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A direct derivation of the solution of the discounted exponential linear quadratic Gaussian optimal control problem is presented. Some extensions to partial observations problems are also given. The solution involves time-varying Riccati equations, providing an example of an infinite horizon, constant coefficient, regulator problem with a time-varying optimal control.

This note presents frequency-domain characterizations of exponential stability and stabilizability of neutral systems based on transfer-function matrices and the existence of 'nice' solutions of certain Bezout equations. It turns out that the existence of H infinity-solutions is not sufficient for exponential stabilizability, but that they have to satisfy an additional growth assumption as well. Whilst the proofs of our results are based on an abstract infinite-dimensional representation of the neutral system, we emphasize that the results are expressed in terms of the original parameters of the neutral equation and do not require a reformulation of the system in an abstract state-space form. The sufficiency parts of the results hold even when the delay operator acting on the derivative contains a singular part.

The Bayesian decision process was used in the generalization of cumulative sums (cusums) operations. Cusums, which are highly useful in monitoring model quality and forecasting systems, were studied by using the single-sided monitoring scheme instead of the conventional approach. The single-sided scheme revealed a relationship between the Bayesian decisions and led to the formulation of a new method for generalizing cusum equivalent schemes.

The matrix exponential plays a central role in linear systems and control theory. In this note, we give explicit formulas for computing the exponential of some special matrices.

The effect of background on the statistical precision of exponential decay rates is examined using Monte Carlo calculations and direct evaluation of the least--squares error matrix. For typical observation time ranges, the presence of an unknown background reduces the precision in the decay rate by an order of magnitude. However, the uniform background is practically the worst case, and if the background distribution is 'shaped' and if this shape can be ascertained independent of the decay data, the loss in precision can be greatly reduced. For the uniform background, the strong correlation between the decay parameter and the background parameter can be broken by operating at less than 100% duty cycle, such that the background is recorded independent of the signal some of the time. This approach can yield precision approaching that for the case of an a priori known background., Changes in the statistical precision of exponential decay rates due to background were investigated. The least squares error matrix was subjected to Monte Carlo calculations and direct evaluation. Results indicate that the quality of decay rate estimates is greatly reduced by background counts in exponential decay data. Loss in precision can, however, be neutralized by separate estimation of background.

The statistical properties of maximum likelihood (ML) and least-squares (LS) estimators for the exponential decay parameters [gamma](rate) and [tau] (lifetime) are examined for finite observation time (T) and a small number of counts (N). The ML estimators and their variances are assessed via numerical calculations employing the exact N-point distributions for exponentially distributed random data. Bias is a common feature of all these estimators under these conditions. In addition, the ML lifetime estimator exhibits the interesting property of divergence in its variance, which means that [gamma] is the statistically preferred parameter for characterizing exponential decay for finite observation time. The bias in [gamma] is less than N/(N - 1) for all T and thus can be easily corrected for. The effects of various ways of binning the data are examined for both the ML and LS approaches., The occurence of problems due to sparse data limit was investigated. Finite observation time and small count numbers were used for the characterization of the maximum likelihood and least squares estimators of rate and lifetime as exponential decay parameters. Results indicate that the appropriate binning of data is necessary for least squares to produce results comparable to maximum likelihood methods.

Proposed algorithms based on the polynomial method for parameter estimation of multiple exponential signals to noise are tested for multiple snapshot unconditional, single and multiple snapshot conditional maximum likelihood estimations. Results show that, although cost-effective fast local convergence are allowed, the advantages of the polynomial method in terms of computational capacity are retained. Therefore, the programs are useful in improving global convergent parameter estimates. The need to modify the programs to eliminate nonconvergence did not surface in all cases studied.

Proposed exponential adaptive control techniques are presented. Results are global and show that the techniques provide exponential tracking results without persistent excitation on the desired path by using the direct adaptive method. The algorithms do not need a prediction error since these are based only on the tracking error. The proposed techniques involve bounded parameter estimation error which is adequate for exponential path tracking.

Generalizing the exponential function allows the development of new memory functions to define solid polymeric mechanical behavior. The proposed function or root function is shown to perform well since its incomplete spectrum fM conforms well with the molecular weight distribution (MWD). The MWD curve can be best derived using the general constitutive equations which allow the response of the material in external impulse to be predicted given the rheological parametric values.

