarriers. The identifiability of the proposed scheme is mathematically proved, which implies that minimizing the derived cost function gives an approximate estimate of the CFO. We demonstrate that the proposed scheme provides an excellent trade-off between complexity and performance as compared to prominent existing estimation schemes. Index Terms--Constant modulus (CM), carrier frequency offset (CFO), multiple-input multiple-output (MIMO) systems, orthogonal frequency division multiplexing (OFDM), orthogonal space-time block code (OSTBC).

This investigation deals with some exact solutions of the equations governing the steady plane motions of an incompressible third grade fluid by using complex variables and complex functions. Some of the solutions admit, as particular cases, all the solutions of Moro et al. [1990, 'Steady Flows of a Third Grade Fluid by Transformation Methods,' ZAMM, 70(3), pp. 189-198] [DOI: 10.1115/1.2928407] Keywords: vorticity function, third grade fluid, stream function

We present a mathematical analysis of the common-mode instability and power back-off feature of a transformer-coupled Class-AB differential power amplifier (PA). The efficient impedance matching of the transformer combiner and efficiency improvement at power back-off, a major benefit of this structure, are illustrated. In addition, an analytical model is derived to predict the common-mode oscillations in PA. The analytical results, based on a simple hand-calculation model for the transistor, show good agreement with simulation results using complete 90-nm models. Two methods to suppress the common-mode oscillations are investigated and analyzed in detail. Index Terms--CMOS, power amplifiers (PAs), power combining, stability.

In this paper, the recurrent neural networks (RNNs) with a generalized activation function class is proposed. In this proposed model, every component of the neuron's activation function belongs to a convex hull which is bounded by two odd symmetric piecewise linear functions that are convex or concave over the real space. All of the convex hulls are composed of generalized activation function classes. The novel activation function class is not only with a more flexible and more specific description of the activation functions than other function classes but it also generalizes some traditional activation function classes. The absolute exponential stability (AEST) of the RNN with a generalized activation function class is studied through three steps. The first step is to demonstrate the global exponential stability (GES) of the equilibrium point of original RNN with a generalized activation function being equivalent to that of RNN under all vertex functions of convex hull. The second step transforms the RNN under every vertex activation function into neural networks under an array of saturated linear activation functions. Because the GES of the equilibrium point of three systems are equivalent, the next stability analysis focuses on the GES of the equilibrium point of RNN system under an array of saturated linear activation functions. The last step is to study both the existence of equilibrium point and the GES of the RNN under saturated linear activation functions using the theory of M-matrix. In the end, a two-neuron RNN with a generalized activation function is constructed to show the effectiveness of our results. Index Terms--Absolute exponential stability (AEST), convex hull, generalized activation function class, piecewise linear function, recurrent neural networks (RNNs).

The learning capability of neural networks is equivalent to modeling physical events that occur in the real environment. Several early works have demonstrated that neural networks belonging to some classes are universal approximators of input-output deterministic functions. Recent works extend the ability of neural networks in approximating random functions using a class of networks named stochastic neural networks (SNN). In the language of system theory, the approximation of both deterministic and stochastic functions falls within the identification of nonlinear no-memory systems. However, all the results presented so far are restricted to the case of Gaussian stochastic processes (SPs) only, or to linear transformations that guarantee this property. This paper aims at investigating the ability of stochastic neural networks to approximate nonlinear input-output random transformations, thus widening the range of applicability of these networks to nonlinear systems with memory. In particular, this study shows that networks belonging to a class named non-Gaussian stochastic approximate identity neural networks (SAINNs) are capable of approximating the solutions of large classes of nonlinear random ordinary differential transformations. The effectiveness of this approach is demonstrated and discussed by some application examples. Index Terms--Approximation, Lee-Schetzen method, neural computation, neural networks, nonlinear systems, stochastic processes (SPs).

