Richard E. Crandall

QUANTITATIVE PREDICTIONS OF DELAYED MATURITY

What are the selective pressures that have driven the evolution of delayed maturity? K-selection (selection operating at high population densities) has been suggested as an explanation, but it does not fit all the data (Wilbur et al., 1974; Stearns, 1977) and it oversimplifies (Whittaker and Goodman, 1979). In a declining population, the rate of decrease is minimized when age at maturity is maximized (Hamilton, 1966; Mertz, 1971), but that explanation is probably not general, for such populations have an increased probability of extinction. Four other hypotheses, which we regard as more plausible, have been suggested: a delay in maturity results (a) in a gain in fecundity (Tinkle, 1969; Gadgil and Bossert, 1970; Tinkle et al., 1970; Wiley, 1974; Schaffer and Elson, 1975; Bell, 1977), (b) in an improvement in the juvenile survival of the offspring produced (Wiley, 1974; Hirshfield and Tinkle, 1977), (c) in a reduction in the cost of reproduction and an increase in adult survival rates (Schaffer, 1972; Wittenberger, 1979), or (d) in increased fitness in the face of unpredictable, catastrophic larval mortality, where the delay comes in the egg stage (Livdahl, 1979). These hypotheses are not mutually exclusive; all four could shape the life-history of some organism. We have built optimality models in which reproduction and survival depend on the age at maturity of the organism. To keep the models tractable, we assumed that we could deal with a population as though it were a set of asexually reproducing, haploid clones, with each clone endowed with a different life-history. The optimization procedure tells us which clone will win the intraspecific competition for numerical dominance in future generations. To do this, we have assumed that the population is in stable age distribution, and that the definition of fitness is the rate at which a new allele grows in the population, where that allele affects age at maturity, survival, and fecundity. Thus the unit of selection in these models is the gene, not the individual or the population, but because we conceive the population as a set of haploid clones, the gene is effectively equivalent to the individual. For those who are uncomfortable with this simplification, we note that this analysis holds for the marginal effect of a gene substitution on phenotypes in a haploid, asexual population (cf. Charlesworth and Williamson, 1975). We also note that a clone can be growing exponentially in a stationary population. In these models, clones grow exponentially at a rate implied by the particular age at maturity and survivorship and fecundity schedules with which they are endowed. Thus the technical definition of fitness is the Malthusian parameter (Fisher, 1930), with neither frequency-dependence nor density-dependence, but we denote it as r-the intrinsic rate of increase of a clone. Our models assume either the fecundity-gain hypothesis (a), the quality-ofyoung hypothesis (b), or both. We have found that either fecundity-gain or quality-of-young assumptions can yield predictions of a considerable delay in maturity when r is maximized. Moreover, when we used published data to predict optimal ages at maturity on the assumption that one, the other, or both effects were at work, the predictions were surprisingly good.

A search for Wieferich and Wilson primes

An odd prime p is called a Wieferich prime if 2 P-1 = 1 (mod p 2 ) alternatively, a Wilson prime if (p - 1)|= -1 (mod p 2 ). To date, the only known Wieferich primes are p = 1093 and 3511, while the only known Wilson primes are p = 5,13, and 563. We report that there exist no new Wieferich primes p < 4 x 10 12 , and no new Wilson primes p < 5x 10 8 . It is elementary that both defining congruences above hold merely (mod p), and it is sometimes estimated on heuristic grounds that the probability that p is Wieferich (independently: that p is Wilson) is about 1/p. We provide some statistical data relevant to occurrences of small values of the pertinent Fermat and Wilson quotients (mod p).

On the Random Character of Fundamental Constant Expansions

We propose a theory to explain random behavior for the digits in the expansions of fundamental mathematical constants. At the core of our approach is a general hypothesis concerning the distribution of the iterates generated by dynamical maps. On this main hypothesis, one obtains proofs of base-2 normality—namely bit randomness in a specific technical sense—for a collection of celebrated constants, including π, log 2, ζ(3), and others. Also on the hypothesis, the number ζ(5) is either rational or normal to base 2. We indicate a research connection between our dynamical model and the theory of pseudorandom number generators.

