H. O. Pollak

Steiner Minimal Trees

A Steiner minimal tree for given points

Some remarks on the Steiner problem

A_1 , \cdots ,A_n

An analysis of parallel-read sequential-write systems

in the plane is a tree which interconnects these points using lines of shortest possible total length. In order to achieve minimum lengt...

Energy distribution of band-limited functions whose samples on a half line vanish

Abstract Let Σ be a set of n points in the plane. The minimal network for Σ is the tree of shortest total length LM(Σ) whose vertices are exactly the points of S. The Steiner minimal network for Σ is the tree of shortest possible total length LS(Σ) when the vertices are allowed to be any set Σ′ ⊇ Σ. Clearly LS(Σ) ⩽ LM(Σ), since the minimization in LS is over a larger set. It has long been conjectured that, conversely, L S (Σ) ⩾ ( 3 1 2 2 ) L M (Σ) , but this has previously been proved only if n = 3. In this paper, among other results, this is proved for n = 4. Unfortunately the proof is sufficiently complicated that immediate generalization to arbitrary n, no matter how desirable, is unlikely.

The Effects of Technology on the Mathematics Curriculum

Abstract We consider a classical problem in the parallel access of processes to a common non-shareable resource. Our purpose is to evaluate the effective read throughput of a set of independent processes in the presence of write operations carried out by the same processes. Read operations are assumed to be executable in parallel, while writes have to be strictly sequential. Explicit expressions are obtained for the distributions of effective read times in terms of their Laplace transforms, under general assumptions concerning the access times and assuring a Poisson stream of write accesses.

Mathematical Modelling — a Conversation with Henry Pollak

Let S be the class of band-limited functions f(t) of bandwidth π such that ∝−∞∞f2(t) dt < ∞ and f(n) = 0 for n = 1, 2, 3, …. A study is made of the proportion of energy which such a function may have to the right of some given point N. Let φ(N)=supfϵS∫N∞f2(t)dt∫−∞∞f2(t)dt It is shown, among other results, that φ(N) = 12if N ⩾ 34, that φ(N)12if N ⩽ 14, and that φ(N) < 1 for all real N. The first result has the rather surprising consequence that φ(N) does not tend to 0 as N → ∞. It is shown that this conclusion is still valid if we restrict ourselves to finite sampling series; other restrictions on S which do succeed in forcing φ(N) to 0 for large N are also discussed.

Proportional allocation schemes for tour costs

Mathematics is one of a very small number of subjects which is, in most countries, taught to all students every year throughout many years of schooling. Why does society give us all this time — when it will do the same for almost no other subject? A variety of reasons is given for this unexpected phenomenon, and we wish to begin by mentioniing several of them. Mathematics is an essential part of human culture which the educational system is designed to transmit. This reason fits well into the traditional liberal education of Western Europe, as designed in the 17th and 18th centuries, with an emphasis on classics, literature, natural philosophy and mathematics. Mathematics is the best way to teach youngsters how to think; this purpose fits well into the ideal of encouraging students to go as far in their education as their talents and motivation will permit. Mathematics is beautiful; a personal aesthetic experience of mathematics which we fervently hope will become real to more than a small percentage of our children, associates well with progressive education, discovery learning, and individualised instruction.

Photoelastic calculations by a complex variable method

Henry Pollak is, without any doubt, one of the pioneers in the field of applications and modelling in mathematics education. As early as in the sixties of the last century he pleaded for an integration of applications and modelling into mathematics teaching (for instance, Pollak, 1969). He was able to do that particularly competently and credibly since he was not part of the educational system itself but a leading member of Bell Laboratories. The educational scene of the sixties and the early seventies was shaped by the New Maths movement which had, in contrast to its own intentions, led in many countries to an emphasis on intra-mathematical aspects (see Pollak, 2003, for the situation in the USA). Internationally, Henry’s engagement for applications and modelling became particularly visible at ICME-3, 1976, where he gave the survey lecture on “The Interaction between Mathematics and Other School Subjects” (Pollak, 1979). Also in the first ICTMAs, Henry was active on a prominent position, for instance as a plenary speaker at ICTMA-3, 1987 (Pollak, 1989).

The Place of Mathematical Modelling in the System of Mathematics Education: Perspective and Prospect

Abstract This paper examines the problem of allocating intercity travel costs for a multi-city trip to sponsors at the different cities in as fair a manner as possible when each sponsor is willing to pay as much as the round-trip cost between his city and the travelers home. Intercity travel costs are presumed to satisfy the usual triangle inequality. The paper first considers a fixed-route situation and shows that four axioms—including full cost recovery, monotonicity and additivity conditions—delineate a simple class of allocation schemes. All possible tours through the cities are then admitted under a fifth axion which says that minimum-cost tours are Pareto optimal for the sponsors. This efficiency axiom further restricts the class of allocation schemes, but it does not force a unique result. Additional conditions that do in fact identify a unique scheme are mentioned. A relaxation of the triangle inequality for costs is also discussed.

Pure and Applied Mathematics From an Industrial Perspective

ZusammenfassungDie komplexe Darstellung der allgemeinen Lösung für den ebenen elastischen Spannungszustand führt zu einer einfachen Formel für den Mittelwert der Normalspannung im Prüfstück, die sich spannungsoptischer Messungen auf zwei Kreisen bedient. Diese Formel vermeidet die Messfehler betonende numerische Differentiation und benötigt die Kenntnis der Randspannungen an jedem Randpunkte nicht.