Sliding in rough convex bowls

It is an elementary fact that a particle sliding on a rough horizontal table experiences a deceleration of magnitude μg, where μ is the coefficient of friction between the particle and the table. Thus, if the initial speed u of the particle and the braking distance d are both known, then μ can be determined by the formulaThe idea of using braking distances to determine the coefficient of friction led the second author to ask the following question:A particle slides on the interior surface of a rough hemispherical bowl starting from rest. If the starting point of the particle and the point at which it comes to instantaneous rest after its slide are both known, is it possible to determine the coefficient of friction between the particle and the bowl?

On the structure of Ext(A,Z) in ZFC+

A fundamental problem in the theory of abelian groups is to determine the structure of Ext(A, Z) for arbitrary abelian groups A. This problem was raised by L. Fuchs in 1958, and since then has been the center of considerable activity and progress.We briefly summarize the present state of this problem. It is a well-known fact thatwhere tA denotes the torsion subgroup of A. Thus the structure problem for Ext(A, Z) breakdown to the two distinct cases, torsion and torsion free groups. For a torsion group T,which is compact and reduced, and its structure is known explicitly [12].For torsion free A, Ext(A, Z) is divisible; hence it has a unique representationThus Ext(A, Z) is characterized by countably many cardinal numbers, which we denote as follows: ν0(A) is the rank of the torsion free part of Ext(A, Z), and νp(A) are the ranks of the p-primary parts of Ext(A, Z), Extp(A, Z).If A is free it is an elementary fact that Ext(A, Z) = 0. The second named author has shown [16] that in the presence of V = L the converse is also true. For countable torsion free, nonfree A, C. Jensen [13] has shown that νp(A) is either finite or and νp(A) ≤ ν0(A). Therefore, the case for uncountable, nonfree, torsion free groups A remains to be studied.

A Banach Space which is not Equivalent to an Adjoint Space

A Banach space which is not reflexive may or may not be equivalent (in Banach's sense) to an adjoint space. For example, it is an elementary fact that the space (l), though not reflexive, is equivalent to (co)*, where (co) is the space of all sequences that converge to zero, normea in the usual way. On the other hand, (co) itself is not equivalent to any adjoint space : this can be proved by means of the Krein-Milman theorem, but here we obtain the result by an elementary argument which is scarcely more complicated than the standard proof that (co) is not reflexive.

The Rational and Empirical Elements in Physics

Itis a platitude that thought implies a subject and an object: the subject is the thinker, or the thinking mind, and the object is that which is thought about. This is probably the most elementary fact of consciousness, comprehensible alike to the child, the unreflecting man of affairs, and the philosopher, and it forms the natural startingpoint for philosophy. And indeed, one of the great divisions between philosophical systems is that which separates subjectivism on one hand from objectivism (more often called by the indefinite and overburdened name, realism) on the other. Subjectivism professes to interpret the object in terms of the subject, and objectivism professes to interpret the subject in terms of the object.

Extension of a set of theorems in circle geometry

The fascinating chain of theorems due to Clifford and recently extended by Mr F. P. White originates in the elementary fact that the circumscribing circles of the triangles formed by four lines meet in a point. From a like simple germ, namely the fact that the centres of the above-mentioned circles lie on a circle, an infinite chain of theorems was first evolved by Pesci and may be enunciated as follows:(i) The centres of the circumcircles of the triangles formed by four lines lie on a circle; thus from four lines we derive a point, namely the centre of the latter circle.(ii) Five lines give five sets of four and the five derived points lie on a circle; thus from five lines we derive a point, namely the centre of the latter circle.(iii) Six lines give six sets of five and the six derived points lie on a circle.