Topics in advanced scientific computation

1) Selected Numerical algorithms, 2) evaluation of constants & Functions, 3) Number-theoretical Algorithms, 4) Transforms, 5) Nonlinear & Complex Systems, 6) Data manipulation

Irregular primes and cyclotomic invariants to 12 million

Computations of irregular primes and associated cyclotomic invariants were extended to all primes up to 12 million using multisectioning/convolution methods and a novel approach which originated in the study of Stickelberger codes Shokrollahi (1996). The latter idea reduces the problem to that of finding zeros of a polynomial overFpof degree < (p? 1)/2 among the quadratic nonresidues mod p. Use of fast polynomial gcd-algorithms gives anO (p log2p loglog p)-algorithm for this task. A more efficient algorithm, with comparable asymptotic running time, can be obtained by using Schonhage?Strassen integer multiplication techniques and fast multiple polynomial evaluation algorithms; this approach is particularly efficient when run on primes p for whichp? 1 has small prime factors. We also give some improvements on previous implementations for verifying the Kummer?Vandiver conjecture and for computing the cyclotomic invariants of a prime.

On the evaluation of Euler sums

Euler studied double sums of the form for positive integers r and s, and inferred, for the special cases r = 1 or r + s odd, elegant identities involving values of the Riemann zeta function. Here we establish various series expansions of ζ(r, s) for real numbers r and s. These expansions generally involve infinitely many zeta values. The series of one type terminate for integers r and s with r + s odd, reducing in those cases to the Euler identities. Series of another type are rapidly convergent and therefore useful in numerical experiments.

On the binary expansions of algebraic numbers

Employing concepts from additive number theory, together with results on binary evaluations and partial series, we establish bounds on the density of 1s in the binary expansions of real algebraic numbers. A central result is that if a real y has algebraic degree D > 1, then the number {number_sign}(|y|, N) of 1-bits in the expansion of |y| through bit position N satisfies {number_sign}(|y|, N) > CN{sup 1/D} for a positive number C (depending on y) and sufficiently large N. This in itself establishes the transcendency of a class of reals {summation}{sub n{ge}0} 1/2{sup f(n)} where the integer-valued function f grows sufficiently fast; say, faster than any fixed power of n. By these methods we re-establish the transcendency of the Kempner--Mahler number {summation}{sub n{ge}0}1/2{sup 2{sup n}}, yet we can also handle numbers with a substantially denser occurrence of 1s. Though the number z = {summation}{sub n{ge}0}1/2{sup n{sup 2}} has too high a 1s density for application of our central result, we are able to invoke some rather intricate number-theoretical analysis and extended computations to reveal aspects of the binary structure of z{sup 2}.

