Wallace's unfinished business: The “Other Man” in evolutionary theory

After a century in the shadows, Alfred Russel Wallace (1823–1913) has recently become the subject of increasing attention. It is suggested here, expanding on observations made by anthropologist Gregory Bateson some years ago, that Wallace's cybernetics‐like view of the operation of natural selection—as a governor‐like principle tending to keep species unvarying—can be expanded to a more complete evolutionary understanding by exploring in modern context Wallace's idea that “more recondite forces” are driving the process. Specifically, when the environment is regarded as a final cause (but not a deterministic force), individual adaptations may be viewed as entropy‐relaying structures (acting in response to, and as a part of, larger scale biogeochemical agenda), whereas negentropy is accumulated by nonrandomly directed organism‐ and population‐level forms of ecological engagement. Thus, range change in particular is viewed as a process that is both driven and nonrandom, and ultimately connected to the derivation of more and more organized individual, population, and community structures. © 2004 Wiley Periodicals, Inc. Complexity 10:25–32, 2004     

The “golden” algebraic equations

The special case of the (p + 1)th degree algebraic equations of the kind x^p+1 = x^p + 1 (p = 1, 2, 3, …) is researched in the present article. For the case p = 1, the given equation is reduced to the well-known Golden Proportion equation x² = x + 1. These equations are called the golden algebraic equations because the golden p-proportions τ_p, special irrational numbers that follow from Pascal’s triangle, are their roots. A research on the general properties of the roots of the golden algebraic equations is carried out in this article. In particular, formulas are derived for the golden algebraic equations that have degree greater than p + 1. There is reason to suppose that algebraic equations derived by the authors in the present article will interest theoretical physicists. For example, these algebraic equations could be found in the research of the energy relationships within the structures of many compounds and physical particles. For the case of butadiene (C₄H₆), this fact is proved by the famous physicist Richard Feynman.     

The “Error” in the Indian “Taylor Series Approximation” to the Sine

It has been repeatedly noted, but not discussed in detail, that certain so-called “third-order Taylor series approximations” found in the school of the medieval Keralese mathematician M dhava are inaccurate. That is, these formulas, unlike the other series expansions brilliantly developed by M dhava and his followers, do not correspond exactly to the terms of the power series subsequently discovered in Europe, by whose name they are generally known. We discuss a Sanskrit commentary on these rules that suggests a possible derivation explaining this discrepancy, and in the process re-emphasize that the Keralese work on such series was rooted in geometric approximation rather than in analysis per se. © 2001 Elsevier Science (USA).Es ist mehrfach festgestellt bisher aber nicht ausführlich diskutiert worden, daß einige sogenannte Taylor-reihennäherungswerte dritter Ordnung, die in der mittelalterlichen Schule keralesischen M dhava gefunden werden, ungenau sind. Das heißt, diesc Formeln sind den Termen der Potenzreihe, die später in Europa entwickelt wurde und unter dem Namen Taylorreihe bekannt ist, nicht äquivalent, im Gegensatz zu den anderen Entwicklungen von Reihen, die glänzend von M dhava und seinen Nachfolgern entwickelt werden. Wir behandeln einen Sanskritkommentar zu den Regeln, der eine mögliche Herleitung suggeriert, die diese Diskrepanz erklärt. Dabei betonen wir nochmals, daß die keralesische Arbeit über solche Reihen eher in geometrischen Näherungen als in der Analysis an sich ihre Wurzeln hat. © 2001 Elsevier Science (USA).MSC subject classification: 01A32.     

On a vector-valued integral in the “noncommutative” statistical decision theory

A generalization of a vector-valued integral arising in the noncommutative (quantum) statistical decision theory is considered.     

Self-excited oscillations of a two-mass oscillator with dry “stick-slip” friction

A system of two masses, moving along a single straight line, is considered. The first is connected by a spring to a fixed point, while the second is connected by a spring to the first and is in contact with a belt with dry friction moving with constant velocity. A piecewise-constant model of dry friction with different coefficients of friction, sliding and at rest, is used. The limit “stick-slip” type cycles are investigated analytically. It is shown numerically that in the case of equal masses there are forward and reverse limit cycles. The period of the oscillations of the forward and reverse cycles increases as the ratio of the stick and slip coefficients of friction increases, and decreases when the velocity of the belt increases. The reverse cycle exists for all values of the parameters of the problem, while the forward cycle exists up to a certain critical value of the ratio of the stick and slip coefficients of friction, and this critical value increases when the velocity of the belt increases.     

On “bent” functions

Let P(x) be a function from GF(2ⁿ) to GF(2). P(x) is called “bent” if all Fourier coefficients of (−1)^P(x) are ±1. The polynomial degree of a bent function P(x) is studied, as are the properties of the Fourier transform of (−1)^P(x), and a connection with Hadamard matrices.     

An efficient algorithm for the “stable roommates” problem

The stable marriage problem is that of matching n men and n women, each of whom has ranked the members of the opposite sex in order of preference, so that no unmatched couple both prefer each other to their partners under the matching. At least one stable matching exists for every stable marriage instance, and efficient algorithms for finding such a matching are well known. The stable roommates problem involves a single set of even cardinality n, each member of which ranks all the others in order of preference. A stable matching is now a partition of this single set into n/2 pairs so that no two unmatched members both prefer each other to their partners under the matching. In this case, there are problem instances for which no stable matching exists. However, the present paper describes an O(n²) algorithm that will determine, for any instance of the problem, whether a stable matching exists, and if so, will find such a matching.     

Reading “A variant of the Hales–Jewett theorem” on its anniversary

In a recent paper “A variant of the Hales–Jewett theorem”, M. Beiglböck provides a version of the classic coloring result in which an instance of the variable in a word giving rise to a monochromatic combinatorial line can be moved around in a finite structure of specified type (for example, an arithmetic progression). We give an elementary proof and infinitary extensions.