In this paper, we study the robust stability of uncertain time-delay systems. We consider uncertain quasi-polynomials whose coefficients may vary in a certain prescribed range. Our goal is to derive necessary and sufficient conditions for such uncertain quasi-polynomials to maintain stability independent of delay parameters. Our primary contributions are frequency-sweeping conditions for interval, diamond, and spherical quasi-polynomial families, which can be readily checked, requiring only the computation of two simple frequency-dependent functions. Additionally, we also obtain vertex- and edge-type results in the spirit of the Kharitonov approach known in robust stability analysis, showing that the stability of interval and diamond quasi-polynomials can be ascertained by checking the stability of certain special vertex and/or edge members in those families. Both type of results provide necessary and sufficient conditions for the quasi-polynomial families to be robustly stable independent of delay. Index Terms--Frequency-sweeping tests, robust stability, time-delay systems, uncertain quasi-polynomials, vertex tests.

This paper considers the problem of obtaining good attitude estimates from measurements obtained from typical low cost inertial measurement units. The outputs of such systems are characterized by high noise levels and time varying additive biases. We formulate the filtering problem as deterministic observer kinematics posed directly on the special orthogonal group SO(3) driven by reconstructed attitude and angular velocity measurements. Lyapunov analysis results for the proposed observers are derived that ensure almost global stability of the observer error. The approach taken leads to an observer that we term the direct complementary filter. By exploiting the geometry of the special orthogonal group a related observer, termed the passive complementary filter, is derived that decouples the gyro measurements from the reconstructed attitude in the observer inputs. Both the direct and passive filters can be extended to estimate gyro bias online. The passive filter is further developed to provide a formulation in terms of the measurement error that avoids any algebraic reconstruction of the attitude. This leads to an observer on SO(3), termed the explicit complementary filter, that requires only accelerometer and gyro outputs; is suitable for implementation on embedded hardware; and provides good attitude estimates as well as estimating the gyro biases online. The performance of the observers are demonstrated with a set of experiments performed on a robotic test-bed and a radio controlled unmanned aerial vehicle. Index Terms--Attitude estimates, complementary filter, nonlinear observer, special orthogonal group.

Analytical analysis of fluid flow in an annular microchannel subjected to uniform wall injection at various Reynolds numbers is presented. The classical Navier-Stokes equations are used in the present study. Mathematically, using an appropriate change of variable, Navier-Stokes equations are transformed to a set of nonlinear ordinary differential equations. The governing equations are analytically solved using series solution method. Some analytical results are given for the prediction of velocity profiles and pressure distributions in annular microchannels. The agreement between the computational fluid dynamics and the analytical predictions is very good. However, the analytical results are valid in a limited range of radius ratios and needs more study. Keywords: laminar and incompressible fluid flow, annular microchannel, Navier-Stokes equations, analytical method

Abstract Let [p.sub.1] and [p.sub.2] are two primes with [p.sub.2] < [p.sub.2] and [p.sub.2] = [p.sub.2] - 2, we call such pairs of primes are twin primes. About the elementary [...]

The epistemological foundation of unity of knowledge is used to formulate a system-model of participatory socioeconomic development. The micro-properties of such a participatory development approach are deeply ethical in nature. In order to bring out the endogenous role of ethics derived from the moral law in reference to the epistemic foundation, and thereby explain their impact on the socioeconomic development experience, the methods of topological space and topological mappings are found to be appropriate for formalizing the complex nature of participatory dynamics. They emerge in the midst of institutional and organic interaction, integration, and evolution. The application of topological methods to participatory socioeconomic development thus opens up a field that is rarely used by mathematical economists. In this sense this article is an original contribution to the area of ethical development theory and its topological explanation and application. KEYWORDS: Evolutionary epistemology, mathematical model, participatory development.

