On the use of computational tools to promote students’ mathematical thinking

This column will publish short (from just a few paragraphs to ten or so pages), lively and intriguing computer-related mathematics vignettes. These vignettes or snapshots should illustrate ways in which computer environments have transformed the practice of mathematics or mathematics pedagogy. They could also include puzzles or brain-teasers involving the use of computers or computational theory. Snapshots are subject to peer review. In this snapshot students employ dynamic geometry software to find great mathematical richness around a seemingly simple question about rectangles.

Editor: Uri Wilensky

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Hybrid synthesis of the optimal discrete nonlinear filter

It has been known for many years that an optimal discrete nonlinear filter may be synthesized for systems whose plant dynamics, sensor characteristics and signal statistics are known by applying Bayes' Rule to sequentially update the conditional probability density function from the latest data. However, it was not until 1969 that a digital computer algorithm implementing the theory for a one-state variable one-step predictor appeared in the literature. This delay and the continuing scarcity of multidimensional nonlinear filters result from the overwhelming computational task which leads to unrealistic data processing times. For many nonlinear filtering problems analog and digital computers (a hybrid computation) combine to yield a higher data rate than can be obtained by con¬ventional digital methods. This paper describes an implementation of the theory by means of a hybrid computer algorithm for the optimal nonlinear one-step predictor.

The hybrid computer algorithm presented reduces the overall solution time per prediction because:

1) Many large computations of identical form are executed on the analog computer in parallel.

2) The discrete running variable in the digital algorithm may be replaced with a continuous analog computer variable in one or more dimensions leading to increased computational speed and finer resolution of the exponential transformation.

3) The modern analog computer is well suited to generate functions such as the expo¬nential at high speed with modest equipment.

4) The arithmetic, storage, and control functions performed rapidly by the digital computer are utilized without introducing extensive auxiliary calculations.

To illustrate pertinent aspects of the algorithm developed, the scalar cubed sensor problem previously described by Bucy is treated extensively. The hybrid algorithm is described. Problems associated with partitioning of equations between analog and digital computers, machine representations of variables, setting of initial conditions and floating of grid base are discussed. The effects of analog component bandwidths, digital-to-analog and analog-to-digital conversion times, analog computer mode switching times and digital computer I/O data rates on overall processing time are examined. The effect of limited analog computer dynamic range on accuracy is discussed. Results from a simulation of this optimal predictor using MOBSSL, a continuous system simulation language, are given. Timing estimates are presented and compared against similar estimates for the all digital algorithm.

For example, given a four-state variable optimal 1-step predictor utilizing 7 discrete points in each dimension, the hybrid algorithm can be used to generate predictions accurate to 2 decimal places once every 10 seconds. An analog computer complement of 250 integra¬tors and multipliers and a high-speed 3rd generation digital computer such as the CDC 6600 or IBM 360/85 are required. This compares with a lower bound of about 3 seconds per all digital prediction which would require 49 CDC 6600's operating in parallel. Analytical and simulation work quantifying errors in one state variable filters is presented. Finally, the use of an interactive graphic system for real time display and for filter evaluation is described.     

Zur Theorie und Praxis der dynamischen Optimierung

Zusammenfassung Für eine große Klasse kontinuierlicher dynamischer Optimierungsprobleme, bei denen es sich um die Optimierung eines Integralausdrucks unter gewissen nichtholonomen Nebenbedingungen handelt, wird nach der Untersuchung der Existenz von Lösungen und der Ableitung einer Funktionalgeichung in Teil I dieser Arbeit in einem II. Teil (veröffentlicht im nächsten Heft dieser Zeitschrift) ein Verfahren zur Lösung mit Hilfe eines iterativen elektronischen Analogrechners einschließlich der zugehörigen Konvergenzbetrachtungen angegeben. Das Lösungsverfahren ist auch für Digitalrechner verwendbar, als besonders geeignet erscheinen die gerade jetzt in der Entwicklung begriffenen hybriden Rechenanlagen.

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Summary We treat a large class of problems of continuous dynamic programming. We have to optimize a certain integral expression restricted by some nonholonomic secondary conditions. We investigate the existence of solutions, and derive a functional equation. In a second part of this paper (being published in the next number of this journal) we give a method for the solution of this problem by means of an iterative analogue computer, and study the convergence of this method. For our procedure we can also use digital computers, especially hybrid computers.

