The catwalk task: Reflections and synthesis: Part 1

In this article I recount my experiences with a series of encounters with the catwalk task and reflect on the professional growth that these opportunities afforded. First, I reflect on my own mathematical work on the catwalk task, including my efforts to fit various algebraic models to the data. Second, I reflect on my experiences working with a group of high school students on the catwalk task and my interpretations of their mathematical thinking. Finally, I reflect on the entire experience with the catwalk problem, as a mathematics learner, as a teacher, and as a professional.     

IMPACTS OF RECREATIONAL FISHING ON THE COMMERCIAL SECTOR: AN EMPIRICAL ANALYSIS OF ATLANTIC MACKEREL

There are often tensions between recreational and commercial uses of marine fisheries. The Atlantic mackerel along the coast of the United States and Canada is one such fishery. Although recreational catch is only a small percent of total catch, it has been suggested that recreational use be incorporated into a broad management scheme for the fishery. This paper measures the costs domestic commercial fishermen incur from the recreational harvest of Atlantic mackerel. The results indicate that the domestic commercial fishery, which takes only a small portion of total commercial catch, would see no significant gain from limiting recreational fishing. An earlier version of this paper was presented at the AERE Workshop on Sport and Commercial Fisheries. Participants at that conference, anonymous referees, and Peter Berck offered valuable suggestions which improved this paper. Any errors are my own.     

An infinite horizon mathematical programming model of a multicohort single species fishery

Fishery policy evaluation should take account of the initial state of the fishery and the population dynamics of the fish stock. Although multicohort bioeconomic fishery policy evaluation models have been developed, the results from these models depend on the choice of planning period and the desired state of the stock at the end of this period. In this paper it is noted that these limitations can be overcome by evaluating fishery policy over an infinite time horizon, and a mixed integer programming (MIP) model is developed for carrying out this form of analysis in a multicohort single species fishery. This new MIP model allows policies to be evaluated over an infinite horizon by incorporating results from a steady state fishery model into a multiperiod framework. The use of this MIP model in determining policies for reaching and maintaining a steady state is illustrated.     

Mathematics in economics: Achievements,difficulties, perspectives

I am deeply moved by this high honor which has fallen my lot to receive and I am happy to have the opportunity to appear here as a participant in this honorable series of lectures.In our time mathematics has penetrated into economics so solidly and widely and my theme is connected with such a variety of facts and problems that it brings me to cite the words of Kozma Prutkov, which are very popular in my country: One cannot embrace the unembraceable. The appropriateness of this wise sentence is not diminished by the fact that that great thinker is only a pen name.So, I want to restrict my theme to topics which are nearer to me, mainly to optimization models and their use in the control of the economy in order to best use resources for obtaining best results. I shall touch mainly on the problems and experiences of a planned economy, especially of the Soviet economy. Even within these limits I will succeed to consider only a few problems.Copyright © the Nobel Foundation 1975.     

An epistemic model of an agent who does not reflect on reasoning processes

This paper introduces an epistemic model of a boundedly rational agent under the two assumptions that (i) the agent’s reasoning process is in accordance with the model but (ii) the agent does not reflect on these reasoning processes. For such a concept of bounded rationality a semantic interpretation by the possible world semantics of the Kripke (1963) type is no longer available because the definition of knowledge in these possible world semantics implies that the agent knows all valid statements of the model. The key to my alternative semantic approach is the extension of the method of truth tables, first introduced for the propositional logic by Wittgenstein (1922), to an epistemic logic so that I can determine the truth value of epistemic statements for all relevant truth conditions. In my syntactic approach I define an epistemic logic–consisting of the classical calculus of propositional logic plus two knowledge axioms–that does not include the inference rule of necessitation, which claims that an agent knows all theorems of the logic. As my main formal result I derive a determination theorem linking my semantic with my syntactic approach. The difference between my approach and existing knowledge models is illustrated in a game-theoretic application concerning the epistemic justification of iterative solution concepts.     

STEADY STATE AND INTERTEMPORAL MANAGEMENT MODELS FOR MARINE FISHERIES

Abstract Steady state and dynamic management models are developed for analyzing the Malaysian marine fisheries. These models originate from the theoretical concepts of the natural resource economics namely the open access, limited entry and the intertemporal fishery models. Such management models are deemed necessary because of the need to sustain the depleting resource and degrading environment. Marine fisheries had been managed under open access for a long time before government intervention took effect sometime during the 1960s. Open access and government intervention during the earlier phase of economic development contributed to the immediate pressure on fisheries. Community development programs geared to alleviate poverty among the fishermen apparently contradicted the effort of sustaining fisheries. Even today this fundamental management objective of sustainable development of fishery resource is not fully adhered to. This study suggests that ability to sustain fishery requires government intervention that can direct resource use to steady state or intertemporal optimal levels.     

