The Wayside Mechanic: An Analysis of Skill Acquisition in Ghana

This is a study of skill learning in an informal learning setting in Africa. The purpose of the study is to describe an analyze the nature of the skill acquisition process in one indigenous training system: the apprenticeship of the wayside mechanics workshops in Koforidua, Ghana. The study first examines informal skill training from a broad perspective. The history of the West African craft workshop and its associated apprenticeship is traces. Several major themes in the literature on informal skill training systems such as apprenticeships. The second part of the study describes the specific setting of the wayside mechanics workshops and the general features of the apprenticeship system. Subsequent sections consider contextual factors which may have a bearing on apprentice skill acquisition. Case studies of several apprentice and master artisans are presented to illustrate personal experiences at various levels of the mechanics profession. Finally, the learning/teaching process in the wayside workshop is analyzed intensively using data gathered from general observation, structured interveiws, and structured observation instruments. Apprentice skill competence is assessed through self-reports and administration of mechanics skill test. Results are used to evaluate the effectiveness of apprenticeship training in fostering diagnostic skills and higher-order theoretical understanding. Several possible ways of enhancing apprenticeship training through supplementary training programs are suggested. The implications of the study are of interest to educational anthropologists who are concerned with learning in traditional naturalistic settings. The study is also significant for educational planners in that it calls attention to the strengths and limitations of building nonformal educational programs around indigenous learning systems

The Convergence and Divergence of q-Continued Fractions outside the Unit Circle

We consider two classes of q-continued fraction whose odd and even parts are limit 1-periodic for |q| \u3e 1, and give theorems which guarantee the convergence of the continued fraction, or of its odd- and even parts, at points outside the unit circle

Polynomial Continued Fractions

Continued fractions whose elements are polynomial sequences have been carefully studied mostly in the cases where the degree of the numerator polynomial is less than or equal to two and the degree of the denominator polynomial is less than or equal to one. Here we study cases of higher degree for both numerator and denominator polynomials, with particular attention given to cases in which the degrees are equal. We extend work of Ramanujan on continued fractions with rational limits and also consider cases where the limits are irrational

The Convergence behavior of q-Continued Fractions on the Unit Circle

In a previous paper, we showed the existence of an uncountable set of points on the unit circle at which the Rogers-Ramanujan continued fraction does not converge to a finite value. In this present paper, we generalise this result to a wider class of qcontinued fractions, a class which includes the Rogers-Ramanujan continued fraction and the three Ramanujan-Selberg continued fractions. We show, for each q-continued fraction, G(q), in this class, that there is an uncountable set of points, YG, on the unit circle such that if y ∈ YG then G(y) does not converge to a finite value. We discuss the implications of our theorems for the convergence of other q-continued fractions, for example the G¨ollnitz-Gordon continued fraction, on the unit circle

A Theorem on Divergence in the General Sense for Continued Fractions

If the odd and even parts of a continued fraction converge to different values, the continued fraction may or may not converge in the general sense. We prove a theorem which settles the question of general convergence for a wide class of such continued fractions. We apply this theorem to two general classes of q continued fraction to show, that if G(q) is one of these continued fractions and |q| \u3e 1, then either G(q) converges or does not converge in the general sense. We also show that if the odd and even parts of the continued fraction K∞n=1an/1 converge to different values, then limn→∞ |an| = ∞

A Theorem on Divergence in the General Sense for Continued Fractions

If the odd and even parts of a continued fraction converge to different values, the continued fraction may or may not converge in the general sense. We prove a theorem which settles the question of general convergence for a wide class of such continued fractions. We apply this theorem to two general classes of q continued fraction to show, that if G(q) is one of these continued fractions and |q| \u3e 1, then either G(q) converges or does not converge in the general sense. We also show that if the odd and even parts of the continued fraction K∞n=1an/1 converge to different values, then limn→∞ |an| = ∞

On the Divergence of the Rogers-Ramanujan Continued Fraction on the Unit Circle

This paper is an intensive study of the convergence of the Rogers-Ramanujan continued fraction. Let the continued fraction expansion of any irrational number t ∈ (0, 1) be denoted by [0, a1(t), a2(t), · · · ] and let the i-th convergent of this continued fraction expansion be denoted by ci(t)/di(t). Let S = {t ∈ (0, 1) : ai+1(t) ≥ φ di(t) infinitely often}, where φ = (√ 5 + 1)/2. Let YS = {exp(2πit) : t ∈ S}. It is shown that if y ∈ YS then the Rogers-Ramanujan continued fraction, R(y), diverges at y. S is an uncountable set of measure zero. It is also shown that there is an uncountable set of points, G ⊂ YS, such that if y ∈ G, then R(y) does not converge generally. It is further shown that R(y) does not converge generally for |y| \u3e 1. However we show that R(y) does converge generally if y is a primitive 5m-th root of unity, some m ∈ N so that using a theorem of I. Schur, it converges generally at all roots of unity

On The Divergence in the General Sense of q-Continued Fractions on the Unit Circle

We show, for each q-continued fraction G(q) in a certain class of continued fractions, that there is an uncountable set of points on the unit circle at which G(q) diverges in the general sense. This class includes the Rogers-Ramanujan continued fraction and the three Ramanujan-Selberg continued fraction. We discuss the implications of our theorems for the general convergence of other q-continued fractions, for example the G¨ollnitz-Gordon continued fraction, on the unit circle

Collective Bargaining: The Next Twenty Years

Collective bargaining will evolve in the next 20 years in response to the changes taking place in the world of work and union reaction to those changes. Job security will be a central issue, with increased emphasis on reducing work time to create more jobs. We also foresee more union mergers and increased inclusion in the labor movement of workers not traditionally a part of the union's constituency. That, in turn, will further erode the effectiveness of centralized bargaining. Moreover, unions will attempt to alter their traditional role as reactors to managements' unilateral actions and see themselves as partners with man agement. The role of the government and the courts may alter the collective bargaining process, but whether it will erode or strengthen that process is still an open question.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/67844/2/10.1177_000271628447300104.pd

Continued Fractions with Multiple Limits

For integers m ≥ 2, we study divergent continued fractions whose numerators and denominators in each of the m arithmetic progressions modulo m converge. Special cases give, among other things, an infinite sequence of divergence theorems, the first of which is the classical Stern-Stolz theorem. We give a theorem on a class of Poincar´e type recurrences which shows that they tend to limits when the limits are taken in residue classes and the roots of their characteristic polynomials are distinct roots of unity. We also generalize a curious q-continued fraction of Ramanujan’s with three limits to a continued fraction with k distinct limit points, k ≥ 2. The k limits are evaluated in terms of ratios of certain q series. Finally, we show how to use Daniel Bernoulli’s continued fraction in an elementary way to create analytic continued fractions with m limit points, for any positive integer m ≥ 2