Mathematical Heuristics Research Articles

Freshman STEM students’ misconceptions in a basic limit theorem of [formula omitted]

This study examined learning difficulties among three consecutive cohorts of freshmen engineering and technology students (n = 251; 17 females) in solving problems in the basic limit theorem. The students were enrolled in a four-credit hour calculus course delivered in 16 weeks. Then, they completed a 1-h six-item test after they covered the topics of the basic limit theorem. Answer papers were scored using an explicit guide. Interviews were also conducted after the test with 15 randomly selected students without and with cognitive supports (cues) in topics related to the test. The study demonstrated that the students have difficulties understanding the surface structure of the problems, identifying the mathematical heuristics to solve the problems, and solving the problems by properly employing procedures, rules, and conventions. Examination of students' test results showed that for every student earning a correct answer via valid problem-processing pathways, there were about two students earning two correct answers via invalid pathways. Interview results showed the students were incapable of fixing their limitations without cognitive support and cues. They needed an average of 7.7–9.3 question-response dialogues to fix their limitations. Such learning difficulties were due to lack and misapplication of solid prior knowledge. Identifying and compiling lessons that cause learning difficulties due to misapplication of prior knowledge can help to adopt better instructional approaches. Cognitive supports and cues can be used as research tools to extract the reasons and sources of students’ learning difficulties and devise innovative instructional methods for better learning.

What is worthy of investigation?

We describe in dialogue form a possible way of discovering and investigating 10-adic numbers starting from the naive question about a “largest natural number”. Among the topics we pursue are possibilities of extensions to transfinite 10-adic numbers, 10-adic representations of rational numbers, zero divisors, square roots and 10-adic roots of higher degree of natural numbers, and applications of 10-adic number representation in computer arithmetic. The participants of the dialogue are idealized embodiments of different philosophical attitudes towards mathematics. The article aims at illustrating how these attitudes interact, in both jarring and stimulating ways, and how they impact mathematical development. Moreover, the article demonstrates how different attitudes translate into mathematical heuristics. In our opinion this is also relevant for teaching mathematics as it raises consciousness for reflections on how mathematical investigations work.

A Mixed-Integer Program for Drawing Orthogonal Hyperedges in a Hierarchical Hypergraph

This paper presents a new formulation and solution of a mixed-integer program for the hierarchical orthogonal hypergraph drawing problem, and the number of hyperedge crossings is minimized. The novel feature of the model is in combining several stages of the Sugiyama framework for graph drawing: vertex ordering, the assignment of vertices’ x-coordinates, and orthogonal hyperedge routing. The hyperedges of a hypergraph are assumed to be multi-source and multi-target, and vertices are depicted as rectangles with ports on their top and bottom sides. Such hypergraphs are used in data-flow diagrams and in a scheme of cooperation. The numerical results demonstrate the correctness and effectiveness of the proposed approach compared to mathematical heuristics. For instance, the proposed exact approach yields a 67.3% reduction of the number of crossings compared to that obtained by using a mathematical heuristic for a dataset of non-planar graphs.

Explaining Yang Mills Theory and Mass Gap and Proton Mass Based on Mathematical Heuristics

Hydrodynamic of vortex flows can explain quark gluon plasma fluid flows as an highly idealized liquid which has transition to a hadronic phase to account for nuclear matter components like protons.The mass of quarks are only in a tiny part derived from the Higgs field mechanisms. Strong forces in the atomic nucleus hold together quarks and gluons in the protons as components of nuclear matter. QCD theories are sought to account for the strong fields and hadron mass. There are possible correspondence and analogies with quantum hydrodynamic and its conformal thermodynamics,which might explain the mass gap based on turbulence of vortex flows and its shock wave like energy dissipation leading to a thermodynamics and its momentum or mass derivations. The yang mills theory emerges as an interphase of quantum electrodynamics,vortex hydrodynamics and conformal thermodynamics,so that the nuclear matter and hadronic phase has entropy and information transfers to experience a Higgs like mechanism and phonon modes for self interactions leading to mass and momentum effects. The mathematical models of Hamilton Jacobi equations for positive and negative mass ensembles approximate to a Stochastic NLSE as a Schrodinger expression derivation of Lindblad master equation for heavy quark QCD or QED models. The negative mass like effects can occur in spin orbit coupled high angular momentum idealized fluids at relativistic limits and quantum limits ,explained by GP equations as NLSE approximations for hydrodynamic plasma vortex flows. The hydrodynamics has analogies with QED,and conform to thermodynamics in the entropic limits,and its dissipative turbulence and fluctuations in energy. The Higgs like phonon modes are the result of angular momentum and mass or energy distribution and equilibrium processes described by LG Boltzman equations corresponding to Schrodinger equations in the non linear limits. The self interaction potential can explain mass gap and gluon condensates,and yang mills theory based topological phases with quark hadron transitions and symmetry breaking,that is a combination of the QED,hydrodynamic and thermodynamic energy and mass- momentum transfers,and confining hadronic pressures due to negative mass effects.This paper presents a heuristic model to examine this.