An epistemic model of an agent who does not reflect on reasoning processes

This paper introduces an epistemic model of a boundedly rational agent under the two assumptions that (i) the agent’s reasoning process is in accordance with the model but (ii) the agent does not reflect on these reasoning processes. For such a concept of bounded rationality a semantic interpretation by the possible world semantics of the Kripke (1963) type is no longer available because the definition of knowledge in these possible world semantics implies that the agent knows all valid statements of the model. The key to my alternative semantic approach is the extension of the method of truth tables, first introduced for the propositional logic by Wittgenstein (1922), to an epistemic logic so that I can determine the truth value of epistemic statements for all relevant truth conditions. In my syntactic approach I define an epistemic logic–consisting of the classical calculus of propositional logic plus two knowledge axioms–that does not include the inference rule of necessitation, which claims that an agent knows all theorems of the logic. As my main formal result I derive a determination theorem linking my semantic with my syntactic approach. The difference between my approach and existing knowledge models is illustrated in a game-theoretic application concerning the epistemic justification of iterative solution concepts.     

Stochastic models of HIV epidemic in homosexual populations—the effects of mixing patterns

Extending my previous work [1–3], in this paper I proceed to develop a stochastic model for HIV epidemic in a homosexual population under general conditions. Through computer generated data, I assess various deterministic models as compared with the expected numbers of the stochastic models. It is shown that different mixing patterns have significant impacts on the HIV epidemic except possibly restricted mixing. Thus, in populations with preferred mixing (mixing proportion less than 1) and with proportional mixing, the numbers of S people, L people, I people and A people differ significantly from the corresponding expected numbers of the stochastic models. For the L people, I people and A people, the numbers of the deterministic models first appear to be smaller and later appear to be larger than the corresponding mean numbers of the stochastic models, indicating that while in the short run the deterministic models would underestimate the true numbers, in the long run the deterministic models would overestimate the true numbers.     

Reasoning about relationships between quantities to reorganize inverse function meanings: The case of Arya

Researchers have argued high school students, college students, pre-service teachers, and in-service teachers do not construct productive inverse function meanings. In this report, I first summarize the literature examining students’ and teachers’ inverse function meanings. I then provide my theoretical perspective, including my use of the terms understanding and meaning and my operationalization of productive inverse function meanings. I describe a conceptual analysis of ways students may reorganize their limited inverse function meanings into productive meanings via reasoning about relationships between covarying quantities. I then present one pre-service teacher’s activity in a semester long teaching experiment to characterize how her quantitative, covariational, and bidirectional reasoning supported her in reorganizing her limited inverse function meanings into more productive meanings. I describe how this reorganization required her to reconstruct her meanings for various related mathematical ideas. I conclude with research and pedagogical implications stemming from this work and directions for future research.     

Note on models

The first part of this note deals with the importance of the polytechnical tradition for mathematical models.¹ In the second part I discuss the question of the “semantic shift” in the mathematical usage of the word “model” (M. Epple),² that means, the passage from concrete material objects to models in the modern sense of the word – abstract entities satisfying a certain system of axioms. My aim is to provide some new information on the theme and to open a new and in my opinion essential aspect – the polytechnical tradition – to look at it. The two parts are only in a loose connection.     

A quote a day educates

Conclusion  I often ponder on my duties as a teacher of the subject I love. I feel I am responsible for more than simply transmitting knowledge. I wish I could help my students see mathematics from various vantage points. One of these should be from a point high enough to afford a full, sweeping view of the mathematical valley below—maybe missing the details we strive to convey in class-but seeing thelandscape of mathematics. Claude Bragdon said, “Mathematics is the handwriting on the human consciousness of the very Spirit of Life itself.” I want my students to consider that such a bold statement might actually be true.     

Different approaches to efficiency analysis. An application to the Spanish Trawl fleet operating in Moroccan waters

Several approaches exist for the estimation of technical efficiency. In this paper the four most relevant ones (data envelopment analysis, stochastic production frontiers, panel data and distance functions) have been compared. The different approaches have been applied to the to the Spanish Trawl fishery that operated in Moroccan waters. The study highlights some of the problems arising from the wrong management of the fishery. It can also be concluded that in most cases, the multi- vs single-output feature is determinant in producing higher differences in the efficiency estimates, though the effect of either ignoring or taking into account random error also influences the results.     