Usability assessment plays a fundamental role in discovering usability problems and the determination of the level of usability for a given software product. One crucial aspect in every usability assessment is the estimation of the sample size desired for a software product. Once we start estimating the sample size needed for a usability assessment, a baffling question comes in mind: 'how many users are enough for a given software product?'. To the best of our knowledge, the majority of existing models estimates the sample size needed for a usability assessment, based on historical data. A better estimation should be based on both historical data, which provide an initial idea from previous software products and based on present data, which provide a practical idea for a given software product. Therefore, in this paper we have proposed an adjustable sample size estimation model for usability assessment, which enhances the estimation process by using two factors: Alpha factor ([alpha]), which estimates the problem discovery rate ([[lambda].sub.[alpha]]) from historical data and the Beta factor ([beta]), which estimates the problem discovery rate ([[lambda].sub.[alpha]]) by using the complexity of the software product. [[lambda].sub.[beta]] is taken as vertical and domain wise for a given software product, to adjust the alpha factor appropriately to give the desired confidence level in the results. An illustrative case study has been provided at the end of the paper. Key words: Usability Asssessment, Sample Size Estimation Models, Adjustable Sample Size Estimation Model, Alpha Factor ([[lambda].sub.[alpha]]), Beta Factor ([[lambda].sub.[beta]]), Ilustrative Case Study., INTRODUCTION With the increase demand for software products and websites, the focus on the usability for these needs becomes essential (1),(2). Several studies in the field of usability engineering, have [...]

A novel structured grid approach which provides an efficient way of treating a class of complex geometries is proposed. The incompressible Navier-Stokes equations are formulated in a two-dimensional, generalized curvilinear coordinate system complemented by a third quasi-curvilinear coordinate. By keeping all two-dimensional planes defined by constant third coordinate values parallel to one another, the proposed approach significantly reduces the memory requirement in fully three-dimensional geometries, and makes the computation more cost effective. The formulation can be easily adapted to an existing flow solver based on a two-dimensional generalized coordinate system coupled with a Cartesian third direction, with only a small increase in computational cost. The feasibility and efficiency of the present method have been assessed in a simulation of flow over a tapered cylinder. [DOI: 10.1115/1.2354533]

A study proposes the multiclass proximal support vector machine (MPSVM) classifier, which extends the binary PSVM to the multiclass case. The effectiveness of MPSVM is demonstrated by both simulation studies and applications to cancer classifications using microarray data.

A mathematical solution to surveillance routing problem in maritime surface surveillance is evaluated.

Fractal symbolic analysis is a symbolic analysis technique for verifying the legality of program transformations. It is strictly more powerful than dependence analysis; for example, it can be used to verify the legality of blocking LU factorization with pivoting, a task for which dependence analysis is inadequate. In this paper, we show how fractal symbolic analysis can be used to convert between left- and right-looking versions of three kernels of central importance in computational science: triangular solve, Cholesky factorization, and LU factorization with pivoting. KEY WORDS: Program restructuring; symbolic analysis; numerical methods; high-performance computing.

Usage of probability theory for constraint propagation during study uncertainty of scheduling intervals in manufacturing facilities is described.

Empirical results of duality theorems, which are used to examine second order F-convexity problems in mathematical programming, are presented.

Application of complexity theory for analysis of semidefinite programming relaxation, by using optimization techniques to solve nonconvex quadratic programs, is described.

Description of classical short-step interior point method, which is used for solving convex optimation problems, is provided. Optimization of parameters of this method to provide appropriate iteration bounds for these parameters is described.

Application of polyhedral theory to set limits for central path of linear programming, by using interior point methods for performing sensitivity analysis and ensuring strict complimentarity of variables, is described.

Development of alternative model for data envelopment analysis (DEA) with imprecise data is discussed.

The 'Cav u' theorem, introduced by R.J. Aumann and M Maschler, asserts that as the number of n repetitions goes to infinity, the values v(sub n)(p) of these games converge with convergence rate n(super -1/2), to the concavification Cav u(p) of the value u(p) of the game where none of the players has any information. An alternative proof for this theorem was developed using Fenchel duality techniques.

A new procedure for determining eigenvector derivatives of a complex structural system has been developed. The novel scheme uses the dynamic flexibility method and is based on the practical complete modal space approach. It can be used on systems with or without repeated eigenvalues while the numerical results obtained are precise. There is also a reduction in computer processing time while engineers would find that it is easier to do theoretical computations using the proposed method.

An efficient method of computing the least-squares optimal basis functions to steer any function locally is presented. The method combines the Lie group-theoretic and the singular value decomposition approaches. Its efficiency is demonstrated with the design of basis functions to steer a Gabor function under the four-parameter linear transformation group. Index Terms - Steerable filters, filter design, low-level vision, pattern analysis.