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Diese Arbeit ist ein Auszug aus der von der Fakultät für Allgemeine Wissenschaften der Technischen Hochschule München genehmigten Dissertation des Dipl.-Math.Klaus Neumann, Institut für Angewandte Mathematik der Technischen Hochschule München, Dir. Prof. Dr.J. Heinhold, 8 München 2, Arcisstraße 21. Tag der Promotion: 18. 12. 1964.

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Vorgel. v.:J. Heinhold     

Statistical and Numerical Methods of Processing Examination Results by Computer

The need to examine the average, the standard deviation and the distribution of a set of raw examination marks is discussed.

A small computer program is described which reads a set of marks and names and calculates and prints the average mark, the standard deviation, a histogram and a list of names and marks in order of merit. Further marks lists are printed (a) with the average mark modified to a predetermined value, (b) with both the average mark and the standard deviation adjusted (Zscores) and (c) Tscores in which the order of merit is transformed into numerical scores assuming a normal distribution. The several forms of marks lists are produced automatically in one run, and should encourage users to examine their own methods, and choose the one most appropriate for their purpose.

The program is written in FORTRAN, requires a minimum of 10 K of core, and should be easily implemented on most computers.     

Convex-cone-based comparisons of and difference evaluations for fuzzy sets

Abstract

This paper focuses on how to compare two fuzzy sets and, from the viewpoint of set optimization, proposes eight types of fuzzy-set relations based on a convex cone as new comparison criteria of fuzzy sets. Then, difference evaluation functions for fuzzy sets are introduced. Under suitable assumptions of certain compactness and stability of fuzzy sets, we show that these functions correspond well to the fuzzy-set relations. In addition, through transforming these functions stepwise, we deal with numerical calculation methods of them in particular cases. Consequently, we can judge whether each fuzzy-set relation holds or not for given two fuzzy sets with the aid of computers.     

An Experiment in Teaching College Mathematics†

Kac has observed that the ideal preparation in mathematics, especially for non‐mathematicians, should focus not on acquiring skills but on acquiring certain attitudes. We administered a special attitude questionnaire to a sample of graduate students in mathematics and undergraduate speech majors. We found significant differences on 10 of 27 items on this test. We then administered this test to a mixed group of undergraduates at the beginning and at the end of a special experimental mathematics ‘course’ designed to modify and shape attitudes. We found changes in attitudes in the intended direction. The primary aims of the experimental course were to:

1. Get students without any prior acquaintance with mathematics or a fear thereof to approach their studies more analytically.

2. Acquire orientation to and acquaintance with 25‐75 basic concepts and methods covering sets, algebra, logic, computers, analysis, probability, math‐statistics and topology in an over‐all map of how they logically fit together and how they relate to problems of modern life.

3. Read, with appreciation, mathematical literature previously incomprehensible to them. These aims were met.

     

An academic experiment on the use of computers in elementary school mathematics classrooms

The role of computers in elementary school math classrooms is still being determined. Although computers are promised effective visual tools to promote independent work and study; many educators neglect to use them. Since there are varying points of view, individual teachers generally decide whether to incorporate computers into their methods.Purpose: My experiment analyzes and quantifies the value of computers in elementary school math classrooms.Method: Over a course of 11 weeks, my first grade class worked with the teaching software “Mathematikus 1” (Lorenz, 2000). Using both interpersonal and video observation, I completed written evaluations of each pair of my students' will and ability to cooperate, communicate and independently solve mathematical problems.Conclusion: My results show that it is generally beneficial to use computers in elementary school math lessons. However, some elements of said software leave room for improvement.     

Kostenanalyse im streckenvortrieb

Zusammenfassung Es wird untersucht, durch welche Faktoren die Kosten bei der Auffahrung einer Strecke beeinflußt werden, wenn die Streckenvortriebsgeschwindigkeit gesteigert wird, und in welcher Weise sich diese Faktoren auswirken. Je nach den im Einzelfall vorliegenden Bedingungen ergeben Variationen bei den Kostenfaktoren teils erhebliche Einsparungen, teils erhebliche Erhöhungen der Streckengesamtkosten. Die komplizierten Zusammenhänge machen es unmöglich, eine streng mathematische Kostenfunktion aufzustellen und zu optimieren. Durch den Einsatz einer elektronischen Rechenanlage läßt sich jedoch auf dem Wege der Simulation des Betriebsablaufes eine befriedigende Aussage über die zu erwartenden Kosten und ihre Veränderungen bei unterschiedlichen Einflußgrößen erstellen.

