Mark J. Christensen

Stability of the roots of random algebraic polynomials

In this paper we consider the behavior of the roots of random algebraic polynomials. A code was developed which generates a sample of random algebraic polynomials, calculates the roots of each sample polynomial, and then calculates the averages of the roots. Finally, the roots of the deterministic algebraic polynomial whose coefficients are the averages of the sample coefficients are calculated. These data are then tabulated and graphically displayed. The relationship between the averages of the roots of the sample polynomials and the roots of the average polynomial is discussed.

Extension theorems for operator‐valued measures

Benioff [J. Math. Phys. 13 (1972)] has shown that every consistent, countable family of positive, normalized operator‐valued (PNOV) measures μn over Rn can be extended to a PNOV measure over R∞. In this paper we show that the same result holds for arbitrary, consistent families of PNOV measures over complete, separable metric spaces. Further we show that, while there may be no extension if the topological conditions are relaxed, it is always possible to construct a related family of PNOV measure spaces which: (1) Are measures theoretically indistinguishable from the original spaces; (2) Have an extending PNOV measure. These results use developments in the theory of algebraic models of measures as initiated by Dinculeanu and Foias [Ill. J. Math. 12 (1968)] and applied to stochastic processes by Schreiber, Sun, and Bharucha‐Reid [Trans. AMS 158 (1973)].

Lag time in microbe growth as an age-dependent branching process with two phase types.

Abstract We study a two-type, age-dependent branching process in which the branching probabilities of one of the types may vary with time. Specifically this modification of the Bellman-Harris process starts with a Type I particle which may either die or change to a Type II particle depending upon a time varying probability. A Type II particle may either die or reproduce with fixed probabilities but may not return to a particle of Type I. In this way the process models the lag phenomenon observed in microbe growth subsequent to transfer to a new culture medium while the organism is adapting to its new environment. We show that if the mean reproduction rate of Type II particles exceeds 1, then the population size grows exponentially. Further the extinction probability for this process is related to that of the Bellman-Harris process. Finally the governing equations are solved for several choices of the growth parameters and the solutions are graphically displayed showing that a wide variety of behavior can be modeled by this process.

Gleason measures on infinite tensor products of Hilbert spaces

Nowak [Bull. Acad. Pol. Sci. Ser. Sci. Math. Astron. Phys. 22, 393–5 (1974)] has given an example of a consistent (in the sense of Kolmogorov) family of Gleason measures [A. M. Gleason, J. Math. Mech. 6, 885–94 (1957)] {mn} defined over ⊗ni=1Hi which do not extend to a Gleason measure on ⊙∞i=1 ΦHi for a given construction of the infinite tensor product. In this paper we show: (1) In the example of Nowak it is not necessary to assume, as is done, that the Hi are infinite dimensional. (2) That every consistent family developed from pure states, which is the type considered by Nowak, extends over the complete infinite tensor product of von Neumann [Compositio. Math. 6, 1–77 (1938)]. (3) Even if each Hi is two‐dimensional and the complete infinite tensor product of von Neumann is used, it is possible to give a simple counterexample to the conjecture that every consistent family of Gleason measures extends by the use of nonpure states.

Quantum dynamics of Hamiltonians perturbed by pulses

There has been a recent controversy between Gzyl [J. Math Phys. 18, 1327 (1977)] and Blume [J. Math. Phys. 19, 2004 (1978)] concerning the correct way to integrate the equation i∂ψ/∂t=H0ψ+δ (t) Vψ where δ is the Dirac delta function. In this paper we: (1) Suggest some physical reasons for rejecting Gzyl’s procedure and (2) Extend the limiting procedure suggested the Blume to unbounded Hamiltonians.

APPROXIMATION OF FUNCTIONS

This chapter discusses the ways to generate interpolating polynomials of arbitrary degree. An interpolating polynomial is one that passes through a given set of points. One of the main advantages of the Taylor estimate is that the error is easier to calculate. The main disadvantage of the Taylor method is that it requires the reader to know the formulas for the derivatives of the function that is being approximated. In general, the interpolating polynomial of a given degree will be more beneficial than the Taylor polynomial of the same degree. However, with the interpolating polynomial, the points through which the polynomial passes through must be decided. With the Taylor polynomial, only a single point can be specified. To calculate the Taylor polynomial of degree N, all the derivatives up to and including the Nth must be calculated.

ESTIMATION OF SEQUENCES AND SERIES

This chapter describes the ways to estimate the limit of a sequence and infinite series. The chapter presents a sequence of numbers by means of a formula A(N) = 1/N or A(N) = (-1)↑N or A(N) = (1 + l/N)↑N and discusses what happens to the sequence as N gets very large. In the first example, as N gets big, 1/N gets small; therefore, that sequence should tend to 0. The limit of the sequence, as N goes to infinity, is 0. In the second case, as the sequence consists of alternating plus and minus ones, there will be no limit.

THE DEFINITE INTEGRAL

This chapter describes the definite integral and elementary estimation of areas. Aside from the derivative, the calculus has only one other key idea: the definite integral. The definite integral of a function F on the interval [A,B] is defined to be the area under the graph of F and above the x-axis between A and B. Using the computer, this area can be numerically estimated to a high degree of accuracy. For example, if one cuts the interval [A,B] into N equal subintervals, then the sum of the areas above these little intervals must be equal to the area above the entire interval [A,B]. Each of the resulting narrow columns can then be approximated by a rectangle.

THREE DIMENSIONAL GRAPHICS

This chapter describes level curves of a surface and the central projection of surfaces. The cross-sectional views of a surface, such as level curves, are very useful in determining the way a surface looks. The chapter focuses on the development of a complete three-dimensional view of a surface. There are many ways to represent, or map, a three-dimensional object onto a two-dimensional chart. In attempting to graph surfaces in three dimensions, the same problem is faced. The chapter presents one of the most common methods for projecting a three-dimensional object onto a two-dimensional surface: the central projection. In this projection, a three-dimensional object is looked at from some point in space, which is called the view point.

IMPLICIT FUNCTIONS AND DIFFERENTIATION

This chapter describes implicit functions and implicit differentiation. Not every equation, even of polynomial type, can be solved directly for its unknown. If two variables are related by an equation that cannot be solved algebraically for one of the variables, then one can only discover the relationship between the two variables numerically. To be specific, if X and Y are related by the equation Y↑2 + 3 * X * Y − 6 * X↑3 * Y = 5, then by treating this as a quadratic in Y and applying the quadratic formula, one has Y = (-3 * X + -SQR(9 * X↑2 + 4 * (5 + 6 * X↑3)))/2. Thus, in this case, one is able to solve the equation relating X and Y to obtain an explicit relationship between the two variables. If the polynomial relating X and Y had been of degree 5 or greater in both of the variables, then it would in general be impossible to obtain an explicit relationship between the two variables. In that case, the relationship can be called implicit.