Interval arithmetic deals with intervals that include their endpoints. Computers have performed interval arithmetic easily since the early 1960s, when Ramon Moore produced a version of interval Fortran. The need to develop interval arithmetic has come from industry, which requires solutions to systems of nonlinear algebraic equations and finite dimensional optimization problems.

The two-branch tournament genetic algorithm (GA) is used in deriving a set of Pareto-optimal solutions in a 10-bar truss problem. A single-run of the two-branch GA generated 224 different designs. With a population size of 180 runs over 300 generations, the GA needs 54,000 function evaluations. Two-hundred twenty-four runs would have been required if a method which can produce a single solution at a time was used instead. A calculus-based approach might have used 50 to 80 function evaluations including evaluations needed for a line search.

Mathematical problems can only be solved in the event that the problems were definitely understood. Mathematics teachers can teach their students an alternative method in solving problems through the utilization of manipulatives and the graphing calculator. Pattern exploration involved in mathematical summation may be represented by manipulatives. Oftentimes, students use graphical approaches in solving mathematical relationships and this can be facilitated with the use of graphing calculators such as the TI-80s series.

Two algorithms for determining the optimal weights in finite difference formulas are presented. The two new methods are not computationally slow or are restricted in their usage. The first algorithm is for equispaced grids and provides the exact weights for explicit and implicit approximations. The second algorithm is for irregular grid spacings but is to be used mainly for explicit approximations. Solutions for implicit functions may be done through a simple extension of the derivation.

Transfer principle is among the basic tenets of nonstandard analysis. This principle refers to the assumption that all propositions accepted as true in classical set theory are also considered true in nonstandard set theory, and vise versa. As such, nonstandard analysis seems to be merely another model under segment of analytic phenomena and within relations under such class which serve as the basis for the development of classical analysis. In fact, the inability of classical analysis to deal with actual infinitesimals lead to the development of nonstandard analysis.

Simulation techniques using the device simulator SPECTRA and optics simulator TOCCATA have identified the three-dimensional optical and electrical characteristics of an interline-transfer charge-coupled device (CCD). A fast, modified version of Gummel's technique is used by SPECTRA to solve Poisson's and time dependent current continuity equations. Layer shapes are assessed using TOCCATA calculations.

In this paper, a new technique for assessing the accuracy of analytical methods using linear regression is reported. The results of newly developed analytical methods are regressed against the results obtained using reference methods. The new test is based on the joint confidence interval for the slope and the intercept of the regression line, which is calculated taking the uncertainties in both axes into account. The slope, intercept, and variances which are associated with the regression coefficients are calculated with bivariate least-squares regression (BLS). The new technique was validated using three simulated and five real data sets. The Monte Carlo method was applied to obtain 100 000 data sets for each of the initial simulated data sets to show the correctness of the new technique. The application of the new technique to five real data sets enables differences to be detected between the results of the joint confidence interval based on the BLS method and the results of the commonly used tests based on ordinary least-squares or weighted least-squares regression.

Abstract algebra and topology are two methods of key importance for the in depth understanding of mathematics in a simplified way. Their applications in the development and research of complex problems in various streams of mathematics are given. The abstract algebraic and topological approaches converge in classical algebraic function theory but are otherwise disparate approaches.

The pore size distribution in mesoporous materials is linked to a simple relationship between the pore radius and the freezing point depression of confined pore water using 1H NMR spectroscopy, HREM and N2 adsorption. The formula based on this relationship also involves the thickness of nonfreezing pore surface water which is considered a constant along with temperature. Catalysis and separation applications are likely to be generated from this formula because of its ability to investigate mesoporous materials.

Aminiature perifusion system, comprising of two pH sensors, is able to indicate modifications in extracellular protons following the application of metabolic perturbants. A convolution technique applied to the perifusion system facilitates the analysis of the response of HeLa cells to carbonyl cyanide m-chlorophenyl hydrazone (CCCP). Convolution evaluation can give new insights into metabolism and cellular responsiveness.