FISHING YIELD,CURVATURE AND SPATIAL BEHAVIOR: IMPLICATIONS FOR MODELING MARINE RESERVES

ABSTRACT. Given a paucity of empirical data, policymakers are forced to rely on modeling to assess potential impacts of creating marine reserves to manage fisheries. Many modeling studies of reserves conclude that fishing yield will increase (or decrease only modestly) after creating a reserve in a heavily exploited fishery. However, much of the marine reserves modeling ignores the spatial heterogeneity of fishing behavior. Contrary to empirical findings in fisheries science and economics, most models assume explicitly or implicitly that fishing effort is distributed uniformly over space. This paper demonstrates that by ignoring this heterogeneity, yield‐per‐recruit models systematically overstate the yield gains (or understate the losses) from creating a reserve in a heavily exploited fishery. Conversely, at very low levels of exploitation, models that ignore heterogeneous fishing effort overstate the fishing yield losses from creating a reserve. Starting with a standard yield‐per‐recruit model, the paper derives a yield surface that maps spatially differentiated fishing effort into total long‐run fishing yield. It is the curvature of this surface that accounts for why the spatial distribution of fishing effort so greatly affects predicted changes from forming a reserve. The results apply generally to any model in which the long‐run fishing yield has similar curvature to a two‐patch Beverton‐Holt model. A simulation of marine reserve formation in the California red sea urchin fishery with Beverton‐Holt recruitment, eleven patches, and common larval pool dispersal dynamics reinforces these results.     

Markov chains,Gauss soups,and compromise dynamics

It must have been around 1966 when I first met Klaus Krickeberg at a conference in Greece. In my memory it was Konrad Jacobs who encouraged some young people from Erlangen to attend. I still remember when Klaus took us, M. Sieveking, J. Köhn and me, to a quite exciting tour by car on the Peloppenese. Later on, times were exciting too when I was around 1970 assistent to Klaus in Heidelberg. I also remember David G. Kendall at the Greece-conference, telling us about the Delphi method. About this I had forgotten until recently when writing this paper I noticed that my current work on opinion dynamics is related to the Delphi method.     

Faces of mathematical modeling

In this paper I will discuss and exemplify my perspectives on how to teach mathematical modeling, as well as discuss quite different faces of mathematical modeling. The field of mathematical modeling is so enormous and vastly outspread and just not possible to comprehend in one single paper; or in one single book, or even in one single book shelf. Nevertheless, I have found that the more I can illuminate some of the various interpretations and perceptions of mathematical modeling which exists in the world around us when introducing and starting a course in mathematical modeling, the more benefit I will have during the course when discussing the need and purpose of mathematical modeling with the students. The fact that only some models fit within the practical teaching and assessing of a course in mathematical modeling, does not exclude the importance to illustrate that the world of today cannot go on without mathematical modeling. Students are nevertheless much more charmed with some models of reality than others.     

A MODEL OF THE NORTH CAROLINA BAY SCALLOP FISHERY WITH ENDOGENOUS FISHING EFFORT AND ENTRY

The principal purpose of this model is to aid in the evaluation and design of regulations that affect the fishery. It differs from most previous models with a similar purpose in three ways. (1) Daily fishing costs are assumed to vary among fishermen. (2) Daily fishing effort is determined endogenously rather than being treated as a control variable. (3) Entry into the fishery is determined endogenously up to a cap imposed by the regulatory agency. The model explains the adverse reaction to a proposed attempt to increase the value of the fishery by delaying the opening date. The model is used to predict the economic consequences of four feasible sets of regulations.     

The role of mathematics in educational systems

In this paper, I will explain my views on education, particularly mathematics education, stressing my critique on the socio-cultural dimensions involved in mathematics education. The state of the world and ethical behavior play an important role in my critique. In exposing my views, the importance of my relationship with Hans-Georg Steiner will be clearly stated. In my memory, our discussions on these issues were very influential to my thinking. The importance of the support given by Hans-Georg to the development of mathematics education worldwide will be underlined. Both in the position of the chairman of the International Program Committee of ICME 3 and in the early years of the IDM in Bielefeld, Hans-Georg provided the material and intellectual space for the strengthening of Didactics of Mathematics as a research field. Following these preliminary remarks, I will briefly present the Program Ethnomathematics and explain how the program fits into what I call the modern trivium of literacy, matheracy and technoracy, which is a curriculum alternative for contemporary education.     

Noncommutative Batalin–Vilkovisky geometry and matrix integrals

I study the new type of supersymmetric matrix models associated with any solution to the quantum master equation of the noncommutative Batalin–Vilkovisky geometry. The asymptotic expansion of the matrix integrals gives homology classes in the Kontsevich compactification of the moduli spaces, which I associated with the solutions to the quantum master equation in my previous paper. I associate with the Bernstein–Leites matrix superalgebra equipped with an odd differentiation, whose square is nonzero, the family of cohomology classes of the compactification. This family is the generating function for the products of the tautological classes. The simplest example of my matrix integrals in the case of dimension zero is a supersymmetric extension of the Kontsevich model of 2-dimensional gravity.