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Summary The factors which have an influence on the costs of driving roadways are investigated for the case that velocity of drift is increased. Depending on the given conditions for everey special case the variations of those factors cause partly an important cut or partly a considerable raising of the drift total. — The rather complicated connections do not allow any specification and optimalisation of strict mathematical cost functions. Only by application of electronic computers, a satisfying statement of the expected costs and their variations caused by differed influences is possible in simulating the way of working.

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Mitteilung aus dem Institut für Bergbaukunde und Bergwirtschaftslehre (Institutsdirektor: Prof. Dr.-Ing.G. Dorstewitz) der Bergakademie Clausthal.

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Vorgel. v.:G. Dorstewitz     

The Wave Function of the Lyman-Alpha Photon Part II. The Shape,Position, and Trajectory of the Photon

In Part I of the present paper we derived the wave function of the Lyman-a \alpha photon in both the linear and angular momentum bases using relativistic concepts for the photon wave function. In the present paper, Part II, we derive two [(X)\vec] \vec{X} -representations. In the first we assume one-particle theory for the photon wave function and the usual commutation rules for the position operators X_i X_i and linear momentum operators P_i P_i . The second representation employs the quantized photon field to derive an [(X)\vec] \vec{X} -representation. The stress, energy tensor density is used to provide a probability density in [(x)\vec] \vec{x} -space which is relativistic. The two methods of defining [(x)\vec] \vec{x} -space are compared.¶It is found in the present case that, despite the use of particle operators, the photon resembles a field far more than it does a particle.     

Huge Data Sets and the Frontiers of Computational Feasibility

Abstract

Recently, Huber offered a taxonomy of data set sizes ranging from tiny (10² bytes) to huge (10¹⁰ bytes). This taxonomy is particularly appealing because it quantifies the meaning of tiny, small, medium, large, and huge. Indeed, some investigators consider 300 small and 10,000 large while others consider 10,000 small. In Huber's taxonomy, most statistical and visualization techniques are computationally feasible with tiny data sets. With larger data sets, however, computers run out of computational horsepower and graphics displays run out of resolution fairly quickly. In this article, I discuss aspects of data set size and computational feasibility for general classes of algorithms in the context of CPU performance, memory size, hard disk capacity, screen resolution and massively parallel architectures. I discuss some strategies such as recursive formulations that mitigate the impact of size. I also discuss the potential for scalable parallelization that will mitigate the effects of computational complexity.     

Analogrechner und ihre Anwendung auf ein volkswirtschaftliches Modell

Zusammenfassung Nach kurzer Erläuterung der verschiedenen Arten von Analogrechnern und ihrer Arbeitsweise wird die Aufstellung eines Kopplungsplanes gezeigt. An Hand des Beispiels Wechselwirkung zwischen Angebot und Nachfrage wird diskutiert, wie ein volkswirtschaftliches Modell auf einem Analogrechner nachgebildet und gelöst werden kann.

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Summary After explaining the various types of analog computers and their operation mode the set up for differential equations is described. An example of interactions between supply and demand is discussed showing the possibility for simulation and solution of an economic model with an analog computer.

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Mitteilung aus dem Institut für Praktische Mathematik (IPM), der Technischen Hochschule Darmstadt, Prof. Dr. A. Walther.     

Modern factorization methods

In this expository paper the progress in factorization of large integers since the introduction of computers is reported. Thanks to theoretical advances and refinements, as well as to more powerful computers, the practical limit of integers possible to factor has been raised considerably during the past 20 years. The present practical limit is around 10⁷⁵ if supercomputers are used and if much computer time is available.     

A Method of Systematic Search for Optimal Multipliers in Congruential Random Number Generators

This paper presents a method of systematic search for optimal multipliers for congruential random number generators. The word-size of computers is a limiting factor for development of random numbers. The generators for computers up to 32 bit word-size are already investigated in detail by several authors. Some partial works are also carried out for moduli of 2⁴⁸ and higher sizes. Rapid advances in computer technology introduced recently 64 bit architecture in computers. There are considerable efforts to provide appropriate parameters for 64 and 128 bit moduli. Although combined generators are equivalent to huge modulus linear congruential generators, for computational efficiency, it is still advisable to choose the maximum moduli for the component generators. Due to enormous computational price of present algorithms, there is a great need for guidelines and rules for systematic search techniques. Here we propose a search method which provides ‘fertile’ areas of multipliers of perfect quality for spectral test in two dimensions. The method may be generalized to higher dimensions. Since figures of merit are extremely variable in dimensions higher than two, it is possible to find similar intervals if the modulus is very large. This revised version was published online in July 2006 with corrections to the Cover Date.     