Projects in scientific computation

1 Numbers everywhere Selected topics in numerical analysis.- 1.1 Numerical evaluation.- 1.1.1 Evaluation of famous constants.- 1.1.2 Evaluation of elementary functions.- 1.1.3 Special functions.- 1.2 Equation solving.- 1.2.1 Matrix algebra.- 1.2.2 Non-linear equation systems.- 1.2.3 Differential equations.- 1.3 Random numbers and Monte Carlo.- 1.3.1 Generating Random Numbers.- 1.3.2 Numerical integration and Monte Carlo.- 2 Exploratory computation Collected intra- and interdisciplinary projects.- 2.1 Mathematical problems.- 2.1.1 Planar geometry problems.- 2.1.2 Symbolic manipulation.- 2.1.3 Real and complex analysis.- 2.2 Nature-motivated models.- 2.2.1 Neural network experiments.- 2.2.2 Genetic algorithms and artificial life.- 2.3 Projects from biology.- 2.3.1 Population models.- 2.3.2 Physiology, neurobiology, and medicine.- 2.3.3 Molecular biology.- 2.4 Projects from physics and chemistry.- 2.4.1 Classical physics.- 2.4.2 Quantum theory.- 2.4.3 Molecules and structure.- 2.4.4 Relativity.- 3 The lure of large numbers Projects in number theory.- 3.1 Large-integer arithmetic.- 3.1.1 Testing the operations.- 3.2 Prime numbers.- 3.2.1 Mersenne primes.- 3.2.2 Primes in general.- 3.3 Fast algorithms.- 3.3.1 Fast multiplication.- 3.3.2 Fast mod, division, inversion.- 3.3.3 Other fast algorithms.- 3.4 Factoring.- 3.4.1 Factoring algorithms.- 3.4.2 Status of Fermat numbers.- 4 The FFT forest-The ubiquitous FFT and its relatives.- 4.1 Discrete Fourier transform.- 4.1.1 Fundamental DFT manipulations.- 4.1.2 Algebraic aspects of the DFT.- 4.1.3 DFT test signals.- 4.1.4 Direct DFT software.- 4.2 FFT algorithms.- 4.2.1 Recursive FFTs.- 4.2.2 FFT indexing and butterflies.- 4.2.3 Complex FFTs, N a power of 2.- 4.2.4 Real-signal FFTs.- 4.2.5 FFTs for other radices.- 4.2.6 FFTs in higher dimensions.- 4.2.7 Applications of the FFT.- 4.3 Real-valued transforms.- 4.3.1 Hartley transform.- 4.3.2 Discrete cosine transform.- 4.3.3 Walsh-Hadamard transform.- 4.3.4 Square-wave transform.- 4.4 Number-theoretic transforms.- 4.4.1 Exploring number-theoretic transforms.- 5 Wavelets Young arrivals in the transform family.- 5.1 Chords, notes, and little waves.- 5.1.1 Windowed Fourier transform.- 5.1.2 Continuous wavelet transform.- 5.2 Discrete wavelet bases.- 5.2.1 Example wavelet expansions.- 5.2.2 Mother function and its wavelet.- 5.2.3 Wavelets of compact support.- 5.3 Discrete wavelet transform.- 5.3.1 Fast wavelet transform algorithms.- 5.3.2 Applications of fast wavelet transforms.- 6 Complexity reigns Chaos & fractals & such.- 6.1 Chaos.- 6.1.1 Quadratic map algebra.- 6.1.2 Bifurcation and chaos.- 6.1.3 Chaos models.- 6.1.4 Chaos, stability, and Lyapunov exponents.- 6.1.5 Applications of chaos theory.- 6.2 Fractals.- 6.2.1 Theory of fractals.- 6.2.2 Visualization of fractals.- 6.2.3 Fractal Brownian noise.- 6.2.4 Measurement of fractal dimension.- 7 Signals from the real world Projects in signal processing.- 7.1 Data compression.- 7.1.1 Tour of lossless data compressors.- 7.2 Sound.- 7.2.1 Examples of sound processing.- 7.2.2 Examples of sound compression.- 7.3 Images.- 7.3.1 Examples of image processing.- 7.3.2 Image compression.- Appendix Support code for the book Projects.- References.

Obstacles to the adoption of radio frequency identification technology in the emergency rooms of hospitals.

The Emergency Room (ER) receives patients in critical conditions. The operation of many emergency service chains is hampered because the required medical equipment is not always conveniently available and patient vital signs are manually monitored, but not automatically tracked on a real-time basis. This has resulted in medical errors, increased stress levels of medical teams, and poor utilisation of staff and equipment. Our research investigates factors that contribute to the adoption of Radio Frequency Identification (RFID) technologies by the medical team in ERs. We propose a theoretical framework to address this issue based upon the Unified Theory of Acceptance and Use of Technology (UTAUT) theory.

New representations for the Madelung constant

From a modern theta-function identity of G. E. Andrews we derive new representations for the celebrated Madelung constant and various of its analytic relatives. The method leads to connections with the modern theory of multiple zeta sums, generates an apparently entire “η series” representation, and, for the Madelung constant in particular, yields a finite-integral representation. These analyses suggest variants of the Andrews identity, leading in turn to number-theoretical results concerning sums of three squares.