We compare the (asymptotic) efficiencies of the sample mean and the sample median in the exponential power family. This single family of densities is an ideal one to illustrate the effect of 'heavy tails' on the two estimators. We find the density for which the mean and median are equally efficient, and observe what happens for limiting cases within the family. Only basic tools are needed in the analysis and statistical development. KEY WORDS: Asymptotic; Efficiency; Sample mean; Sample median., 1. INTRODUCTION In introductory mathematical statistics courses the concept of (asymptotic) efficiency of an estimator is frequently discussed. The efficiency of the ubiquitous estimators, the sample mean [Mathematical Expression Omitted] [...]

Chemistry textbooks often fail to inform students when chemical data describes a hyperbolic relationship, leading to confusion. An example of this problem is the presentation of linear titration curves, which are often not linear, because textbooks lead students to believe the curves are just segments of other curve types rather than hyperbolas. The treatment of Michaelis-Menten kinetics is another example in that textbooks often exclude negative substrate concentrations because of their physical insignificance but which could clarify the hyperbolic nature of the plot.

Light cones are generated by the exact world line of a photon. This theoretically corresponds to a situation where the distributional function of the wave equation can be applied. The wave equation is an important model equation in mathematical physics and is physically relevant. No solution exists for this wave equation as it excludes several solutions such as 'photon torpedoes' and 'microphotons. ' The hyperbolity of the wave operator and the non-characteristic features also contribute to the null effect of the wave equation.

This paper exhibits exponentially converging observers for a large class of multiinput multioutput (MIMO) nonlinear systems. Potential applications of these observers to mechanical systems and distillation columns are presented.

The amount of money a person was willing to forgo in order to give $75 to another person decreased as a hyperbolic function of the perceived social distance between them. Similar hyperbolic functions have previously been shown to describe both time and probability discounting.

A formal analysis of the Guggenheim and other difference methods shows that the correlation is not a result of the subtraction but of the multiple uses of the original data in computing the differences. The assumption of random error in the raw data is reasonable in most cases, which makes correlation a nonproblem in the direct fitting approaches.

A study was conducted to show the variety of ways in which hyperbolic functions arise in the solution of the barge problem. The result reveals that the barge problem offers an easily understood application of hyperbolic functions and illustrates many properties of these functions and their inverses.

In a recent paper, Fang and Kincaid proposed an open problem about the relationship between the local stability of the unique equilibrium point and the global stability for a Hopfield-type neural network with continuously differentiable and monotonically increasing activation functions. As a partial answer to the question, in the two-neuron case it is proved that for each given specific interconnection weight matrix, a Hopfield-type neural network has a unique equilibrium point which is also locally exponentially stable for any activation functions and for any other network parameters if and only if the network is globally asymptotically stable for any activation functions and for any other network parameters. If the derivatives of the activation functions of the network are bounded, then the network is globally exponentially stable for any activation functions and for any other network parameters. Index Terms--Global stability, Hopfield-type neural network, local exponential stability.

Global exponential stability is a desirable property for dynamic systems. This paper studies the global exponential stability of several existing recurrent neural networks for solving linear programming problems, convex programming problems with interval constraints, convex programming problems with nonlinear constraints, and monotone variational inequalities. In contrast to the existing results on global exponential stability, the present results do not require additional conditions on the weight matrices of recurrent neural networks and improve some existing conditions for global exponential stability. Therefore, the stability results in this paper further demonstrate the superior convergence properties of the existing neural networks for optimization. Index Terms--Global exponential stability, optimization problems, recurrent neural networks.

The freshman exponentiation law, a common mistake made by elementary algebra students, sometimes actually turns out correct. The n-domain, the domain of this rare case, is defined and analyzed. Ad hoc calculations show its characteristics while congruences from elementary group theory are able to prove its range of 2

The cost variance investigation problem studied by Baiman and Demski (BD) (1980) is reexamined. Their model is further developed to optimize a 'two-tailed' investigations model with two non-HARA utility functions assumed. The BD models, the optimal investigation rules and investigation regions are investigated. The results show that it is possible to investigate both favorable and unfavorable cost variances.

The Bieberbach conjecture defining Taylor coefficients of univalent analytic functions led to the Loewner parametric method, which had useful applications in univalent function theory. I.M. Milin developed a model of functionals related to exponentiation and the conjecture, where nonpositive functionals implied the Bieberbach conjecture. Final confirmation of the Bieberbach conjecture came in 1984, when L. de Branges proved that Milin's functionals were indeed nonpositive.

For a recently derived general exponential formula the computational form of the expansion suitable for machine computation is given.

Another obvious 'truth' that has posed a long-standing problem for mathematicians is Fermat's Last Theorem - the famous assertion that the equation [x.sup.n] + [y.sup.n] = [z.sup.n] has no solution [...]