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R. E. Maute and coworkers have proposed a new method to estimate the Archie parameters (cementation exponent and saturation exponent) by minimizing the error between water saturation measured from core and water saturation computed by the Archie equation. To make this principle applicable utilizing only well-logs, the authors proposed to use a dielectric log and minimize the error (constant m and n method) between measured water saturation (Sxodielectric) and calculated water saturation (SxoArchie). In order to reduce further the error between Sxo (dielectric) and Sxo (Archie), m and n were approximated by mathematical functions (polynomial and Fourier series) to model the vertical variation of m and n in the reservoir (variable m and n method). This variable m and n method based on a Fourier series resulted in the greatest error-reduction when compared to the constant m and n method. After determining m and n values that result in the minimum error between Sxo (dielectric) and Sxo (Archie), these values then can be used to calculate the water saturation in the uninvaded zone (Sw).

A subbundle of variable dimension inside the tangent bundle of a smooth manifold is called a smooth distribution if it is the pointwise span of a family of smooth vector fields. We prove that all such distributions are finitely generated, meaning that the family may be taken to be a finite collection. Further, we show that the space of smooth sections of such distributions need not be finitely generated as a module over the smooth functions. Our results are valid in greater generality, where the tangent bundle may be replaced by an arbitrary vector bundle., 13 pages

At the Georgia Institute of Technology, a computer program is used in freshman calculus which graphically illustrates upper and lower Riemann sums and generates values of their differences. The students often observe that the differences $\Delta _n $ seem to be proportional to ${1 / n}$, where n is the number of subdivisions; but this is only approximate.We make this rigorous by showing that \[ \Delta _n = \frac{V} {n} + O\left( {\frac{1} {{n^3 }}} \right)\quad {\text{ as }}n \to \infty ,\] for nice functions, where V is the total variation. The proof is simple, and is a nice illustration of the ideas of asymptotic analysis, and several other techniques of analysis.

It is shown that discrete time systems whose dynamics are governed by piecewise monotone maps of the interval are globally observable under a general class of observations. The results rest on the fact that the dynamics are ergodic.

We study the linear system ⨰=Ax+Bu from a differential geometric point of view. It is well-known that controllability of the system is related to the one-parameter family of operators eΛtB. We use this to give a proof of the classical controllability conditions in terms of the differential geometry of certain curves in ℝn. We then view γ(t)=Im(eΛtB) as a curve in appropriate Grassmannian and see that, in local coordinates, γ is an integral curve of the flow induced by a matrix Riccati equation. We obtain qualitative geometric conditions on γ that are equivalent to the controllability of the system. To get quantitiative results, we lift γ to a curve l' in a splitting space, a generalized Grassmannian, which has the advantage of being a reductive homogeneous space of the general linear group, GL(ℝn). Explicit and simple expressions concerning the geometry of Γ are computed in terms of the Lie algebra of GL(ℝn), and these are related to the controllability of the system.

Kronecker's theorem is used to show that the irrational flows on then-dimensional torus are globally observed by a large class of continuous functions. These results are used to study the observability of Riccati flows on the Grassman manifolds.

We study the non-linear delay differential equation x′(t) + g(x(t), x (t−τ))= f(t) under a non-resonance condition which assures the existence of a unique bounded solution. Using the algebra structure of the space of bounded continuous functions we investigate the properties of this solution. We discuss some generalizations and the initial value problem.

Publisher Summary This chapter describes some results concerning the qualitative properties of bounded solutions of the nonlinear (scalar) delay differential equation. The chapter extends these results and places them in a general framework by considering certain subalgebras of the algebra of bounded continuous functions on the line.

Aspects of global observability (GO) for some standard examples of chaotic systems are considered, the underlying philosophy being that such systems should be easy to observe and should be observed by large classes of scalar functions. Consideration is given to GO for piecewise monotone maps of the interval (a standard example of a chaotic system). Attention is then given to GO for ergodic translations on compact (metrizable) topological groups; the most familiar examples are the irrational translation flows on the n-torus.

Harmonic analysis is used to show that an ergodic translation on a compact abelian group is observed by almost all continuous scalar functions.

Doug McMahon was killed in a climbing accident in late 1986. He was a member of the Department of Mathematics at Arizona State University. The first two authors feel fortunate to have known Doug and to have worked with him. We would also like to thank Cris Brynes and Clyde Martin for getting us together with Doug.

We consider the evolution equation (E) du/dt = au + f(t,u), u(0) = u0 in a Hilbert space H, where A is assumed to generate a C0 semigroup. The asymptotic-in-time behavior of solutions to (E) is studied together with the asymptotic-in-time behavior of methods for approximating (E).

(1986). The Contraction Mapping Lemma and the Inverse Function Theorem in Advanced Calculus. The American Mathematical Monthly: Vol. 93, No. 1, pp. 52-54.