To speed up transient stability assessment of direct methods we use power system reduction techniques, combining the composite electromechanical distance (CED) approach with node reduction: the CED approach to identify and ultimately aggregate machines (almost) irrelevant to the transient phenomena; node reduction to eliminate irrelevant nodes, while possibly taking advantage of techniques such as sparsity preserving and connectivity based ones. The resulting approach is illustrated on the French EHV power system. It uses direct methods of the extended equal area criterion type, together with a CED appropriately tailored to the system specifics and with various node reduction techniques. Many variants so devised are compared. It is found that the CED approach contributes to significantly speed up the direct methods while fully preserving accuracy.

Traditional eigenvalue sensitivity for power systems requires the formulation of the system matrix, which lacks sparsity. In this paper, a new sensitivity analysis, derived for a sparse formulation, is presented. Variables that are computed as intermediate results in established eigenvalue programs for power systems, but not used further, are given a new interpretation. The effect of virtually any control action can be assessed based on a single eigenvalue-eigenvector calculation. In particular, the effect of active and reactive power modulation can be found as a multiplication of two or three complex numbers. The method is illustrated in an example for a large power system when applied to the control design for an HVDC-link.

An interesting behaviour observed in ants is the almost straight trails they create from the anthill to the location of the food found by the pioneer ant. The ant-pursuit problem is presented and solved to demonstrate how a simple interaction of players can locate the easiest route between a source and the destination. It is also shown that a simple method of circular pursuit can ensure that all the players eventually converge together.

Two topics are discussed: the industrious ant game and the use of a markable straightedge. In the Ant, a plane where the insect supposedly lives is divided by a square grid into black and white cells. The game starts out with a certain configuration of the black and while cells and the behaviour of the ant is predicted. Straightedge constructions uses only a pencil, eraser and a markable/eraseable rectangular strip and illustrates how one can make constructions which are not possible using the ordinary ruler and compass.

Mathematical equations may be solved by alternative graphical and numeric methods with the help of technological advances. Function plotters for graphical representations require knowledge of toolkit functions, which include sketching of graphs of basic constant, linear, polynomial, logarithmic, nth root, reciprocal, exponential and trigonometric functions. Novel strategies affect a solution's accuracy since these may be approximations rather than exact solutions.

Solutions to a pair of recreational mathematics problems, tiling of rectangular surfaces by unequal squares and dividing a cake, are presented. The first problem is solved with an assumption that no two squares have the same size. It is shown that the rectangular surface can be completely tiled by no less than nine squares. The fixed point theorem and the finite algorithm are presented for arriving at an envy-free allocation in the cake-division problem.

White noise analyis was used to study physiological processes with recordings less than 30 seconds. It serves as a probe which facilitates comparisons between systems. This was applied to the visual system where input of light to the retina was transformed into electrical signals in photoreceptors and measured using white noise analysis. The mathematical computations involved present an enormous task to the neurophysiologist, but it is an effective tool especially when used intracellularly.

During the perinatal period the newborn must adapt to a different environment from that of the womb. After birth, maternal supplies of nutrient ceases requiring changes in glucose and fatty acid metabolism stimulated by alterations in hormone secretion. The suckling-weaning transition also changes nutritional patterns in the neonate resulting in further metabolic adaptations of glucose and fat metabolism.

State-space techniques are applied to continuous-time spectral factorization to derive an approach for discrete-time spectral factorization. A method is developed for solving the problem. It entails direct sum decomposition of linear equations of the Toeplitz plus Hankel form and the solution of a modified algebraic Riccati equation (MARE). Meanwhile, recently developed techniques for the MARE solution are adapted and discussed.

The classical Foster's reactance theorem is generalized to complex frequency functions. The result sheds light on the stability of complex polynomials, which is of interest in the system theory and applied mathematics. In Assertion 1, a novel method for the generation and the analysis of a Complex Reactance Function (CRF) is presented. Assertion 2 deals with the expansion and synthesis of CRF. Assertion 3 relates this note to a more basic problem of stability and applied mathematics. Assertion 4 gives an example of application by focusing on the two-dimensional homogeneous reactance functions.