Die Rolle der ETH bei der Computerentwicklung

Zusammenfassung Nach dem 2. Weltkreige gehörte Professor Stiefel zu den aufgeschlossenen Mathematikern, die die grossen Möglichkeiten der Computer erkannten. Nachdem er die bekannten in den USA gelaufenen Entwicklungen studiert hatte, besuchte er auch ein kleines Alpendorf in Bayern. Dort wurde ihm das aus Berlin gerettete Gerät Z4 vorgeführt, das er dann für die ETH in Zürich mietete. Das führte zu einer freundschaftlichen Zusammenarbeit, und manche Erinnerung an diese schöne Pionierzeit mit all ihren Schwierigkeiten und Problemen gibt den Beteiligten heute das befriedigende Gefühl, mit an der vordersten Front der technischen Entwicklung gestanden zu haben.

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Summary After World War II Professor Stiefel was one of the open-minded mathematicians who realized the importance of the computers. After having studied the well-known developments in the USA he visited also a small Bavarian village. There he was shown the model Z4 which had been rescued from Berlin; Stiefel then rented it for the ETH Zürich. This was the beginning of a friendly cooperation. Remembering this pioneer time with all the difficulties and problems gives the participants the feeling of having influenced the technical advancements of our time.

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Dem Andenken an Eduard Stiefel     

The Wave Function of the Lyman-Alpha Photon Part I. The Wave Function in the Angular and Linear Momentum Representations

To the best of the writer‘s knowledge no one has given the wave function of a photon emitted in an atomic, molecular, or nuclear transition. In the present paper we derive the wave function in the angular momentum and linear momentum representations for the photon emitted by a non-relativistic hydrogen atom, when the electron of the atom falls from the first excited state to the ground state. This is the simplest transition which produces a photon. A two level model for the atom is used, in which the lower level (the ground state energy) is associated with a non-degenerate wave function and the upper level (the energy of the first excited state) is associated with wave functions corresponding to the four-fold degeneracy of that state. We use a generalization of Dirac‘s method for finding the eigenfunctions in resonance scattering. We find the exact solution of the two-level problem using the exact matrix elements of the interaction. The calculations are finite without renormalization. In the next paper we shall introduce the [(x)\vec] \vec{x} -representation and thereby obtain the "position", "shape", and "trajectory" of the photon.     

Reconstructing the Adjusted Census for Florida: A Case Study in Data Examination

Abstract

This article is a case study in the value of close examination of available data when one is attempting to make inferences. The article is also an example of what can be done to reconstruct government files when data have been concealed, whether for administrative or political reasons, as here, or to preserve confidentiality. The various methods currently used are under review by a panel under the aegis of the Committee on National Statistics of the National Research Council. The upshot of our work is that even suppressing half of the microlevel data in a file need not daunt a determined and reasonably adroit reconstructor equipped with modern computers. The particular problem discussed here arose due to a bizarre set of political and legal circumstances, explained in Section 1, whereby the U.S. Department of Commerce released a tape consisting of the adjusted population (cross-classified by age, race, and gender) of half the Census Blocks in the nation, together with certain summary adjusted information. This article concerns techniques used to fill in the other half of the adjusted block data for possible use in reapportionment.     

Große oder kleine programmgesteuerte Rechengeräte?