The case of the composite Higgs: The model as a “Rosetta stone” in contemporary high-energy physics

This paper analyses the practice of model-building “beyond the Standard Model” in contemporary high-energy physics and argues that its epistemic function can be grasped by regarding models as mediating between the phenomenology of the Standard Model and a number of “theoretical cores” of hybrid character, in which mathematical structures are combined with verbal narratives (“stories”) and analogies referring back to empirical results in other fields (“empirical references”). Borrowing a metaphor from a physics research paper, model-building is likened to the search for a Rosetta stone, whose significance does not lie in its immediate content, but rather in the chance it offers to glimpse at and manipulate the components of hybrid theoretical constructs. I shall argue that the rise of hybrid theoretical constructs was prompted by the increasing use of nonrigorous mathematical heuristics in high-energy physics. Support for my theses will be offered in form of a historical–philosophical analysis of the emergence and development of the theoretical core centring on the notion that the Higgs boson is a composite particle. I will follow the heterogeneous elements which would eventually come to form this core from their individual emergence in the 1960s and 1970s, through their collective life as a theoretical core from 1979 until the present day.

Optimal Product Line Design: Genetic Algorithm Approach to Mitigate Cannibalization

In this marketing-oriented era where manufacturers maximize profits through customer satisfaction, there is an increasing need to design a product line rather than a single product. By offering a product line, the manufacturer can customize his or her products to the needs of a variety of segments in order to maximize profits by satisfying more customers than a single product would. When the amount of data on customer preferences or possible product configurations is large and no analytical relations can be established, the problem of an optimal product line design becomes very difficult and there are no traditional methods to solve it. In this paper, we show that the usage of genetic algorithms, a mathematical heuristics mimicking the process of biological evolution, can solve efficiently the problem. Special domain operators were developed to help the genetic algorithm mitigate cannibalization and enhance the algorithm’s local search abilities. Using manufacturer’s profits as the criteria for fitness in evaluating chromosomes, the usage of domain specific operators was found to be highly beneficial with better final results. Also, we have hybridized the genetic algorithm with a linear programming postprocessing step to fine tune the prices of products in the product line. Attacking the core difficulty of cannibalization in the algorithm, the operators introduced in this work are unique.

Automated search for Gödel’s proofs

We present strategies and heuristics underlying a search procedure that finds proofs for Gödel’s incompleteness theorems at an abstract axiomatic level. As axioms we take for granted the representability and derivability conditions for the central syntactic notions as well as the diagonal lemma for constructing self-referential sentences. The strategies are logical ones and have been developed to search for natural deduction proofs in classical first-order logic. The heuristics are mostly of a very general mathematical character and are concerned with the goal-directed use of definitions and lemmata. When they are specific to the meta-mathematical context, these heuristics allow us, for example, to move between the object- and meta-theory. Instead of viewing this work as high-level proof search, it can be regarded as a first step in a proof-planning framework: the next refining steps would consist in verifying the axiomatically given conditions. Comparisons with the literature are detailed in Section 4. (The general mathematical heuristics are indeed general: in Appendix B we show that they, together with two simple algebraic facts and the logical strategies, suffice to find a proof of “ √ 2 is not rational”.)

On the legacy of G. Pólya

In this article are given some new aspects of mathematical problem solving. A framework is presented by three main resources: (1) Pólya's studies about mathematical heuristics are augmented by information drawn from a study of the history of mathematical problem solving. (2) Connections are presented between mathematical problem solving and mathematical beliefs. (3) Experience with a special program for mathematical talented students is sketched. On this background a new textbook-series has been developed and some teaching examples are taken from this context. An outlook is given on some new research on teaching of problem solving, including possible relations to modern brain research.

Decision Support for Bus Operations in a Mass Transit System*

ABSTRACTA computerized system has been developed for assigning, parking, and dispatching buses so that each operator in the St. Louis metropolitan transit system generally drives the same bus each day. Combinations of optimizing routines, mathematical heuristics, decision rules from experienced personnel, and human intervention are used to manage the bus operations efficiently. This paper describes the tools created for this task and documents significant improvements in maintenance that occurred after the system became operational.

Einstein's revolution: a study in heuristic

The author applies the methodology of scientific research programmes to the origins of relativity, showing how Eddington, Lorentz, Poincare, Planck and Weyl were driven by mathematical heuristics to make their various contributions to Einstein's programme.