Zusammenfassung Der wirtschaftliche Einsatz großer (Preis: 1 Mill. und mehr) und kleiner (DM 100 000 – 300 000) programmgesteuerter Rechenanlagen wird diskutiert. Zunächst stand der Einsatz von Rechenanlagen für wissenschaftliche und technische Aufgaben im Vordergrund: die mathematisch-arithmetischen Vorrichtungen hatten gegenüber Anschlußeinrichtungen (Ein-, Ausgabe usw.) den Vorrang. Später trat die Anwendung auf kaufmännische und betriebswirtschaftliche Probleme in den Vordergrund: wegen der Massen zu verarbeitender Daten bekamen die Anschlußeinrichtungen das Übergewicht — die Gesamtanlagen wurden z.T. außerordentlich umfangreich und teuer, gleichzeitig aber immer mehr in das Blickfeld der Allgemeinheit gerückt. Die Vorstellung kam auf, nur universell eingerichtete Großanlagen seien allen Anforderungen gewachsen: diese wird diskutiert und widerlegt. Es wird untersucht, wann und wo Einsatz von Groß-bzw. Kleinanlagen wirtschaftlich ist, sowie die Gründe hierfür. Vor- und Nachteile beider Typen werden miteinander verglichen und in einer Tabelle zusammen- und gegenübergestellt.

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Summary The economical application of small-sized (prize: ca 100.000.–300.000.-DM) and large-scale (ca. one million and more) program-controlled computers is discussed. Initially, the application of computers for solving technical and scientifical problems has been preeminent: terminal organs and similar equipment has been less important then mathematical and arithmetical structural computer elements. Later, the importance of the former increased rapidly because of the bulk of data to be processed with computer application in buisiness an economics etc: these organs has been continually increasing in size and prize; thus, a broad general interest in computers raised. Moreover, as a result, the idea of universality of large-scale program-controlled computers in application, reliability and efficiency becames almost generally a believe. This idea is discussed and disproved: the application (when, where) of large-scale and of small-sized computers is classified; advantages an disadvantages of either types are discussed, compared and tabulated.

     

Nonlinear Effects Induced by Vacuum Polarization by a Strong Electromagnetic Field in a Weak Static Electromagnetic Field

In the one-loop approximation of quantum electrodynamics, we study the Faraday rotation effect in the propagation of a weak linearly polarized wave through the field of a strong plane electromagnetic wave and also processes induced by a strong wave: the emission of a photon whose energy is an integer multiple of the strong-wave quantum and photon absorption with the emission of several photons identical (coherent) to the strong-wave quanta. The induced processes should occur in a vacuum in a superposition of external fields, namely, the field of a strong plane electromagnetic wave and a static spatially inhomogeneous electromagnetic field. We show that the induced photon emission in the field of a strong electromagnetic wave with circular polarization and in a static inhomogeneous field is equivalent to the merging of two strong-wave quanta with the production of one photon and that the induced photon absorption in the same combination of fields is equivalent to the decay (splitting) of a photon into two photons completely identical to the strong-wave quanta. All these effects are induced by the phenomenon of the vacuum polarization by a strong external time-dependent electromagnetic field. The probabilities of the induced processes are nontrivial nonlinear functions of the squared strength of the time-dependent field also depending on other parameters.     

The k‐ary Montgomery modular inverse over nonbinary computers

This paper presents a k‐ary Montgomery modular inverse algorithm over nonbinary computers by using Sedjelmaci's right shift k‐ary greatest common divisor scheme. Over traditional binary computers, Kaliski's scheme can output Montgomery modular inverse Q ^{? 1}2ⁿ mod P, where P is coprime to Q and n is the bit length of P. Over k‐ary computers, our algorithm can discover the k‐ary Montgomery inverse Q ^{? 1}k^m mod P, where P, Q, and k are pairwise relatively prime positive integers and m = log _kP. In the worst case, the computational cost of our algorithm is O(m²)k‐ary digit operations. Copyright © 2013 John Wiley & Sons, Ltd.     

A-stable diagonally implicit Runge-Kutta-Nyström methods for parallel computers

In this paper, we study diagonally implicit Runge-Kutta-Nyström methods (DIRKN methods) for use on parallel computers. These methods are obtained by diagonally implicit iteration of fully implicit Runge-Kutta-Nyström methods (corrector methods). The number of iterations is chosen such that the method has the same order of accuracy as the corrector, and the iteration parameters serve to make the method at least A-stable. Since a large number of the stages can be computed in parallel, the methods are very efficient on parallel computers. We derive a number of A-stable, strongly A-stable and L-stable DIRKN methods of orderp withs ^* (p) sequential, singly diagonal-implicit stages wheres ^*(p)=[(p+1)/2] ors ^* (p)=[(p+1)/2]+1,[°] denoting the integer part function.These investigations were supported by the University of Amsterdam with a research grant to enable the author to spend a total of two years in Amsterdam.